  Magnetic monopoles in Yang-Mills-Higgs theory with a non-abelian unbroken
gauge group are classified by holomorphic charges in addition to the
topological charges familiar from the abelian case. As a result the moduli
spaces of monopoles of given topological charge are stratified according to the
holomorphic charges. Here the physical consequences of the stratification are
explored in the case where the gauge group SU(3) is broken to U(2). The
description due to A. Dancer of the moduli space of charge two monopoles is
reviewed and interpreted physically in terms of non-abelian magnetic dipole
moments. Semi-classical quantisation leads to dyonic states which are labelled
by a magnetic charge and a representation of the subgroup of U(2) which leaves
the magnetic charge invariant (centraliser subgroup). A key result of this
paper is that these states fall into representations of the semi-direct product
$U(2) \semidir R^4$. The combination rules (Clebsch-Gordan coefficients) of
dyonic states can thus be deduced. Electric-magnetic duality properties of the
theory are discussed in the light of our results, and supersymmetric dyonic BPS
states which fill the SL(2,Z)-orbit of the basic massive W-bosons are found.
