  The structure of the neutrino mixing matrix is indicative of an underlying
family symmetry that interrelates the three generations of fermions in the
Standard Model. We systematically scan the parameter space of 76 discrete
non-Abelian family symmetries and construct all models with the Standard Model
particle content and up to three flavon fields where we include
non-renormalizable interactions of mass dimension five and six. We find that of
the 76 groups that we considered, 44 groups can accommodate models that are
consistent with experiment at 3sigma, and 38 groups can have models that are
tribimaximal. One immediate consequence is that A4 is not "special", but should
be considered on equal footing with other groups such as T7 that is the
smallest group for which we find tribimaximal mixing, and T13 that has the
largest fraction of TBM models. We present the details of a model with
theta12=33.9, theta23=49.1, theta13=5.1 to show that a non-zero theta13 can
easily be accommodated.
