Finding all normal subgroups of dihedral group of order eight $D_8$

Question

Let $D_8:=\{e,a,a^2,a^3,b,ab,a^2b,a^3b\}$, and $a^4=e$, $b^2=e$, $ba=a^{-1}b$. Find all normal subgroups of $D_8$.

What the technique to approach this? I've found the center $Z(D_8)$, and it's a normal subgroup. Are there any others?

Answer 1

Hint: Start by finding conjugacy classes, that is find the set $\{h^{-1}gh\ |\ h\in G\}$ for each $g\in G$. Normal subgroups must be unions of these classes.