There is only one 6-quartic graph: the octahedron. This polyhedron has 6 vertices, 12 edges, and 8 faces. Thus, there is a corresponding 8-loop planar Feynman diagram.
If each body is labeled with an integer from 1 to 6, then one contribution to this 6-body planar potential includes three sets of bonds. The first set describes a hexagon cycle:
$$G_{12} \quad G_{23} \quad G_{34} \quad G_{45} \quad G_{56} \quad G_{61}$$
The second set describes a triangle cycle:
$$G_{13} \quad G_{35} \quad G_{51}$$
The third set also describes a triangle cycle:
$$G_{24} \quad G_{46} \quad G_{62}$$
One can think of the 6 bodies as forming a necklace with 6 beads. There are 60 distinct possible necklaces, so one would need to sum over all possible permutations to obtain the potential. Good luck with that!