File:Z2^3; Lattice of subgroups Hasse diagram adjacency matrix (sequence).svg

Among the 8x8 matrices in files like there are sixteen, that describe equivalence relations.
(The power of two is a coincidence. There are five 4x4 matrices of this kind, and sixty seven 16x16 matrices.)
This matrix shows, how they are ordered by the refinement of relation,
and thus it's substantially equivalent to a Hasse diagram:

The strong red fields correspond to the edges in a Hasse diagram.
There are 35 strong red fields and 66 red fields in all.
This matrix is symmetrical to the subdiagonal.

The small black numbers over the 8x8 matrices are their bottom rows, read as binary numbers. Their numerical order justifies the bigger gray numbers, starting from 0, which simply denote the matrices' position in this sequence.
This matrix is contained in it's bigger equivalent File:Boolean partition lattice 4.svg.

These 8x8 matrices' top rows:

1 0 0 0 0 0 0 0 (1)
1 1 0 0 0 0 0 0 (2)
1 0 1 0 0 0 0 0 (2)
1 0 0 1 0 0 0 0 (2)
1 1 1 1 0 0 0 0 (4)
1 0 0 0 1 0 0 0 (2)
1 0 0 0 0 1 0 0 (2)
1 1 0 0 1 1 0 0 (4)
1 0 0 0 0 0 1 0 (2)
1 0 1 0 1 0 1 0 (4)
1 0 0 1 0 1 1 0 (4)
1 0 0 0 0 0 0 1 (2)
1 0 0 1 1 0 0 1 (4)
1 0 1 0 0 1 0 1 (4)
1 1 0 0 0 0 1 1 (4)
1 1 1 1 1 1 1 1 (8)

(The binary digit sum is shown in parentheses.)

The elements can be ordered in different ways. The probably best way is shown to the right.
There's a bigger equivalent of this lattice, shown in the line below.

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