Is there a nice description of all divisors of $n!$ which can be realized as cardinals of permutation groups acting transitively on $\{1,\dots,n\}$?

A necessary condition is of course that such a divisor is a multiple of $n$.

$n$ (cyclic) and $(2n)$ dihedral are always possible but, without mistake on my behalf, $3n$ is not feasible unless $n\equiv 1\pmod 3$ (it can then be realized as a semi-direct product analogous to the dihedral group). can only be realized if $3$ divides the number of invertible integers modulo $n$.