First principles calculations are preformed to systematically investigate the electronic structures, elastic and thermodynamic properties of the monoclinic and orthorhombic phases of SiC2N4 under pressure. The calculated structural parameters and elastic moduli are in good agreement with the available theoretical values at zero pressure. The elastic constants of the two phases under pressure are calculated by stress-strain method. It is found that both phases satisfy the mechanical stability criteria within 60 GPa. With the increase of pressure, the degree of the anisotropy decreases rapidly in the monoclinic phase, whereas it remains almost constant in the orthorhombic phase. Furthermore, using the hybrid density-functional theory, the monoclinic and orthorhombic phases are found to be wide band-gap semiconductors with band gaps of about 2.85 eV and 3.21 eV, respectively. The elastic moduli, ductile or brittle behaviors, compressional and shear wave velocities as well as Debye temperatures as a function of pressure in both phases are also investigated in detail.