Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
584 conjugacy classes of subgroups; only the 155 normal subgroups are shown.
This group has 584 conjugacy classes of subgroups — too many to draw. Only its 155 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 | C23×D21 |
| commutator, socle, cosocle | C22×C3×C7 | C24 |
| Fitting | C3×C4×C7×D4 | C2 |
| hypercenter | C4×D4 | D21 |
| hypocenter | C3×C7 | #547 |
| LCS | G ▷ C22×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C22 ◁ C4×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#66728 | C2 | semidirect |
| 2. C3×C4×C7×D4 | C2 | semidirect |
| 3. #10621421 | C2 | semidirect |
| 4. #10621407 | C2 | semidirect |
| 5. C4⋊C4×D21 | C2 | semidirect |
| 6. #10621366 | C2 | semidirect |
| 7. C4×D84 | C2 | semidirect |
| 8. #10621352 | C2 | semidirect |
| 28. C4×C7×D4 | D3 | semidirect |
| 47. D84 | D4 | semidirect |
| 69. C3×C4×D4 | D7 | semidirect |
| 86. C3×C4×C7 | #41 | semidirect |
| 133. C4×D4 | D21 | semidirect |
| 144. C4×C7 | #235 | semidirect |
| 163. C3×C7 | #547 | semidirect |
| 195. C3×C4 | #1157 | semidirect |
| 214. C7 | #5551 | semidirect |
| 231. C4 | #66728 | 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×#66728 | 2 |
| C3×C4×C7×D4 | 2 |
| #10621421 | 2 |
| #10621407 | 2 |
| C4⋊C4×D21 | 2 |
| #10621366 | 2 |
| C4×D84 | 2 |
| #10621352 | 2 |
| #90242 | 3 |
| #5551 | 7 |