Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
845 conjugacy classes of subgroups; only the 187 normal subgroups are shown.
This group has 845 conjugacy classes of subgroups — too many to draw. Only its 187 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×C22⋊C4 |
| commutator | C23 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C2×#343 | C2 | semidirect |
| 4. #1728 | C2 | semidirect |
| 16. C2×#100 | C4 | semidirect |
| 41. C2×M16 | C2×C4 | semidirect |
| 150. C2×C4 | #100 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C4×D4 | 2 |
| C22×#100 | 2 |
| C2×#343 | 2 |
| #1728 | 2 |