Finite Group DatabasebetaFinite Group DatabasebetaOrder 6976.
292 conjugacy classes of subgroups; only the 166 normal subgroups are shown.
This group has 292 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×C109 | C24 |
| commutator, Frattini, cosocle | C22 | C24×C109 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C109 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#41×C109 | C2 | semidirect |
| 2. C4⋊Q8×C109 | C2 | semidirect |
| 3. #123×C109 | C2 | semidirect |
| 4. C22⋊Q8×C109 | C2 | semidirect |
| 5. #121×C109 | C2 | semidirect |
| 6. C4×D4×C109 | C2 | semidirect |
| 7. C4×D4×C109 | C2 | semidirect |
| 8. C2×C4⋊C4×C109 | C2 | semidirect |
| 31. D4×C109 | D4 | semidirect |
| 53. C4×C109 | #41 | semidirect |
| 59. C109 | #547 | direct (N × H) |
| 60. #547 | C109 | direct (N × H) |
| 61. C2×#41 | C2×C109 | semidirect |
| 62. C4⋊Q8 | C2×C109 | semidirect |
| 63. #123 | C2×C109 | semidirect |
| 64. C22⋊Q8 | C2×C109 | semidirect |
| 65. #121 | C2×C109 | semidirect |
| 66. C4×D4 | C2×C109 | semidirect |
| 67. C4×D4 | C2×C109 | semidirect |
| 68. C2×C4⋊C4 | C2×C109 | semidirect |
| 91. D4 | D4×C109 | semidirect |
| 113. C4 | #41×C109 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#41×C109 | 2 |
| C4⋊Q8×C109 | 2 |
| #123×C109 | 2 |
| C22⋊Q8×C109 | 2 |
| #121×C109 | 2 |
| C4×D4×C109 | 2 |
| C2×C4⋊C4×C109 | 2 |
| #547 | 109 |