Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
324 conjugacy classes of subgroups; only the 243 normal subgroups are shown.
This group has 324 conjugacy classes of subgroups — too many to draw. Only its 243 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C42 | D21 |
| commutator, hypocenter | C3×C7 | C2×C4×C8 |
| Frattini | C2×C4 | C22×D21 |
| Fitting | C2×C3×C42×C7 | C2 |
| socle, prime radical | C23×C3×C7 | C2×C4 |
| cosocle | C2×C3×C4×C7 | C23 |
| LCS | G ▷ C3×C7 | |
| UCS | C1 ◁ C2×C42 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C4×C3⋊(C7⋊C8) | C2 | direct (N × H) |
| 9. C2×C3⋊(C7⋊C8) | C4 | direct (N × H) |
| 16. C2×C3×C4×C7 | C8 | semidirect |
| 21. C3⋊(C7⋊C8) | C2×C4 | direct (N × H) |
| 35. C3×C4×C7 | C2×C8 | semidirect |
| 39. C2×C4×C7 | C3⋊C8 | semidirect |
| 51. C2×C3×C7 | C4×C8 | semidirect |
| 60. C4×C7 | C2×C3⋊C8 | semidirect |
| 64. C2×C3×C4 | C7⋊C8 | semidirect |
| 70. C3×C7 | C2×C4×C8 | semidirect |
| 77. C2×C7 | C4×C3⋊C8 | semidirect |
| 83. C3×C4 | C2×C7⋊C8 | semidirect |
| 87. C2×C4 | C3⋊(C7⋊C8) | direct (N × H) |
| 93. C7 | C2×C4×C3⋊C8 | semidirect |
| 94. C2×C3 | C4×C7⋊C8 | semidirect |
| 100. C4 | C2×C3⋊(C7⋊C8) | direct (N × H) |
| 103. C3 | C2×C4×C7⋊C8 | semidirect |
| 104. C2 | C4×C3⋊(C7⋊C8) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×C42×C7 | 2 |
| C22×C3⋊(C7⋊C8) | 2 |
| C4×C3⋊(C7⋊C8) | 2 |
| C2×C4×C7⋊C8 | 3 |
| C2×C4×C3⋊C8 | 7 |