Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
250 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 250 conjugacy classes of subgroups — too many to draw. Only its 94 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. C2×#135 | C2 | semidirect |
| 3. #547 | C2 | semidirect |
| 4. #546 | C2 | semidirect |
| 5. #497 | C2 | semidirect |
| 6. #470 | C2 | semidirect |
| 7. #465 | C2 | semidirect |
| 8. #463 | C2 | semidirect |
| 9. #461 | C2 | semidirect |
| 10. #452 | C2 | semidirect |
| 11. #452 | C2 | semidirect |
| 12. #450 | C2 | semidirect |
| 13. #449 | C2 | semidirect |
| 14. #442 | C2 | semidirect |
| 15. #435 | C2 | semidirect |
| 18. C2×#41 | C22 | semidirect |
| 22. C2×Q16 | C22 | semidirect |
| 23. C2×Q16 | C22 | semidirect |
| 28. C2×SD16 | C22 | semidirect |
| 32. C22×C8 | C22 | semidirect |
| 33. C4⋊Q8 | C22 | semidirect |
| 40. C22⋊Q8 | C22 | semidirect |
| 47. C4×D4 | C22 | semidirect |
| 50. #106 | C22 | semidirect |
| 51. C4⋊C8 | C22 | semidirect |
| 52. #103 | C22 | semidirect |
| 53. #103 | C22 | semidirect |
| 54. #103 | C22 | semidirect |
| 57. #102 | C22 | semidirect |
| 59. C22⋊C8 | C22 | semidirect |
| 60. #97 | C22 | semidirect |
| 97. D8 | D4 | 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 |
| C2×#135 | 2 |
| #547 | 2 |
| #546 | 2 |
| #497 | 2 |
| #470 | 2 |
| #465 | 2 |
| #463 | 2 |
| #461 | 2 |
| #452 | 2 |
| #450 | 2 |
| #449 | 2 |
| #442 | 2 |
| #435 | 2 |