Stock Based Compensation Plans (Additional Information) (Detail) - USD ($)	3 Months Ended
Dec. 31, 2018	Dec. 31, 2017
Share-based Compensation Arrangement by Share-based Payment Award, Number of Shares Available for Grant	8,658
Fair value of restricted shares	$ 3,653	$ 3,453
Employee Service Share-based Compensation, Nonvested Awards, Total Compensation Cost Not yet Recognized, Share-based Awards Other than Options	231,000
Stock or Unit Option Plan Expense	$ 18,000	16,000
Equity Incentive Plan 2016 [Member]
Share-based Compensation Arrangement by Share-based Payment Award, Number of Shares Available for Grant	88,000
Share-based Compensation Arrangement by Share-based Payment Award, Options, Vested in Period, Fair Value	$ 100,000
Stock Option Plan 2016 [Member]
Share-based Compensation Arrangement by Share-based Payment Award, Number of Shares Available for Grant	11,661
Employee Stock Option [Member]
Share-based Compensation Arrangement by Share-based Payment Award, Options, Exercises in Period, Intrinsic Value	$ 470,000	355,000
Employee Stock Option [Member] \| Equity Incentive Plan 2016 [Member]
Share-based Compensation Arrangement by Share-based Payment Award, Number of Shares Available for Grant	66,000
Restricted Stock [Member]
Fair value of restricted shares	$ 216,000	195,000
Employee Service Share-based Compensation, Nonvested Awards, Total Compensation Cost Not yet Recognized, Share-based Awards Other than Options	$ 580,000
Share-based Compensation Arrangement by Share-based Payment Award, Award Vesting Period	4 years 10 months 24 days
Employee Stock Ownership Plan (ESOP), Compensation Expense	$ 41,000	$ 37,000
Restricted Stock [Member] \| Equity Incentive Plan 2016 [Member]
Share-based Compensation Arrangement by Share-based Payment Award, Number of Shares Available for Grant	22,000
Equity Option [Member]
Share-based Compensation Arrangement by Share-based Payment Award, Number of Shares Available for Grant	10,255
Employee Service Share-based Compensation, Nonvested Awards, Total Compensation Cost Not yet Recognized, Period for Recognition	4 years 10 months 24 days
Equity Option [Member] \| Restricted Stock [Member]
Share-based Compensation Arrangement by Share-based Payment Award, Number of Shares Available for Grant	1,406

Weighted average period over which unrecognized compensation is expected to be recognized for equity-based compensation plans, in 'PnYnMnDTnHnMnS' format, for example, 'P1Y5M13D' represents the reported fact of one year, five months, and thirteen days.

Reference 1: http://fasb.org/us-gaap/role/ref/legacyRef
-Publisher FASB
-Name Accounting Standards Codification
-Topic 718
-SubTopic 10
-Section 50
-Paragraph 2
-Subparagraph (i)
-URI http://asc.fasb.org/extlink&oid=109197908&loc=d3e5070-113901

+ Details

Name:	us-gaap_EmployeeServiceShareBasedCompensationNonvestedAwardsTotalCompensationCostNotYetRecognizedPeriodForRecognition1
Namespace Prefix:	us-gaap_
Data Type:	xbrli:durationItemType
Balance Type:	na
Period Type:	duration

- Definition

Unrecognized cost of unvested share-based awards, other than options, awarded to employees as compensation.

+ References

+ Details

Name:	us-gaap_EmployeeServiceShareBasedCompensationNonvestedAwardsTotalCompensationCostNotYetRecognizedShareBasedAwardsOtherThanOptions
Namespace Prefix:	us-gaap_
Data Type:	xbrli:monetaryItemType
Balance Type:	debit
Period Type:	instant

- Definition

The amount of plan compensation cost recognized during the period.

+ References

Reference 1: http://fasb.org/us-gaap/role/ref/legacyRef
-Publisher FASB
-Name Accounting Standards Codification
-Topic 718
-SubTopic 40
-Section 50
-Paragraph 1
-Subparagraph (c)
-URI http://asc.fasb.org/extlink&oid=109244661&loc=d3e17540-113929

+ Details

Name:	us-gaap_EmployeeStockOwnershipPlanESOPCompensationExpense
Namespace Prefix:	us-gaap_
Data Type:	xbrli:monetaryItemType
Balance Type:	debit
Period Type:	duration

- Definition

Period which an employee's right to exercise an award is no longer contingent on satisfaction of either a service condition, market condition or a performance condition, in 'PnYnMnDTnHnMnS' format, for example, 'P1Y5M13D' represents the reported fact of one year, five months, and thirteen days.

