Strong light-matter coupling is demonstrated at low temperature in an ultrathin GaN microcavity fabricated using two silica/zirconia Bragg mirrors, in addition to a three-period epitaxial (Al,Ga)N mirror serving as an etch stop and assuring good quality of the overgrown GaN. The lambda/2 cavity is grown by molecular beam epitaxy on a Si substrate. Analysis of angle-resolved data reveal key features of the strong coupling regime in both reflectivity and transmission spectra at 5 K: anticrossing with a normal mode splitting of 43 +/- 2 meV and 56 +/- 2 meV for reflectivity and transmission, respectively, and narrowing of the lower polariton linewidth near resonance.