inversion symmetry, local Chern invariants have the values of ±1/2 at around K and K' valleys, leading to the existence of valley-locked edge states at the boundaries between two valley-Hall insulators with valley-Chern numbers of opposite signs at K (K') valleys [20,21] . Due to the robustness of the edge states against the structural disorder and the feasibility of fabricating such structures, valley-Hall photonic topological insulators have found promising applications for beam splitting, [31] topological lasing, [32][33][34][35] and frequency conversion. [36] Although the various hexagonal photonic lattices have been revealed to exhibit valley-Hall topological effects, such effects have been rarely investigated in square lattices, since they have non-zero Berry curvatures at non-symmetric points in the first Brillouin zone.Recently, higher-order topological insulating phases have emerged as an important research topic in topological physics, including topological photonics [37][38][39][40][41] and phononics. [42] When a parameter of a photonic topological insulator is changed, gapless edge bands are open above a certain value. For further parameter change, gapped edge states appear accompanying with corner or hinge states, which have dimensions lower than the edge states. As an example, the 2D second-order topological insulators generate zero-dimensional corner states as well as 1D edge states immune to the structural defects, enabling robust photonic nanocavities [43] and nanolasers. [44] In photonic practices, one needs multiband edge and corner states, in particular for nonlinear topological photo nics which frequently requires operating over wide spectral ranges. Up to date, multiband edge states [45,46] in photonic topological systems and their applications for nonlinear optical frequency conversion [36] have been reported, while multiband corner states have still not been investigated. In this work, we reveal 2D photonic crystals with square lattices of triangular dielectric rods to exhibit multiband edge and corner states depending on the structural parameters. For small constituent dielectric rods, there exist gapless multiband edge states immune to the structural disorder. With increasing their sizes, multiple photonic bandgaps appear, resulting in simultaneous occurrence of multiband edge and corner states. By evaluating the eigenstates and their excitation characteristics, we show that the multiband corner states are robust against structural defects. The results presented in this work can find important applications for nonlinear topological frequency conversion.