Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1101 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 1101 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 |
| 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 |
|---|---|---|
| 1. C22×#127 | C2 | semidirect |
| 2. C2×#459 | C2 | semidirect |
| 3. C2×#447 | C2 | semidirect |
| 4. C4×#121 | C2 | semidirect |
| 5. #2041 | C2 | semidirect |
| 6. #2011 | C2 | semidirect |
| 7. #1982 | C2 | semidirect |
| 8. #1797 | C2 | semidirect |
| 9. #1786 | C2 | semidirect |
| 13. C22×D8 | C22 | semidirect |
| 21. C2×C4×D4 | C22 | semidirect |
| 25. C2×C4⋊C8 | C22 | semidirect |
| 26. C2×#102 | C22 | semidirect |
| 27. C2×#102 | C22 | semidirect |
| 28. C2×#102 | C22 | semidirect |
| 30. #390 | C22 | semidirect |
| 31. #389 | C22 | semidirect |
| 35. C4×C4⋊C4 | C22 | semidirect |
| 37. #336 | C22 | semidirect |
| 69. #127 | D4 | semidirect |
| 71. #121 | D4 | semidirect |
| 138. C2×D4 | C2×D4 | semidirect |
| 152. C22×C4 | D8 | semidirect |
| 169. C4⋊C4 | C2×D4 | semidirect |
| 276. C22 | #459 | semidirect |
| 291. C4 | #447 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#127 | 2 |
| C2×#459 | 2 |
| C2×#447 | 2 |
| C4×#121 | 2 |
| #2041 | 2 |
| #2011 | 2 |
| #1982 | 2 |
| #1797 | 2 |
| #1786 | 2 |