Finite Group DatabasebetaFinite Group DatabasebetaOrder 1296.
714 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 714 conjugacy classes of subgroups — too many to draw. Only its 71 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #10620150 | C2 | 1 | 3 | yes | maximal |
| 2 | C3×#6296 | C2 | 1 | 4 | yes | maximal, complemented |
| 3 | #88958 | — | 3 | 4 | no | maximal |
| 4 | C2×C3⋊(C3⋊(C3⋊D3)) | C22 | 1 | 1 | yes | characteristic |
| 5 | C32×C3⋊Dic3 | C22 | 1 | 6 | yes | — |
| 6 | #6296 | D3 | 1 | 4 | yes | complemented |
| 7 | #6289 | — | 3 | 12 | no | — |
| 8 | #6289 | — | 6 | 6 | no | — |
| 9 | C3×#627 | — | 3 | 12 | no | — |
| 10 | C2×C34 | C23 | 1 | 1 | yes | characteristic |
| 11 | C3⋊(C3⋊(C3⋊D3)) | Q8 | 1 | 2 | yes | complemented |
| 12 | #3797 | — | 9 | 6 | no | — |
| 13 | C2×C3⋊(C3⋊D3) | — | 3 | 4 | no | — |
| 14 | C2×C3⋊(C3⋊D3) | — | 6 | 6 | no | — |
| 15 | C2×C3⋊(C3⋊D3) | — | 12 | 4 | no | — |
| 16 | C2×C3⋊(C3⋊D3) | — | 12 | 2 | no | — |
| 17 | C3×C3⋊Dic3 | — | 2 | 6 | no | — |
| 18 | C3×C3⋊Dic3 | C2×D3 | 1 | 12 | yes | — |
| 19 | C32×Dic3 | — | 3 | 12 | no | — |
| 20 | C32×Dic3 | — | 6 | 6 | no | — |
| 21 | C34 | C2×Q8 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 22 | C4×C3⋊D3 | — | 9 | 6 | no | — |
| 23 | C3×Dic6 | — | 9 | 12 | no | — |
| 24 | #627 | — | 3 | 12 | no | — |
| 25 | #624 | — | 9 | 12 | no | — |
| 26 | #624 | — | 18 | 12 | no | — |
| 27 | #624 | — | 18 | 6 | no | — |
| 28 | C2×C33 | — | 2 | 6 | no | — |
| 29 | C2×C33 | — | 4 | 2 | no | — |
| 30 | C2×C33 | — | 4 | 4 | no | — |
| 31 | C2×C33 | C22×D3 | 1 | 4 | yes | — |
| 32 | C3⋊(C3⋊D3) | — | 3 | 8 | no | — |
| 33 | C3⋊(C3⋊D3) | — | 6 | 12 | no | — |
| 34 | C3⋊(C3⋊D3) | — | 12 | 8 | no | — |
| 35 | C3⋊(C3⋊D3) | — | 12 | 4 | no | — |
| 36 | D3×Q8 | — | 27 | 4 | no | — |
| 37 | C2×C3⋊D3 | — | 9 | 6 | no | — |
| 38 | C2×C3⋊D3 | — | 18 | 12 | no | — |
| 39 | C2×C3⋊D3 | — | 18 | 6 | no | — |
| 40 | C2×C3⋊D3 | — | 36 | 4 | no | — |
| 41 | C2×C3⋊D3 | — | 36 | 4 | no | — |
| 42 | C2×C3⋊D3 | — | 36 | 12 | no | — |
| 43 | C2×C3⋊D3 | — | 36 | 2 | no | — |
| 44 | C32×C4 | — | 9 | 6 | no | — |
| 45 | C3⋊Dic3 | D32 | 1 | 6 | yes | — |
| 46 | C3×Dic3 | — | 3 | 24 | no | — |
| 47 | C3×Dic3 | — | 6 | 12 | no | — |
| 48 | C3×Dic3 | — | 6 | 12 | no | — |
| 49 | C3×Dic3 | — | 6 | 12 | no | — |
| 50 | C33 | — | 2 | 6 | no | — |
| 51 | C33 | — | 4 | 2 | no | — |
| 52 | C33 | — | 4 | 4 | no | — |
| 53 | C33 | D3×Q8 | 1 | 4 | yes | complemented |
