  We classify irreducible II_1 subfactors A \subset B such that B \ominus A is
reducible as an A-A bimodule, with index at most 6+1/5, leaving aside the
composite subfactors at index exactly 6. Previous work has already achieved
this up to index 3+\sqrt{5} \approx 5.23. We find there are exactly three such
subfactors with index in (3+\sqrt{5}, 6+1/5], all with index 3+2\sqrt{2}. One
of these comes from SO(3)_q at a root of unity, while the other two appear to
be closely related, and are `braided up to a sign'.
