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We have performed unconstrained search for low-lying structures of medium-sized silicon clusters Si31–Si40 and Si45, by means of the minimum-hopping global optimization method coupled with a density-functional based tight-binding model of silicon. Subsequent geometric optimization by using density-functional theory with the PBE, BLYP, and B3LYP functionals was carried out to determine the relative stability of various candidate low-lying silicon clusters obtained from the unconstrained search. The low-lying characteristics of these clusters can be affirmed by comparing the binding energies per atom of these clusters with previously determined lowest-energy clusters (Sin) in the size range of 21≤n≤30. In view of the fact that there exist numerous low-lying “endohedral fullerenelike” isomers for each size in the range 30≤n≤40, we used the homologue carbon-fullerene cage to classify different families of isomers. This structural classification allows us to focus on generic features of various isomers and to group many apparently different isomers into a single family. In addition, we report a new family of low-lying clusters which have “Y-shaped three-arm” structures. Isomers in this “handmade” family can be energetically competitive as the endohedral fullerene isomers when the total energies are calculated with the BLYP or B3LYP functional.