Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
338 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 338 conjugacy classes of subgroups — too many to draw. Only its 92 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 | #466 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1970 |
| prime radical | #33733 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C8×M16 ◁ 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 |
|---|---|---|
| 2. C4×#366 | C2 | semidirect |
| 3. #11742 | C2 | semidirect |
| 4. C16×M16 | C2 | semidirect |
| 5. #9719 | C2 | semidirect |
| 7. #2252 | C4 | semidirect |
| 8. #2246 | C4 | semidirect |
| 9. #2240 | C4 | semidirect |
| 117. M16 | D16 | semidirect |
| 142. C16 | #104 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33733 | 2 |
| C4×#366 | 2 |
| #11742 | 2 |
| C16×M16 | 2 |
| #9719 | 2 |