Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
424 conjugacy classes of subgroups; only the 264 normal subgroups are shown.
This group has 424 conjugacy classes of subgroups — too many to draw. Only its 264 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×C3 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×C42 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×#500 | C2 | direct (N × H) |
| 5. C2×#500 | C3 | direct (N × H) |
| 16. #500 | C2×C3 | direct (N × H) |
| 25. C2×C3×C8 | Q8 | semidirect |
| 52. C3×C8 | C2×Q8 | semidirect |
| 59. C2×C8 | C3×Q8 | semidirect |
| 86. C8 | C2×C3×Q8 | semidirect |
| 87. C2×C3 | #500 | direct (N × H) |
| 100. C3 | C2×#500 | direct (N × H) |
| 101. C2 | C3×#500 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×C4⋊Q8 | 2 |
| C3×#500 | 2 |
| C2×C3×#107 | 2 |
| C2×C3×C4×C8 | 2 |
| C2×#500 | 3 |