Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1479 conjugacy classes of subgroups; only the 823 normal subgroups are shown.
This group has 1479 conjugacy classes of subgroups — too many to draw. Only its 823 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 |
|---|---|---|
| 3. #2327 | C2 | direct (N × H) |
| 9. C2×C4⋊Q8 | C4 | semidirect |
| 37. C4⋊Q8 | C2×C4 | semidirect |
| 128. C2 | #2327 | 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 |
| C22×C4×Q8 | 2 |
| #2327 | 2 |
| C2×#391 | 2 |
| C2×#387 | 2 |
| C2×#382 | 2 |
| C2×#376 | 2 |