Infrared multiple photon dissociation (IRMPD) spectra of Rh6Om+ (m = 4-10) are obtained in the 300-1000 cm(-1) spectral range using the free electron laser for infrared experiments (FELIX) via dissociation of Rh6Om+ or Rh6Om+-Ar complexes. The spectra are compared with the calculated spectra of several stable geometries obtained by density functional theory (DFT) structural optimization. The spectrum for Rh6O4+ shows prominent bands at 620 and 690 cm(-1) and is assigned to a capped-square pyramidal Rh atom geometry with three bridging O atoms and one O atom in a hollow site. Rh6O5+ displays bands at 460, 630, 690, and 860 cm(-1) and has a prismatic Rh geometry with three bridging O atoms and two O atoms in a hollow site. Rh6O6+ shows three intense bands around 600-750 cm(-1) and multiple weak bands in the range of 350-550 cm(-1). This species has a prismatic Rh geometry with four bridging O atoms and two O atoms in a hollow site. Considering that Rh6Om+ (m <= 3) adopts tetragonal bipyramidal Rh-6 structures, the change at m = 4 to capped bipyramidal and at m = 5 to prismatic geometries results in a reduction of the number of triangular hollow sites. Since NO preferentially binds on a triangular hollow site through the N atom, the geometry change lowers the possibility of NO dissociative adsorption.