Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
464 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 464 conjugacy classes of subgroups — too many to draw. Only its 118 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 | #3109 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #3054 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #33216 | C2 | semidirect |
| 3. #13508 | C2 | semidirect |
| 4. #13501 | C2 | semidirect |
| 5. #13484 | C2 | semidirect |
| 6. #13458 | C2 | semidirect |
| 8. #13372 | C2 | semidirect |
| 9. #13118 | C2 | semidirect |
| 10. #12955 | C2 | semidirect |
| 11. #12850 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33216 | 2 |
| #21550 | 2 |
| #13508 | 2 |
| #13501 | 2 |
| #13484 | 2 |
| #13458 | 2 |
| #13386 | 2 |
| #13372 | 2 |
| #13118 | 2 |
| #12955 | 2 |
| #12850 | 2 |