Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
710 conjugacy classes of subgroups; only the 198 normal subgroups are shown.
This group has 710 conjugacy classes of subgroups — too many to draw. Only its 198 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 | C2×#493 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| prime radical | #33713 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1450 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #33718 | C2 | semidirect |
| 3. C2×#2247 | C2 | semidirect |
| 4. #19662 | C2 | semidirect |
| 5. #12520 | C2 | semidirect |
| 6. #12518 | C2 | semidirect |
| 7. #12516 | C2 | semidirect |
| 8. #11399 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33718 | 2 |
| #33713 | 2 |
| C2×#2247 | 2 |
| #19662 | 2 |
| #12520 | 2 |
| #12518 | 2 |
| #12516 | 2 |
| #11399 | 2 |