Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
781 conjugacy classes of subgroups; only the 183 normal subgroups are shown.
This group has 781 conjugacy classes of subgroups — too many to draw. Only its 183 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 | C2×C4 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #1736 | C2 | semidirect |
| 5. C2×#339 | C2 | semidirect |
| 10. #518 | C4 | semidirect |
| 19. C2×#100 | C4 | semidirect |
| 21. #390 | C4 | semidirect |
| 44. C2×M16 | C2×C4 | semidirect |
| 55. C2×C4⋊C4 | C2×C4 | semidirect |
| 67. C2×C42 | C2×C4 | semidirect |
| 128. C4⋊C4 | C22⋊C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#518 | 2 |
| C22×#100 | 2 |
| C2×#390 | 2 |
| #1736 | 2 |
| C2×#339 | 2 |