In the $2$-group $G$ of order $32$ which has Id$ [32,5]$, and could be describe as:
$$\begin{align} G &\cong \langle {a,b,x:a^8,b^2,x^2,[a,b]=[a,x], ...} \rangle \\ &\cong (\mathbb Z_8 \times \mathbb Z_2) \rtimes \mathbb Z_2. \end{align}$$
I am trying to show that one of the non-normal subgroups of order $4$ is permutable in $G$. So, I am trying to identify possible subgroups that need to be checked. My question is: Is there any information on how to calculate the number of subgroups of such $G$?
Thank you.