Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
584 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 584 conjugacy classes of subgroups — too many to draw. Only its 144 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D3×D7 |
| commutator, socle, cosocle | C22×C3×C7 | C24 |
| Fitting | C3×C42×C7 | C22 |
| hypercenter | C42 | D3×D7 |
| hypocenter | C3×C7 | #547 |
| LCS | G ▷ C22×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C22 ◁ C42 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#66673 | C2 | semidirect |
| 2. C4×D84 | C2 | semidirect |
| 3. C4×C7×D12 | C2 | semidirect |
| 4. C3×#6432 | C2 | semidirect |
| 5. #10621028 | C2 | semidirect |
| 6. #10621000 | C2 | semidirect |
| 7. #10620999 | C2 | semidirect |
| 8. #10620931 | C2 | semidirect |
| 34. #6432 | D3 | semidirect |
| 42. D84 | D4 | semidirect |
| 75. C4×D12 | D7 | semidirect |
| 85. C3×C4×C7 | #41 | semidirect |
| 145. C4×C7 | #239 | semidirect |
| 163. C3×C7 | #547 | semidirect |
| 195. C3×C4 | #1157 | semidirect |
| 214. C7 | #5726 | semidirect |
| 231. C4 | #66673 | semidirect |
| 233. C3 | #90242 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#66673 | 2 |
| C4×D84 | 2 |
| C4×C7×D12 | 2 |
| C3×#6432 | 2 |
| #10621028 | 2 |
| #10621000 | 2 |
| #10620999 | 2 |
| #10620931 | 2 |
| #90242 | 3 |
| #5726 | 7 |