Order 512.
2618 conjugacy classes of subgroups.
The subgroup lattice has 379 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C2×D4 |
| commutator | C4 | C2×C43 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C24 | C4×D4 |
| prime radical | C25 | #41 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 6. #10707 | C2 | direct (N × H) |
| 14. C2×#405 | C4 | semidirect |
| 18. C22×#97 | C4 | semidirect |
| 21. C2×#341 | C4 | semidirect |
| 23. #1735 | C4 | semidirect |
| 25. C2×#340 | C4 | semidirect |
| 26. C2×#339 | C4 | semidirect |
| 72. #405 | C2×C4 | semidirect |
| 77. C2×#97 | C2×C4 | semidirect |
| 80. #341 | C2×C4 | semidirect |
| 81. #340 | C2×C4 | semidirect |
| 82. #339 | C2×C4 | semidirect |
| 374. C2 | #10707 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#405 | 2 |
| C23×#97 | 2 |
| C22×#341 | 2 |
| C2×#1735 | 2 |
| C22×#339 | 2 |
| #10707 | 2 |