Finite Group DatabasebetaFinite Group DatabasebetaOrder 2016.
780 conjugacy classes of subgroups; only the 312 normal subgroups are shown.
This group has 780 conjugacy classes of subgroups — too many to draw. Only its 312 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C32×C7 | C23 |
| commutator, Frattini, cosocle | C22 | C23×C32×C7 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C32×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C24×C32×C7 | C2 | semidirect |
| 2. C2×C32×C7×D4 | C2 | semidirect |
| 3. C32×C7×C22⋊C4 | C2 | semidirect |
| 4. C3×C7×#120 | C3 | direct (N × H) |
| 7. C23×C32×C7 | C22 | semidirect |
| 9. C2×C32×C4×C7 | C22 | semidirect |
| 10. C24×C3×C7 | C2×C3 | semidirect |
| 11. C2×C3×C7×D4 | C2×C3 | semidirect |
| 12. C3×C7×C22⋊C4 | C2×C3 | semidirect |
| 13. C32×#120 | C7 | direct (N × H) |
| 17. C22×C32×C7 | D4 | semidirect |
| 20. C7×#120 | C32 | direct (N × H) |
| 23. C23×C3×C7 | C22×C3 | semidirect |
| 25. C2×C3×C4×C7 | C22×C3 | semidirect |
| 26. C24×C32 | C2×C7 | semidirect |
| 27. C2×C32×D4 | C2×C7 | semidirect |
| 28. C32×C22⋊C4 | C2×C7 | semidirect |
| 32. C24×C7 | C2×C32 | semidirect |
| 33. C2×C7×D4 | C2×C32 | semidirect |
| 34. C7×C22⋊C4 | C2×C32 | semidirect |
| 35. C3×#120 | C3×C7 | direct (N × H) |
| 39. C22×C3×C7 | C3×D4 | semidirect |
| 44. C23×C32 | C22×C7 | semidirect |
| 46. C2×C32×C4 | C22×C7 | semidirect |
| 47. C32×C7 | #120 | direct (N × H) |
| 50. C23×C7 | C22×C32 | semidirect |
| 52. C2×C4×C7 | C22×C32 | semidirect |
| 53. C24×C3 | C2×C3×C7 | semidirect |
| 54. C2×C3×D4 | C2×C3×C7 | semidirect |
| 55. C3×C22⋊C4 | C2×C3×C7 | semidirect |
| 62. C22×C32 | C7×D4 | semidirect |
| 65. #120 | C32×C7 | direct (N × H) |
| 69. C22×C7 | C32×D4 | semidirect |
| 74. C23×C3 | C22×C3×C7 | semidirect |
| 76. C2×C3×C4 | C22×C3×C7 | semidirect |
| 77. C3×C7 | C3×#120 | direct (N × H) |
| 81. C24 | C2×C32×C7 | semidirect |
| 82. C2×D4 | C2×C32×C7 | semidirect |
| 83. C22⋊C4 | C2×C32×C7 | semidirect |
| 90. C22×C3 | C3×C7×D4 | semidirect |
| 93. C32 | C7×#120 | direct (N × H) |
| 96. C23 | C22×C32×C7 | semidirect |
| 98. C2×C4 | C22×C32×C7 | semidirect |
| 99. C7 | C32×#120 | direct (N × H) |
| 106. C22 | C32×C7×D4 | semidirect |
| 109. C3 | C3×C7×#120 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×C32×C7 | 2 |
| C2×C32×C7×D4 | 2 |
| C32×C7×C22⋊C4 | 2 |
| C3×C7×#120 | 3 |
| C32×#120 | 7 |