Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
284 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 284 conjugacy classes of subgroups — too many to draw. Only its 96 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. C2×AGL(1,ℤ/8) | C2 | semidirect |
| 2. C22×D8 | C2 | semidirect |
| 3. #546 | C2 | semidirect |
| 4. D42 | C2 | semidirect |
| 5. #496 | C2 | semidirect |
| 6. #469 | C2 | semidirect |
| 7. #465 | C2 | semidirect |
| 8. #463 | C2 | semidirect |
| 9. #459 | C2 | semidirect |
| 10. #450 | C2 | semidirect |
| 11. #449 | C2 | semidirect |
| 12. #447 | C2 | semidirect |
| 13. #447 | C2 | semidirect |
| 14. #442 | C2 | semidirect |
| 15. #435 | C2 | semidirect |
| 16. C2×#41 | C22 | semidirect |
| 19. C22×D4 | C22 | semidirect |
| 20. C22×D4 | C22 | semidirect |
| 22. C2×SD16 | C22 | semidirect |
| 23. C2×SD16 | C22 | semidirect |
| 24. C2×SD16 | C22 | semidirect |
| 28. C2×D8 | C22 | semidirect |
| 29. C2×D8 | C22 | semidirect |
| 30. C2×D8 | C22 | semidirect |
| 31. C2×D8 | C22 | semidirect |
| 32. C2×M16 | C22 | semidirect |
| 33. C22×C8 | C22 | semidirect |
| 34. #127 | C22 | semidirect |
| 35. #124 | C22 | semidirect |
| 37. C22⋊Q8 | C22 | semidirect |
| 41. #121 | C22 | semidirect |
| 42. #121 | C22 | semidirect |
| 47. C4×D4 | C22 | semidirect |
| 48. C4×D4 | C22 | semidirect |
| 50. #106 | C22 | semidirect |
| 51. C4⋊C8 | C22 | semidirect |
| 52. #103 | C22 | semidirect |
| 53. #102 | C22 | semidirect |
| 54. #102 | C22 | semidirect |
| 55. #102 | C22 | semidirect |
| 56. #102 | C22 | semidirect |
| 57. #102 | C22 | semidirect |
| 58. C22⋊C8 | C22 | semidirect |
| 59. C22⋊C8 | C22 | semidirect |
| 60. #97 | C22 | semidirect |
| 102. D8 | D4 | semidirect |
| 161. D4 | C2×D4 | semidirect |
| 180. C8 | C2×D4 | semidirect |
| 204. C22 | AGL(1,ℤ/8) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×AGL(1,ℤ/8) | 2 |
| C22×D8 | 2 |
| #546 | 2 |
| D42 | 2 |
| #496 | 2 |
| #469 | 2 |
| #465 | 2 |
| #463 | 2 |
| #459 | 2 |
| #450 | 2 |
| #449 | 2 |
| #447 | 2 |
| #442 | 2 |
| #435 | 2 |