Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
346 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 346 conjugacy classes of subgroups — too many to draw. Only its 138 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 | C22×D16 |
| commutator, Frattini, cosocle | C2×C16 | C24 |
| prime radical | C2×#508 | C22 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #117 ◁ #405 ◁ #2108 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #33729 | C2 | semidirect |
| 3. #13708 | C2 | semidirect |
| 4. #13707 | C2 | semidirect |
| 5. #13705 | C2 | semidirect |
| 6. #13704 | C2 | semidirect |
| 7. #13634 | C2 | semidirect |
| 8. C2×#1424 | C2 | semidirect |
| 9. #13628 | C2 | semidirect |
| 45. #508 | Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33729 | 2 |
| #33459 | 2 |
| #13708 | 2 |
| #13707 | 2 |
| #13705 | 2 |
| #13704 | 2 |
| #13634 | 2 |
| C2×#1424 | 2 |
| #13628 | 2 |
| #13616 | 2 |
| #13615 | 2 |