Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
197 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 197 conjugacy classes of subgroups — too many to draw. Only its 88 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 |
| prime radical | C22×D4 | C22 |
| 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. #541 | C2 | semidirect |
| 3. #485 | C2 | semidirect |
| 4. #478 | C2 | semidirect |
| 5. #461 | C2 | semidirect |
| 7. #440 | C2 | semidirect |
| 8. C4×SD16 | C2 | semidirect |
| 9. #433 | C2 | semidirect |
| 13. C4⋊Q8 | C22 | semidirect |
| 16. #125 | C22 | semidirect |
| 18. C4×Q8 | C22 | semidirect |
| 25. #107 | C22 | semidirect |
| 26. #106 | C22 | semidirect |
| 27. C4⋊C8 | C22 | semidirect |
| 28. #103 | C22 | semidirect |
| 29. #103 | C22 | semidirect |
| 33. #97 | C22 | semidirect |
| 34. C4×C8 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #541 | 2 |
| #518 | 2 |
| #485 | 2 |
| #478 | 2 |
| #461 | 2 |
| #450 | 2 |
| #440 | 2 |
| C4×SD16 | 2 |
| #433 | 2 |