The dual of the tiling represents the fundamental domains of (*∞44) orbifold symmetry.
From [(∞,4,4)] (*∞44) symmetry, there are 15 small index subgroup (11 unique) by mirror removal and alternation operators.
Mirrors can be removed if its branch orders are all even, and cuts neighboring branch orders in half.
In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors.
The symmetry can be doubled to *∞42 by adding a bisecting mirror across the fundamental domains.