Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
283 conjugacy classes of subgroups; only the 85 normal subgroups are shown.
This group has 283 conjugacy classes of subgroups — too many to draw. Only its 85 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator | C2×C4 | C22×C8 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #386 |
| prime radical | C2×AGL(1,ℤ/8) | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #2988 | C2 | semidirect |
| 3. #1954 | C2 | semidirect |
| 4. #1478 | C2 | semidirect |
| 5. #1476 | C2 | semidirect |
| 6. #1283 | C2 | semidirect |
| 7. #1278 | C2 | semidirect |
| 37. AGL(1,ℤ/8) | C8 | semidirect |
| 97. C2×D4 | C2×C8 | semidirect |
| 114. M16 | C2×C8 | semidirect |
| 152. Q8 | C22⋊C8 | semidirect |
| 160. D4 | C22⋊C8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #2988 | 2 |
| C4×AGL(1,ℤ/8) | 2 |
| #1954 | 2 |
| #1478 | 2 |
| #1476 | 2 |
| #1283 | 2 |
| #1278 | 2 |