Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
707 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 707 conjugacy classes of subgroups — too many to draw. Only its 150 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×D4 | D8 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C8×D4 ◁ 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. #33796 | C2 | semidirect |
| 3. #33664 | C2 | semidirect |
| 5. #12398 | C2 | semidirect |
| 6. #12346 | C2 | semidirect |
| 7. #11972 | C2 | semidirect |
| 8. #11658 | C2 | semidirect |
| 24. #2252 | C22 | semidirect |
| 25. #2241 | C22 | semidirect |
| 29. #2227 | C22 | semidirect |
| 30. #2209 | C22 | semidirect |
| 40. #1365 | C22 | semidirect |
| 41. #1332 | C22 | semidirect |
| 217. C2×D4 | Q32 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#3338 | 2 |
| #33796 | 2 |
| #33664 | 2 |
| #19256 | 2 |
| #12398 | 2 |
| #12346 | 2 |
| #11972 | 2 |
| #11658 | 2 |
| #11629 | 2 |