Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
1780 conjugacy classes of subgroups; only the 619 normal subgroups are shown.
This group has 1780 conjugacy classes of subgroups — too many to draw. Only its 619 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23 | C22×D21 |
| commutator, cosocle | C2×C3×C7 | C25 |
| Frattini | C2 | C24×D21 |
| Fitting | C22×C3×C7×Q8 | C2 |
| socle | C23×C3×C7 | C23 |
| hypercenter | C22×Q8 | D21 |
| hypocenter | C3×C7 | C22×#41 |
| LCS | G ▷ C2×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C23 ◁ C22×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C3×C7×Q8 | C2 | semidirect |
| 2. C2×#66732 | C2 | direct (N × H) |
| 3. C22×D84 | C2 | semidirect |
| 4. C22×C4×D21 | C2 | semidirect |
| 7. C2×C3×C7×Q8 | C22 | semidirect |
| 10. #66732 | C22 | direct (N × H) |
| 11. C2×D84 | C22 | semidirect |
| 12. C2×C4×D21 | C22 | semidirect |
| 13. C22×C7×Q8 | D3 | semidirect |
| 21. C3×C7×Q8 | C23 | semidirect |
| 24. D84 | C23 | semidirect |
| 25. C4×D21 | C23 | semidirect |
| 27. C2×C7×Q8 | C2×D3 | semidirect |
| 33. C22×C3×Q8 | D7 | semidirect |
| 37. C22×C3×C7 | #41 | semidirect |
| 47. C7×Q8 | C22×D3 | semidirect |
| 53. C2×C3×Q8 | C2×D7 | semidirect |
| 59. C2×C3×C7 | C2×#41 | semidirect |
| 63. C22×Q8 | D21 | semidirect |
| 66. C22×C7 | #239 | semidirect |
| 75. C3×Q8 | C22×D7 | semidirect |
| 80. C3×C7 | C22×#41 | semidirect |
| 83. C2×Q8 | C2×D21 | semidirect |
| 88. C2×C7 | C2×#239 | semidirect |
| 91. C22×C3 | #1161 | semidirect |
| 100. Q8 | C22×D21 | semidirect |
| 105. C7 | C22×#239 | semidirect |
| 106. C2×C3 | C2×#1161 | semidirect |
| 111. C22 | #66732 | direct (N × H) |
| 115. C3 | C22×#1161 | semidirect |
| 117. C2 | C2×#66732 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C3×C7×Q8 | 2 |
| C2×#66732 | 2 |
| C22×D84 | 2 |
| C22×C4×D21 | 2 |
| C22×#1161 | 3 |
| C22×#239 | 7 |