Minimal nonmodular finite p-groups

We describe first the structure of finite minimal nonmodular 2-groups G. We show that in case |G| > 25, each proper subgroup of G is Q8-free and G/(G) is minimal nonabelian of order 24 or 25. If |G/(G)| = 24, then the structure of G is determined up to isomorphism (Propositions 2.4 and 2.5). If |G/(G)| = 25, then (G) E8 and G/(G) is metacyclic (Theorem 2.8). Then we classify finite minimal nonmodular p-groups G with p > 2 and |G| > p4 (Theorems 3.5 and 3.7). We show that G/(G) is nonabelian of order p3 and exponent p and (G) is metacyclic. Also, G/(G) Ep and G/(G) is metacyclic