Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
338 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 338 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 | C2×#331 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C82 | C23 |
| prime radical | C2×#493 | C4 |
| LCS | G ▷ C4×C8 ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #12012 | C2 | semidirect |
| 3. #12004 | C2 | semidirect |
| 4. #7094 | C2 | semidirect |
| 18. #1370 | C22 | semidirect |
| 19. #1367 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#1706 | 2 |
| #12012 | 2 |
| #12004 | 2 |
| #7094 | 2 |