Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
227 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 227 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24 |
| socle | C22 | C22×D4 |
| prime radical | #447 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #529 | C2 | semidirect |
| 2. C2×C4×D4 | C2 | semidirect |
| 3. #484 | C2 | semidirect |
| 4. #481 | C2 | semidirect |
| 5. #478 | C2 | semidirect |
| 6. #463 | C2 | semidirect |
| 7. #450 | C2 | semidirect |
| 9. C4×SD16 | C2 | semidirect |
| 10. C4×D8 | C2 | semidirect |
| 11. #431 | C2 | semidirect |
| 17. #125 | C22 | semidirect |
| 18. #124 | C22 | semidirect |
| 22. C4×Q8 | C22 | semidirect |
| 28. C4×D4 | C22 | semidirect |
| 33. #107 | C22 | semidirect |
| 34. #106 | C22 | semidirect |
| 35. C4⋊C8 | C22 | semidirect |
| 36. #103 | C22 | semidirect |
| 39. #102 | C22 | semidirect |
| 42. C4×C8 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #529 | 2 |
| C2×C4×D4 | 2 |
| #484 | 2 |
| #481 | 2 |
| #478 | 2 |
| #463 | 2 |
| #450 | 2 |
| #447 | 2 |
| C4×SD16 | 2 |
| C4×D8 | 2 |
| #431 | 2 |