Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
565 conjugacy classes of subgroups; only the 191 normal subgroups are shown.
This group has 565 conjugacy classes of subgroups — too many to draw. Only its 191 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #2061 | C2 | direct (N × H) |
| 34. C2×M16 | Q8 | semidirect |
| 63. C2×C42 | Q8 | semidirect |
| 94. M16 | C2×Q8 | semidirect |
| 113. C42 | C2×Q8 | semidirect |
| 174. C2 | #2061 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#544 | 2 |
| C2×#428 | 2 |
| C2×#382 | 2 |
| #2061 | 2 |
| C2×#343 | 2 |
| C2×#339 | 2 |