Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
802 conjugacy classes of subgroups; only the 234 normal subgroups are shown.
This group has 802 conjugacy classes of subgroups — too many to draw. Only its 234 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C2×#493 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ #1744 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #22260 | C2 | semidirect |
| 3. #9642 | C2 | semidirect |
| 4. #8947 | C2 | semidirect |
| 5. #8785 | C2 | semidirect |
| 6. #8133 | C2 | semidirect |
| 28. #493 | C8 | semidirect |
| 55. C2×D8 | C2×C8 | semidirect |
| 64. #127 | C2×C8 | semidirect |
| 87. C4×C8 | C2×C8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #22260 | 2 |
| C4×#493 | 2 |
| #9642 | 2 |
| #8947 | 2 |
| #8785 | 2 |
| #8133 | 2 |