We study photonic crystal (PhC)-assisted light extraction from gallium nitride (GaN) light-emitting diodes (LEDs). We focus on the issue of omnidirectional extraction, and we introduce a complex crystal lattice, namely the Archimedean tiling, which enables efficient and omnidirectional light extraction. PhC LEDs with regular triangular lattices and Archimedean lattices are studied experimentally by angle-resolved luminescence, and the theoretical properties of the Archimedean tiling are confirmed experimentally.