In many cases, optical metasurfaces are studied in the single-resonant regime. However, a multiresonant behavior can enable multiband devices with reduced footprint, and is desired for applications such as display pixels, multispectral imaging and sensing. Multiresonances are typically achieved by engineering the array lattice (e.g., to obtain several surface lattice resonances), or by adopting a unit cell hosting one (or more than one) nanostructure with some optimized geometry to support multiple resonances. Here, we present a study on how to achieve multiresonant metasurfaces in the visible spectral range by exploiting high-order multipoles in dielectric (e.g., diamond or titanium dioxide) nanostructures. We show that in a simple metasurface (for a fixed particle and lattice geometry) one can achieve triple resonance occurring nearly at RGB (red, green, and blue) wavelengths. Based on analytical and numerical analysis, we demonstrate that the physical mechanism enabling the multiresonance behavior is the lattice induced coupling (energy exchange) between high-order Mie-type multipoles moments of the metasurface’s particles. We discuss the influence on the resonances of the metasurface’s finite size, surrounding material, polarization, and lattice shape, and suggest control strategies to enable the optical tunability of these resonances.