Three-dimensional turbulent magnetoconvection at a Rayleigh number of Ra = 10 7 in liquid gallium at a Prandtl number P r = 0.025 is studied in a closed square cell for very strong external vertical magnetic fields B 0 in direct numerical simulations which apply the quasistatic approximation. As B 0 or equivalently the Hartmann number Ha are increased, the convection flow that is highly turbulent in the absence of magnetic fields crosses the Chandrasekhar linear stability limit for which thermal convection is ceased in an infinitely extended layer and which can be assigned with a critical Hartmann number Ha c . Similar to rotating Rayleigh-Bénard convection, our simulations reveal subcritical sidewall modes that maintain a small but finite convective heat transfer for Ha > Ha c . We report a detailed analysis of the complex two-layer structure of these wall modes, their extension into the cell interior and a resulting sidewall boundary layer composition that is found to scale with the Shercliff layer thickness.