Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
447 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 447 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, socle | C22 | #1961 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#493 | C4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1726 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #12541 | C2 | semidirect |
| 3. #11599 | C2 | semidirect |
| 4. #11593 | C2 | semidirect |
| 5. #8780 | C2 | semidirect |
| 54. #346 | D4 | semidirect |
| 56. C4×C16 | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #21697 | 2 |
| #12541 | 2 |
| #11599 | 2 |
| #11593 | 2 |
| #8780 | 2 |