Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
967 conjugacy classes of subgroups; only the 255 normal subgroups are shown.
This group has 967 conjugacy classes of subgroups — too many to draw. Only its 255 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 | C25 | D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C2×#419 | C2 | semidirect |
| 5. #1873 | C2 | semidirect |
| 6. #1872 | C2 | semidirect |
| 7. #1736 | C2 | semidirect |
| 14. C2×C22⋊Q8 | C4 | semidirect |
| 29. C2×#103 | C22 | semidirect |
| 33. #340 | C22 | semidirect |
| 39. C22×Q8 | C2×C4 | semidirect |
| 63. C22⋊Q8 | C2×C4 | semidirect |
| 76. C2×C4⋊C4 | C2×C4 | semidirect |
| 237. C22 | #419 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C22⋊Q8 | 2 |
| C22×#117 | 2 |
| C2×#419 | 2 |
| C2×#407 | 2 |
| #1873 | 2 |
| #1872 | 2 |
| #1736 | 2 |