Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
561 conjugacy classes of subgroups; only the 390 normal subgroups are shown.
This group has 561 conjugacy classes of subgroups — too many to draw. Only its 390 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C22 | C25 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #557 | C2 | semidirect |
| 2. #553 | C2 | semidirect |
| 3. #550 | C2 | semidirect |
| 4. #550 | C2 | semidirect |
| 5. #550 | C2 | semidirect |
| 6. #547 | C2 | semidirect |
| 7. #540 | C2 | semidirect |
| 8. #532 | C2 | semidirect |
| 9. #529 | C2 | semidirect |
| 10. #520 | C2 | semidirect |
| 11. C4×#41 | C2 | semidirect |
| 86. Q8 | #41 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #557 | 2 |
| #553 | 2 |
| #550 | 2 |
| #547 | 2 |
| #540 | 2 |
| #532 | 2 |
| #529 | 2 |
| #520 | 2 |
| C4×#41 | 2 |