Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
801 conjugacy classes of subgroups; only the 223 normal subgroups are shown.
This group has 801 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 | C22 | C22×C22⋊C4 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C24 | C24 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #1820 | C2 | semidirect |
| 4. #1819 | C2 | semidirect |
| 5. C2×#342 | C2 | semidirect |
| 8. C2×C4×D4 | C4 | semidirect |
| 24. C23×C4 | C2×C4 | semidirect |
| 28. C2×C4⋊C4 | C2×C4 | semidirect |
| 35. C2×C22⋊C4 | C2×C4 | semidirect |
| 38. C2×C42 | C2×C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#518 | 2 |
| #2308 | 2 |
| #1820 | 2 |
| #1819 | 2 |
| C2×#342 | 2 |