Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
317 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 317 conjugacy classes of subgroups — too many to draw. Only its 78 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 | #2033 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#493 | C22 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #2960 ◁ 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. #13265 | C2 | semidirect |
| 4. #12009 | C2 | semidirect |
| 6. #8685 | C2 | semidirect |
| 7. #7277 | C2 | semidirect |
| 109. M32 | SD16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #19203 | 2 |
| #13265 | 2 |
| #13213 | 2 |
| #12009 | 2 |
| C2×#1361 | 2 |
| #8685 | 2 |
| #7277 | 2 |