Superexchange effects play an important role in the determination of crystal structures; however, there has been much less reported on how they determine the stability of clusters. Using evolutionary search strategies and DFT+U (density functional theory with the Hubbard U correction) calculations, we investigate the global minimum-energy structures of Fe₁₂O_n clusters. Among predicted Fe₁₂O_n clusters, a cage-shaped Fe₁₂O₁₂ cluster with unexpected stability was observed. In addition, the bare Fe₁₂O₁₂ cluster is shown to possess an extremely large energy gap (2.00 eV), which is greater than that of C₆₀, Au₂₀ and Al_13-clusters. Using a Heisenberg model, we traced the origin of the unexpected stability of the bare Fe₁₂O₁₂ cluster to magnetic competition between the nearest-neighbor exchange constant J₁ and the next-nearest neighbor exchange constant J₂ that was induced by the superexchange interactions. The bare Fe₁₂O₁₂ cluster is thus a unique molecule that is stable and chemically inert.