Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
307 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 307 conjugacy classes of subgroups — too many to draw. Only its 82 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 | #1347 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C2×M32 | C23 |
| prime radical | C24 | D16 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ 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 |
|---|---|---|
| 4. #7423 | C2 | direct (N × H) |
| 5. C22×Q32 | C4 | semidirect |
| 12. #2219 | C4 | semidirect |
| 14. C2×#367 | C4 | semidirect |
| 27. C2×Q32 | C2×C4 | semidirect |
| 52. #367 | C2×C4 | semidirect |
| 232. C2 | #7423 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2219 | 2 |
| C2×#1400 | 2 |
| C2×#1384 | 2 |
| #7423 | 2 |