Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
443 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 443 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #459 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1879 |
| prime radical | #3169 | C22 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C42×Q16 | C2 | semidirect |
| 4. #11658 | C2 | semidirect |
| 6. #7296 | C2 | semidirect |
| 7. #7148 | C2 | semidirect |
| 13. #2225 | C4 | semidirect |
| 14. #2207 | C4 | semidirect |
| 165. Q16 | #104 | semidirect |
| 203. C42 | SD32 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33663 | 2 |
| C42×Q16 | 2 |
| #11742 | 2 |
| #11658 | 2 |
| #9844 | 2 |
| #7296 | 2 |
| #7148 | 2 |