In this Letter, we present hybrid strong coupling between multiple photonic modes and excitons in an organic-dye-attached photonic quasicrystal. The excitons effectively interact with the photonic modes offered by the photonic quasicrystal, and multiple hybrid polariton bands are demonstrated in both experiments and calculations. Furthermore, by retrieving the measured dispersion map, we get the mixing fractions of photonic modes and excitons, and show that the polariton bands inherit not only the energy dispersion features, but also the damping behaviors from both the photonic modes and the excitons. Our investigation may inspire related studies on multimode light-matter interactions and achieve some potential applications for multimode sensors.