  We develop methods to study the scalar sector of multi-Higgs models with
large discrete symmetry groups that are softly broken. While in the exact
symmetry limit, the model has very few parameters and can be studied
analytically, proliferation of quadratic couplings in the most general softly
broken case makes the analysis cumbersome. We identify two sets of soft
breaking terms which play different roles: those which preserve the symmetric
vacuum expectation value alignment, and the remaining terms which shift it.
Focusing on alignment preserving terms, we check which structural features of
the symmetric parent model are conserved and which are modified. We find
remarkable examples of structural features which are inherited from the parent
symmetric model and which persist even when no exact symmetry is left. The
general procedure is illustrated with the example of the three-Higgs-doublet
model with the softly broken symmetry group $\Sigma(36)$.
