Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
450 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 450 conjugacy classes of subgroups — too many to draw. Only its 138 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 | #1677 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| prime radical | C2×#2213 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#2244 | C2 | semidirect |
| 2. C2×#2231 | C2 | semidirect |
| 4. C2×#2171 | C2 | semidirect |
| 5. #12465 | C2 | direct (N × H) |
| 6. C2×#1369 | C2 | semidirect |
| 7. C2×#1366 | C2 | semidirect |
| 8. C2×#1357 | C2 | semidirect |
| 15. #2244 | C22 | semidirect |
| 17. #2231 | C22 | semidirect |
| 21. #2171 | C22 | semidirect |
| 32. #1369 | C22 | semidirect |
| 33. #1366 | C22 | semidirect |
| 34. #1357 | C22 | semidirect |
| 285. C2 | #12465 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2244 | 2 |
| C2×#2231 | 2 |
| C2×#2213 | 2 |
| C2×#2171 | 2 |
| #12465 | 2 |
| C2×#1369 | 2 |
| C2×#1366 | 2 |
| C2×#1357 | 2 |