Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
467 conjugacy classes of subgroups; only the 87 normal subgroups are shown.
This group has 467 conjugacy classes of subgroups — too many to draw. Only its 87 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #447 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C2×#576 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#361 | C2 | semidirect |
| 3. C2×#359 | C2 | semidirect |
| 4. #2139 | C2 | semidirect |
| 5. C2×#357 | C2 | semidirect |
| 6. C2×#350 | C2 | semidirect |
| 7. #1816 | C2 | semidirect |
| 12. #576 | C4 | semidirect |
| 20. C2×M32 | C22 | semidirect |
| 21. C22×C16 | C22 | semidirect |
| 22. C2×#108 | C22 | semidirect |
| 23. C2×#107 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#576 | 2 |
| C2×#361 | 2 |
| C2×#359 | 2 |
| #2139 | 2 |
| C2×#357 | 2 |
| C2×#350 | 2 |
| #1816 | 2 |