This question arose while I was trying to work out examples for the second question of this thread: Reconstruction Conjecture: Group theoretic formulation?
In the beginning, I considered some computable properties of groups and wondered whether two groups of the same order having equal value for that computable property would necessarily be isomorphic. For instance, take centers of groups and it is not difficult to find many specific examples where two groups have the same order and isomorphic centers but then the two groups are not necessarily isomorphic. Considering lattice of groups, Scott Carnahan has already given a counterexample there.
Are there any two finite groups of the same order that have the same number of subgroups?