Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
556 conjugacy classes of subgroups; only the 162 normal subgroups are shown.
This group has 556 conjugacy classes of subgroups — too many to draw. Only its 162 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 | #459 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| prime radical | C2×#493 | C22 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C82 ◁ 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×#2209 | C2 | semidirect |
| 4. C2×C8×Q16 | C2 | semidirect |
| 5. #12429 | C2 | direct (N × H) |
| 6. C2×#1370 | C2 | semidirect |
| 7. C2×#1362 | C2 | semidirect |
| 18. #2209 | C22 | semidirect |
| 28. C8×Q16 | C22 | semidirect |
| 32. #1370 | C22 | semidirect |
| 33. #1362 | C22 | semidirect |
| 296. C2 | #12429 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2209 | 2 |
| C2×#2207 | 2 |
| C2×#1709 | 2 |
| C2×C8×Q16 | 2 |
| #12429 | 2 |
| C2×#1370 | 2 |
| C2×#1362 | 2 |
| C2×#1361 | 2 |