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 ◁ C2×#41 ◁ 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×#131 | C2 | semidirect |
| 3. #537 | C2 | semidirect |
| 4. #535 | C2 | semidirect |
| 5. #469 | C2 | semidirect |
| 6. #468 | C2 | semidirect |
| 7. #466 | C2 | semidirect |
| 8. #465 | C2 | semidirect |
| 9. #428 | C2 | semidirect |
| 10. #416 | C2 | semidirect |
| 11. C2×#102 | C2 | semidirect |
| 20. C2×M16 | C22 | semidirect |
| 22. C22×C8 | C22 | semidirect |
| 26. C22⋊Q8 | C22 | semidirect |
| 31. #121 | C22 | semidirect |
| 35. #107 | C22 | semidirect |
| 36. #106 | C22 | semidirect |
| 37. #103 | C22 | semidirect |
| 38. #103 | C22 | semidirect |
| 41. #102 | C22 | semidirect |
| 42. #102 | C22 | semidirect |
| 68. M16 | 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×#131 | 2 |
| #537 | 2 |
| #535 | 2 |
| #469 | 2 |
| #468 | 2 |
| #466 | 2 |
| #465 | 2 |
| #428 | 2 |
| #416 | 2 |
| C2×#102 | 2 |