Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1080 conjugacy classes of subgroups; only the 258 normal subgroups are shown.
This group has 1080 conjugacy classes of subgroups — too many to draw. Only its 258 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#127 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#1712 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #22269 | C2 | semidirect |
| 2. C4×#493 | C2 | semidirect |
| 4. #9832 | C2 | semidirect |
| 5. #9824 | C2 | semidirect |
| 6. #8816 | C2 | semidirect |
| 7. #7841 | C2 | semidirect |
| 20. #1712 | C4 | semidirect |
| 31. #493 | C2×C4 | semidirect |
| 32. #493 | C2×C4 | semidirect |
| 51. C82 | C2×C4 | semidirect |
| 84. C4×C8 | D8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #22269 | 2 |
| C4×#493 | 2 |
| C2×#1712 | 2 |
| #9832 | 2 |
| #9824 | 2 |
| #8816 | 2 |
| #7841 | 2 |