In plain English
FAR 52.227-22, Major System-Minimum Rights, is a contract clause prescribed at FAR 27.409(k), most recently dated June 1987. 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 27.409(k), insert the following clause:
Prescribing reference: FAR 27.409(k).
The official text, verbatim
As prescribed in 27.409(k), insert the following clause:
Major System-Minimum Rights (June 1987)
Notwithstanding any other provision of this contract, the Government shall have unlimited rights in any technical data, other than computer software, developed in the performance of this contract and relating to a major system or supplies for a major system procured or to be procured by the Government, to the extent that delivery of such technical data is required as an element of performance under this contract. The rights of the Government under this clause are in addition to and not in lieu of its rights under the other provisions of this contract.
(End of clause)