Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
643 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 643 conjugacy classes of subgroups — too many to draw. Only its 146 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 | #2537 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #28174 | C2 | semidirect |
| 2. #19255 | C2 | semidirect |
| 3. #17956 | C2 | semidirect |
| 4. C2×C8×D8 | C2 | semidirect |
| 5. #10457 | C2 | semidirect |
| 6. #8824 | C2 | semidirect |
| 7. #8688 | C2 | semidirect |
| 8. #8616 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #28174 | 2 |
| #19255 | 2 |
| #17956 | 2 |
| C2×C8×D8 | 2 |
| #10457 | 2 |
| #8824 | 2 |
| #8688 | 2 |
| #8616 | 2 |