Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
469 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 469 conjugacy classes of subgroups — too many to draw. Only its 142 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#476 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#135 | C23 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #33313 | C2 | semidirect |
| 3. #13542 | C2 | semidirect |
| 5. #13457 | C2 | semidirect |
| 7. #13184 | C2 | semidirect |
| 8. #12883 | C2 | semidirect |
| 41. C4×Q16 | Q8 | semidirect |
| 43. C4×D8 | Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33313 | 2 |
| #21292 | 2 |
| #13542 | 2 |
| #13534 | 2 |
| #13457 | 2 |
| #13371 | 2 |
| #13184 | 2 |
| #12883 | 2 |
| #12817 | 2 |