+ References

Reference 1: http://fasb.org/us-gaap/role/ref/legacyRef
-Publisher FASB
-Name Accounting Standards Codification
-Topic 718
-SubTopic 10
-Section 50
-Paragraph 2
-Subparagraph (a)(1)
-URI http://asc.fasb.org/extlink&oid=109197908&loc=d3e5070-113901

+ Details

Name:	us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardAwardVestingPeriod1
Namespace Prefix:	us-gaap_
Data Type:	xbrli:durationItemType
Balance Type:	na
Period Type:	duration

- Definition

Fair value of share-based awards for which the grantee gained the right by satisfying service and performance requirements, to receive or retain shares or units, other instruments, or cash.

+ References

Reference 1: http://fasb.org/us-gaap/role/ref/legacyRef
-Publisher FASB
-Name Accounting Standards Codification
-Topic 718
-SubTopic 10
-Section 50
-Paragraph 2
-Subparagraph (d)(2)
-URI http://asc.fasb.org/extlink&oid=109197908&loc=d3e5070-113901

+ Details

Name:	us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsVestedInPeriodTotalFairValue
Namespace Prefix:	us-gaap_
Data Type:	xbrli:monetaryItemType
Balance Type:	debit
Period Type:	duration

- Definition

The difference between the maximum number of shares (or other type of equity) authorized for issuance under the plan (including the effects of amendments and adjustments), and the sum of: 1) the number of shares (or other type of equity) already issued upon exercise of options or other equity-based awards under the plan; and 2) shares (or other type of equity) reserved for issuance on granting of outstanding awards, net of cancellations and forfeitures, if applicable.

+ References

Reference 1: http://fasb.org/us-gaap/role/ref/legacyRef
-Publisher FASB
-Name Accounting Standards Codification
-Topic 718
-SubTopic 10
-Section 50
-Paragraph 2
-URI http://asc.fasb.org/extlink&oid=109197908&loc=d3e5070-113901

+ Details

Name:	us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardNumberOfSharesAvailableForGrant
Namespace Prefix:	us-gaap_
Data Type:	xbrli:sharesItemType
Balance Type:	na
Period Type:	instant

- Definition

Amount of accumulated difference between fair value of underlying shares on dates of exercise and exercise price on options exercised (or share units converted) into shares.

+ References

Reference 1: http://fasb.org/us-gaap/role/ref/legacyRef
-Publisher FASB
-Name Accounting Standards Codification
-Topic 718
-SubTopic 10
-Section 50
-Paragraph 2
-Subparagraph (d)(2)
-URI http://asc.fasb.org/extlink&oid=109197908&loc=d3e5070-113901

+ Details

Name:	us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsExercisesInPeriodTotalIntrinsicValue
Namespace Prefix:	us-gaap_
Data Type:	xbrli:monetaryItemType
Balance Type:	debit
Period Type:	duration

- Definition

Fair value of options vested. Excludes equity instruments other than options, for example, but not limited to, share units, stock appreciation rights, restricted stock.

+ References

Reference 1: http://fasb.org/us-gaap/role/ref/legacyRef
-Publisher FASB
-Name Accounting Standards Codification
-Topic 718
-SubTopic 10
-Section 50
-Paragraph 2
-Subparagraph (c)
-URI http://asc.fasb.org/extlink&oid=109197908&loc=d3e5070-113901

+ Details

Name:	us-gaap_SharebasedCompensationArrangementBySharebasedPaymentAwardOptionsVestedInPeriodFairValue1
Namespace Prefix:	us-gaap_
Data Type:	xbrli:monetaryItemType
Balance Type:	credit
Period Type:	duration

- Definition

The noncash expense that accounts for the value of stock or unit options distributed to employees as compensation.

+ References

Reference 1: http://fasb.org/us-gaap/role/ref/legacyRef
-Publisher FASB
-Name Accounting Standards Codification
-Topic 230
-SubTopic 10
-Section 45
-Paragraph 28
-Subparagraph (b)
-URI http://asc.fasb.org/extlink&oid=109223946&loc=d3e3602-108585

+ Details

Name:	us-gaap_StockOptionPlanExpense
Namespace Prefix:	us-gaap_
Data Type:	xbrli:monetaryItemType
Balance Type:	debit
Period Type:	duration

- Details

Name:	us-gaap_PlanNameAxis=fsfg_EquityIncentivePlan2016Member
Namespace Prefix:
Data Type:	na
Balance Type:
Period Type:

- Details

Name:	us-gaap_PlanNameAxis=fsfg_StockOptionPlan2016Member
Namespace Prefix:
Data Type:	na
Balance Type:
Period Type:

- Details

Name:	us-gaap_OptionIndexedToIssuersEquityEquityAxis=us-gaap_EmployeeStockOptionMember
Namespace Prefix:
Data Type:	na
Balance Type:
Period Type:

- Details

Name:	us-gaap_AwardTypeAxis=us-gaap_RestrictedStockMember
Namespace Prefix:
Data Type:	na
Balance Type:
Period Type:

- Details

Name:	us-gaap_FinancialInstrumentAxis=us-gaap_StockOptionMember
Namespace Prefix:
Data Type:	na
Balance Type:
Period Type: