Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
3192 conjugacy classes of subgroups.
The subgroup lattice has 806 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×C22⋊C4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C22 | C23×C22⋊C4 |
| prime radical | C2×#3444 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#2920 | C2 | semidirect |
| 3. #17757 | C2 | semidirect |
| 4. #17752 | C2 | semidirect |
| 5. #17748 | C2 | semidirect |
| 6. #17744 | C2 | semidirect |
| 7. #17739 | C2 | semidirect |
| 8. #17738 | C2 | semidirect |
| 9. #17716 | C2 | semidirect |
| 38. #2920 | C22 | semidirect |
| 39. #2920 | C22 | semidirect |
| 43. C2×#431 | C22 | semidirect |
| 44. C2×#423 | C22 | semidirect |
| 47. #1561 | C22 | semidirect |
| 48. #1560 | C22 | semidirect |
| 49. #1558 | C22 | semidirect |
| 50. #1556 | C22 | semidirect |
| 51. #1555 | C22 | semidirect |
| 52. #1554 | C22 | semidirect |
| 53. #1553 | C22 | semidirect |
| 54. #1550 | C22 | semidirect |
| 55. #1547 | C22 | semidirect |
| 56. #1544 | C22 | semidirect |
| 57. #1543 | C22 | semidirect |
| 225. #333 | C23 | semidirect |
| 226. #332 | C23 | semidirect |
| 227. #331 | C23 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#3444 | 2 |
| C2×#2920 | 2 |
| #17757 | 2 |
| #17752 | 2 |
| #17748 | 2 |
| #17744 | 2 |
| #17739 | 2 |
| #17738 | 2 |
| #17716 | 2 |