Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
278 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 278 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. #550 | C2 | semidirect |
| 3. D42 | C2 | semidirect |
| 4. #493 | C2 | semidirect |
| 5. #469 | C2 | semidirect |
| 6. #460 | C2 | semidirect |
| 7. #459 | C2 | semidirect |
| 8. #449 | C2 | semidirect |
| 9. #447 | C2 | semidirect |
| 10. C4×SD16 | C2 | semidirect |
| 11. #436 | C2 | semidirect |
| 12. C2×#41 | C22 | semidirect |
| 17. C2×SD16 | C22 | semidirect |
| 19. C2×D8 | C22 | semidirect |
| 20. C2×D8 | C22 | semidirect |
| 21. C2×M16 | C22 | semidirect |
| 24. #127 | C22 | semidirect |
| 28. #121 | C22 | semidirect |
| 30. C4×Q8 | C22 | semidirect |
| 34. #106 | C22 | semidirect |
| 35. C4⋊C8 | C22 | semidirect |
| 36. #103 | C22 | semidirect |
| 38. #102 | C22 | semidirect |
| 39. #102 | C22 | semidirect |
| 40. C22⋊C8 | C22 | semidirect |
| 41. C4×C8 | C22 | semidirect |
| 65. SD16 | D4 | semidirect |
| 91. Q8 | C2×D4 | semidirect |
| 120. C8 | C2×D4 | semidirect |
| 140. C4 | 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 |
| #550 | 2 |
| D42 | 2 |
| #493 | 2 |
| #469 | 2 |
| #460 | 2 |
| #459 | 2 |
| #449 | 2 |
| #447 | 2 |
| C4×SD16 | 2 |
| #436 | 2 |