Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1091 conjugacy classes of subgroups; only the 239 normal subgroups are shown.
This group has 1091 conjugacy classes of subgroups — too many to draw. Only its 239 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 | C25 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×Q16 | C2 | semidirect |
| 3. C22×#103 | C2 | semidirect |
| 5. #2001 | C2 | direct (N × H) |
| 6. C2×#340 | C2 | semidirect |
| 11. C22×Q16 | C22 | semidirect |
| 22. C2×#103 | C22 | semidirect |
| 25. #340 | C22 | semidirect |
| 71. C24 | Q16 | semidirect |
| 137. C23 | C2×Q16 | semidirect |
| 196. C2 | #2001 | 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×C22⋊Q8 | 2 |
| C22×#103 | 2 |
| C22×C22⋊C8 | 2 |
| #2001 | 2 |
| C2×#340 | 2 |