Finite Group DatabasebetaFinite Group Databasebeta← C2×C3⋊(C3⋊SL(2,ℤ/4)) #11724472
Order 864.
634 conjugacy classes of subgroups; only the 173 normal subgroups are shown.
This group has 634 conjugacy classes of subgroups — too many to draw. Only its 173 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | C3⋊(C3⋊S4) |
| commutator, hypocenter | C32×A4 | C2×C4 |
| Frattini | C2 | C2×C3⋊(C3⋊S4) |
| Fitting, socle | C24×C32 | D3 |
| prime radical | C22×C32×A4 | C2 |
| cosocle | C2×C32×A4 | C22 |
| LCS | G ▷ C32×A4 | |
| UCS | C1 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3⋊(C3⋊SL(2,ℤ/4)) | C2 | direct (N × H) |
| 5. C2×C32×A4 | C4 | semidirect |
| 16. C32×A4 | C2×C4 | semidirect |
| 23. C23×C32 | Dic3 | semidirect |
| 25. C2×C3×A4 | Dic3 | semidirect |
| 50. C22×C32 | C2×Dic3 | semidirect |
| 52. C3×A4 | C2×Dic3 | semidirect |
| 61. C23×C3 | C3⋊Dic3 | semidirect |
| 63. C2×A4 | C3⋊Dic3 | semidirect |
| 76. C2×C32 | SL(2,ℤ/4) | semidirect |
| 91. C22×C3 | C2×C3⋊Dic3 | semidirect |
| 93. A4 | C2×C3⋊Dic3 | semidirect |
| 98. C32 | C2×SL(2,ℤ/4) | semidirect |
| 102. C23 | C3⋊(C3⋊Dic3) | semidirect |
| 114. C2×C3 | C3⋊SL(2,ℤ/4) | semidirect |
| 123. C22 | C2×C3⋊(C3⋊Dic3) | semidirect |
| 128. C3 | C2×C3⋊SL(2,ℤ/4) | semidirect |
| 132. C2 | C3⋊(C3⋊SL(2,ℤ/4)) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C32×A4 | 2 |
| C3⋊(C3⋊SL(2,ℤ/4)) | 2 |
| C2×C3⋊SL(2,ℤ/4) | 3 |
| C2×#3757 | 3 |
| C2×C3⋊(C3⋊Dic3) | 4 |