Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1942 conjugacy classes of subgroups.
The subgroup lattice has 499 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | C25 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#407 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #22608 | C2 | semidirect |
| 4. #22607 | C2 | semidirect |
| 5. #22606 | C2 | semidirect |
| 6. #21165 | C2 | semidirect |
| 7. #21042 | C2 | semidirect |
| 10. #20445 | C2 | semidirect |
| 21. #2911 | C4 | semidirect |
| 22. #2911 | C4 | semidirect |
| 27. C2×#428 | C4 | semidirect |
| 28. C2×#428 | C4 | semidirect |
| 36. #1946 | C22 | semidirect |
| 37. #1945 | C22 | semidirect |
| 38. #1922 | C22 | semidirect |
| 39. #1922 | C22 | semidirect |
| 45. #1773 | C22 | semidirect |
| 46. #1772 | C22 | semidirect |
| 100. #428 | C2×C4 | semidirect |
| 101. #428 | C2×C4 | semidirect |
| 102. #428 | C2×C4 | semidirect |
| 103. #428 | C2×C4 | semidirect |
| 104. #427 | C2×C4 | semidirect |
| 105. #427 | C2×C4 | semidirect |
| 106. #427 | C2×C4 | semidirect |
| 110. C2×#107 | C2×C4 | semidirect |
| 111. C2×#107 | C2×C4 | semidirect |
| 112. C2×#106 | C2×C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #60124 | 2 |
| C2×#2911 | 2 |
| #22608 | 2 |
| #22607 | 2 |
| #22606 | 2 |
| #21165 | 2 |
| #21042 | 2 |
| C2×#1826 | 2 |
| #20667 | 2 |
| #20445 | 2 |
| C22⋊C4×M16 | 2 |