Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
616 conjugacy classes of subgroups; only the 392 normal subgroups are shown.
This group has 616 conjugacy classes of subgroups — too many to draw. Only its 392 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C22 | C25 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #559 | C2 | semidirect |
| 2. #550 | C2 | semidirect |
| 3. #546 | C2 | semidirect |
| 4. #540 | C2 | semidirect |
| 5. #535 | C2 | semidirect |
| 6. #533 | C2 | semidirect |
| 7. #532 | C2 | semidirect |
| 8. C2×#127 | C2 | semidirect |
| 9. #518 | C2 | semidirect |
| 10. C4×#41 | C2 | semidirect |
| 12. C2×#41 | C22 | semidirect |
| 15. C4⋊Q8 | C22 | semidirect |
| 19. #127 | C22 | semidirect |
| 20. #125 | C22 | semidirect |
| 21. #124 | C22 | semidirect |
| 22. #124 | C22 | semidirect |
| 23. #123 | C22 | semidirect |
| 25. #121 | C22 | semidirect |
| 26. #121 | C22 | semidirect |
| 27. #121 | C22 | semidirect |
| 29. C4×Q8 | C22 | semidirect |
| 30. C4×Q8 | C22 | semidirect |
| 32. C4×D4 | C22 | semidirect |
| 33. C4×D4 | C22 | semidirect |
| 34. C4×D4 | C22 | semidirect |
| 35. C4×D4 | C22 | semidirect |
| 38. #117 | C22 | semidirect |
| 41. C2×C42 | C22 | semidirect |
| 95. C2×C4 | #41 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #559 | 2 |
| #550 | 2 |
| #546 | 2 |
| #540 | 2 |
| #535 | 2 |
| #533 | 2 |
| #532 | 2 |
| C2×#127 | 2 |
| #518 | 2 |
| C4×#41 | 2 |