This is already sort of in the comments, but just to say it clearly:
In the paper Enumerating Finite Groups of Given Order Author(s): L. Pyber Source: Annals of Mathematics, Second Series, Vol. 137, No. 1 (Jan., 1993), pp. 203-220
Pyber proves that the number of subgroups of $S_n$ is bounded below by $2^{(1/16 + o(1))n^2}.$ He shows this by noting that $S_n$ contains $C_2^{[\frac{n}2]},$ and the latter has the specified number of subgroups. He then conjectures that the lower bound is, in fact, sharp. In other words, modulo this conjecture, at least morally, the maximal order of a subgroup is, indeed, a power of two, and the actual power of two is presumably $2^{[[\frac{n}2]/2]}.$ It seems that Pyber's conjecture is still open.