Symmetry

The dual of the tiling represents the fundamental domains of (*664) orbifold symmetry. From [(6,6,4)] (*664) 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. Removing two mirrors leaves a half-order gyration point where the removed mirrors met. 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 862 symmetry by adding a bisecting mirror across the fundamental domains. The subgroup index-8 group, [(1⁺,6,1⁺,6,1⁺,4)] (332332) is the commutator subgroup of [(6,6,4)].

A large subgroup is constructed [(6,6,4^*)], index 8, as (4*33) with gyration points removed, becomes (*3⁸), and another large subgroup is constructed [(6,6^*,4)], index 12, as (6*32) with gyration points removed, becomes (*(32)⁶).

Small index subgroups of [(6,6,4)] (*664)

Fundamental domains
Subgroup index	1	2			4
Coxeter	[(6,6,4)]	[(1+,6,6,4)]	[(6,6,1+,4)]	[(6,1+,6,4)]	[(1+,6,6,1+,4)]	[(6+,6+,4)]
Orbifold	*664	*6362		*4343	2*3333	332×
Coxeter		[(6,6+,4)]	[(6+,6,4)]	[(6,6,4+)]	[(6,1+,6,1+,4)]	[(1+,6,1+,6,4)]
Orbifold		6*32		4*33	3*3232
Direct subgroups
Subgroup index	2	4			8
Coxeter	[(6,6,4)]+	[(1+,6,6+,4)]	[(6+,6,1+,4)]	[(6,1+,6,4+)]	[(6+,6+,4+)] = [(1+,6,1+,6,1+,4)] =
Orbifold	664	6362		4343	332332