  An attractive feature of New Physics models with multiple Higgs fields is
that they are equipped with discrete symmetry groups in the Higgs and flavour
sectors. These symmetry groups are often broken at the global minimum of the
Higgs potential, either completely or to a proper subgroup, with certain
phenomenological consequences. Here, we systematically explore these symmetry
breaking patterns in the scalar sector of the three-Higgs-doublet model (3HDM).
We use the full list of discrete symmetry groups allowed in 3HDM, and for each
group we find all possible ways it can break by the Higgs vacuum expectation
value alignment. We also discuss the interplay between these symmetry groups
and various forms of CP-violation in the scalar sector of 3HDM. Not only do our
results solve the problem for 3HDM, but they also hint at several general
features in multi-scalar sectors.
