In plain English
FAR 52.236-24, Work Oversight in Architect-Engineer Contracts, is a contract clause prescribed at FAR 36.609-3, most recently dated Apr 1984. 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 36.609-3 , insert the following clause:
Prescribing reference: FAR 36.609-3.
The official text, verbatim
As prescribed in 36.609-3 , insert the following clause:
Work Oversight in Architect-Engineer Contracts (Apr 1984)
The extent and character of the work to be done by the Contractor shall be subject to the general oversight, supervision, direction, control, and approval of the Contracting Officer.
(End of clause)