The emergent magnetism in close-packed assemblies of interacting superparamagnetic particles is commonly referred to as supermagnetism. The magnetic characteristics of such systems are determined by the dipolar coupling between the nanomagnets, rather than the exchange interaction responsible for ferro- and antiferromagnetism in continuous material. The dipolar coupling facilitates tuning of the magnetism, which renders supermagnetic ensembles suitable model systems for exploration of new physics. In this work, we discuss micromagnetic simulations of regular arrays of thin film nanomagnets, with magnetic material parameters typical of the ferromagnetic oxide La0.7Sr0.3MnO3. The ground state supermagnetic order in these systems is primarily determined by the lattice configuration, in that a square lattice results in antiferromagnetic order, whereas a triangular lattice shows ferromagnetic order. We found that a square lattice of circular nanomagnets may be switched from superferromagnetic to superantiferromagnetic order by a small external field applied in the appropriate direction.