Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
338 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 338 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #357 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C22×C16 | C23 |
| prime radical | C24 | D16 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C23×C8 ◁ C23×C16 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #7419 | C2 | direct (N × H) |
| 5. C22×Q32 | C4 | semidirect |
| 12. #2216 | C4 | semidirect |
| 14. C2×#366 | C4 | semidirect |
| 26. C2×Q32 | C2×C4 | semidirect |
| 47. #366 | C2×C4 | semidirect |
| 204. C2 | #7419 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2216 | 2 |
| C2×C22⋊C32 | 2 |
| C2×#1380 | 2 |
| #7419 | 2 |