  We study the problem of realising modular invariants by braided subfactors
and the related problem of classifying nimreps. We develop the fusion rule
structure of these modular invariants. This structure is useful tool in the
analysis of modular data from quantum double subfactors, particularly those of
the double of cyclic groups, the symmetric group on 3 letters and the double of
the subfactors with principal graph the extended Dynkin diagram D_5^(1). In
particular for the double of S_3, 14 of the 48 modular modular invariants are
nimless, and only 28 of the remaining 34 nimble invariants can be realised by
subfactors.