| 54 | C3×Q8 | — | 27 | 4 | no | — |
| 55 | C4×D3 | — | 27 | 12 | no | — |
| 56 | C4×D3 | — | 54 | 6 | no | — |
| 57 | Dic6 | — | 9 | 12 | no | — |
| 58 | C2×C32 | — | 2 | 6 | no | — |
| 59 | C2×C32 | — | 2 | 12 | no | — |
| 60 | C2×C32 | — | 4 | 4 | no | — |
| 61 | C2×C32 | — | 4 | 12 | no | — |
| 62 | C2×C32 | — | 4 | 2 | no | — |
| 63 | C2×C32 | — | 4 | 4 | no | — |
| 64 | C2×C32 | C2×D32 | 1 | 6 | yes | — |
| 65 | C3⋊D3 | — | 9 | 12 | no | — |
| 66 | C3⋊D3 | — | 18 | 24 | no | — |
| 67 | C3⋊D3 | — | 18 | 12 | no | — |
| 68 | C3⋊D3 | — | 36 | 8 | no | — |
| 69 | C3⋊D3 | — | 36 | 8 | no | — |
| 70 | C3⋊D3 | — | 36 | 24 | no | — |
| 71 | C3⋊D3 | — | 36 | 4 | no | — |
| 72 | C2×Q8 | — | 81 | 1 | no | Sylow |
| 73 | C2×D3 | — | 27 | 4 | no | — |
| 74 | C2×D3 | — | 54 | 6 | no | — |
| 75 | C2×D3 | — | 108 | 4 | no | — |
| 76 | C2×D3 | — | 108 | 2 | no | — |
| 77 | C3×C4 | — | 9 | 12 | no | — |
| 78 | C3×C4 | — | 18 | 6 | no | — |
| 79 | Dic3 | — | 3 | 12 | no | — |
| 80 | Dic3 | — | 6 | 6 | no | — |
| 81 | C32 | — | 2 | 6 | no | — |
| 82 | C32 | — | 2 | 12 | no | — |
| 83 | C32 | — | 4 | 4 | no | — |
| 84 | C32 | — | 4 | 12 | no | — |
| 85 | C32 | — | 4 | 2 | no | — |
| 86 | C32 | — | 4 | 4 | no | — |
| 87 | C32 | #3797 | 1 | 6 | yes | complemented |
| 88 | Q8 | — | 27 | 4 | no | — |
| 89 | C2×C4 | — | 81 | 3 | no | — |
| 90 | C2×C3 | — | 2 | 6 | no | — |
| 91 | C2×C3 | — | 4 | 2 | no | — |
| 92 | C2×C3 | — | 4 | 4 | no | — |
| 93 | C2×C3 | D33 | 1 | 4 | yes | — |
| 94 | D3 | — | 27 | 8 | no | — |
| 95 | D3 | — | 54 | 12 | no | — |
| 96 | D3 | — | 108 | 8 | no | — |
| 97 | D3 | — | 108 | 4 | no | — |
| 98 | C22 | — | 81 | 1 | no | — |
| 99 | C4 | — | 9 | 6 | no | — |
| 100 | C3 | — | 2 | 6 | no | — |
| 101 | C3 | — | 4 | 2 | no | — |
| 102 | C3 | — | 4 | 4 | no | — |
| 103 | C3 | #88958 | 1 | 4 | yes | complemented |
| 104 | C2 | — | 81 | 2 | no | — |
| 105 | C2 | #10620215 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | #10620215 |
| commutator, Fitting, socle, cosocle | C2×C34 | C23 |
| prime radical | C2×C3⋊(C3⋊(C3⋊D3)) | C22 |
| hypocenter | C34 | C2×Q8 |
| LCS | G ▷ C2×C34 ▷ C34 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×#6296 | C2 | semidirect |
| 6. #6296 | D3 | semidirect |
| 11. C3⋊(C3⋊(C3⋊D3)) | Q8 | semidirect |
| 21. C34 | C2×Q8 | semidirect |
| 53. C33 | D3×Q8 | semidirect |
| 87. C32 | #3797 | semidirect |
| 103. C3 | #88958 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10620150 | 2 |
| C3×#6296 | 2 |
| #88958 | 3 |