Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
526 conjugacy classes of subgroups; only the 388 normal subgroups are shown.
This group has 526 conjugacy classes of subgroups — too many to draw. Only its 388 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 |
| prime radical | #534 | C2 |
| 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. #552 | C2 | semidirect |
| 2. #551 | C2 | semidirect |
| 3. #543 | C2 | semidirect |
| 4. #539 | C2 | semidirect |
| 6. #529 | C2 | semidirect |
| 7. C2×#123 | C2 | semidirect |
| 8. #518 | C2 | semidirect |
| 9. C2×#117 | C2 | semidirect |
| 12. #126 | C22 | semidirect |
| 13. #126 | C22 | semidirect |
| 14. #125 | C22 | semidirect |
| 15. #125 | C22 | semidirect |
| 16. #124 | C22 | semidirect |
| 19. #123 | C22 | semidirect |
| 20. #123 | C22 | semidirect |
| 21. #123 | C22 | semidirect |
| 22. C22⋊Q8 | C22 | semidirect |
| 23. C22⋊Q8 | C22 | semidirect |
| 24. C22⋊Q8 | C22 | semidirect |
| 27. C4×Q8 | C22 | semidirect |
| 29. C4×D4 | C22 | semidirect |
| 30. C4×D4 | C22 | semidirect |
| 31. #117 | C22 | semidirect |
| 32. #117 | C22 | semidirect |
| 33. #117 | C22 | semidirect |
| 34. C2×C4⋊C4 | C22 | semidirect |
| 35. C2×C4⋊C4 | C22 | semidirect |
| 39. C2×C22⋊C4 | C22 | semidirect |
| 40. C2×C22⋊C4 | C22 | semidirect |
| 41. C2×C42 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #552 | 2 |
| #551 | 2 |
| #543 | 2 |
| #539 | 2 |
| #534 | 2 |
| #529 | 2 |
| C2×#123 | 2 |
| #518 | 2 |
| C2×#117 | 2 |