I am trying to solve this:
Find all semidirect products $\mathbb{Z}_{8} \rtimes \mathbb{Z}_{4}$ up to isomorphism
First I tried to find all homomorphisms $\varphi:\mathbb{Z}_{4} \to{\rm Aut}(\mathbb{Z}_{8})$. As $\varphi$ is only decided by the value $\varphi(\overline{1})(\overline{1})$, I find there are 4 different homomorphisms $\varphi$.They are $\varphi_{1}(\overline{1})(\overline{1})=\overline{1}$, $\varphi_{2}(\overline{1})(\overline{1})=\overline{3}$, $\varphi_{3}(\overline{1})(\overline{1})=\overline{5}$ and $\varphi_{4}(\overline{1})(\overline{1})=\overline{7}$. $\varphi_{1}$ is the identity mapping so $\mathbb{Z}_{8} \rtimes_{\varphi_{1}} \mathbb{Z}_{4}$ is not isomorphic to the other three. But I don't know what to do next. I have no idea about whether $\mathbb{Z}_{8} \rtimes_{\varphi_{2}} \mathbb{Z}_{4}$, $\mathbb{Z}_{8} \rtimes_{\varphi_{3}} \mathbb{Z}_{4}$ and $\mathbb{Z}_{8} \rtimes_{\varphi_{4}} \mathbb{Z}_{4}$ are isomorphic. I tried to prove they are isomorphic but failed. So maybe they are not isomorphic, but I also do not know how to prove two groups are not isomorphic. Is there a better way to solve this? Thank you for help.