Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
475 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 475 conjugacy classes of subgroups — too many to draw. Only its 120 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 | C2×#99 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×D4 | C23 |
| socle | C22 | C4×#99 |
| prime radical | C2×#127 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C8×#99 | C2 | semidirect |
| 5. #7205 | C2 | semidirect |
| 6. C2×C8×D4 | C4 | semidirect |
| 10. #1954 | C4 | semidirect |
| 14. #1831 | C4 | semidirect |
| 17. #1744 | C4 | semidirect |
| 20. #1325 | C4 | semidirect |
| 21. #1318 | C4 | semidirect |
| 48. #353 | C8 | semidirect |
| 97. #99 | C2×C8 | semidirect |
| 118. C2×D4 | C4×C8 | semidirect |
| 172. C42 | C4×C8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #22260 | 2 |
| C4×#353 | 2 |
| #11475 | 2 |
| C8×#99 | 2 |
| #7205 | 2 |