  We investigate the breaking of SU(3) into its subgroups from the viewpoints
of explicit and spontaneous breaking. A one-to-one link between these two
approaches is given by the complex spherical harmonics, which form a complete
set of SU(3)-representation functions. An invariant of degrees p and q in
complex conjugate variables corresponds to a singlet, or vacuum expectation
value, in a (p,q)-representation of SU(3). We review the formalism of the
Molien function, which contains information on primary and secondary
invariants. Generalizations of the Molien function to the tensor generating
functions are discussed. The latter allows all branching rules to be deduced.
We have computed all primary and secondary invariants for all proper finite
subgroups of order smaller than 512, for the entire series of groups
\Delta(3n^2), \Delta(6n^2), and for all crystallographic groups. Examples of
sufficient conditions for breaking into a subgroup are worked out for the
entire T_{n[a]}-, \Delta(3n^2)-, \Delta(6n^2)-series and for all
crystallographic groups \Sigma(X). The corresponding invariants provide an
alternative definition of these groups. A Mathematica package, SUtree, is
provided which allows the extraction of the invariants, Molien and generating
functions, syzygies, VEVs, branching rules, character tables, matrix
(p,q)_{SU(3)}-representations, Kronecker products, etc. for the groups
discussed above.
