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In this article, we describe some maximal $3$-local subgroups of the Monstersimple group using vertex operator algebras (VOA). We first study theholomorphic vertex operator algebra obtained by applying the orbifoldconstruction to the Leech lattice vertex operator algebra and a lift of afixed-point free isometry of order $3$ of the Leech lattice. We also considersome of its special subVOAs and study their stabilizer subgroups using thesymmetries of the subVOAs. It turns out that these stabilizer subgroups are$3$-local subgroups of its full automorphism group. As one of our main results,we show that its full automorphism group is isomorphic to the Monster simplegroup by using a $3$-local characterization and that the holomorphic VOA isisomorphic to the Moonshine VOA. This approach allows us to obtain relativelyexplicit descriptions of two maximal $3$-local subgroups of the shape$3^{1+12}.2.\Suz{:}2$ and $3^8.\Omega^-(8,3).2$ in the Monster simple group.

$\Z_3$-orbifold construction of the Moonshine vertex operator algebra and some maximal $3$-local subgroups of the Monster