Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
542 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 542 conjugacy classes of subgroups — too many to draw. Only its 106 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 | C2×#99 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C22×M16 | C23 |
| socle | C22 | C2×#323 |
| prime radical | C2×#532 | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ 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. #12785 | C2 | semidirect |
| 3. C4×#349 | C2 | semidirect |
| 4. C2×#1321 | C2 | semidirect |
| 60. #349 | D4 | semidirect |
| 303. C4 | #1321 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #21189 | 2 |
| #12785 | 2 |
| C4×#349 | 2 |
| C2×#1321 | 2 |