Subfactors of von Neumann factors have a rich representation theory that gives rise to interesting mathematical structures such as fusion categories, planar algebras or link invariants. They are highly noncommutative algebras of operators and in general not determined by their representations. It is open how to distinguish them. I will explain a natural notion of noncommutativity for a subfactor and illustrate it with a theorem that provides the first examples of very noncommutative, irreducible subfactors. This notion might also be of interest in quantum information theory.