Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
174 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 174 conjugacy classes of subgroups — too many to draw. Only its 86 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 | #121 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ 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. #533 | C2 | semidirect |
| 3. #479 | C2 | semidirect |
| 4. #477 | C2 | semidirect |
| 5. #476 | C2 | semidirect |
| 6. #470 | C2 | semidirect |
| 8. #464 | C2 | semidirect |
| 9. #462 | C2 | semidirect |
| 10. #461 | C2 | semidirect |
| 11. #423 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #533 | 2 |
| #529 | 2 |
| #479 | 2 |
| #477 | 2 |
| #476 | 2 |
| #470 | 2 |
| #468 | 2 |
| #464 | 2 |
| #462 | 2 |
| #461 | 2 |
| #423 | 2 |