Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
736 conjugacy classes of subgroups; only the 162 normal subgroups are shown.
This group has 736 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 | #447 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| prime radical | C22×D4 | D8 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#2241 | C2 | semidirect |
| 3. C2×#2209 | C2 | semidirect |
| 5. #12345 | C2 | direct (N × H) |
| 6. C2×#1364 | C2 | semidirect |
| 7. C2×#1332 | C2 | semidirect |
| 25. #2241 | C22 | semidirect |
| 27. #2209 | C22 | semidirect |
| 36. #1364 | C22 | semidirect |
| 37. #1332 | C22 | semidirect |
| 347. C2 | #12345 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#3342 | 2 |
| C2×#2241 | 2 |
| C2×#2209 | 2 |
| C2×#1669 | 2 |
| #12345 | 2 |
| C2×#1364 | 2 |
| C2×#1332 | 2 |
| C2×D4⋊C16 | 2 |