Easy question. How do I find the subgroups of $\displaystyle D_8$?
$\displaystyle D_8=<r,s:r^4=s^2=1, \ rs=sr^{-1}>$
$\displaystyle D_8=\{1,r,r^2,r^3,s,sr,sr^2,sr^3\}$
Just by checking the subgroups generated by the elements concerned - no smart method - just working through the elements of the group checking the subgroups generated by each element. If the group concerned was very much bigger you would have to be more analytic I guess.
Another possibility is <$\displaystyle r^2 $> = {e, $\displaystyle r^2 $} and of course there is {e}.
Peter