Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1059 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 1059 conjugacy classes of subgroups — too many to draw. Only its 167 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×#127 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#460 | C2 | semidirect |
| 3. C2×#450 | C2 | semidirect |
| 4. C4×C22⋊Q8 | C2 | semidirect |
| 5. #2041 | C2 | semidirect |
| 6. #2014 | C2 | semidirect |
| 7. #1982 | C2 | semidirect |
| 8. #1797 | C2 | semidirect |
| 9. #1786 | C2 | semidirect |
| 13. C22×SD16 | C22 | semidirect |
| 20. C2×C4×Q8 | C22 | semidirect |
| 24. C2×C4⋊C8 | C22 | semidirect |
| 25. C2×#102 | C22 | semidirect |
| 26. C2×#102 | C22 | semidirect |
| 27. C2×#102 | C22 | semidirect |
| 29. #391 | C22 | semidirect |
| 30. #389 | C22 | semidirect |
| 35. C4×C4⋊C4 | C22 | semidirect |
| 37. #336 | C22 | semidirect |
| 69. C22⋊Q8 | D4 | semidirect |
| 108. C2×Q8 | C2×D4 | semidirect |
| 145. C22×C4 | SD16 | semidirect |
| 164. C4⋊C4 | C2×D4 | semidirect |
| 256. C22 | #460 | semidirect |
| 272. C4 | #450 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#127 | 2 |
| C2×#460 | 2 |
| C2×#450 | 2 |
| C4×C22⋊Q8 | 2 |
| #2041 | 2 |
| #2014 | 2 |
| #1982 | 2 |
| #1797 | 2 |
| #1786 | 2 |