Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
553 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 553 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, socle | C22 | C2×#327 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C22×M16 | C23 |
| prime radical | #33175 | C2 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ 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. C4×#349 | C2 | semidirect |
| 3. C2×#1339 | C2 | semidirect |
| 4. #10837 | C2 | semidirect |
| 5. C2×#493 | C4 | semidirect |
| 14. #1972 | C4 | semidirect |
| 16. #1844 | C4 | semidirect |
| 55. #349 | D4 | semidirect |
| 267. C4 | #1339 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33175 | 2 |
| C4×#349 | 2 |
| C2×#1339 | 2 |
| #10837 | 2 |