Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
548 conjugacy classes of subgroups; only the 166 normal subgroups are shown.
This group has 548 conjugacy classes of subgroups — too many to draw. Only its 166 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×D16 |
| commutator, Frattini, cosocle | C2×C16 | C24 |
| socle | C22 | C22×D16 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#2262 | C2 | semidirect |
| 2. #33713 | C2 | semidirect |
| 3. #13728 | C2 | semidirect |
| 4. #13727 | C2 | semidirect |
| 5. #13726 | C2 | semidirect |
| 6. #13725 | C2 | semidirect |
| 7. C2×C4×C32 | C2 | semidirect |
| 47. C2×C32 | D4 | semidirect |
| 153. C4 | #2262 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2262 | 2 |
| #33713 | 2 |
| #13728 | 2 |
| #13727 | 2 |
| #13726 | 2 |
| #13725 | 2 |
| C2×C4×C32 | 2 |