Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
246 conjugacy classes of subgroups; only the 140 normal subgroups are shown.
This group has 246 conjugacy classes of subgroups — too many to draw. Only its 140 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 | C4 | C23×C4 |
| Frattini, cosocle | C2×C4 | C24 |
| prime radical | C22×D4 | C22 |
| LCS | G ▷ 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 |
|---|---|---|
| 2. #519 | C2 | semidirect |
| 4. #440 | C2 | semidirect |
| 5. C4×SD16 | C2 | semidirect |
| 6. #427 | C2 | semidirect |
| 7. #419 | C2 | semidirect |
| 9. #405 | C2 | semidirect |
| 12. C2×SD16 | C4 | semidirect |
| 18. C4×Q8 | C22 | semidirect |
| 26. #107 | C22 | semidirect |
| 27. #106 | C22 | semidirect |
| 28. #103 | C22 | semidirect |
| 30. #97 | C22 | semidirect |
| 31. C4×C8 | C22 | semidirect |
| 35. C2×Q8 | C2×C4 | semidirect |
| 44. SD16 | C2×C4 | semidirect |
| 45. SD16 | C2×C4 | semidirect |
| 47. C2×C8 | C2×C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×SD16 | 2 |
| #519 | 2 |
| #518 | 2 |
| #440 | 2 |
| C4×SD16 | 2 |
| #427 | 2 |
| #419 | 2 |
| #418 | 2 |
| #405 | 2 |