Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
412 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 412 conjugacy classes of subgroups — too many to draw. Only its 98 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 | #1961 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #19661 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1450 ◁ 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. #13213 | C2 | semidirect |
| 3. #13211 | C2 | semidirect |
| 4. #13209 | C2 | semidirect |
| 5. #13206 | C2 | semidirect |
| 6. #11405 | C2 | semidirect |
| 7. #8793 | C2 | semidirect |
| 20. #1712 | C4 | semidirect |
| 21. #1709 | C4 | semidirect |
| 104. C2×C16 | D8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #19661 | 2 |
| #13213 | 2 |
| #13211 | 2 |
| #13209 | 2 |
| #13206 | 2 |
| #11405 | 2 |
| #8793 | 2 |