Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
252 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 252 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×#135 | C2 | semidirect |
| 2. C22×SD16 | C2 | semidirect |
| 3. #548 | C2 | semidirect |
| 4. #546 | C2 | semidirect |
| 5. #495 | C2 | semidirect |
| 6. #465 | C2 | semidirect |
| 7. #465 | C2 | semidirect |
| 8. #464 | C2 | semidirect |
| 9. #463 | C2 | semidirect |
| 10. #452 | C2 | semidirect |
| 11. #451 | C2 | semidirect |
| 12. #449 | C2 | semidirect |
| 13. #447 | C2 | semidirect |
| 14. C4×SD16 | C2 | semidirect |
| 15. C8×D4 | C2 | semidirect |
| 17. C2×#41 | C22 | semidirect |
| 21. C2×Q16 | C22 | semidirect |
| 22. C2×Q16 | C22 | semidirect |
| 28. C2×SD16 | C22 | semidirect |
| 32. C22×C8 | C22 | semidirect |
| 35. #124 | C22 | semidirect |
| 39. C22⋊Q8 | C22 | semidirect |
| 45. C4×Q8 | 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. C4×C8 | C22 | semidirect |
| 95. SD16 | D4 | semidirect |
| 187. C22 | #135 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#135 | 2 |
| C22×SD16 | 2 |
| #548 | 2 |
| #546 | 2 |
| #495 | 2 |
| #465 | 2 |
| #464 | 2 |
| #463 | 2 |
| #452 | 2 |
| #451 | 2 |
| #449 | 2 |
| #447 | 2 |
| C4×SD16 | 2 |
| C8×D4 | 2 |