Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
246 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 246 conjugacy classes of subgroups — too many to draw. Only its 98 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 |
| 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. #532 | C2 | semidirect |
| 2. C2×C4×D4 | C2 | semidirect |
| 3. #480 | C2 | semidirect |
| 4. #474 | C2 | semidirect |
| 5. #459 | C2 | semidirect |
| 6. #447 | C2 | semidirect |
| 7. C4×D8 | C2 | semidirect |
| 8. #431 | C2 | semidirect |
| 13. C4⋊Q8 | C22 | semidirect |
| 14. #127 | C22 | semidirect |
| 23. C4×D4 | C22 | semidirect |
| 27. #107 | C22 | semidirect |
| 28. C4⋊C8 | C22 | semidirect |
| 30. #102 | C22 | semidirect |
| 32. C4×C8 | C22 | semidirect |
| 78. D4 | #41 | semidirect |
| 87. C2×C4 | D8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #532 | 2 |
| C2×C4×D4 | 2 |
| #480 | 2 |
| #474 | 2 |
| #459 | 2 |
| #447 | 2 |
| C4×D8 | 2 |
| #431 | 2 |