{abstract} The supercritical flow states of the spherical Couette flow between two concentric spheres with the inner sphere rotating are investigated via direct numerical simulation using a three-dimensional finite difference method. For comparison with experiments of Nakabayashi \emph{et al.} and Wimmer, a narrow gap and a medium gap with clearance ratio $\beta=0.06$ and 0.18 respectively are considered for the Reynolds number range covering the first Hopf bifurcation point. With adequate initial conditions and temporary imposition of small wave-type perturbation, multiple periodic flow states with three different pair numbers of spiral Taylor-G\"{o}rtler (TG) vortices have been simulated successfully for $\beta=0.06$, of which the 1-pair and 2-pair of spiral TG vortices are newly obtained. Three different periodic flow states with shear waves, Stuart vortices or wavy outflow boundary, have been obtained for $\beta=0.18$. Analysis of the numerical results reveals these higher flow modes in terms of fundamental frequency, wave number and spatial structure.