TITLE: Groups with Same Number Of Elements and Subgroups
QUESTION [8 upvotes]: Say I have a finite group and I tell you how many elements and how many subgroups it has. Do you now know what group it is? Or could two groups which are not isomorphic to each other have the same amount of elements and subgroups? In other words, does the amount of elements and subgroups of a group define it completely? It is interesting because if I give you just the number of elements there are several possibilities and if I give you the number of subgroups also but perhaps both combined are enough to define the group.

REPLY [11 votes]: 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.