Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
185 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 185 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 | #532 | C2 |
| 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. #500 | C2 | semidirect |
| 4. #498 | C2 | semidirect |
| 5. #486 | C2 | semidirect |
| 6. #486 | C2 | semidirect |
| 7. #482 | C2 | semidirect |
| 8. #478 | C2 | semidirect |
| 9. #431 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #533 | 2 |
| #532 | 2 |
| #500 | 2 |
| #498 | 2 |
| #486 | 2 |
| #482 | 2 |
| #478 | 2 |
| #431 | 2 |