The Group T

Up to M-symmetry there are 12 cubes, which exactly have the symmetries of this subgroup. All cubes are generated in 19 moves or less.

The next pretty trivial table gives the number of cubes up to M-symmetry which exactly have T-symmetry, the average maneuver length is 17.1:

Distance
Number
Distance
Number
12f
2
18f
5
17f
2
19f
3

If you are interested in a list of all optimal maneuvers for this subgroup, they are included in the file T.zip.
Name
Generator
 B F L R B' F' D' U' L R D U (12f*)
Name
Generator
 B F L' R' B F D' U' L R D' U' (12f*)
Name
  
Generator
 D B L D L U R F L U F2 R F' D U' R U (17f*)
Name
Generator
 R2 F L R' B F D' L' R' D' F2 L D' U' L' F2 U' (17f*)
Name
Generator
 D' U' L' D' U R2 F U2 B D2 F U2 L' R U R2 F2 U' (18f*)
Name
Generator
 B2 L' R2 D2 B' L' R' B D2 L' B' F' L' D U B F' U' (18f*)
Name
Generator
 U B2 F2 L2 R2 U B F U2 L R U' B2 F2 U2 L2 R2 U' (18f*)
Name
Generator
 D' B' F' D' L R D' B2 F2 L2 R2 U' B F U' L' R' U' (18f*)
Name
Generator
 D U L B' F L' R B2 F' R2 D U L U2 B' F R2 U' (18f*)
Name
Generator
 B D R F R D2 U R B2 U' F' L U B' R2 B' U' R' U (19f*)
Name
Generator
 R' U L R' B2 D2 U2 R' B L F R U F R U' F' R' U' (19f*)
Name
Generator
 F' D' F2 L2 B2 R2 U L' D' U' L' D' U2 B2 R2 U' F' D' U' (19f*)