Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
276 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 276 conjugacy classes of subgroups — too many to draw. Only its 66 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2218 |
| commutator, Frattini, cosocle | C22×C16 | C23 |
| socle | C2 | #13381 |
| prime radical | #33600 | C2 |
| LCS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×C8 ◁ C23×C16 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #13638 | C2 | semidirect |
| 3. #13625 | C2 | semidirect |
| 4. #13620 | C2 | semidirect |
| 5. #13553 | C2 | semidirect |
| 6. #12837 | C2 | semidirect |
| 7. #7480 | C2 | semidirect |
| 58. C2×C32 | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33600 | 2 |
| #13638 | 2 |
| #13625 | 2 |
| #13620 | 2 |
| #13553 | 2 |
| #12837 | 2 |
| #7480 | 2 |