Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
948 conjugacy classes of subgroups; only the 258 normal subgroups are shown.
This group has 948 conjugacy classes of subgroups — too many to draw. Only its 258 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×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 ◁ C23 ◁ C43 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#2177 | C2 | semidirect |
| 3. C4×#493 | C2 | semidirect |
| 4. #13239 | C2 | semidirect |
| 5. #13202 | C2 | semidirect |
| 6. #12906 | C2 | semidirect |
| 7. C4×#346 | C2 | semidirect |
| 13. #2246 | C4 | semidirect |
| 33. C2×D16 | C2×C4 | semidirect |
| 37. #493 | C2×C4 | semidirect |
| 57. #346 | D4 | semidirect |
| 59. C4×C16 | C2×C4 | semidirect |
| 151. C16 | C4×D4 | semidirect |
| 195. C4 | #2177 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2246 | 2 |
| C2×#2177 | 2 |
| C4×#493 | 2 |
| #13239 | 2 |
| #13202 | 2 |
| #12906 | 2 |
| C4×#346 | 2 |