Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
202 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 202 conjugacy classes of subgroups — too many to draw. Only its 92 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 |
| prime radical | #535 | C2 |
| 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. #544 | C2 | semidirect |
| 3. #490 | C2 | semidirect |
| 4. #489 | C2 | semidirect |
| 5. #488 | C2 | semidirect |
| 6. #486 | C2 | semidirect |
| 7. #418 | C2 | semidirect |
| 8. #417 | C2 | semidirect |
| 9. #416 | C2 | semidirect |
| 10. C2×#102 | C2 | semidirect |
| 11. #405 | C2 | semidirect |
| 17. C4⋊Q8 | C22 | semidirect |
| 20. #125 | C22 | semidirect |
| 29. #103 | C22 | semidirect |
| 30. #103 | C22 | semidirect |
| 31. #103 | C22 | semidirect |
| 36. #102 | C22 | semidirect |
| 37. #102 | C22 | semidirect |
| 38. #102 | C22 | semidirect |
| 39. #97 | 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 |
|---|---|
| #544 | 2 |
| #535 | 2 |
| #490 | 2 |
| #489 | 2 |
| #488 | 2 |
| #486 | 2 |
| #418 | 2 |
| #417 | 2 |
| #416 | 2 |
| C2×#102 | 2 |
| #405 | 2 |