Groups with Same Number Of Elements and Subgroups
No finite list of numerical inariants is sufficient to classify finite groups up to isomorphism (as far as anyone knows).
Subgroups of groups of order 16 are tabulated here. Note that groups 2 and 4 both have 15 subgroups: 3 of order 2, 6 cyclic of order 4, 1 Klein-4 group, 3 of type $C_4\oplus C_2$ (and the one element group, and the whole group). So, not just the same number of subgroup, the same number of each type of (proper) subgroup. Similarly for groups 5 and 6.