Neutron-diffraction experiments have been performed on a single crystal of alpha-U in the phase below 43 K, in which a charge-density wave (CDW) exists. We report on the identification of the superspace group in the CDW state. This is primitive monoclinic, in agreement with recent theoretical predictions. The CDW state below 37 K consists of two coherent q vectors existing within the same crystal volume, and this coexistence gives rise to additional interference effects. At 22 K we find that qy (the component of the CDW in the [010] direction) locks in to a commensurate value of 1/6. Associated with this lockin, we find higher-order satellites of third, fifth, seventh, and ninth harmonics of the total CDW distortion appearing below 30 K. Elastic diffraction effects extend over six decades of intensity. The presence of higher odd-order harmonics indicates a squaring of the CDW displacement, but this is not complete even at the lowest temperatures. Unusual domain effects are found near the 22-K lock-in phase transition.