Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
697 conjugacy classes of subgroups; only the 199 normal subgroups are shown.
This group has 697 conjugacy classes of subgroups — too many to draw. Only its 199 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 | C2×C22⋊C4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C42 | C24 |
| prime radical | C23×D4 | C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#432 | C2 | semidirect |
| 3. C2×C4×M16 | C2 | semidirect |
| 4. #1527 | C2 | direct (N × H) |
| 5. C2×#329 | C2 | semidirect |
| 21. #432 | C22 | semidirect |
| 25. C4×M16 | C22 | semidirect |
| 26. C2×#97 | C22 | semidirect |
| 27. C2×#97 | C22 | semidirect |
| 29. #329 | C22 | semidirect |
| 79. #97 | C23 | semidirect |
| 80. #97 | C23 | semidirect |
| 211. C2 | #1527 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#124 | 2 |
| C2×#432 | 2 |
| C2×C4×M16 | 2 |
| #1527 | 2 |
| C2×#329 | 2 |