TITLE: I need to determine the subgroups of the dihedral group of order 8, $D_4$.
QUESTION [19 upvotes]: I need to determine the subgroups of the dihedral group of order 4, $D_4$.
I know that the elements of $D_4$ are $\{1,r,r^2,r^3, s,rs,r^2s,r^3s\}$
But I don't understand how to get the subgroups..

REPLY [18 votes]: The notes by K. Conrad have a nice answer: the dihedral group $D_n$ is generated by a rotation $r$ and a reflection $s$ subject to the relations $r^n=s^2=1$ and $(rs)^2=1$.
Proposition: Every subgroup of $D_n$ is cyclic or dihedral. A complete listing of the subgroups (including $1$ and $D_n$) is as follows:
$(1)$ $\langle r^d \rangle$ for all divisors $d\mid n$.

$(2)$ $\langle r^d,r^is \rangle$, where $d\mid n$ and $0\le i\le d-1 $.
Very nice pictures of the subgroup diagram of $D_4$ can be found here.