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We have studied the excitation of higher-order modes and their role in supercontinuum generation in a three-hole silica suspended-core fiber, both experimentally and numerically. We find that pump coupling optimized to highest transmission can yield substantial excitation of higher order modes. With up to about 40% of the pump power coupled to higher order modes, we have studied supercontinuum generation in this fiber. In agreement with experiments, simulation results based on the multimode generalized nonlinear Schrödinger equation confirm that the spectral width is determined by spectral broadening in the fundamental mode, whereas the numerical analysis reveals that intermodal nonlinear interactions are strongly suppressed.