In plain English
FAR 52.217-8, Option to Extend Services, is a contract clause prescribed at FAR 17.208(f), most recently dated Nov 1999. The complete official text is reproduced below, verbatim, from GSA's published FAR source files.
Its text contains no sentence requiring insertion into subcontracts; see the flowdown section below for what that does and does not mean.
Does it flow down to subcontracts?
No flowdown mandate found in the clause text
We scanned the full clause text and found no sentence directing the contractor to insert this clause into subcontracts. That is a statement about the text, not legal advice: a prime contractor may still flow terms down contractually, an agency supplement (DFARS, VAAR, …) may add requirements, and clauses listed below (if any) may order this clause into subcontracts from the outside.
Where it's prescribed
As prescribed in 17.208(f), insert a clause substantially the same as the following:
Prescribing reference: FAR 17.208(f).
The official text, verbatim
As prescribed in 17.208(f), insert a clause substantially the same as the following:
Option to Extend Services (Nov 1999)
The Government may require continued performance of any services within the limits and at the rates specified in the contract. These rates may be adjusted only as a result of revisions to prevailing labor rates provided by the Secretary of Labor. The option provision may be exercised more than once, but the total extension of performance hereunder shall not exceed 6 months. The Contracting Officer may exercise the option by written notice to the Contractor within _____ [insert the period of time within which the Contracting Officer may exercise the option].
(End of clause)