In this paper we analyze in detail the secondary bifurcations of stationary hexagonal patterns in a prototype model of nonlinear optics. Hexagonal pattern solutions with all allowed wavenumbers are computed and their linear stability is studied by means of a Bloch analysis. Depending on the wavenumber of the selected pattern we predict and numerically observe phase instabilities, amplitude instabilities, both stationary and oscillatory, and oscillatory finite wavelength bifurcations. The results presented here illustrate a typical bifurcation scenario for patterns with different wavenumbers in self-focusing systems.