We are trying to find all Sylow 2-subgroups of an arbitrary dihedral group $D_n$ of order $2n$.
In the case that $n$ is odd, any Sylow 2-subgroup must have order 2, and it is fairly easy to deduce the set of elements in $D_n$ of order 2.
However, in the case that $n=2m$ is even, we see that $|D_n|=2\cdot 2m$ is divisible by $2^2$ (but not $2^3$), so that every Sylow 2-subgroup has order 4. We are not having luck in finding all subgroups of order 4 in $D_n$, and are wondering if anyone has any hints or suggestions. Thank you.