Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
591 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 591 conjugacy classes of subgroups — too many to draw. Only its 152 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #3341 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | #28227 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×#443 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #33617 | C2 | semidirect |
| 2. C2×#2179 | C2 | semidirect |
| 3. C2×#2171 | C2 | semidirect |
| 4. #28249 | C2 | semidirect |
| 5. #13392 | C2 | semidirect |
| 6. #12910 | C2 | semidirect |
| 7. #12840 | C2 | semidirect |
| 8. #12825 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33617 | 2 |
| C2×#2179 | 2 |
| C2×#2171 | 2 |
| #28249 | 2 |
| #13392 | 2 |
| #12910 | 2 |
| #12840 | 2 |
| #12825 | 2 |