Perpendicularly magnetized materials have attracted significant interest owing to their high anisotropy, which gives rise to extremely narrow, nanosized domain walls. As a result, the recently studied current-induced domain wall motion (CIDWM) in these materials promises to enable a new class of data, memory and logic devices. Here we propose the spin Hall effect as an alternative mechanism for CIDWM. We are able to carefully tune the net spin Hall current in depinning experiments on Pt/Co/Pt nanowires, offering unique control over CIDWM. Furthermore, we determine that the depinning efficiency is intimately related to the internal structure of the domain wall, which we control by the application of small fields along the nanowire. This manifestation of CIDWM offers an attractive degree of freedom for manipulating domain wall motion by charge currents, and sheds light on the existence of contradicting reports on CIDWM in perpendicularly magnetized materials.