Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
470 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 470 conjugacy classes of subgroups — too many to draw. Only its 90 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 | #120 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C23 | #385 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #8833 | C2 | semidirect |
| 4. #8832 | C2 | semidirect |
| 6. #7028 | C2 | semidirect |
| 7. #7028 | C2 | semidirect |
| 27. #1272 | C4 | semidirect |
| 28. #1272 | C4 | semidirect |
| 29. #1270 | C4 | semidirect |
| 30. #1270 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #21909 | 2 |
| #11053 | 2 |
| #8833 | 2 |
| #8832 | 2 |
| #7563 | 2 |
| #7028 | 2 |