  We answer three related questions concerning the Haagerup subfactor and its
even parts, the Haagerup fusion categories. Namely we find all simple module
categories over each of the Haagerup fusion categories (in other words, we find
the `"quantum subgroups" in the sense of Ocneanu), we find all subfactors whose
principal even part is one of the Haagerup fusion categories, and we compute
the Brauer-Picard groupoid of Morita equivalences of the Haagerup fusion
categories. In addition to the two even parts of the Haagerup subfactor, there
is exactly one more fusion category which is Morita equivalent to each of them.
This third fusion category has six simple objects and the same fusion rules as
one of the even parts of the Haagerup subfactor, but has not previously
appeared in the literature. We also find the full lattice of intermediate
subfactors for every subfactor whose even part is one of these three fusion
categories, and we discuss how our results generalize to Izumi subfactors.
