Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
156 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 156 conjugacy classes of subgroups — too many to draw. Only its 72 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 | C22×C3 | #121 |
| commutator, Frattini, cosocle | C42 | C23×C3 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×C42 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 6. #1670 | C3 | direct (N × H) |
| 51. C3×Q8 | Q16 | semidirect |
| 88. Q8 | C3×Q16 | semidirect |
| 111. C3 | #1670 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#494 | 2 |
| C3×#475 | 2 |
| C3×C8×Q8 | 2 |
| C3×#335 | 2 |
| C3×#332 | 2 |
| #1670 | 3 |