Plasmonic oligomer clusters are assemblies of closely packed metallic nanoparticles. They provide a rich set of spectral features such as Fano lineshapes and a simultaneous tunability of the supported resonances in the optical wavelength regime. In this study, we investigate numerically and experimentally clusters of plasmonic nanoparticles that exhibit multiple Fano resonances due to the interference of one broad superradiant mode and multiple narrow subradiant modes. In particular we investigate oligomers with multiple ring modes and elongated chains of nanoparticles surrounded by one ring of nanoparticles. We show that the number of nanoparticles and their respective arrangement in the cluster strongly influence the spectral position and modulation depth of the spectral signature of the supported modes. Our study opens up the pathway to "plasmonic super molecules" that show unprecedented tunability, which renders them highly suitable for applications such as multiwavelength surface-enhanced Raman scattering.