Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
719 conjugacy classes of subgroups; only the 247 normal subgroups are shown.
This group has 719 conjugacy classes of subgroups — too many to draw. Only its 247 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 | #121 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | #41 |
| LCS | G ▷ C2×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 |
|---|---|---|
| 6. #1979 | C2 | direct (N × H) |
| 12. C22×Q16 | C4 | semidirect |
| 47. C2×Q16 | C2×C4 | semidirect |
| 198. C2 | #1979 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×Q16 | 2 |
| C22×#108 | 2 |
| C2×#419 | 2 |
| C22×#101 | 2 |
| C2×#405 | 2 |
| #1979 | 2 |