Minimal non-quaternion-free finite 2-groups. (Q2472714)

Let \(G\) be a \(2\)-group all of whose proper subgroups are quaternion-free but \(G\) is not quaternion-free (= minimal non-quaternion-free \(2\)-group). Then there is \(N\triangleleft G\) such that \(G/N\cong Q_8\), \(N<\Phi(G)\) so that \(d(G)=2\). If \(R\) is any \(G\)-invariant subgroup of index \(2\) in \(N\), then \(G/R\) is (nonabelian) metacyclic of exponent \(4\). It follows that \(G\) has an epimorphic image \(\cong D_8\), so \(G\) is nonmodular. Using this result, the author classifies the minimal non-quaternion-free \(2\)-groups containing a nonmodular proper subgroup.