Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
351 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 351 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1530 |
| commutator | C42 | C22×C8 |
| Frattini, cosocle | #376 | C23 |
| socle | C2 | #7587 |
| prime radical | #3147 | C4 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ #1726 ◁ G | |
| LECS | G ▷ #376 ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #8780 | C2 | semidirect |
| 3. #8137 | C2 | semidirect |
| 4. #7115 | C2 | semidirect |
| 26. #494 | C8 | semidirect |
| 27. #493 | C8 | semidirect |
| 28. #492 | C8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #21707 | 2 |
| #8780 | 2 |
| #8137 | 2 |
| #7115 | 2 |