Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1363 conjugacy classes of subgroups; only the 903 normal subgroups are shown.
This group has 1363 conjugacy classes of subgroups — too many to draw. Only its 903 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C24 |
| commutator | C22 | C24×C4 |
| Frattini, cosocle | C23 | C25 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #2369 | C2 | direct (N × H) |
| 8. C2×C4⋊Q8 | C4 | semidirect |
| 32. C4⋊Q8 | C2×C4 | semidirect |
| 36. C2×C4⋊C4 | Q8 | semidirect |
| 77. C4⋊C4 | C2×Q8 | semidirect |
| 123. C2 | #2369 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C4⋊Q8 | 2 |
| #2369 | 2 |
| C2×#391 | 2 |
| C2×#389 | 2 |
| C2×C4×C4⋊C4 | 2 |