Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
227 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 227 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 | C2 | D42 |
| commutator, Frattini, cosocle | C2×C4 | C24 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #578 | C2 | semidirect |
| 2. #577 | C2 | semidirect |
| 3. #576 | C2 | semidirect |
| 4. #495 | C2 | semidirect |
| 5. #473 | C2 | semidirect |
| 6. #472 | C2 | semidirect |
| 7. #471 | C2 | semidirect |
| 8. #455 | C2 | semidirect |
| 9. #454 | C2 | semidirect |
| 10. #443 | C2 | semidirect |
| 11. #443 | C2 | semidirect |
| 12. 2−1+4 | C22 | semidirect |
| 17. #137 | C22 | semidirect |
| 21. AGL(1,ℤ/8) | C22 | semidirect |
| 28. C2×Q16 | C22 | semidirect |
| 30. C2×SD16 | C22 | semidirect |
| 34. #131 | C22 | semidirect |
| 35. #131 | C22 | semidirect |
| 36. #124 | C22 | semidirect |
| 37. #108 | C22 | semidirect |
| 38. #108 | C22 | semidirect |
| 40. #104 | C22 | semidirect |
| 41. #104 | C22 | semidirect |
| 42. #101 | C22 | semidirect |
| 43. #100 | C22 | semidirect |
| 44. C4×C8 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #578 | 2 |
| #577 | 2 |
| #576 | 2 |
| #495 | 2 |
| #473 | 2 |
| #472 | 2 |
| #471 | 2 |
| #455 | 2 |
| #454 | 2 |
| #443 | 2 |