A two-dimensional chessboard of non-Bravais lattice is designed in this paper: the square columns in photonic structure of a two-dimensional square lattice are rotated by 45°,and a cylinder is introduced into the center of each lattice unit cell. The complete photonic band gap in two-dimensional chessboard of non-Bravais lattice is calculated by the plane wave expansion method. The result shows that a gap width to midgap frequency ratio Δω/ω of chessboard non-Bravais lattice is almost 5 times that of the ordinary chessboard lattice, and the number of complete photonic band gaps is increased. Comparaed with other compound lattices, the optimal Δω/ω of chessboard non-Bravais lattice is 2.1 times that of a kind of compound lattices.