Finite Group DatabasebetaFinite Group DatabasebetaOrder 1536.
174 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 174 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | #1333 |
| commutator | C16 | C2×C3×C16 |
| Frattini, cosocle | #1311 | C22×C3 |
| socle | C2×C3 | #7070 |
| prime radical | C3×#508 | C8 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C2×C3×C8 ◁ C3×C4×C16 ◁ C3×#1311 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×#7456 | C2 | semidirect |
| 3. C3×#7053 | C2 | semidirect |
| 4. #93759 | C3 | direct (N × H) |
| 13. #7456 | C2×C3 | semidirect |
| 14. #7053 | C2×C3 | semidirect |
| 36. C3×Q32 | C16 | semidirect |
| 38. C3×D16 | C16 | semidirect |
| 84. Q32 | C3×C16 | semidirect |
| 86. D16 | C3×C16 | semidirect |
| 157. C3 | #93759 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#12047 | 2 |
| C3×#7456 | 2 |
| C3×#7053 | 2 |
| #93759 | 3 |