Crystal structures are very different from the atomic structure in the gaseous state, so the high-order harmonic generation (HHG) from the crystal irradiated by an in intense laser is also different from that of an atom exposed to a strong laser field. By simulating the dynamics of a single active electron in periodic potentials based on the expansion method of the basis functions, we study the HHG in crystals and find, in certain wave lengths and intensity of the laser, that solid harmonic generation exhibits the characteristics of double plateaus. After analyzing the induced electric current of laser field, which is the source of HHG in the crystal, we find that the first plateau of HHG arises chiefly from the electric current between the lowest conduction band and the valence band (electron-hole recollision), and the second plateau is predominantly due to electric current between higher conduction bands and the valence band (electron-hole recollision). The cutoff energies of the two plateaus vary approximately linearly with the laser field strength. Furthermore, by considering the crystal driven by the few-cycle laser pulse, the cutoff energy of the second plateau changes monotonously with carrier-envelope phases. Based on this phenomenon, it can be a way to measure the carrier-envelope phases of the few-cycle laser pulse. Finally, we study further the HHG from crystals driven by the chirped laser and find that it has a great influence on the HHG, and the second plateau of HHG is sensitive to the chirp parameter. According to this phenomenon, we propose a novel way that is capable of greatly improving the emission efficiency of the second plateau by changing the chirp parameter of the driving laser.