The centrifugal instability of a Stokes layer has been investigated by Seminara & Hall (1976, 1977). It was found that the Stokes layer on a torsionally oscillating circular cylinder is unstable to perturbations periodic along the axis of the cylinder when the Taylor number for the flow exceeds a certain critical value. The weakly nonlinear theory given by Seminara & Hall showed that, if nonlinear effects are considered, at this Taylor number a stable axially periodic equilibrium flow bifurcates from the basic circumferential flow. It is known experimentally that this equilibrium flow becomes unstable to disturbances having a longer axial wavelength at a second critical Taylor number about 10% greater than the first critical value. Moreover it is known that, in the initial stages of this destabilization, a mode having twice the axial wavelength of the fundamental is present. In this paper we investigate the linear stability of the bifurcating solution to such a subharmonic mode. An approximate solution of the linear stability problem shows that the subharmonic becomes unstable at a Taylor number remarkably close to the experimentally measured second critical Taylor number.