Finite Group DatabasebetaFinite Group DatabasebetaOrder 8576.
656 conjugacy classes of subgroups; only the 165 normal subgroups are shown.
This group has 656 conjugacy classes of subgroups — too many to draw. Only its 165 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D4×D67 |
| commutator | C2×C4×C67 | C24 |
| Frattini | C2×C4 | C23×D67 |
| Fitting | #492×C67 | C2 |
| socle | C22×C67 | C22×D4 |
| hypercenter | #492 | D67 |
| hypocenter | C67 | C2×#492 |
| LCS | G ▷ C2×C4×C67 ▷ C2×C67 ▷ C67 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #492 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×SD16×D67 | C2 | semidirect |
| 2. C4⋊Q8×D67 | C2 | semidirect |
| 4. #16863770 | C2 | semidirect |
| 5. #16863769 | C2 | semidirect |
| 6. #16863768 | C2 | semidirect |
| 7. #16863767 | C2 | semidirect |
| 8. C4×C8×D67 | C2 | semidirect |
| 9. #492×C67 | C2 | semidirect |
| 13. C2×#11761617 | C22 | semidirect |
| 14. C2×#11761616 | C22 | semidirect |
| 15. C2×#11761615 | C22 | semidirect |
| 18. #14524043 | C22 | semidirect |
| 20. #14524041 | C22 | semidirect |
| 26. C4×C67⋊C8 | C22 | semidirect |
| 27. C8×Dic67 | C22 | semidirect |
| 28. C2×SD16×C67 | C22 | semidirect |
| 29. C4⋊Q8×C67 | C22 | semidirect |
| 31. C4×C8×C67 | C22 | semidirect |
| 49. C8×D67 | D4 | semidirect |
| 80. C4×D67 | SD16 | semidirect |
| 84. C8×C67 | C2×D4 | semidirect |
| 85. C67⋊C8 | C2×D4 | semidirect |
| 95. C4×C67 | C2×SD16 | semidirect |
| 99. Dic67 | C2×SD16 | semidirect |
| 104. D67 | #492 | direct (N × H) |
| 106. C67 | C2×#492 | semidirect |
| 114. #492 | D67 | direct (N × H) |
| 123. C2×SD16 | C2×D67 | semidirect |
| 128. C4⋊Q8 | C2×D67 | semidirect |
| 137. C4×C8 | C2×D67 | semidirect |
| 191. C8 | D4×D67 | semidirect |
| 204. C4 | SD16×D67 | semidirect |