Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
639 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 639 conjugacy classes of subgroups — too many to draw. Only its 142 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 | C2×#476 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #33208 | C2 | semidirect |
| 3. #13534 | C2 | semidirect |
| 4. #13467 | C2 | semidirect |
| 5. #13372 | C2 | semidirect |
| 6. #12928 | C2 | semidirect |
| 7. #12788 | C2 | semidirect |
| 10. C2×#499 | C22 | semidirect |
| 16. C2×#365 | C22 | semidirect |
| 17. C2×#364 | C22 | semidirect |
| 27. #1941 | C22 | semidirect |
| 28. #1846 | C22 | semidirect |
| 31. #1383 | C22 | semidirect |
| 32. #1376 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33208 | 2 |
| C2×#1974 | 2 |
| #13534 | 2 |
| #13467 | 2 |
| #13372 | 2 |
| #12928 | 2 |
| #12788 | 2 |
| #12735 | 2 |