Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
442 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 442 conjugacy classes of subgroups — too many to draw. Only its 130 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#102 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C2×D4⋊C8 |
| prime radical | #3147 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #12900 | C2 | semidirect |
| 3. C2×#1336 | C2 | semidirect |
| 4. #7841 | C2 | semidirect |
| 27. #495 | C8 | semidirect |
| 28. #494 | C8 | semidirect |
| 29. #493 | C8 | semidirect |
| 56. #322 | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #21705 | 2 |
| #12900 | 2 |
| C2×#1336 | 2 |
| #7841 | 2 |