Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
867 conjugacy classes of subgroups; only the 223 normal subgroups are shown.
This group has 867 conjugacy classes of subgroups — too many to draw. Only its 223 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C42 | C24 |
| socle | C23 | C2×C22⋊C4 |
| prime radical | C23×D4 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#432 | C2 | semidirect |
| 3. C2×C4×M16 | C2 | semidirect |
| 4. #1490 | C2 | direct (N × H) |
| 5. C2×D4⋊C8 | C2 | semidirect |
| 21. #432 | C22 | semidirect |
| 22. C2×C4⋊C8 | C22 | semidirect |
| 25. C4×M16 | C22 | semidirect |
| 28. C2×C4×C8 | C22 | semidirect |
| 29. D4⋊C8 | C22 | semidirect |
| 76. C4⋊C8 | C23 | semidirect |
| 81. C4×C8 | C23 | semidirect |
| 94. C2×D4 | M16 | semidirect |
| 158. D4 | C2×M16 | semidirect |
| 213. C2 | #1490 | 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×#432 | 2 |
| C2×C4×M16 | 2 |
| #1490 | 2 |
| C2×D4⋊C8 | 2 |