Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
430 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 430 conjugacy classes of subgroups — too many to draw. Only its 69 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×A4×D7 |
| commutator, socle, cosocle | C23×C7 | C23×C3 |
| Fitting | C22×C7×Q8 | C2×C3 |
| hypercenter | Q8 | A4×D7 |
| hypocenter | C22×C7 | C3×#41 |
| LCS | G ▷ C23×C7 ▷ C22×C7 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×Q8×A4 | C2 | semidirect |
| 2. A4×D28 | C2 | semidirect |
| 3. C4×A4×D7 | C2 | semidirect |
| 4. C22×#1161 | C3 | semidirect |
| 9. C22×C7×Q8 | C2×C3 | semidirect |
| 14. C22×D28 | C2×C3 | semidirect |
| 15. C22×C4×D7 | C2×C3 | semidirect |
| 33. #1161 | A4 | direct (N × H) |
| 41. Q8×A4 | D7 | semidirect |
| 45. C7×A4 | #41 | semidirect |
| 56. C7×Q8 | C2×A4 | semidirect |
| 64. D28 | C2×A4 | semidirect |
| 69. C4×D7 | C2×A4 | semidirect |
| 80. C22×Q8 | C3×D7 | semidirect |
| 85. C22×C7 | C3×#41 | semidirect |
| 127. A4 | #1161 | direct (N × H) |
| 137. Q8 | A4×D7 | semidirect |
| 151. C7 | A4×#41 | semidirect |
| 162. C22 | C3×#1161 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×Q8×A4 | 2 |
| A4×D28 | 2 |
| C4×A4×D7 | 2 |
| C22×#1161 | 3 |
| C3×#1161 | 4 |
| A4×#41 | 7 |