8 Clarified explanation in response to a comment
the nonempty vertices. Since sections are equivalent under permutations of the coordinates, we should consider these to be 8 identical elements (so the only thing that matters is how many elements are on a vertex). There are four cases.

7 fixed minor mistake
{$5,6$}, the vertex $\bar{a}\bar{b}\bar{c}$ would contain {$8$}, $4,8$}, and the