\lambda(tst^{-1})\rho(t) = \lambda(s^{-1})\rho(t).\end{aligned}$$ Similar calculations apply to the other elements, and so $E_{s,t}$ is normalized by $\lambda(G)$. Finally, we have $E_{s,t}\neq E_{t,s}$ since $ \lambda(t)\rho(s) $ lies in $ E_{t,s} $ but not in $ E_{s,t} $. Referring to we see that $ E_{\sigma,\tau} $ and $ E_{\tau,\sigma} $ are the underlying groups of the two Hopf Galois structures of type $C_2\times C_2\times C_2$ on $ L/F $.

We now find the subgroups corresponding to the Hopf-Galois structures of type $C_4\times C_2$ using a similar technique.

\[lemma\_Ast\] Let $s,t\in\{\sigma,\tau,\sigma\tau\} $ with $ s\neq t$ and let $A_{s,t}$ be generated by the permutations $\lambda(s)$ and $\rho(t)$. Then $A_{s,t}$ is a regular subgroup of ${\text{Perm}}(G)$ that is normalized by $\lambda(G)$ and isomorphic to $C_4\times C_2$. The 6 choices of the pair $ s,t $ yield distinct groups, and these are the underlying groups of the 6 structures of type $C_4\times C_2$ on $ L/F $.

We have $\langle\rho(t),\lambda(s)\rangle\cong C_4\times C_2$ since $\rho(t)$ and $\lambda(s)$ are both of order 4, commute with each other, and share the same square. It is clear that $ A_{s,t} \subset {\text{Perm}}(G) $ and that for $g,h\in G$ we have $\lambda(g)\rho(h)\cdot1_{G}=gh^{-1}$; hence $A_{s,t}$ is a regular subgroup of 