Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
531 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 531 conjugacy classes of subgroups — too many to draw. Only its 143 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| socle | C23 | C22×D4 |
| prime radical | #1999 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C4×D8 | C2 | semidirect |
| 2. C2×#431 | C2 | semidirect |
| 3. C4×#121 | C2 | semidirect |
| 4. #2086 | C2 | semidirect |
| 5. #2075 | C2 | semidirect |
| 6. #2067 | C2 | semidirect |
| 7. #2054 | C2 | semidirect |
| 8. #2033 | C2 | semidirect |
| 10. C4×#107 | C2 | semidirect |
| 11. C4×#102 | C2 | semidirect |
| 84. C4×C8 | D4 | semidirect |
| 113. C22×C4 | D8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4×D8 | 2 |
| C2×#431 | 2 |
| C4×#121 | 2 |
| #2086 | 2 |
| #2075 | 2 |
| #2067 | 2 |
| #2054 | 2 |
| #2033 | 2 |
| #1999 | 2 |
| C4×#107 | 2 |
| C4×#102 | 2 |