Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
527 conjugacy classes of subgroups; only the 243 normal subgroups are shown.
This group has 527 conjugacy classes of subgroups — too many to draw. Only its 243 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×D4 | D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #1948 | C2 | semidirect |
| 4. #1853 | C2 | semidirect |
| 5. #1834 | C2 | semidirect |
| 6. #1807 | C2 | semidirect |
| 7. C4×#106 | C2 | semidirect |
| 8. C2×#340 | C2 | semidirect |
| 13. C8×D4 | C4 | semidirect |
| 33. C22×C8 | C2×C4 | semidirect |
| 55. C4⋊C8 | C2×C4 | semidirect |
| 56. C22⋊C8 | C2×C4 | semidirect |
| 57. C4×C8 | C2×C4 | semidirect |
| 105. D4 | #106 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C8×D4 | 2 |
| D4×C4⋊C4 | 2 |
| #1948 | 2 |
| #1853 | 2 |
| #1834 | 2 |
| #1807 | 2 |
| C4×#106 | 2 |
| C2×#340 | 2 |