Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
282 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 282 conjugacy classes of subgroups — too many to draw. Only its 94 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 | #1762 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #466 | C2×C4 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×#322 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #11415 | C2 | semidirect |
| 5. #8287 | C2 | semidirect |
| 6. #7301 | C2 | semidirect |
| 7. #7294 | C2 | semidirect |
| 14. #1594 | C4 | semidirect |
| 15. #1593 | C4 | semidirect |
| 17. #1580 | C4 | semidirect |
| 18. #1579 | C4 | semidirect |
| 24. #1363 | C4 | semidirect |
| 25. #1360 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #17963 | 2 |
| #11945 | 2 |
| #11746 | 2 |
| #11415 | 2 |
| #8287 | 2 |
| #7301 | 2 |
| #7294 | 2 |