Because of my interest in [this question](https://mathoverflow.net/q/108530), I listed the subgroups of ${\frak S}\_n$ for $1\le n\le4$. I found that the number of subgroups are, respectively, $1,2,6,24$. It might be a coincidence, or it could reveal a deep connection.



> 
> Is it always the case that ${\frak S}\_n$ has exactly $n!$ (non-proper) subgroups ?
> 
> 
> 


**Edit**. Because of my erroneous claim about ${\frak S}\_4$, I tried to delete the question. But this cannot be done once an answer has been posted. So I asked the help of the front desk. Decision is pending.



The number of subgroups of ${\frak S}_n$
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](https://homepages.warwick.ac.uk/~mareg/download/papers/symsubs/symsubs.pdf)


Also, according to [**this**](https://math.stackexchange.com/q/79139) 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.


