= Subfields of the splitting field of x to the fourth minus seven
Put $\alpha=\sqrt[4]{7}$ and $L=\mathbb Q(\alpha,i)$. The <dihedral group> $D_8=\langle r,s:r^4=s^2=1,\ srs=r^{-1}\rangle$ acts by
$$
r(\alpha)=i\alpha,\quad r(i)=i,\qquad
s(\alpha)=\alpha,\quad s(i)=-i.
$$
The ten subgroups and their fixed fields are
$$
\begin{array}{c|c}
D_8&\mathbb Q\\
\langle r\rangle&\mathbb Q(i)\\
\langle r^2,s\rangle&\mathbb Q(\sqrt7)\\
\langle r^2,rs\rangle&\mathbb Q(\sqrt{-7})\\
\langle r^2\rangle&\mathbb Q(\sqrt7,i)\\
\langle s\rangle&\mathbb Q(\alpha)\\
\langle r^2s\rangle&\mathbb Q(i\alpha)\\
\langle rs\rangle&\mathbb Q((1+i)\alpha)\\
\langle r^3s\rangle&\mathbb Q((1-i)\alpha)\\
\{1\}&L.
\end{array}
$$
The normal subgroups are $D_8$, $\langle r\rangle$, the two displayed Klein four-groups, $\langle r^2\rangle$, and $\{1\}$.
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