Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
710 conjugacy classes of subgroups; only the 194 normal subgroups are shown.
This group has 710 conjugacy classes of subgroups — too many to draw. Only its 194 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#2246 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #1772 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #33456 | C2 | semidirect |
| 3. #23201 | C2 | semidirect |
| 4. #13261 | C2 | semidirect |
| 5. #13245 | C2 | semidirect |
| 6. #13017 | C2 | semidirect |
| 7. #12670 | C2 | semidirect |
| 8. #12664 | C2 | semidirect |
| 14. #2246 | C4 | semidirect |
| 40. C2×D16 | C2×C4 | semidirect |
| 41. C2×D16 | C2×C4 | semidirect |
| 46. #493 | C2×C4 | semidirect |
| 62. #367 | D4 | semidirect |
| 63. #366 | D4 | semidirect |
| 69. C4×C16 | C2×C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2246 | 2 |
| #33456 | 2 |
| #23201 | 2 |
| #13261 | 2 |
| #13245 | 2 |
| #13017 | 2 |
| #12670 | 2 |
| #12664 | 2 |