Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
247 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 247 conjugacy classes of subgroups — too many to draw. Only its 70 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #459 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1925 |
| prime radical | #3169 | C22 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C42×C8 ◁ 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 |
|---|---|---|
| 1. #12036 | C2 | semidirect |
| 5. #8266 | C2 | semidirect |
| 6. #7291 | C2 | semidirect |
| 7. #7148 | C2 | semidirect |
| 20. #1365 | C4 | semidirect |
| 21. #1365 | C4 | semidirect |
| 22. #1361 | C4 | semidirect |
| 23. #1361 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #12036 | 2 |
| #11948 | 2 |
| #11649 | 2 |
| #9844 | 2 |
| #8266 | 2 |
| #7291 | 2 |
| #7148 | 2 |