Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
867 conjugacy classes of subgroups; only the 231 normal subgroups are shown.
This group has 867 conjugacy classes of subgroups — too many to draw. Only its 231 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C4⋊Q8 | C2 | semidirect |
| 4. #2023 | C2 | direct (N × H) |
| 5. C2×#340 | C2 | semidirect |
| 6. C2×#336 | C2 | semidirect |
| 12. C2×C4⋊Q8 | C22 | semidirect |
| 23. #340 | C22 | semidirect |
| 24. #336 | C22 | semidirect |
| 34. C22×D4 | Q8 | semidirect |
| 85. C2×D4 | C2×Q8 | semidirect |
| 204. C2 | #2023 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C4⋊Q8 | 2 |
| C22×#102 | 2 |
| C2×#390 | 2 |
| #2023 | 2 |
| C2×#340 | 2 |
| C2×#336 | 2 |