Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
209 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 209 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 ◁ #117 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #540 | C2 | semidirect |
| 2. #518 | C2 | semidirect |
| 3. #482 | C2 | semidirect |
| 4. #478 | C2 | semidirect |
| 5. #459 | C2 | semidirect |
| 6. #447 | C2 | semidirect |
| 7. #442 | C2 | semidirect |
| 8. C4×D8 | C2 | semidirect |
| 9. #433 | C2 | semidirect |
| 13. #127 | C22 | semidirect |
| 14. #125 | C22 | semidirect |
| 21. C4×D4 | C22 | semidirect |
| 25. #107 | C22 | semidirect |
| 26. #106 | C22 | semidirect |
| 27. C4⋊C8 | C22 | semidirect |
| 30. #102 | C22 | semidirect |
| 31. #102 | C22 | semidirect |
| 33. #97 | C22 | semidirect |
| 34. C4×C8 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #540 | 2 |
| #518 | 2 |
| #482 | 2 |
| #478 | 2 |
| #459 | 2 |
| #447 | 2 |
| #442 | 2 |
| C4×D8 | 2 |
| #433 | 2 |