Finite Group DatabasebetaFinite Group DatabasebetaOrder 12352.
290 conjugacy classes of subgroups; only the 166 normal subgroups are shown.
This group has 290 conjugacy classes of subgroups — too many to draw. Only its 166 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C193 | C24 |
| commutator, Frattini, cosocle | C22 | C24×C193 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C193 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#41×C193 | C2 | semidirect |
| 2. C22×Q8×C193 | C2 | semidirect |
| 3. #124×C193 | C2 | semidirect |
| 4. C22⋊Q8×C193 | C2 | semidirect |
| 5. #121×C193 | C2 | semidirect |
| 6. C4×Q8×C193 | C2 | semidirect |
| 7. C4×D4×C193 | C2 | semidirect |
| 25. Q8×C193 | D4 | semidirect |
| 40. C22×C193 | #41 | semidirect |
| 52. C193 | #548 | direct (N × H) |
| 53. #548 | C193 | direct (N × H) |
| 54. C2×#41 | C2×C193 | semidirect |
| 55. C22×Q8 | C2×C193 | semidirect |
| 56. #124 | C2×C193 | semidirect |
| 57. C22⋊Q8 | C2×C193 | semidirect |
| 58. #121 | C2×C193 | semidirect |
| 59. C4×Q8 | C2×C193 | semidirect |
| 60. C4×D4 | C2×C193 | semidirect |
| 78. Q8 | D4×C193 | semidirect |
| 93. C22 | #41×C193 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#41×C193 | 2 |
| C22×Q8×C193 | 2 |
| #124×C193 | 2 |
| C22⋊Q8×C193 | 2 |
| #121×C193 | 2 |
| C4×Q8×C193 | 2 |
| C4×D4×C193 | 2 |
| #548 | 193 |