Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
669 conjugacy classes of subgroups; only the 183 normal subgroups are shown.
This group has 669 conjugacy classes of subgroups — too many to draw. Only its 183 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 | #99 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C2×D4 | C24 |
| socle | C22 | C2×#99 |
| prime radical | C2×#532 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#410 | C2 | semidirect |
| 3. #2126 | C2 | direct (N × H) |
| 4. C2×#354 | C2 | semidirect |
| 5. C2×#353 | C2 | semidirect |
| 6. C22×#41 | C4 | semidirect |
| 9. #532 | C4 | semidirect |
| 11. C2×C4⋊Q8 | C4 | semidirect |
| 13. C2×#124 | C4 | semidirect |
| 18. C22×C42 | C4 | semidirect |
| 20. #410 | C22 | semidirect |
| 23. #354 | C22 | semidirect |
| 24. #353 | C22 | semidirect |
| 33. C2×#41 | C2×C4 | semidirect |
| 48. C4⋊Q8 | C2×C4 | semidirect |
| 55. #124 | C2×C4 | semidirect |
| 73. C2×C42 | C2×C4 | semidirect |
| 245. C2 | #2126 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#532 | 2 |
| C2×#410 | 2 |
| #2126 | 2 |
| C2×#354 | 2 |
| C2×#353 | 2 |