Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1325 conjugacy classes of subgroups; only the 575 normal subgroups are shown.
This group has 1325 conjugacy classes of subgroups — too many to draw. Only its 575 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C25 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #2857 | C2 | semidirect |
| 3. #2659 | C2 | semidirect |
| 4. #2583 | C2 | semidirect |
| 5. #2502 | C2 | semidirect |
| 6. #2434 | C2 | semidirect |
| 11. C2×C4⋊Q8 | C22 | semidirect |
| 22. #398 | C22 | semidirect |
| 23. #395 | C22 | semidirect |
| 24. #391 | C22 | semidirect |
| 25. #389 | C22 | semidirect |
| 28. #384 | C22 | semidirect |
| 61. C2×C22⋊C4 | Q8 | semidirect |
| 96. C22×C4 | C2×Q8 | semidirect |
| 111. C22⋊C4 | C2×Q8 | semidirect |
| 121. C23 | C4⋊Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C22⋊Q8 | 2 |
| #2857 | 2 |
| #2659 | 2 |
| #2583 | 2 |
| #2502 | 2 |
| #2434 | 2 |
| #2315 | 2 |
| C2×C4×C22⋊C4 | 2 |