We investigate second-order methods for nonconvex optimisation, and propose a Random Subspace Adaptive Cubic Regularisation (R-ARC) method, which we analyse under various assumptions on the objective function and the sketching matrices that generate the random subspaces. We show that, when the sketching matrix achieves a subspace embedding of the augmented matrix of the gradient and the Hessian with sufficiently high probability, then the R-ARC method satisfies, with high probability, a complexity bound of order O(ε^{-3/2}) to drive the (full) gradient norm below ε; matching in the accuracy order its deterministic counterpart (ARC). As an illustration, we particularise our results to the special case of a scaled Gaussian ensemble.