Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
309 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 309 conjugacy classes of subgroups — too many to draw. Only its 80 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1347 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C2×M32 | C23 |
| socle | C2 | C2×#1347 |
| prime radical | #33604 | C2 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×C8 ◁ C22×M32 ◁ G | |
| LECS | G ▷ C2×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#1400 | C2 | semidirect |
| 3. C2×#1384 | C2 | semidirect |
| 4. #7423 | C2 | semidirect |
| 5. #7422 | C2 | semidirect |
| 6. C2×#508 | C4 | semidirect |
| 13. #2219 | C4 | semidirect |
| 15. #2218 | C4 | semidirect |
| 17. #2158 | C4 | semidirect |
| 29. C2×Q32 | C2×C4 | semidirect |
| 31. C2×D16 | C2×C4 | semidirect |
| 57. #367 | C2×C4 | semidirect |
| 58. #367 | C2×C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33604 | 2 |
| C2×#1400 | 2 |
| C2×#1384 | 2 |
| #7423 | 2 |
| #7422 | 2 |