Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
353 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 353 conjugacy classes of subgroups — too many to draw. Only its 121 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 | #3147 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#510 | C2 | semidirect |
| 3. C2×C4×Q16 | C2 | semidirect |
| 4. #2211 | C2 | semidirect |
| 5. #2209 | C2 | semidirect |
| 6. #2207 | C2 | semidirect |
| 7. #2150 | C2 | semidirect |
| 8. #2139 | C2 | semidirect |
| 42. C2×Q16 | D4 | semidirect |
| 175. C4 | #510 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#510 | 2 |
| #3147 | 2 |
| C2×C4×Q16 | 2 |
| #2211 | 2 |
| #2209 | 2 |
| #2207 | 2 |
| #2150 | 2 |
| #2139 | 2 |