Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
394 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 394 conjugacy classes of subgroups — too many to draw. Only its 94 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1964 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C22×M16 | C23 |
| socle | C2 | #9651 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #12791 | C2 | semidirect |
| 3. C4×#349 | C2 | semidirect |
| 4. #11458 | C2 | semidirect |
| 5. #11456 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #21099 | 2 |
| #12791 | 2 |
| C4×#349 | 2 |
| #11458 | 2 |
| #11456 | 2 |
| #11438 | 2 |
| #11436 | 2 |