Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
475 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 475 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1519 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×#97 | C23 |
| socle | C2 | #7572 |
| prime radical | C22×#135 | C4 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #12790 | C2 | semidirect |
| 3. #8156 | C2 | semidirect |
| 4. #7109 | C2 | semidirect |
| 18. #1375 | C22 | semidirect |
| 19. #1281 | C22 | semidirect |
| 27. C2×#135 | C8 | semidirect |
| 29. C22×Q16 | C8 | semidirect |
| 31. C22×D8 | C8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #21240 | 2 |
| #12790 | 2 |
| #8156 | 2 |
| #7109 | 2 |