Order 512.
1112 conjugacy classes of subgroups.
The subgroup lattice has 415 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×C22⋊C4 |
| commutator | C23 | C23×C8 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C22×C22⋊C8 |
| prime radical | C25 | C2×C8 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #33419 | C2 | semidirect |
| 3. #12847 | C2 | semidirect |
| 4. #12786 | C2 | semidirect |
| 5. #12785 | C2 | semidirect |
| 7. #12602 | C2 | semidirect |
| 8. C4×#349 | C2 | semidirect |
| 26. #2150 | C22 | semidirect |
| 30. C4×M32 | C22 | semidirect |
| 31. #1377 | C22 | semidirect |
| 32. #1377 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2873 | 2 |
| #33419 | 2 |
| #12847 | 2 |
| #12786 | 2 |
| #12785 | 2 |
| #12681 | 2 |
| #12602 | 2 |
| C4×#349 | 2 |