Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
198 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 198 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 | #550 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×D4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #556 | C2 | semidirect |
| 3. #486 | C2 | semidirect |
| 4. #482 | C2 | semidirect |
| 5. #475 | C2 | semidirect |
| 6. #465 | C2 | semidirect |
| 7. #460 | C2 | semidirect |
| 8. #452 | C2 | semidirect |
| 9. C4×Q16 | C2 | semidirect |
| 10. C8×D4 | C2 | semidirect |
| 11. #416 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #556 | 2 |
| #550 | 2 |
| #486 | 2 |
| #482 | 2 |
| #475 | 2 |
| #465 | 2 |
| #460 | 2 |
| #452 | 2 |
| C4×Q16 | 2 |
| C8×D4 | 2 |
| #416 | 2 |