Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
6242 conjugacy classes of subgroups.
The subgroup lattice has 463 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C4 | C26 |
| commutator, socle | C22 | C25×C4 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #60251 | C2 | semidirect |
| 3. #60250 | C2 | semidirect |
| 4. #60249 | C2 | semidirect |
| 5. #60248 | C2 | semidirect |
| 6. #60192 | C2 | semidirect |
| 23. #2987 | C22 | semidirect |
| 24. #2986 | C22 | semidirect |
| 25. #2985 | C22 | semidirect |
| 26. #2984 | C22 | semidirect |
| 27. #2983 | C22 | semidirect |
| 28. #2983 | C22 | semidirect |
| 29. #2982 | C22 | semidirect |
| 30. Q8×M16 | C22 | semidirect |
| 31. #2962 | C22 | semidirect |
| 32. #2961 | C22 | semidirect |
| 33. #2960 | C22 | semidirect |
| 118. #446 | C23 | semidirect |
| 119. #446 | C23 | semidirect |
| 120. C8×Q8 | C23 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#533 | 2 |
| #60251 | 2 |
| #60250 | 2 |
| #60249 | 2 |
| #60248 | 2 |
| #60192 | 2 |
| #60124 | 2 |
| C2×#2920 | 2 |