Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1671 conjugacy classes of subgroups; only the 415 normal subgroups are shown.
This group has 1671 conjugacy classes of subgroups — too many to draw. Only its 415 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C24 |
| commutator | C23 | C23×C4 |
| prime radical | C22×#127 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#390 | C2 | semidirect |
| 3. C2×#384 | C2 | semidirect |
| 4. C2×C43 | C2 | semidirect |
| 8. C2×#127 | C4 | semidirect |
| 34. C2×C42 | D4 | semidirect |
| 59. C42 | C22⋊C4 | semidirect |
| 82. C2×C4 | #127 | semidirect |
| 97. C4 | #390 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#127 | 2 |
| C2×#390 | 2 |
| C2×#384 | 2 |
| C2×C43 | 2 |