Using numerical arguments, we find that for a tetrahedral configuration and for a dihedral configuration are likely the global energy minimum for Thomson’s problem of minimizing the energy of unit charges on the surface of a unit conducting sphere. These would be the largest by far, outside of the icosadeltahedral series, for which a global minimum for Thomson’s problem is known. We also note that the current theoretical understanding of Thomson’s problem does not rule out a symmetric configuration as the global minima for and 542. We explicitly find that analogues of the tetrahedral and dihedral configurations for larger than 306 and 542, respectively, are not global minima, thus helping to confirm the theory of Dodgson and Moore [Phys. Rev. B 55, 3816 (1997)] that as grows, dislocation defects can lower the lattice strain of symmetric configurations and concomitantly the energy. As well, making explicit previous work by ourselves and others, for we give a full accounting of icosadeltahedral configurations which are not global minima and those which appear to be, and discuss how this listing and our results for the tetahedral and dihedral configurations may be used to refine theoretical understanding of Thomson’s problem.