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We present a non-minimal renormalizable SUSY $SU(5)$~model, with extendedHiggs sector and right-handed neutrinos, where the flavour sector exhibits a$Q_{6}$ flavour symmetry. We analysed the simplest version of this model, inwhich R-parity is conserved and the right-handed neutrino masses in the flavourdoublet are considered with and without degeneration. We find the generic formof the mass matrices both in the quark and lepton sectors. We reproduce,according to current data, the mixing in the CKM matrix. In the leptonicsector, in the general case where the right-handed neutrino masses are notdegenerate, we find that the values for the solar, atmospheric, and reactormixing angles are in very good agreement with the experimental data, both for anormal and an inverted hierarchy. In the particular case where the right handedneutrinos masses are degenerate, the model predicts a strong inverted hierarchyspectrum and a sum rule among the neutrino masses. In this case the atmosphericand solar angles are in very good agreement with experimental data, and thereactor 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 thegeneral case. We also find the range of values for the neutrino masses in eachcase.

$Q_{6}$ As The Flavour Symmetry in a Non-minimal SUSY $SU(5)$ Model