Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
921 conjugacy classes of subgroups; only the 279 normal subgroups are shown.
This group has 921 conjugacy classes of subgroups — too many to draw. Only its 279 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 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#423 | C2 | semidirect |
| 4. #1800 | C2 | direct (N × H) |
| 5. C2×#343 | C2 | semidirect |
| 18. #423 | C22 | semidirect |
| 23. #343 | C22 | semidirect |
| 157. C2×C4 | #100 | semidirect |
| 197. C4 | C2×#100 | semidirect |
| 204. C2 | #1800 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C4×D4 | 2 |
| C2×#423 | 2 |
| C22×#100 | 2 |
| #1800 | 2 |
| C2×#343 | 2 |