# Maximal subgroup of Wreathed 2-groups

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.

• Do you mean wreathed? – 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
$$\langle (a^2,1), (1,a^2), (a,a^{-1}), (a,1)i \rangle$$.