The threshold for subgroup profiles to agree is $\Omega(\log n)$
  For primes $p,e>2$ there are at least $p^{e-3}/e$ groups of order $p^{2e+2}$ that have equal multisets of isomorphism types of proper subgroups and proper quotient groups, isomorphic character tables, and power maps. This obstructs recent speculation concerning a path towards efficient isomorphism tests for general finite groups. These groups have a special purpose polylogarithmic-time isomorphism test. 
James B. Wilson
math.GR