Definition A $2$-group $S$ is called wreathed if it is isomorphic to $(C\times C)\rtimes \langle i \rangle$ where $C$ is a cyclic group of order $2^n$ and $i$ is an involution with action $(a,b)^i=(b,a)$ for all $(a,b)\in C\times C$.

I wonder about the structure of maximal subgroups of $S$. Clearly, one of them is $C\times C$ and it is abelian.

I think the rest of the maximal subgroups is not abelian when $n\geq 2$. (When $n=1$ we have a dihedral group of order $8$.) However, if $M$ is a maximal subgroup of $S$ then $M$ has an abelian subgroup of index $2$.

Any remark and reference is welcome.

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    $\begingroup$ Do you mean wreathed? $\endgroup$ – J. W. Tanner Feb 4 '19 at 16:14

Maximal subgroups of $p$-groups $G$ have index $p$ and contain $\Phi(G) = [G,G]G^p$.

In this example, with $a$ a generator of $C$, we have $\Phi(G) = \langle (a,a^{-1}), (a^2,1), (1,a^2) \rangle$ has index $4$ in $G$, and there are three maximal subgroups:

$\langle (a,1), (1,a) \rangle = C \times C$;

$\langle (a^2,1), (1,a^2), (a,a^{-1}), i \rangle$; and

$\langle (a^2,1), (1,a^2), (a,a^{-1}), (a,1)i \rangle$.

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