How to prove that $\mathrm{GL}_2(\Bbb Z_2)$ has only six subgroups

Question

I have found the six subgroups that I know that $\mathrm{GL}_2(\Bbb Z_2)$ has, but now I want to prove that this is all. How can I do this? I am currently thinking that I should argue using the fact that if $H \le G$ is a subgroup, then $|H|$ must divide $|G|$ which would imply that I can have no subgroup of order $4$ or $5$. However I don't think my reasoning is going down a conclusive route. Any help appreciated, thanks!

Answer 1

As you wrote in a comment, you're aware that $\mathrm{GL}_2(\Bbb Z_2) \cong S_3$. Subgroups of order $2$ or $3$ must be cyclic and thus generated by one non-identity element; there are five non-identity elements, two of which generate the same subgroup, so that makes $4$ subgroups. The only other possibilities are order $1$ for the trivial subgroup and order $6$ for the group itself, for a total of $6$ subgroups.