Order 512.
1966 conjugacy classes of subgroups.
The subgroup lattice has 416 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C2×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C24 | C22×D4 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C2×C4×#103 | C2 | semidirect |
| 5. C2×C42×C8 | C2 | semidirect |
| 6. C4×#382 | C2 | semidirect |
| 7. #10926 | C2 | semidirect |
| 9. #10842 | C2 | semidirect |
| 19. C4×#124 | C4 | semidirect |
| 28. #1958 | C4 | direct (N × H) |
| 69. C2×#103 | C2×C4 | semidirect |
| 79. C2×C4×C8 | C2×C4 | semidirect |
| 89. #382 | C2×C4 | semidirect |
| 139. #124 | C42 | semidirect |
| 403. C4 | #1958 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4×#124 | 2 |
| C2×#1958 | 2 |
| C2×C4×#103 | 2 |
| C2×C4×#102 | 2 |
| C2×C42×C8 | 2 |
| C4×#382 | 2 |
| #10926 | 2 |
| #10925 | 2 |
| #10842 | 2 |