A finite group G is called a Dedekind group if every subgroup of G is normal. Let G be a finite group which is not a Dedekind group. Then the set of non-normal subgroups of G is a non-empty set. If S is a non-normal subgroup of G, then all conjugates of S are also non-normal, and so G acts on the set of all non-normal subgroups by conjugation. Similarly, G acts on the set of all non-normal non-cyclic subgroups of G by conjugation. We denote by \u_{nc}(G) the number of conjugacy classes of non-normal non-cyclic subgroups of G. We prove that if N \rianglelefteq G, then \u_{nc}(G/N) \\leq \u_{nc}(G), and thus if equality holds, then N is contained in all the non-normal non-cyclic subgroups of G. Our main purpose is to consider the influence of the value of \u_{nc}(G) on the structure of G.