Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
447 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 447 conjugacy classes of subgroups — too many to draw. Only its 128 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×#459 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| prime radical | #33819 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C8×M16 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #33792 | C2 | semidirect |
| 3. #19638 | C2 | semidirect |
| 4. #18084 | C2 | semidirect |
| 5. #12501 | C2 | semidirect |
| 6. #12483 | C2 | semidirect |
| 7. #12449 | C2 | semidirect |
| 8. #12416 | C2 | semidirect |
| 9. #12018 | C2 | semidirect |
| 10. #11948 | C2 | semidirect |
| 11. #11940 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33819 | 2 |
| #33792 | 2 |
| #19638 | 2 |
| #18084 | 2 |
| #12501 | 2 |
| #12483 | 2 |
| #12449 | 2 |
| #12416 | 2 |
| #12018 | 2 |
| #11948 | 2 |
| #11940 | 2 |