They constructed it as the automorphism group of a certain lattice in the 248-dimensional Lie algebra of E8.
The full normalizer of a 3C element in the Monster group is S3 × Th, so Th centralizes 3 involutions alongside the 3-cycle.
Conway and Norton suggested in their 1979 paper that monstrous moonshine is not limited to the monster, but that similar phenomena may be found for other groups.
Larissa Queen and others subsequently found that one can construct the expansions of many Hauptmoduln from simple combinations of dimensions of sporadic groups.
Linton (1989) found the 16 conjugacy classes of maximal subgroups of Th as follows: