The structural and thermodynamic properties of the hexagonal, tetragonal, and orthorhombic phases of SiCN under high pressure are investigated by first-principles study based on the pseudo-potential plane-wave density functional theory method. The calculated equilibrium lattice constants, bulk modulus and elastic constants at zero pressure agree well with the previous theoretical values. The t-SiCN exhibits an indirect band gap with a value of 1.67 eV. It is found that with increasing pressure, the Debye temperature Θ_D of the o-SiCN and h-SiCN increase, whereas the one of the t-SiCN decreases. Furthermore, the o-SiCN is found to be a brittle material up to 60 GPa, while for t-SiCN and h-SiCN, the change from the brittle to ductile state occurs at about 17.04 GPa and 40.55 GPa, respectively. The calculated anisotropy factors demonstrate that both the o-SiCN and h-SiCN have a weak anisotropy up to 60 GPa, while the t-SiCN exhibits a high degree of anisotropy in shear but only a small anisotropy in compressibility. The ideal tensile and shear strength at large strains of the three phases are examined to further understand the microscopic mechanism of the structural deformation. It is found that all the SiCN compounds have a low ideal strength within 40 GPa, revealing that they may not be intrinsically superhard.