Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
592 conjugacy classes of subgroups; only the 192 normal subgroups are shown.
This group has 592 conjugacy classes of subgroups — too many to draw. Only its 192 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C24 |
| commutator, Frattini, cosocle | C23 | C24×C3 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C3×#124 | C2 | semidirect |
| 2. C3×#394 | C2 | semidirect |
| 3. C3×#392 | C2 | semidirect |
| 4. C3×#391 | C2 | semidirect |
| 5. C3×#390 | C2 | semidirect |
| 6. C3×#388 | C2 | semidirect |
| 7. #2733 | C3 | direct (N × H) |
| 23. C2×#124 | C2×C3 | semidirect |
| 24. #394 | C2×C3 | semidirect |
| 25. #392 | C2×C3 | semidirect |
| 26. #391 | C2×C3 | semidirect |
| 27. #390 | C2×C3 | semidirect |
| 28. #388 | C2×C3 | semidirect |
| 32. C2×C3×Q8 | D4 | semidirect |
| 98. C2×Q8 | C3×D4 | semidirect |
| 186. C3 | #2733 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×#124 | 2 |
| C3×#394 | 2 |
| C3×#392 | 2 |
| C3×#391 | 2 |
| C3×#390 | 2 |
| C3×#388 | 2 |
| #2733 | 3 |