We demonstrate all-optical switching using a multi-mode membranized photonic crystal nanocavity exploiting the free-carrier induced dispersion in InP and the sharp asymmetric lineshape of Fano resonances. A multi-mode cavity is designed to sustain two spatially overlapping modes with a spectral spacing of 18 nm. The measured transmission spectrum of the fabricated device shows multiple asymmetric Fano resonances as predicted by optical simulations. The capabilities of the device are benchmarked by comparing a wavelength conversion from 1538.2 nm to 1565.2 nm with a single-mode wavelength conversion at 1566.2 nm on the same device. The results show an improvement in signal quality with a 5.6 dB power penalty reduction at the receiver as well as in energy efficiency with a reduction of the pump power from 534 fJ/bit to 445 fJ/bit.