Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
393 conjugacy classes of subgroups; only the 215 normal subgroups are shown.
This group has 393 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 | C22×C8 | C23 |
| commutator | C22 | C23×C8 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C23 | C23×C4 |
| prime radical | C24 | C22×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #1916 | C2 | semidirect |
| 4. #1839 | C2 | semidirect |
| 5. C8×C4⋊C4 | C2 | semidirect |
| 7. C42×C8 | C2 | semidirect |
| 31. #126 | C8 | direct (N × H) |
| 68. C4⋊C4 | C2×C8 | semidirect |
| 76. C42 | C2×C8 | semidirect |
| 101. C8 | #126 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×#126 | 2 |
| #1923 | 2 |
| #1916 | 2 |
| #1839 | 2 |
| C8×C4⋊C4 | 2 |
| C8×C22⋊C4 | 2 |
| C42×C8 | 2 |