Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
453 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 453 conjugacy classes of subgroups — too many to draw. Only its 167 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 | C22×D8 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #2357 | C2 | semidirect |
| 3. #2349 | C2 | semidirect |
| 4. #1936 | C2 | semidirect |
| 5. #1918 | C2 | semidirect |
| 7. #1835 | C2 | semidirect |
| 8. #1804 | C2 | semidirect |
| 9. #1804 | C2 | semidirect |
| 10. #1770 | C2 | semidirect |
| 11. #1762 | C2 | semidirect |
| 16. C4×D8 | C4 | semidirect |
| 58. C4×D4 | C2×C4 | semidirect |
| 73. #107 | C2×C4 | semidirect |
| 78. #102 | C2×C4 | semidirect |
| 80. C4×C8 | C2×C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4×D8 | 2 |
| #2357 | 2 |
| #2349 | 2 |
| #1936 | 2 |
| #1918 | 2 |
| #1879 | 2 |
| #1835 | 2 |
| #1804 | 2 |
| #1770 | 2 |
| #1762 | 2 |