Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
478 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 478 conjugacy classes of subgroups — too many to draw. Only its 122 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×#102 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C2×#466 | C22 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C4×C22⋊C16 | C2 | semidirect |
| 4. #10822 | C2 | semidirect |
| 5. #7259 | C2 | semidirect |
| 23. #1349 | C4 | semidirect |
| 55. C22⋊C16 | C2×C4 | semidirect |
| 59. #341 | C2×C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×#466 | 2 |
| C4×C22⋊C16 | 2 |
| C2×#1349 | 2 |
| #10822 | 2 |
| #7259 | 2 |