Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
507 conjugacy classes of subgroups; only the 384 normal subgroups are shown.
This group has 507 conjugacy classes of subgroups — too many to draw. Only its 384 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 | #525 | 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. #563 | C2 | semidirect |
| 2. #552 | C2 | semidirect |
| 3. #551 | C2 | semidirect |
| 4. D4×Q8 | C2 | semidirect |
| 5. #544 | C2 | semidirect |
| 6. #541 | C2 | semidirect |
| 7. #539 | C2 | semidirect |
| 8. #536 | C2 | semidirect |
| 10. C2×C22⋊Q8 | C2 | semidirect |
| 13. C22×Q8 | C22 | semidirect |
| 15. C4⋊Q8 | C22 | semidirect |
| 16. C4⋊Q8 | C22 | semidirect |
| 17. C4⋊Q8 | C22 | semidirect |
| 19. #126 | C22 | semidirect |
| 20. #125 | C22 | semidirect |
| 21. #125 | C22 | semidirect |
| 22. #125 | C22 | semidirect |
| 27. #123 | C22 | semidirect |
| 34. C22⋊Q8 | C22 | semidirect |
| 35. C22⋊Q8 | C22 | semidirect |
| 38. C4×Q8 | C22 | semidirect |
| 43. #117 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #563 | 2 |
| #552 | 2 |
| #551 | 2 |
| D4×Q8 | 2 |
| #544 | 2 |
| #541 | 2 |
| #539 | 2 |
| #536 | 2 |
| #525 | 2 |
| C2×C22⋊Q8 | 2 |