Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
2086 conjugacy classes of subgroups; only the 698 normal subgroups are shown.
This group has 2086 conjugacy classes of subgroups — too many to draw. Only its 698 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C2×#2209 | C2 | direct (N × H) |
| 22. #2209 | C22 | direct (N × H) |
| 134. C22×C4 | Q32 | semidirect |
| 182. C2×C4 | C2×Q32 | semidirect |
| 193. C22 | #2209 | direct (N × H) |
| 203. C4 | C22×Q32 | semidirect |
| 206. C2 | C2×#2209 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×Q32 | 2 |
| C22×#494 | 2 |
| C22×C4×Q16 | 2 |
| C2×#2209 | 2 |
| C22×C4⋊C16 | 2 |
| C22×#358 | 2 |