Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
777 conjugacy classes of subgroups; only the 240 normal subgroups are shown.
This group has 777 conjugacy classes of subgroups — too many to draw. Only its 240 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×#102 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| prime radical | #62471 | C2 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ C8×#41 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #33505 | C2 | semidirect |
| 3. #17837 | C2 | semidirect |
| 4. #11986 | C2 | semidirect |
| 5. #11972 | C2 | semidirect |
| 6. #11948 | C2 | semidirect |
| 7. #11941 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #62471 | 2 |
| #33505 | 2 |
| #17837 | 2 |
| #11986 | 2 |
| #11972 | 2 |
| #11948 | 2 |
| #11941 | 2 |