Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1081 conjugacy classes of subgroups; only the 571 normal subgroups are shown.
This group has 1081 conjugacy classes of subgroups — too many to draw. Only its 571 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator | C22 | C22×C42 |
| socle | C23 | C23×C4 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #1729 | C2 | semidirect |
| 4. #1728 | C2 | semidirect |
| 5. C2×#336 | C2 | semidirect |
| 9. C2×#131 | C4 | semidirect |
| 32. C2×M16 | C2×C4 | semidirect |
| 35. C22×C8 | C2×C4 | semidirect |
| 99. C2×C4 | #131 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#131 | 2 |
| C2×C4×#41 | 2 |
| #1729 | 2 |
| #1728 | 2 |
| C2×#336 | 2 |