Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
190 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 190 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 | #466 | 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. #499 | C2 | semidirect |
| 3. #495 | C2 | semidirect |
| 4. #488 | C2 | semidirect |
| 5. #467 | C2 | semidirect |
| 7. #465 | C2 | semidirect |
| 8. C4×Q16 | C2 | semidirect |
| 9. C4×SD16 | C2 | semidirect |
| 10. C4×D8 | C2 | semidirect |
| 11. C2×C4×C8 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #529 | 2 |
| #499 | 2 |
| #495 | 2 |
| #488 | 2 |
| #467 | 2 |
| #466 | 2 |
| #465 | 2 |
| C4×Q16 | 2 |
| C4×SD16 | 2 |
| C4×D8 | 2 |
| C2×C4×C8 | 2 |