Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
275 conjugacy classes of subgroups; only the 83 normal subgroups are shown.
This group has 275 conjugacy classes of subgroups — too many to draw. Only its 83 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #380 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C22×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #1808 | C2 | semidirect |
| 4. #1289 | C2 | semidirect |
| 5. C23×C8 | C4 | semidirect |
| 11. C2×C4⋊C8 | C4 | semidirect |
| 13. C2×C22⋊C8 | C4 | semidirect |
| 15. C2×C4×C8 | C4 | semidirect |
| 96. C23 | #107 | semidirect |
| 122. C2×C4 | #107 | semidirect |
| 130. C8 | #99 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C8×D4 | 2 |
| #1808 | 2 |
| #1799 | 2 |
| #1289 | 2 |