A connection between the number of subgroups and the order of a finite group

For a finite group G, we associate the quantity beta(G) = vertical bar L(G)vertical bar/vertical bar G vertical bar, where L(G) is the subgroup lattice of G. Different properties and problems related to this ratio are studied throughout this paper. We determine the second minimum value of beta on the class of p-groups of order p(n), where n >= 3 is an integer. We show that the set containing the quantities beta(G), where G is a finite (abelian) group, is dense in [0, infinity). Finally, we consider beta to be a function on L(G) and we indicate some of its properties, the main result being the classification of finite abelian p-groups C satisfying beta(H) <= 1, for all H is an element of L(C).