Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1127 conjugacy classes of subgroups; only the 247 normal subgroups are shown.
This group has 1127 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 | 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 |
|---|---|---|
| 2. C22×C4⋊Q8 | C2 | semidirect |
| 3. C22×#103 | C2 | semidirect |
| 5. #2013 | C2 | direct (N × H) |
| 6. C2×#336 | C2 | semidirect |
| 16. C2×C4⋊Q8 | C22 | semidirect |
| 22. C2×#103 | C22 | semidirect |
| 25. #336 | C22 | semidirect |
| 104. C22×C4 | SD16 | semidirect |
| 162. C2×C4 | C2×SD16 | semidirect |
| 191. C2 | #2013 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×SD16 | 2 |
| C22×C4⋊Q8 | 2 |
| C22×#103 | 2 |
| C2×#390 | 2 |
| #2013 | 2 |
| C2×#336 | 2 |