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We study dynamo action using numerical simulations of planar Boussinesq convection at rapid rotation (low Ekman numbers, Ek), focusing on subcritical dynamo action in which the dynamo is sustained for Rayleigh numbers, Ra, below the critical Rayleigh number for the onset of nonmagnetic convection, Ra$_c$. These solutions are found by first investigating the supercritical regime, in which the dynamo is able to generate a large-scale magnetic field that significantly influences the convective motions, with an associated Elsasser number of order Ek$^{1/3}$. Subcritical solutions are then found by tracking this solution branch into the subcritical regime, taking a supercritical solution and then gradually lowering the corresponding Rayleigh number. We show that decreasing the Ekman number leads to an extension of the subcritical range of Ra/Ra$_c$, down to an optimal value of Ek$=10^{-5}$. For magnetic Prandtl numbers of order unity, subcriticality is then hampered by the emergence of a large-scale mode at lower Ekman numbers when the dynamo driven by the smaller scale convection generates relatively stronger large-scale magnetic field. The inability of the large-scale mode to sustain dynamo action leads to an intermittent behaviour that appears to inhibit subcriticality. The subcritical solutions are also sensitive to the value of the magnetic Reynolds number (or equivalently, the magnetic Prandtl number, Pm), as values of the magnetic Reynolds number greater than 70 are required to produce dynamo action, but large values lead to fluctuations that are able to push the system too far from the subcritical branch and towards the trivial conducting state.