Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
799 conjugacy classes of subgroups; only the 160 normal subgroups are shown.
This group has 799 conjugacy classes of subgroups — too many to draw. Only its 160 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×#447 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| prime radical | C22×D8 | D4 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C8×Q8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #33731 | C2 | semidirect |
| 4. #19118 | C2 | semidirect |
| 5. #12408 | C2 | semidirect |
| 6. #12404 | C2 | semidirect |
| 7. #11940 | C2 | semidirect |
| 8. #11607 | C2 | semidirect |
| 25. #2254 | C22 | semidirect |
| 26. #2252 | C22 | semidirect |
| 35. #1676 | C22 | semidirect |
| 36. #1674 | C22 | semidirect |
| 40. #1364 | C22 | semidirect |
| 41. Q8⋊C16 | C22 | semidirect |
| 128. C2×D8 | Q16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×Q8×D8 | 2 |
| #33731 | 2 |
| #33627 | 2 |
| #19118 | 2 |
| #12408 | 2 |
| #12404 | 2 |
| #11940 | 2 |
| #11607 | 2 |
| #11593 | 2 |