1. Normal subgroups of D4

Find subgroups N and H of D4 such that N is normal H and H is normal to D4, but N is NOT a
normal subgroup of D4.

Well D4={e,r,r^2,r^3,f,fr, fr^2, fr^3}
I'm not really sure on all of the subgroups.
I have a couple:
{e,f}
{e,r,r^2,r^3}
I guess the biggest problem is finding the subgroups

2. {e}
{e,r,r^2,r^3}
{e,r^2}
{e,f}
{e,rf}
{e,fr^2}
{e,fr^3}
{r^2,f}
{r^2,fr}
So I have 9 subgroups, but am unsure on how to work with them

3. Originally Posted by kathrynmath
{e}
{e,r,r^2,r^3}
{e,r^2}
{e,f}
{e,rf}
{e,fr^2}
{e,fr^3}
{r^2,f}
{r^2,fr}
So I have 9 subgroups, but am unsure on how to work with them
So you have your subgroups. You now have to work out if they are normal (or, at least, find one which is and one which isn't).

To find one which is, are you familiar with a theorem which says that if $|G:H|=2$ then $H \lhd G$? If so, use this. Otherwise, check by hand that $\{e,r,r^2,r^3\}$ is closed under conjugation. That is, take every element, h, outside of this set and calculate,

$hrh^{-1}$
$hr^2h^{-1}$
$hr^3h^{-1}$

and note that they are all still in your set.

To find a subgroup which isn't normal, you have to find one which is NOT closed under conjugation. That is, do the same thing as above to a few other subgroups and try if you can find one where there exists a $g \in H$ where $H$ is the subgroup you are looking at, and a $h \not\in H$ such that $hgh^{-1} \not\in H$.