Subgroup structure of elementary abelian group:E8

We consider here the elementary abelian group:E8. For notational simplicity, we consider this group as $$\mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2$$ and represente its elements as ordered 3-tuples with entries from the integers mod 2 with coordinate-wise addition mod 2.

Lattice of subgroups
The lattice of subgroups is bounded at both ends by the trivial subgroup and whole group. Ignoring these, the rest of the lattice can be viewed as a bipartite graph between the subgroups of order two and the subgroups of order four. If we think of elementary abelian group:E8 as a vector space over field:F2, then the subgroups of order two are 1-dimensional affine subspaces (lines) and the subgroups of order four are 2-dimensional affine subspaces (planes). The containment relation of these is captured by looking at the projective plane over field:F2, which is a geometry where:


 * the points are the one-dimensional affine subspaces or lines (i.e., the order two subgroups)
 * the lines are the two-dimensional affine subspaces or planes (i.e., the order four subgroups), and
 * the incidence relation is defined by containment of the point (actually, a line) inside a line (actually, a plane)

The projective plane over the field of two elements is a Fano plane and its picture is below. The thickened dots represent the points (one-dimensional affine subspaces) and the lines (including one drawn as a circle) represent the lines. A point is incident to a line if and only if the corresponding order two subgroup is incident to the corresponding order four subgroup.