Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
323 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 323 conjugacy classes of subgroups — too many to draw. Only its 110 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#102 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C4×#102 |
| prime radical | #3169 | C22 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ C2×C82 ◁ 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 |
|---|---|---|
| 4. #11411 | C2 | semidirect |
| 5. #8282 | C2 | semidirect |
| 6. #7296 | C2 | semidirect |
| 7. #7288 | C2 | semidirect |
| 15. #1589 | C4 | semidirect |
| 17. C8×D8 | C4 | semidirect |
| 22. #1364 | C4 | semidirect |
| 23. #1361 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #17956 | 2 |
| #11947 | 2 |
| #11742 | 2 |
| #11411 | 2 |
| #8282 | 2 |
| #7296 | 2 |
| #7288 | 2 |