All subgroups of index 2 Can we construct examples of non-abelian groups G (finite or infinite) such that for all of it's non-trivial subgroups has index 2 in G?.

Not of finite order. Suppose $G$ has subgroup of index 2. Then order of $G$ is even, as it's a union of two cosets of such subgroup. By Cauchy's Theorem, $G$ has member of order two.  If $|G|>4$, index of the subgroup generated by this element is greater than 2. If $G$ has no subgroups at all, it is cyclic (of prime order), hence abelian.
Will think about general case.
