Abstract We investigate the energy spectra of three electrons in SiGe/Si/SiGe equilateral triangular and symmetric linear triple quantum dots in the presence of magnetic (in either Faraday or Voigt configuration) and electric fields with only the lowest valley eigenstate being relevant by using the real-space configuration interaction method. The strong electron–electron Coulomb interaction, which is crucial to the energy spectra, is explicitly calculated whereas the weak spin–orbit coupling is treated perturbatively. In both equilateral triangular and symmetric linear triple quantum dots, we find doublet–quartet transition of ground-state spin configuration by varying dot size or interdot distance in the absence of external fields. This transition has not been reported in the literature on triple quantum dots. In the magnetic-field (Faraday configuration) dependence of energy spectra, we find anticrossings with large energy splittings between the energy levels with the same spin state in the absence of the spin–orbit coupling. This anticrossing behavior originates from the triple quantum dot confinement potential. In addition, with the inclusion of the spin–orbit coupling, we find that all the intersections shown in the equilateral triangular case become anticrossing whereas only part of the intersections in symmetric linear case show anticrossing behavior in the presence of magnetic field in either the Faraday or Voigt configuration. All the anticrossing behaviors are analyzed based on symmetry consideration. Moreover, we show that the electric field can effectively influence the energy levels and the charge configurations.