Determining all the subfields of the splitting field for $x^8-2$ which are Galois over $\mathbb{Q}$

Question

I have found all the subgroups of the Galois group of the splitting field $K=\mathbb{Q}(\sqrt[8]{2},i)$ over $\mathbb{Q}$, and I know that if any of the subgroups $H$ of $G=$Gal$(K/\mathbb{Q})$ are normal in $G$, then $H$ corresponds to a subfield of $K$ that is Galois over $\mathbb{Q}$. There are $16$ subgroups here -- do I have to check each one individually for normality? That seems like it would take ages, and I really have no idea where to start.