Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
9310 conjugacy classes of subgroups; only the 6366 normal subgroups are shown.
This group has 9310 conjugacy classes of subgroups — too many to draw. Only its 6366 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C7 | C24 |
| commutator, Frattini, cosocle | C22 | C26×C7 |
| prime radical | C26×C7 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C7×#543 | C2 | direct (N × H) |
| 2. C22×C7×C22⋊Q8 | C2 | semidirect |
| 5. C23×C7×Q8 | C22 | semidirect |
| 7. C7×#543 | C22 | direct (N × H) |
| 8. C2×C7×C22⋊Q8 | C22 | semidirect |
| 9. C22×C7×C4⋊C4 | C22 | semidirect |
| 12. C22×#543 | C7 | direct (N × H) |
| 14. C25×C7 | Q8 | semidirect |
| 18. C22×C7×Q8 | C23 | semidirect |
| 22. C7×C22⋊Q8 | C23 | semidirect |
| 23. C2×C7×C4⋊C4 | C23 | semidirect |
| 26. C2×#543 | C2×C7 | direct (N × H) |
| 27. C22×C22⋊Q8 | C2×C7 | semidirect |
| 33. C24×C7 | C2×Q8 | semidirect |
| 38. C2×C7×Q8 | C24 | semidirect |
| 43. C7×C4⋊C4 | C24 | semidirect |
| 46. C23×Q8 | C22×C7 | semidirect |
| 48. #543 | C22×C7 | direct (N × H) |
| 49. C2×C22⋊Q8 | C22×C7 | semidirect |
| 50. C22×C4⋊C4 | C22×C7 | semidirect |
| 59. C23×C7 | C22×Q8 | semidirect |
| 68. C25 | C7×Q8 | semidirect |
| 72. C22×Q8 | C23×C7 | semidirect |
| 76. C22⋊Q8 | C23×C7 | semidirect |
| 77. C2×C4⋊C4 | C23×C7 | semidirect |
| 83. C22×C7 | #543 | direct (N × H) |
| 93. C24 | C2×C7×Q8 | semidirect |
| 98. C2×Q8 | C24×C7 | semidirect |
| 103. C4⋊C4 | C24×C7 | semidirect |
| 106. C2×C7 | C2×#543 | direct (N × H) |
| 115. C23 | C22×C7×Q8 | semidirect |
| 123. C7 | C22×#543 | direct (N × H) |
| 127. C22 | C7×#543 | direct (N × H) |
| 134. C2 | C2×C7×#543 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C7×#543 | 2 |
| C22×C7×C22⋊Q8 | 2 |
| C23×C7×C22⋊C4 | 2 |
| C22×#543 | 7 |