Finite Group DatabasebetaFinite Group DatabasebetaOrder 1152.
648 conjugacy classes of subgroups; only the 360 normal subgroups are shown.
This group has 648 conjugacy classes of subgroups — too many to draw. Only its 360 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C32 | C2×D4 |
| commutator | C2×C4 | C22×C32×C4 |
| Frattini, cosocle | C22×C4 | C23×C32 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C32 ◁ C2×C32×C42 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C3×#1835 | C3 | direct (N × H) |
| 8. C32×C4×C8 | C4 | semidirect |
| 21. #1835 | C32 | direct (N × H) |
| 25. C3×C4×C8 | C3×C4 | semidirect |
| 57. C4×C8 | C32×C4 | semidirect |
| 86. C32 | #1835 | direct (N × H) |
| 106. C3 | C3×#1835 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C32×C4×C8 | 2 |
| C32×#382 | 2 |
| C32×#340 | 2 |
| C3×#1835 | 3 |