If $F \subset E$ is Galois and there is a subfield of $E$ thats not normal, describe and find number of subfields

Question

I know [F:E] = 8 and I need to describe the structure of all subfields and then find all subfields that are normal and not equal to $F$ or $E$.

Since [F:E] = 8, I know that $| \operatorname{Gal} (F/E)| = 8$, but I have no idea where to start so any hints would be appreciated

Answer 1

Using the fundamental theorem of Galois theory, there is a one-to-one correspondence between subfields of $E$ containing $F$ and subgroups of $\mathrm{Gal}(E/F)$. Moreover, a subfield $F\subset K\subset E$ is normal if and only if the corresponding subgroup $\mathrm{Gal}(E/K)\leq \mathrm{Gal}(E/F)$ is normal.

Now, there is a very short list of groups of order $8$, and only one of them has a subgroup that is not normal.