The first "non-prime" Jones indices are 4, 3 + $\sqrt{5}$ and 6. All hyperfinite subfactors with index 4 are known, and it follows from work of Nicoara, Popa and myself that the set of subfactors with composite integer index is wild. I'll explain some of the beautiful structures appearing here and will make some comments about the situation of hyperfinite subfactors with index 3 + $\sqrt{5}$.