Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
311 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 311 conjugacy classes of subgroups — too many to draw. Only its 76 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 | #1824 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C22×D4 | SD16 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#435 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #7263 | C2 | semidirect |
| 5. #7259 | C2 | semidirect |
| 6. #7235 | C2 | semidirect |
| 7. #7232 | C2 | semidirect |
| 15. #1838 | C4 | semidirect |
| 16. #1811 | C4 | semidirect |
| 21. #1778 | C4 | semidirect |
| 22. #1738 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #22718 | 2 |
| #11467 | 2 |
| #8610 | 2 |
| #7263 | 2 |
| #7259 | 2 |
| #7235 | 2 |
| #7232 | 2 |