All‐dielectric nanoantennas enable the manipulation and confinement of light on the nanoscale through a variety of resonant excitations. Understanding and controlling the interactions among resonances at a level of individual nanoantennas is required for the development of appropriate design rules for their applications, especially with regard to nonlinear optics, where the nonlinear interaction depends on the resonant behavior in a superlinear manner. Here, cathodoluminescence imaging spectroscopy is used to experimentally investigate the electric and magnetic modes supported by single AlGaAs nanoantennas and their hybridization in a dimer configuration. An interplay is shown between the resonant modes of the nanostructures and interband emission of the semiconductor material, which contrasts with Si nanoantennas, where the bandgap effects are not significant. The measurements are in very good agreement with numerical simulations, which also provided further insight into the nature of the modes. The resonant modes of semiconductor nanoantennas are important for engineering linear and nonlinear metasurfaces, as well as for sensing applications and fluorescence control in nanophotonic environments.