<?xml version="1.0" encoding="us-ascii"?><InstanceReport xmlns:xsd="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"><Version>2.4.0.8</Version><ReportLongName>2416402 - Disclosure - Subsequent Events (Details)</ReportLongName><DisplayLabelColumn>true</DisplayLabelColumn><ShowElementNames>false</ShowElementNames><RoundingOption /><HasEmbeddedReports>false</HasEmbeddedReports><Columns><Column FlagID="0"><Id>1</Id><IsAbstractGroupTitle>false</IsAbstractGroupTitle><LabelSeparator>

</LabelSeparator><CurrencyCode>USD</CurrencyCode><FootnoteIndexer /><hasSegments>false</hasSegments><hasScenarios>false</hasScenarios><MCU><KeyName /><CurrencySymbol>$</CurrencySymbol><contextRef><ContextID>D2014Q1</ContextID><EntitySchema>http://www.sec.gov/CIK</EntitySchema><EntityValue>0000915779</EntityValue><PeriodDisplayName /><PeriodType>duration</PeriodType><PeriodStartDate>2013-04-28T00:00:00</PeriodStartDate><PeriodEndDate>2013-07-27T00:00:00</PeriodEndDate><Segments /><Scenarios /></contextRef><UPS><UnitProperty><UnitID>usdPerShare</UnitID><UnitType>Divide</UnitType><NumeratorMeasure><MeasureSchema>http://www.xbrl.org/2003/iso4217</MeasureSchema><MeasureValue>USD</MeasureValue><MeasureNamespace>iso4217</MeasureNamespace><MeasureDescription /><MeasureType /></NumeratorMeasure><DenominatorMeasure><MeasureSchema>http://www.xbrl.org/2003/instance</MeasureSchema><MeasureValue>shares</MeasureValue><MeasureNamespace>xbrli</MeasureNamespace><MeasureDescription /><MeasureType /></DenominatorMeasure><Scale>0</Scale></UnitProperty></UPS><CurrencyCode>USD</CurrencyCode><OriginalCurrencyCode>USD</OriginalCurrencyCode></MCU><CurrencySymbol>$</CurrencySymbol><Labels><Label Key="CalendarSupplement" Id="0" Label="3 Months Ended" /><Label Key="Calendar" Id="1" Label="Jul. 27, 2013" /></Labels></Column><Column FlagID="0"><Id>2</Id><IsAbstractGroupTitle>false</IsAbstractGroupTitle><LabelSeparator>

</LabelSeparator><CurrencyCode>USD</CurrencyCode><FootnoteIndexer /><hasSegments>false</hasSegments><hasScenarios>false</hasScenarios><MCU><KeyName /><CurrencySymbol>$</CurrencySymbol><contextRef><ContextID>D2013Q1</ContextID><EntitySchema>http://www.sec.gov/CIK</EntitySchema><EntityValue>0000915779</EntityValue><PeriodDisplayName /><PeriodType>duration</PeriodType><PeriodStartDate>2012-04-29T00:00:00</PeriodStartDate><PeriodEndDate>2012-07-28T00:00:00</PeriodEndDate><Segments /><Scenarios /></contextRef><UPS><UnitProperty><UnitID>usdPerShare</UnitID><UnitType>Divide</UnitType><NumeratorMeasure><MeasureSchema>http://www.xbrl.org/2003/iso4217</MeasureSchema><MeasureValue>USD</MeasureValue><MeasureNamespace>iso4217</MeasureNamespace><MeasureDescription /><MeasureType /></NumeratorMeasure><DenominatorMeasure><MeasureSchema>http://www.xbrl.org/2003/instance</MeasureSchema><MeasureValue>shares</MeasureValue><MeasureNamespace>xbrli</MeasureNamespace><MeasureDescription /><MeasureType /></DenominatorMeasure><Scale>0</Scale></UnitProperty></UPS><CurrencyCode>USD</CurrencyCode><OriginalCurrencyCode>USD</OriginalCurrencyCode></MCU><CurrencySymbol>$</CurrencySymbol><Labels><Label Key="CalendarSupplement" Id="0" Label="3 Months Ended" /><Label Key="Calendar" Id="1" Label="Jul. 28, 2012" /></Labels></Column><Column FlagID="0"><Id>3</Id><IsAbstractGroupTitle>false</IsAbstractGroupTitle><LabelSeparator>

