  We propose a predictive model based on the $SU(3)_C\times SU(3)_L\times
U(1)_X$ gauge symmetry, which is supplemented by the $D_4$ family symmetry and
several auxiliary cyclic symmetries whose spontaneous breaking produces the
observed SM fermion mass and mixing pattern. The masses of the light active
neutrinos are produced by an inverse seesaw mechanism mediated by three right
handed Majorana neutrinos. To the best of our knowledge the model corresponds
to the first implementation of the $D_4$ family symmetry in a $SU(3)_C\times
SU(3)_L\times U(1)_X$ theory with three right handed Majorana neutrinos and
inverse seesaw mechanism. Our proposed model successfully accommodates the
experimental values of the SM fermion mass and mixing parameters, the muon
anomalous magnetic moment as well as the Higgs diphoton decay rate and meson
oscillations constraints. The consistency of our model with the muon anomalous
magnetic moment requires charged exotic vector like leptons at the TeV scale.
