In recent computer simulations of a simple monatomic system interacting via the Dzugutov pair potential, freezing of the fluid into an equilibrium dodecagonal quasicrystal has been reported [M. Dzugutov, Phys. Rev. Lett. 70, 2924 (1993)]. Here, using a combination of molecular dynamics simulation and thermodynamic perturbation theory, we conduct a detailed analysis of the relative stabilities of solid-phase structures of the Dzugutov-potential system. At low pressures, the most stable structure is found to be a bcc crystal, which gives way at higher pressures to a fcc crystal. Although a dodecagonal quasicrystal and a sigma-phase crystal compete with the bcc crystal for stability, they always remain metastable.