Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
869 conjugacy classes of subgroups; only the 215 normal subgroups are shown.
This group has 869 conjugacy classes of subgroups — too many to draw. Only its 215 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 | C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#407 | C2 | semidirect |
| 3. #1465 | C2 | direct (N × H) |
| 4. C2×#324 | C2 | semidirect |
| 5. C2×#323 | C2 | semidirect |
| 20. #407 | C22 | semidirect |
| 23. #324 | C22 | semidirect |
| 24. #323 | C22 | semidirect |
| 84. C24 | M16 | semidirect |
| 120. C22×C4 | M16 | semidirect |
| 172. C23 | C2×M16 | semidirect |
| 204. C2×C4 | C2×M16 | semidirect |
| 246. C2 | #1465 | 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×#407 | 2 |
| #1465 | 2 |
| C2×#324 | 2 |
| C2×#323 | 2 |