The plane-wave expansion method is used to calculate the band structure of a two-dimensional photonic crystalformed by a hexagonal structure of anisotropic cylinders. Two cylindrical inclusions in the unit cell have two differentradii, R1 and R2 (Ri＜R2). By reducing the symmetry of the structure and choosing appropriately parameters R2and s = R1/R2 (s＜1), we obtain six large complete bandgaps, among which three are over 0.05 ωe (where ωe = 2πc/a)in the high region of the normalized frequency (however, one of these over 0.065 ωe is not stable). There are two otherstable complete bandgaps in the low-frequency region.