  Symmetries play a crucial role in electroweak symmetry breaking models with
non-minimal Higgs content. Within each class of these models, it is desirable
to know which symmetry groups can be implemented via the scalar sector. In
N-Higgs-doublet models, this classification problem was solved only for N=2
doublets. Very recently, we suggested a method to classify all realizable
finite symmetry groups of Higgs-family transformations in the
three-Higgs-doublet model (3HDM). Here, we present this classification in all
detail together with an introduction to the theory of solvable groups, which
play the key role in our derivation. We also consider generalized-CP
symmetries, and discuss the interplay between Higgs-family symmetries and
CP-conservation. In particular, we prove that presence of the Z_4 symmetry
guarantees the explicit CP-conservation of the potential. This work completes
classification of finite reparametrization symmetry groups in 3HDM.
