Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
368 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 368 conjugacy classes of subgroups — too many to draw. Only its 106 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#120 |
| commutator, Frattini, cosocle | C2×C4 | C24 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #585 | C2 | semidirect |
| 2. C2×AGL(1,ℤ/8) | C2 | semidirect |
| 3. #535 | C2 | semidirect |
| 4. #454 | C2 | semidirect |
| 5. #453 | C2 | semidirect |
| 6. #421 | C2 | semidirect |
| 7. #413 | C2 | semidirect |
| 9. 2−1+4 | C22 | semidirect |
| 12. H(5,𝔽2) | C22 | semidirect |
| 17. C22×D4 | C22 | semidirect |
| 19. AGL(1,ℤ/8) | C22 | semidirect |
| 22. C2×M16 | C22 | semidirect |
| 23. #127 | C22 | semidirect |
| 24. #124 | C22 | semidirect |
| 27. #117 | C22 | semidirect |
| 28. #104 | C22 | semidirect |
| 29. #101 | C22 | semidirect |
| 30. #100 | C22 | semidirect |
| 42. #41 | D4 | semidirect |
| 46. C2×Q8 | D4 | semidirect |
| 58. C2×D4 | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #585 | 2 |
| C2×AGL(1,ℤ/8) | 2 |
| #535 | 2 |
| #454 | 2 |
| #453 | 2 |
| #421 | 2 |
| #413 | 2 |