In geometry, the truncated tetraoctagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one octagon, and one hexakaidecagon on each vertex. It has Schläfli symbol of tr{8,4}.
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<sup>+</sup>,8,1<sup>+</sup>,4,1<sup>+</sup>] (4242) is the commutator subgroup of [8,4].
A larger subgroup is constructed as [8,4*], index 8, as [8,4<sup>+</sup>], (4*4) with gyration points removed, becomes (*4444) or (*4<sup>4</sup>), and another [8*,4], index 16 as [8<sup>+</sup>,4], (8*2) with gyration points removed as (*22222222) or (*2<sup>8</sup>). And their direct subgroups [8,4*]<sup>+</sup>, [8*,4]<sup>+</sup>, subgroup indices 16 and 32 respectively, can be given in orbifold notation as (4444) and (22222222).
From a Wythoff construction there are fourteen hyperbolic uniform tilings that can be based from the regular order-4 octagonal tiling.
Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 7 forms with full [8,4] symmetry, and 7 with subsymmetry.