Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
142 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 142 conjugacy classes of subgroups — too many to draw. Only its 55 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #493 |
| commutator, Frattini, cosocle | C4×C8 | C23 |
| prime radical | #493 | C22 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C82 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #2254 | C2 | semidirect |
| 3. #2241 | C2 | semidirect |
| 5. #1707 | C2 | semidirect |
| 7. C8×C16 | C2 | semidirect |
| 68. C16 | SD16 | semidirect |
| 85. C8 | SD32 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #2254 | 2 |
| #2248 | 2 |
| #2241 | 2 |
| #2240 | 2 |
| #1707 | 2 |
| #1706 | 2 |
| C8×C16 | 2 |