Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
3530 conjugacy classes of subgroups; only the 1058 normal subgroups are shown.
This group has 3530 conjugacy classes of subgroups — too many to draw. Only its 1058 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C24 | C23×C4 |
| prime radical | C2×#2397 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#436 | C2 | semidirect |
| 3. #21684 | C2 | direct (N × H) |
| 4. C2×#1955 | C2 | semidirect |
| 5. C2×#1843 | C2 | semidirect |
| 6. C2×#1726 | C2 | semidirect |
| 10. C2×#436 | C22 | semidirect |
| 20. #1955 | C22 | semidirect |
| 21. #1843 | C22 | semidirect |
| 22. #1726 | C22 | semidirect |
| 44. C2×C4×C8 | D4 | semidirect |
| 85. C4×C8 | C2×D4 | semidirect |
| 162. C2 | #21684 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#436 | 2 |
| C2×#2397 | 2 |
| #21684 | 2 |
| C2×#1955 | 2 |
| C2×#1843 | 2 |
| C2×#1726 | 2 |