Finite p-groups with few orders of non-normal subgroups

Suppose that [Formula: see text] is a finite non-Dedekind [Formula: see text]-group. Let [Formula: see text] denote the number of the orders of the non-normal subgroups of [Formula: see text]. In this paper, we bound the order of [Formula: see text] based on [Formula: see text]. Assume that [Formula: see text]. We show that [Formula: see text] except for the case where [Formula: see text] is cyclic and the nilpotency class of [Formula: see text] is [Formula: see text].