Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
467 conjugacy classes of subgroups; only the 159 normal subgroups are shown.
This group has 467 conjugacy classes of subgroups — too many to draw. Only its 159 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 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#466 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #2324 | C2 | semidirect |
| 3. #1925 | C2 | semidirect |
| 4. #1921 | C2 | semidirect |
| 5. #1895 | C2 | semidirect |
| 6. #1879 | C2 | semidirect |
| 7. #1862 | C2 | semidirect |
| 8. #1808 | C2 | semidirect |
| 9. #1729 | C2 | semidirect |
| 16. #466 | C4 | semidirect |
| 48. C22×C8 | C2×C4 | semidirect |
| 53. #121 | C2×C4 | semidirect |
| 74. #107 | C2×C4 | semidirect |
| 77. #102 | C2×C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#466 | 2 |
| #2324 | 2 |
| #1925 | 2 |
| #1921 | 2 |
| #1895 | 2 |
| #1879 | 2 |
| #1862 | 2 |
| #1808 | 2 |
| #1729 | 2 |