Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
292 conjugacy classes of subgroups; only the 164 normal subgroups are shown.
This group has 292 conjugacy classes of subgroups — too many to draw. Only its 164 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3×C7 | C22⋊C4 |
| commutator | C22 | C22×C3×C4×C7 |
| Frattini, cosocle | C2×C4 | C23×C3×C7 |
| prime radical | C23×C3×C7 | C2×C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C7 ◁ C22×C3×C4×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C3×C7×#101 | C2 | direct (N × H) |
| 4. C2×C7×#101 | C3 | direct (N × H) |
| 16. C7×#101 | C2×C3 | direct (N × H) |
| 17. C2×C3×#101 | C7 | direct (N × H) |
| 40. C3×#101 | C2×C7 | direct (N × H) |
| 48. C2×#101 | C3×C7 | direct (N × H) |
| 71. C2×C3×C7 | #101 | direct (N × H) |
| 75. #101 | C2×C3×C7 | direct (N × H) |
| 94. C3×C7 | C2×#101 | direct (N × H) |
| 106. C2×C7 | C3×#101 | direct (N × H) |
| 126. C7 | C2×C3×#101 | direct (N × H) |
| 129. C2×C3 | C7×#101 | direct (N × H) |
| 138. C3 | C2×C7×#101 | direct (N × H) |
| 141. C2 | C3×C7×#101 | 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×C3×C7×M16 | 2 |
| C3×C7×#101 | 2 |
| C2×C7×#101 | 3 |
| C2×C3×#101 | 7 |