Fragile topology (FT) opens a new direction in topological photonics, but a new type of photonic crystal (PC) with FT remains to be proposed. In this Letter, the double-site honeycomb lattice (DSHL) PC is proposed by rotating the double dielectric rods (DDR) six times, forming unit cell, and then arraying the unit cells in a triangular lattice. Quantum spin Hall effect occurs by manipulating the DDR in the tangential and radial directions of the unit cell. First, the band structures of DSHL PCs with different structural parameters are calculated, and the laws of topological phase transition are analyzed statistically. Then, to prove the FT properties of two groups of topological nontrivial DSHL PCs, the Wannier-center positions of the bulk bands are calculated by the Wilson-Loop method. Finally, the topological edge states and two groups of topological corner states, which are in the same bulk-state bandgap, are realized successfully. The DSHL PC provides good platforms for both the research of topological photonics and the device design and application, which has a broad prospect.