Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
347 conjugacy classes of subgroups; only the 207 normal subgroups are shown.
This group has 347 conjugacy classes of subgroups — too many to draw. Only its 207 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 | D8 |
| commutator | C8 | C2×C42 |
| Frattini, cosocle | C22×C8 | C23 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C23×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #1380 | C2 | direct (N × H) |
| 7. C22×C16 | C4 | semidirect |
| 11. C2×#107 | C4 | semidirect |
| 27. C2×C16 | C2×C4 | semidirect |
| 30. #107 | C2×C4 | semidirect |
| 93. C2 | #1380 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C16 | 2 |
| C22×#107 | 2 |
| #1380 | 2 |