Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
3920 conjugacy classes of subgroups.
The subgroup lattice has 492 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×C22⋊C4 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×#41 | C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #60284 | C2 | semidirect |
| 3. #17370 | C2 | semidirect |
| 4. #17369 | C2 | semidirect |
| 5. #17366 | C2 | semidirect |
| 6. #17357 | C2 | semidirect |
| 7. #17353 | C2 | semidirect |
| 8. #17352 | C2 | semidirect |
| 27. #2982 | C22 | semidirect |
| 28. #2975 | C22 | semidirect |
| 30. #2920 | C22 | semidirect |
| 31. #1465 | C22 | semidirect |
| 33. #1464 | C22 | semidirect |
| 34. #1463 | C22 | semidirect |
| 36. #1461 | C22 | semidirect |
| 37. #1460 | C22 | semidirect |
| 39. #1458 | C22 | semidirect |
| 143. C2×#41 | M16 | semidirect |
| 146. C22×Q8 | M16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#3428 | 2 |
| #60284 | 2 |
| #17370 | 2 |
| #17369 | 2 |
| #17366 | 2 |
| #17357 | 2 |
| #17353 | 2 |
| #17352 | 2 |