On a subfactor analogue of the second cohomology

The set of equivalence classes of Longo's Q-systems is shown to serve as a right subfactor analogue of the second cohomology. This "cohomology" is computed for several classes of subfactors, which tells us if a subfactor is uniquely determined up to inner conjugacy by the associated canonical endomorphism. As a byproduct of our analysis, two different groups of order 64 are shown to possess the same representation category as an abstract tensor category. Thus, this category has two permutation symmetries with non-isomorphic groups via the Doplicher-Roberts duality.