Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1031 conjugacy classes of subgroups; only the 438 normal subgroups are shown.
This group has 1031 conjugacy classes of subgroups — too many to draw. Only its 438 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C42 | C24 |
| commutator, socle | C23 | C22×C42 |
| prime radical | C22×D4 | C42 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#1455 | C2 | semidirect |
| 3. C22×C82 | C2 | semidirect |
| 4. #8262 | C2 | semidirect |
| 5. #8146 | C2 | semidirect |
| 6. #8144 | C2 | semidirect |
| 7. #7846 | C2 | semidirect |
| 57. C82 | D4 | semidirect |
| 66. C22×C8 | M16 | semidirect |
| 168. C8 | #435 | semidirect |
| 174. C22 | #1455 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×#435 | 2 |
| C2×#1455 | 2 |
| C22×C82 | 2 |
| #8262 | 2 |
| #8146 | 2 |
| #8144 | 2 |
| #7846 | 2 |