As Francesco Polizzi mentions, the answer is no alredy for ${\frak S}_4$: there are $30$ subgroups, but $4!=24$.
Here are some more (very small) calculations:
\begin{array}{|c|c|c|c|} \hline \mathrm{group}& \mathrm{\# subgroups} & n! \\ \hline {\frak S}_1 & 1 &1\\ \hline {\frak S}_2 & 2 &2\\ \hline {\frak S}_3 & 6 &6\\ \hline {\frak S}_4 & 30 &24\\ \hline {\frak S}_5 & 156 &120\\ \hline {\frak S}_6 & 1455 & 720\\ \hline {\frak S}_7 & 11300 & 5040\\ \hline {\frak S}_8 & 151221 & 40320\\ \hline {\frak S}_9 & 1694723 & 362880\\ \hline \end{array}
You can find the number of subgroups up to ${\frak S}_{18}$ in here:
- Derek Holt, Enumerating subgroups of the symmetric group
Also, according to this math.stackexchange, the asymptotics for the number of subgroups is $\log(\#\text{sub}) = \Theta(n^2)$.
As for the number of subgroups for arbitrary ${\frak S}_n$, the problem seems wide open.
