Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
644 conjugacy classes of subgroups; only the 318 normal subgroups are shown.
This group has 644 conjugacy classes of subgroups — too many to draw. Only its 318 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#376 |
| prime radical | C24 | #117 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #8142 | C2 | semidirect |
| 4. #8141 | C2 | semidirect |
| 16. #1451 | C4 | semidirect |
| 18. C8×M16 | C4 | semidirect |
| 47. #322 | C2×C4 | semidirect |
| 48. #322 | C2×C4 | semidirect |
| 49. C82 | C2×C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#1746 | 2 |
| #17323 | 2 |
| #8142 | 2 |
| #8141 | 2 |
| #7861 | 2 |
| #7855 | 2 |