Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
699 conjugacy classes of subgroups; only the 223 normal subgroups are shown.
This group has 699 conjugacy classes of subgroups — too many to draw. Only its 223 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 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C4×Q16 | C2 | semidirect |
| 4. #1969 | C2 | semidirect |
| 5. #1878 | C2 | semidirect |
| 11. C22×Q16 | C4 | semidirect |
| 99. Q16 | C22⋊C4 | direct (N × H) |
| 113. C22⋊C4 | Q16 | direct (N × H) |
| 148. C2×C4 | C2×Q16 | semidirect |
| 160. C22 | C4×Q16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×Q16 | 2 |
| C2×C4×Q16 | 2 |
| Q8×C22⋊C4 | 2 |
| #1969 | 2 |
| #1878 | 2 |
| #1808 | 2 |
| #1787 | 2 |
| C8×C22⋊C4 | 2 |