We construct, by numerical means, static solutions of the spherically symmetric Einstein–Vlasov–Maxwell system and investigate various features of the solutions. This extends a previous investigation (Andréasson and Rein 2007 Class. Quantum Grav. 24 1809) of the chargeless case. We study the possible shapes of the energy density profile as a function of the area radius when the electric charge of an individual particle is varied as a parameter. We find profiles which are multi-peaked, where the peaks are separated either by vacuum or a thin atmosphere, and we find that for a sufficiently large charge parameter the solutions break down at a finite radius. Furthermore, we investigate the inequality



which is derived in Andréasson (2009 Commun. Math. Phys. 288 715) for general matter models, and we find that it is sharp for the Einstein–Vlasov–Maxwell system. Here M is the ADM mass, Q is the charge and R is the area radius of the boundary of the static object. We find two classes of solutions with this property, while there is only one in the chargeless case. In particular we find numerical evidence for the existence of arbitrarily thin shell solutions to the Einstein–Vlasov–Maxwell system. Finally, we consider one-parameter families of steady states, and we find spirals in the mass–radius diagram for all examples of the microscopic equation of state which we consider.