How many of subgroups of order $2$ are there in $S_4$? Any subgroup of order $2$ will be cyclic subgroup and so will be generated by single element of order $2$ in $S_4$, so to count number of subgroups of order $2$ we need to count number of elements of order $2$ in $S_4$, I tried counting them but I got answer $8$ but is $9$ actually.

Assume that $\sigma\in S_4$ has order two. Then $\sigma$ changes some element $x$. Let $y=\sigma(x)$. Then $\sigma(y)=x$.
This shows that $\sigma$ is the product of some disjoint $2$-cycles.
The possibilities are:
$$(1,2),(1,3),(1,4),(2,3),(2,4),(3,4),(1,2)(3,4),(1,3)(2,4),(1,4)(2,3)$$
