Symmetry

Truncated tetraoctagonal tiling with *842, , mirror lines

There are 15 subgroups constructed from [8,4] by mirror removal and alternation. 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 subgroup index-8 group, [1⁺,8,1⁺,4,1⁺] (4242) is the commutator subgroup of [8,4].

A larger subgroup is constructed as [8,4*], index 8, as [8,4⁺], (4*4) with gyration points removed, becomes (*4444) or (*4⁴), and another [8*,4], index 16 as [8⁺,4], (8*2) with gyration points removed as (*22222222) or (*2⁸). And their direct subgroups [8,4*]⁺, [8*,4]⁺, subgroup indices 16 and 32 respectively, can be given in orbifold notation as (4444) and (22222222).

Small index subgroups of [8,4] (*842)
Index	1	2			4
Diagram
Coxeter	[8,4] =	[1+,8,4] =	[8,4,1+] = =	[8,1+,4] =	[1+,8,4,1+] =	[8+,4+]
Orbifold	*842	*444	*882	*4222	*4242	42×
Semidirect subgroups
Diagram
Coxeter		[8,4+]	[8+,4]	[(8,4,2+)]	[8,1+,4,1+] = = = =	[1+,8,1+,4] = = = =
Orbifold		4*4	8*2	2*42	2*44	4*22
Direct subgroups
Index	2	4			8
Diagram
Coxeter	[8,4]+ =	[8,4+]+ =	[8+,4]+ =	[8,1+,4]+ =	[8+,4+]+ = [1+,8,1+,4,1+] = = =
Orbifold	842	444	882	4222	4242
Radical subgroups
Index		8	16		32
Diagram
Coxeter		[8,4*] =	[8*,4]	[8,4*]+ =	[8*,4]+
Orbifold		*4444	*22222222	4444	22222222