Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
467 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 467 conjugacy classes of subgroups — too many to draw. Only its 144 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 | #442 |
| commutator | C8 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1762 |
| prime radical | C22×D8 | C2×C4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C4×#97 ◁ 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. #12594 | C2 | semidirect |
| 5. #8268 | C2 | semidirect |
| 6. #7175 | C2 | semidirect |
| 7. #7174 | C2 | semidirect |
| 22. #1333 | C4 | semidirect |
| 23. #1331 | C4 | semidirect |
| 32. #442 | C8 | semidirect |
| 53. #346 | C2×C4 | semidirect |
| 55. C4×C16 | C2×C4 | semidirect |
| 59. #335 | C2×C4 | semidirect |
| 61. #334 | C2×C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×#442 | 2 |
| #12594 | 2 |
| #11599 | 2 |
| #11593 | 2 |
| #8268 | 2 |
| #7175 | 2 |
| #7174 | 2 |