they are the underlying groups of the 2 Hopf-Galois structures of type $Q_{8}$.

Finally, the subgroups corresponding to the Hopf-Galois structures of type $D_4$, the dihedral group of order 8, have a similar description to the groups $ E_{s,t} $ and $ A_{s,t} $.

\[lemma\_Ds\] Let $s,t\in\{\sigma,\tau,\sigma\tau\}$ with $ s \neq t $. Let $D_{s,\lambda}$ be generated by $\lambda(s)$ and $\lambda(t)\rho(s)$, and let $D_{s,\rho}$ be generated by $\rho(s)$ and $\lambda(s)\rho(t)$. Then $D_{s,\lambda}$ and $D_{s,\rho}$ do not depend upon the choice of $t$, and are regular subgroups of ${\text{Perm}}(G)$ that are normalized by $\lambda(G)$ and isomorphic to $D_4$. The $ 3 $ choices of $ s $ yield $ 6 $ distinct groups, and these are the underlying groups of the Hopf Galois structures of type $D_4$ on $ L/F $.

For a fixed choice of $ t $ the elements of $D_{s,\lambda}$ are $$1,\,\lambda(s),\,\lambda(s^2),\,\lambda(s^{-1}),\,\lambda(t)\rho(s),\,\lambda(st)\rho(s),\,\lambda(t^{-1})\rho(s), \,\lambda((st)^{-1})\rho(s).$$ We see immediately that using $ st $ in place of $ t $ yields the same group, that $ \lambda(s) $ has order $ 4 $, $ \lambda(t)\rho(s) $ has order $ 2 $, and that these elements anticommute. Therefore $ D_{s,\lambda} \cong D_{4} $. It is clear that $ D_{s,\lambda} \subset {\text{Perm}}(G) $ 