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This study addresses the problem of secondary bifurcations of hexagonal patterns through performing a numerical stability analysis of hexagonal structures in a model of a nonlinear optical system showing such bifurcations. The system under study is based on a single feedback mirror arrangement. A thin nonlinear optical medium (sodium vapor in a nitrogen buffer gas atmosphere) is irradiated by a laser beam which is homogeneous in amplitude and phase. The transmitted beam is retroreflected into the medium by a plane high-reflectivity mirror placed behind the medium. During the propagation of the light field to the mirror and back, different points in the transverse plane are coupled due to diffraction. If the system is suitably prepared, the decisive dynamical variable is the longitudinally averaged orientation which is the normalized population difference between the two Zeeman sublevels of the ground state.