Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
230 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 230 conjugacy classes of subgroups — too many to draw. Only its 88 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×D4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #555 | C2 | semidirect |
| 2. D42 | C2 | semidirect |
| 3. #486 | C2 | semidirect |
| 4. #482 | C2 | semidirect |
| 5. #476 | C2 | semidirect |
| 6. #469 | C2 | semidirect |
| 7. #459 | C2 | semidirect |
| 8. #447 | C2 | semidirect |
| 9. C4×D8 | C2 | semidirect |
| 10. #436 | C2 | semidirect |
| 11. #418 | C2 | semidirect |
| 15. C2×D8 | C22 | semidirect |
| 16. C2×M16 | C22 | semidirect |
| 17. C4⋊Q8 | C22 | semidirect |
| 23. #121 | C22 | semidirect |
| 27. C4×D4 | C22 | semidirect |
| 29. #117 | C22 | semidirect |
| 30. #107 | C22 | semidirect |
| 31. #106 | C22 | semidirect |
| 32. C4⋊C8 | C22 | semidirect |
| 35. #102 | C22 | semidirect |
| 36. #102 | C22 | semidirect |
| 37. #102 | C22 | semidirect |
| 38. #102 | C22 | semidirect |
| 39. C22⋊C8 | C22 | semidirect |
| 40. C4×C8 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #555 | 2 |
| D42 | 2 |
| #486 | 2 |
| #482 | 2 |
| #476 | 2 |
| #469 | 2 |
| #459 | 2 |
| #447 | 2 |
| C4×D8 | 2 |
| #436 | 2 |
| #418 | 2 |