Rectangular antidot superlattices with ratios of the lattice constants a:a of 2:1 and 10:1 have been investigated by quantum transport calculations. All the features seen in a recent magnetotransport experiment could be reproduced. A strong anisotropy in the longitudinal magnetoresistances and is observed for small magnetic fields: commensurability peaks show up in for current flow through the closely spaced antidots while for the perpendicular direction is flat except for a maximum close to zero magnetic fields. At very low temperatures quantum oscillations out of phase with Shubnikov–de Haas oscillations appear superimposed to the dominant commensurability peak in . The calculations allow us to trace back the strongly anisotropic behavior of and to differences in the magnetic field dependence of band and scattering contributions to the conductivity which are intimately connected with the magnetic field dependent miniband structure. The quantum oscillations are periodic in B for a soft potential with large antidots and periodic in 1/B in the opposite case.