  We present a non-minimal renormalizable SUSY $SU(5)$~model, with extended
Higgs sector and right-handed neutrinos, where the flavour sector exhibits a
$Q_{6}$ flavour symmetry. We analysed the simplest version of this model, in
which R-parity is conserved and the right-handed neutrino masses in the flavour
doublet are considered with and without degeneration. We find the generic form
of the mass matrices both in the quark and lepton sectors. We reproduce,
according to current data, the mixing in the CKM matrix. In the leptonic
sector, in the general case where the right-handed neutrino masses are not
degenerate, we find that the values for the solar, atmospheric, and reactor
mixing angles are in very good agreement with the experimental data, both for a
normal and an inverted hierarchy. In the particular case where the right handed
neutrinos masses are degenerate, the model predicts a strong inverted hierarchy
spectrum and a sum rule among the neutrino masses. In this case the atmospheric
and solar angles are in very good agreement with experimental data, and the
reactor one is different from zero, albeit too small ($\theta^{\ell^{th}}_{13}
\sim 3.38 $). This value constitues a lower bound for $\theta_{13}$ in the
general case. We also find the range of values for the neutrino masses in each
case.
