Galois group of $F/K$ where $F/K$ is galois with $[F:K]=8$ and where there is an intermediate field $E$ with $[E:K]=4$ such that $E/K$ is not galois

Question

Problem. Let $K$ be a field and suppose that $F$ is Galois over $K$ with $[F:K]=8$. Also suppose that $E$ is an intermediate field which is not Galois over $K$ and $[E:K]=4$. Determine the Galois group of $F$ over $K$.

I want to solve this problem. By the main theorem, we know that the group $G(F/K)$ has order $8$, and that the subgroup $G(F/E)$ is not normal in $G(F/K)$. This implies that $G(F/K)$ must be nonabelian, so it follows that $G(F/K)$ is either isomorphic to the quaternion group $Q$ or the dihedral group $D_8$ of order $8$. But I cannot proceed from here. Any hints?

Answer 1

It turns out that all subgroups of $Q$ are normal, even though $Q$ isn't abelian. You can check this by hand. So the answer is $D_8$.