</LabelSeparator><CurrencyCode>USD</CurrencyCode><FootnoteIndexer /><hasSegments>true</hasSegments><hasScenarios>false</hasScenarios><MCU><KeyName /><CurrencySymbol>$</CurrencySymbol><contextRef><ContextID>D2014Q2DividendDeclared_us-gaap_SubsequentEventTypeAxis_us-gaap_SubsequentEventMember</ContextID><EntitySchema>http://www.sec.gov/CIK</EntitySchema><EntityValue>0000915779</EntityValue><PeriodDisplayName /><PeriodType>duration</PeriodType><PeriodStartDate>2013-08-21T00:00:00</PeriodStartDate><PeriodEndDate>2013-08-22T00:00:00</PeriodEndDate><Segments><anyType xsi:type="Segment"><IsDefaultForEntity>false</IsDefaultForEntity><Name /><IsFromEntityGroup>false</IsFromEntityGroup><ValueName>Subsequent Event [Member]</ValueName><ValueType>us-gaap_SubsequentEventTypeAxis</ValueType><Namespace>xbrldi</Namespace><Schema>http://xbrl.org/2006/xbrldi</Schema><DimensionInfo><Id>us-gaap_SubsequentEventMember</Id><dimensionId>us-gaap_SubsequentEventTypeAxis</dimensionId><type>explicitMember</type></DimensionInfo></anyType></Segments><Scenarios /></contextRef><UPS><UnitProperty><UnitID>usdPerShare</UnitID><UnitType>Divide</UnitType><NumeratorMeasure><MeasureSchema>http://www.xbrl.org/2003/iso4217</MeasureSchema><MeasureValue>USD</MeasureValue><MeasureNamespace>iso4217</MeasureNamespace><MeasureDescription /><MeasureType /></NumeratorMeasure><DenominatorMeasure><MeasureSchema>http://www.xbrl.org/2003/instance</MeasureSchema><MeasureValue>shares</MeasureValue><MeasureNamespace>xbrli</MeasureNamespace><MeasureDescription /><MeasureType /></DenominatorMeasure><Scale>0</Scale></UnitProperty></UPS><CurrencyCode>USD</CurrencyCode><OriginalCurrencyCode>USD</OriginalCurrencyCode></MCU><CurrencySymbol>$</CurrencySymbol><Labels><Label Key="CalendarSupplement" Id="0" Label="0 Months Ended" /><Label Key="Calendar" Id="1" Label="Aug. 22, 2013" /><Label Key="us-gaap_SubsequentEventTypeAxis:us-gaap_SubsequentEventMember" Id="2" Label="Subsequent Event [Member]" /></Labels></Column></Columns><Rows><Row FlagID="0"><Id>1</Id><IsAbstractGroupTitle>true</IsAbstractGroupTitle><LabelSeparator>

</LabelSeparator><Level>3</Level><ElementName>us-gaap_SubsequentEventLineItems</ElementName><ElementPrefix>us-gaap_</ElementPrefix><IsBaseElement>true</IsBaseElement><BalanceType>na</BalanceType><PeriodType>duration</PeriodType><IsReportTitle>false</IsReportTitle><IsSegmentTitle>false</IsSegmentTitle><IsCalendarTitle>false</IsCalendarTitle><IsEquityPrevioslyReportedAsRow>false</IsEquityPrevioslyReportedAsRow><IsEquityAdjustmentRow>false</IsEquityAdjustmentRow><IsBeginningBalance>false</IsBeginningBalance><IsEndingBalance>false</IsEndingBalance><IsReverseSign>false</IsReverseSign><PreferredLabelRole>terseLabel</PreferredLabelRole><FootnoteIndexer /><Cells><Cell FlagID="0" ContextID="" UnitID=""><Id>1</Id><IsNumeric>false</IsNumeric><IsRatio>false</IsRatio><DisplayZeroAsNone>false</DisplayZeroAsNone><NumericAmount>0</NumericAmount><RoundedNumericAmount>0</RoundedNumericAmount><NonNumbericText /><FootnoteIndexer /><CurrencyCode /><CurrencySymbol /><IsIndependantCurrency>false</IsIndependantCurrency><ShowCurrencySymbol>false</ShowCurrencySymbol><DisplayDateInUSFormat>false</DisplayDateInUSFormat></Cell><Cell FlagID="0" ContextID="" UnitID=""><Id>2</Id><IsNumeric>false</IsNumeric><IsRatio>false</IsRatio><DisplayZeroAsNone>false</DisplayZeroAsNone><NumericAmount>0</NumericAmount><RoundedNumericAmount>0</RoundedNumericAmount><NonNumbericText /><FootnoteIndexer /><CurrencyCode /><CurrencySymbol /><IsIndependantCurrency>false</IsIndependantCurrency><ShowCurrencySymbol>false</ShowCurrencySymbol><DisplayDateInUSFormat>false</DisplayDateInUSFormat></Cell><Cell FlagID="0" ContextID="" UnitID=""><Id>3</Id><IsNumeric>false</IsNumeric><IsRatio>false</IsRatio><DisplayZeroAsNone>false</DisplayZeroAsNone><NumericAmount>0</NumericAmount><RoundedNumericAmount>0</RoundedNumericAmount><NonNumbericText /><FootnoteIndexer /><CurrencyCode /><CurrencySymbol /><IsIndependantCurrency>false</IsIndependantCurrency><ShowCurrencySymbol>false</ShowCurrencySymbol><DisplayDateInUSFormat>false</DisplayDateInUSFormat></Cell></Cells><ElementDataType>xbrli:stringItemType</ElementDataType><SimpleDataType>string</SimpleDataType><IsTotalLabel>false</IsTotalLabel><UnitID>0</UnitID><Label>Subsequent Event [Line Items]</Label></Row><Row FlagID="0"><Id>2</Id><IsAbstractGroupTitle>false</IsAbstractGroupTitle><LabelSeparator>

