As far as I know, the only groups of order $n$ that aren't subgroups of $S_{n-1}$ are cyclic groups with prime power order and the Klein four-group. Is there any nonabelian group of order $n$ that isn't a subgroup of $S_{n-1}$?
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4https://math.stackexchange.com/questions/1597347/finding-the-minimal-n-such-that-a-given-finite-group-g-is-a-subgroup-of-s-n – Jean-Claude Arbaut Nov 29 '22 at 21:42
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4Yes. In particular for $n=8$, the quaternion group $Q_8$ is not isomorphic to a subgroup of $S_7$ - see this post. – Dietrich Burde Nov 29 '22 at 21:51