N = E_{s,t},  A_{s,t}, D_{s,\lambda}, $ and $ D_{s,\rho} $ (for all valid choices of $ s,t $) from the definitions of these groups. Finally, we can verify that it holds for the groups $ C_{s,t} $ (for all valid choices of $ s,t $) by computing $ \eta_{s,t}^{4} = \rho(\sigma^{2}) $ in these cases.

Hopf algebra isomorphisms
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In this section we determine which of the Hopf algebras giving Hopf-Galois structures on $ L/F $ are isomorphic as $F$-Hopf algebras. In [@TARP2017 Theorem 2.2] Koch, Kohl, Underwood and the second named author outline the following criterion for two Hopf algebras arising from the Greither-Pareigis correspondence to be isomorphic as Hopf algebras: let $N_1$ and $N_2$ be underlying groups of two Hopf Galois structures on $L/F$. Then $L[N_1]^G\cong L[N_2]^G$ as $F$-Hopf algebras if and only if there exists a $G$-equivariant isomorphism $f:N_1\xrightarrow{\sim}N_2$. In particular, no two Hopf algebras of different types may be isomorphic as $F$-Hopf algebras.

We now determine which of our Hopf algebras are isomorphic. We consider the isomorphism classes of the underlying groups individually. We start with the elementary abelian groups.

\[3.1\] The Hopf algebras giving the two Hopf-Galois structures of type $C_2\times C_2\times C_2$ are isomorphic 