Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
934 conjugacy classes of subgroups; only the 238 normal subgroups are shown.
This group has 934 conjugacy classes of subgroups — too many to draw. Only its 238 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×#493 | C22 |
| 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. #21691 | C2 | semidirect |
| 3. #21646 | C2 | semidirect |
| 5. #9839 | C2 | semidirect |
| 6. #9838 | C2 | semidirect |
| 8. #8815 | C2 | semidirect |
| 9. #7846 | C2 | semidirect |
| 30. #1709 | C4 | semidirect |
| 42. #494 | C2×C4 | semidirect |
| 46. #486 | C2×C4 | semidirect |
| 74. C82 | C2×C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #21691 | 2 |
| #21690 | 2 |
| #21646 | 2 |
| C2×#1709 | 2 |
| #9839 | 2 |
| #9838 | 2 |
| #9837 | 2 |
| #8815 | 2 |
| #7846 | 2 |