Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
630 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 630 conjugacy classes of subgroups — too many to draw. Only its 126 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #392 |
| commutator, Frattini, cosocle | C43 | C23 |
| prime radical | #2867 | C22 |
| LCS | G ▷ C43 ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C43 ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #28511 | C2 | semidirect |
| 4. #9896 | C2 | semidirect |
| 5. #9870 | C2 | semidirect |
| 7. #7015 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33224 | 2 |
| #28749 | 2 |
| #28511 | 2 |
| #9896 | 2 |
| #9870 | 2 |
| #9695 | 2 |
| #7015 | 2 |