Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
332 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 332 conjugacy classes of subgroups — too many to draw. Only its 117 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 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| prime radical | C2×D4 | D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×Q32 | C2 | semidirect |
| 3. #2247 | C2 | semidirect |
| 4. #2216 | C2 | semidirect |
| 5. #2209 | C2 | semidirect |
| 6. #2188 | C2 | semidirect |
| 7. C4×Q32 | C2 | semidirect |
| 47. Q32 | D4 | direct (N × H) |
| 96. D4 | Q32 | direct (N × H) |
| 113. C22 | C2×Q32 | semidirect |
| 120. C4 | C2×Q32 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×Q32 | 2 |
| D4×Q16 | 2 |
| #2247 | 2 |
| #2216 | 2 |
| #2209 | 2 |
| #2188 | 2 |
| C4×Q32 | 2 |
| D4×C16 | 2 |