Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1719 conjugacy classes of subgroups; only the 583 normal subgroups are shown.
This group has 1719 conjugacy classes of subgroups — too many to draw. Only its 583 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C24 |
| commutator, Frattini, cosocle | C23 | C25 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C22⋊Q8 | C2 | semidirect |
| 2. #2683 | C2 | direct (N × H) |
| 3. C2×#396 | C2 | semidirect |
| 4. C2×#395 | C2 | semidirect |
| 6. C2×#389 | C2 | semidirect |
| 12. C2×C22⋊Q8 | C22 | semidirect |
| 22. #396 | C22 | semidirect |
| 23. #395 | C22 | semidirect |
| 25. #389 | C22 | semidirect |
| 35. C22×D4 | Q8 | semidirect |
| 50. C2×C4⋊C4 | D4 | semidirect |
| 79. C2×D4 | C2×Q8 | semidirect |
| 106. C4⋊C4 | C2×D4 | semidirect |
| 171. C2 | #2683 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C22⋊Q8 | 2 |
| #2683 | 2 |
| C2×#396 | 2 |
| C2×#395 | 2 |
| C2×#390 | 2 |
| C2×#389 | 2 |
| C2×#386 | 2 |