Finite p-groups Which Have Many Normal Subgroups

Normal subgroups of a group play an important role in determining the structure of a group. A Dedekindian group is the group all of whose subgoups are normal. The classification of such finite groups has been completed in 1897 by Dedekind. And Passman gave a classification of finite p-groups all of whose nonnormal subgroups are of order p. Above such two finite groups have many normal subgroups. Alone this line, to study the finite p-groups all of whose nonnormal subgroups are of order p or p(2), that is, its subgroups of order >= p(3) are normal. According to the order of the derived subgroups, divide into two cases expression and give all non-isomophic groups.