Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
252 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 252 conjugacy classes of subgroups — too many to draw. Only its 100 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 ◁ #117 ◁ 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. #535 | C2 | semidirect |
| 4. #520 | C2 | semidirect |
| 5. #464 | C2 | semidirect |
| 6. #463 | C2 | semidirect |
| 7. #460 | C2 | semidirect |
| 8. #459 | C2 | semidirect |
| 9. #424 | C2 | semidirect |
| 10. #417 | C2 | semidirect |
| 11. C2×#102 | C2 | semidirect |
| 17. C2×Q16 | C22 | semidirect |
| 19. C2×SD16 | C22 | semidirect |
| 20. C2×SD16 | C22 | semidirect |
| 22. C2×D8 | C22 | semidirect |
| 30. C4×Q8 | C22 | semidirect |
| 32. C4×D4 | C22 | semidirect |
| 37. C4⋊C8 | C22 | semidirect |
| 38. C4⋊C8 | C22 | semidirect |
| 39. #103 | C22 | semidirect |
| 42. #102 | C22 | semidirect |
| 47. #41 | 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 |
| #535 | 2 |
| #520 | 2 |
| #464 | 2 |
| #463 | 2 |
| #460 | 2 |
| #459 | 2 |
| #424 | 2 |
| #417 | 2 |
| C2×#102 | 2 |