Finite Group DatabasebetaFinite Group DatabasebetaOrder 1152.
672 conjugacy classes of subgroups; only the 168 normal subgroups are shown.
This group has 672 conjugacy classes of subgroups — too many to draw. Only its 168 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C32 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×Q8 | C23×C32 |
| prime radical | C32×2−1+4 | C22 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32 ◁ C22×C32 ◁ C2×C32×Q8 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C32×Q82 | C2 | semidirect |
| 3. C32×#356 | C2 | semidirect |
| 4. C3×#2205 | C3 | direct (N × H) |
| 12. C3×Q82 | C2×C3 | semidirect |
| 14. C3×#356 | C2×C3 | semidirect |
| 29. #2205 | C32 | direct (N × H) |
| 49. Q82 | C2×C32 | semidirect |
| 51. #356 | C2×C32 | semidirect |
| 114. C32 | #2205 | direct (N × H) |
| 136. C3 | C3×#2205 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×Q82 | 2 |
| C32×#456 | 2 |
| C32×#356 | 2 |
| C3×#2205 | 3 |