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We continue the program to make a moonshine path between a node of theextended $E_8$-diagram and the Monster. Our theory is a concrete modelexpressing some of the mysterious connections identified by John McKay, GeorgeGlauberman and Simon Norton. In this article, we treat the 3A and 6A-nodes. Wedetermine the orbits of triples $(x,y,z)$ in the Monster where $z\in 2B$, $x, y\in 2A \cap C(z)$ and $xy\in 3A \cup 6A$. Such $x, y$ correspond to a rootless$EE_8$-pair in the Leech lattice. For the 3A and 6A cases, we shall saysomething about the "half Weyl groups", which are proposed in theGlauberman-Norton theory. Most work in this article is with lattices, due totheir connection with dihedral subgroups of the Monster. These lattices are$M+N$, where $M, N$ is the relevant pair of $EE_8$-sublattices, and theirannihilators in the Leech lattice. The isometry groups of these four latticesare analyzed.

Moonshine paths for 3A and 6A nodes of the extended E8-diagram