  Discrete flavour symmetries have been proven successful in explaining the
leptonic flavour structure. To account for the observed mixing pattern, the
flavour symmetry has to be broken to different subgroups in the charged and
neutral lepton sector. However, cross-couplings via non-trivial contractions in
the scalar potential force the group to break to the same subgroup. We present
a solution to this problem by extending the flavour group in such a way that it
preserves the flavour structure, but leads to an 'accidental' symmetry in the
flavon potential. We have searched for symmetry groups up to order 1000, which
forbid all dangerous cross-couplings and extend one of the interesting groups
A4, T7, S4, T' or \Delta(27). We have found a number of candidate groups and
present a model based on one of the smallest extension of A4, namely Q8 \rtimes
A4. We show that the most general non-supersymmetric potential allows for the
correct vacuum alignment. We investigate the effects of higher dimensional
operators on the vacuum configuration and mixing angles, and give a
see-saw-like UV completion. Finally, we discuss the supersymmetrization of the
model. Additionally, we release the Mathematica package "Discrete" providing
various useful tools for model building such as easily calculating invariants
of discrete groups and flavon potentials.