</LabelSeparator><Level>4</Level><ElementName>us-gaap_CommonStockDividendsPerShareDeclared</ElementName><ElementPrefix>us-gaap_</ElementPrefix><IsBaseElement>true</IsBaseElement><BalanceType>na</BalanceType><PeriodType>duration</PeriodType><IsReportTitle>false</IsReportTitle><IsSegmentTitle>false</IsSegmentTitle><IsCalendarTitle>false</IsCalendarTitle><IsEquityPrevioslyReportedAsRow>false</IsEquityPrevioslyReportedAsRow><IsEquityAdjustmentRow>false</IsEquityAdjustmentRow><IsBeginningBalance>false</IsBeginningBalance><IsEndingBalance>false</IsEndingBalance><IsReverseSign>false</IsReverseSign><PreferredLabelRole>terseLabel</PreferredLabelRole><FootnoteIndexer /><Cells><Cell FlagID="0" ContextID="D2014Q1" UnitID="usdPerShare"><Id>1</Id><IsNumeric>true</IsNumeric><IsRatio>false</IsRatio><DisplayZeroAsNone>false</DisplayZeroAsNone><NumericAmount>0.120</NumericAmount><RoundedNumericAmount>0.120</RoundedNumericAmount><NonNumbericText /><FootnoteIndexer /><CurrencyCode>USD</CurrencyCode><CurrencySymbol>$</CurrencySymbol><IsIndependantCurrency>false</IsIndependantCurrency><ShowCurrencySymbol>true</ShowCurrencySymbol><DisplayDateInUSFormat>false</DisplayDateInUSFormat></Cell><Cell FlagID="0" ContextID="D2013Q1" UnitID="usdPerShare"><Id>2</Id><IsNumeric>true</IsNumeric><IsRatio>false</IsRatio><DisplayZeroAsNone>false</DisplayZeroAsNone><NumericAmount>0.115</NumericAmount><RoundedNumericAmount>0.115</RoundedNumericAmount><NonNumbericText /><FootnoteIndexer /><CurrencyCode>USD</CurrencyCode><CurrencySymbol>$</CurrencySymbol><IsIndependantCurrency>false</IsIndependantCurrency><ShowCurrencySymbol>true</ShowCurrencySymbol><DisplayDateInUSFormat>false</DisplayDateInUSFormat></Cell><Cell FlagID="0" ContextID="D2014Q2DividendDeclared_us-gaap_SubsequentEventTypeAxis_us-gaap_SubsequentEventMember" UnitID="usdPerShare"><Id>3</Id><IsNumeric>true</IsNumeric><IsRatio>false</IsRatio><DisplayZeroAsNone>false</DisplayZeroAsNone><NumericAmount>0.09</NumericAmount><RoundedNumericAmount>0.09</RoundedNumericAmount><NonNumbericText /><FootnoteIndexer /><CurrencyCode>USD</CurrencyCode><CurrencySymbol>$</CurrencySymbol><IsIndependantCurrency>false</IsIndependantCurrency><ShowCurrencySymbol>true</ShowCurrencySymbol><DisplayDateInUSFormat>false</DisplayDateInUSFormat></Cell></Cells><ElementDataType>num:perShareItemType</ElementDataType><SimpleDataType>decimal</SimpleDataType><ElementDefenition>Aggregate dividends declared during the period for each share of common stock outstanding.</ElementDefenition><ElementReferences>Reference 1: http://www.xbrl.org/2003/role/presentationRef

 -Publisher SEC

 -Name Regulation S-X (SX)

 -Number 210

 -Section 04

 -Article 3



Reference 2: http://www.xbrl.org/2003/role/presentationRef

 -Publisher FASB

 -Name Accounting Standards Codification

 -Topic 505

 -SubTopic 10

 -Section S99

 -Paragraph 1

 -Subparagraph (SX 210.3-04)

 -URI http://asc.fasb.org/extlink&amp;oid=6959260&amp;loc=d3e187085-122770



</ElementReferences><IsTotalLabel>false</IsTotalLabel><UnitID>3</UnitID><Label>Cash dividend declared per share (in dollars per share)</Label></Row></Rows><Footnotes /><IsEquityReport>false</IsEquityReport><ReportName>Subsequent Events (Details) (USD $)</ReportName><MonetaryRoundingLevel>UnKnown</MonetaryRoundingLevel><SharesRoundingLevel>UnKnown</SharesRoundingLevel><PerShareRoundingLevel>NoRounding</PerShareRoundingLevel><ExchangeRateRoundingLevel>UnKnown</ExchangeRateRoundingLevel><HasCustomUnits>true</HasCustomUnits><IsEmbedReport>false</IsEmbedReport><IsMultiCurrency>false</IsMultiCurrency><ReportType>Sheet</ReportType><RoleURI>http://www.daktronics.com/role/SubsequentEventsDetails</RoleURI><NumberOfCols>3</NumberOfCols><NumberOfRows>2</NumberOfRows></InstanceReport>
