Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1058 conjugacy classes of subgroups; only the 218 normal subgroups are shown.
This group has 1058 conjugacy classes of subgroups — too many to draw. Only its 218 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 | C2×#121 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #33301 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #28441 | C2 | semidirect |
| 3. #21712 | C2 | semidirect |
| 4. C2×#1688 | C2 | semidirect |
| 5. #10262 | C2 | semidirect |
| 6. #9695 | C2 | semidirect |
| 7. C4×#334 | C2 | semidirect |
| 66. #334 | D4 | semidirect |
| 120. C4×C8 | SD16 | semidirect |
| 218. C8 | #492 | semidirect |
| 233. C4 | #1688 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33301 | 2 |
| #28441 | 2 |
| #21712 | 2 |
| C2×#1688 | 2 |
| #10262 | 2 |
| #9695 | 2 |
| C4×#334 | 2 |