One generalization of a minimal noncyclic groups

A finite groups, in which maximal length of antichains of noncyclic subgroups(dn-width) of the group P is more than maximal (dn-width Sn(P) of its propere subgroups are considered. Such an abelian p-groups with Sn(P) ≤ 5 and finite p-groups, which is minimal regarding property "any two noncyclic subgroups is incident" are described