Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
5060 conjugacy classes of subgroups.
The subgroup lattice has 565 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#2906 | C2 | semidirect |
| 4. #20630 | C2 | semidirect |
| 5. #20629 | C2 | semidirect |
| 6. #20628 | C2 | semidirect |
| 7. #20626 | C2 | semidirect |
| 8. C2×#1797 | C2 | semidirect |
| 9. #20336 | C2 | semidirect |
| 28. C2×#424 | C22 | semidirect |
| 29. C2×#423 | C22 | semidirect |
| 30. C22×C4⋊C8 | C22 | semidirect |
| 35. #1798 | C22 | semidirect |
| 40. #1797 | C22 | semidirect |
| 41. #1797 | C22 | semidirect |
| 42. #1797 | C22 | semidirect |
| 43. #1797 | C22 | semidirect |
| 44. #1729 | C22 | semidirect |
| 45. #1728 | C22 | semidirect |
| 46. C2×#336 | C22 | semidirect |
| 363. C22×C4 | #131 | semidirect |
| 533. C22 | #2906 | semidirect |
| 551. C4 | #2880 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C4×#41 | 2 |
| C2×#2906 | 2 |
| C2×#2880 | 2 |
| #20630 | 2 |
| #20629 | 2 |
| #20628 | 2 |
| #20626 | 2 |
| C2×#1797 | 2 |
| #20336 | 2 |