Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
160 conjugacy classes of subgroups; only the 29 normal subgroups are shown.
This group has 160 conjugacy classes of subgroups — too many to draw. Only its 29 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 | #64267 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #64265 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C3×#131⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #5406 | — | 3 | 3 | no | maximal |
| 5 | #5141 | — | 3 | 1 | no | maximal |
| 6 | C3×#41⋊C3 | C22 | 1 | 1 | yes | characteristic |
| 7 | C3⋊GL(2,𝔽3) | — | 2 | 1 | no | — |
| 8 | C3⋊#226 | — | 2 | 1 | no | — |
| 9 | #3744 | — | 4 | 1 | no | maximal |
| 10 | #986 | — | 3 | 3 | no | — |
| 11 | #984 | — | 3 | 3 | no | — |
| 12 | C3×#131 | D3 | 1 | 1 | yes | characteristic, complemented |
| 13 | #951 | — | 3 | 1 | no | — |
| 14 | #949 | — | 3 | 1 | no | — |
| 15 | C2×#204 | — | 3 | 1 | no | — |
| 16 | #131⋊C3 | D3 | 1 | 3 | yes | complemented |
| 17 | #824 | — | 3 | 1 | no | — |
| 18 | #823 | — | 3 | 1 | no | — |
| 19 | #819 | — | 3 | 1 | no | — |
| 20 | C4⋊(C3⋊D3) | — | 4 | 1 | no | — |
| 21 | #634 | — | 4 | 1 | no | — |
| 22 | C3×SL(2,𝔽3) | D4 | 1 | 1 | yes | characteristic |
| 23 | C32×C8 | — | 4 | 1 | no | — |
| 24 | #471 | — | 9 | 1 | no | Sylow |
| 25 | C3×#41 | C2×D3 | 1 | 1 | yes | characteristic |
| 26 | C2×D12 | — | 3 | 1 | no | — |
| 27 | C2×Dic6 | — | 3 | 1 | no | — |
| 28 | #41⋊C3 | C2×D3 | 1 | 3 | yes | — |
| 29 | GL(2,𝔽3) | — | 6 | 3 | no | — |
| 30 | #226 | — | 6 | 3 | no | — |
| 31 | C3×M16 | — | 3 | 1 | no | — |
| 32 | C2×C3×C8 | — | 3 | 1 | no | — |
| 33 | #216 | — | 6 | 1 | no | — |
| 34 | #215 | — | 6 | 1 | no | — |
| 35 | #214 | — | 6 | 1 | no | — |
| 36 | #213 | — | 6 | 1 | no | — |
| 37 | #208 | — | 3 | 1 | no | — |
| 38 | #208 | — | 3 | 1 | no | — |
| 39 | #204 | — | 6 | 1 | no | — |
| 40 | #204 | — | 6 | 1 | no | — |
| 41 | #204 | — | 12 | 3 | no | — |
| 42 | C2×C3⋊D3 | — | 8 | 1 | no | — |
| 43 | C32×C4 | — | 4 | 1 | no | — |
| 44 | C3⋊Dic3 | — | 8 | 1 | no | — |
| 45 | #137 | — | 9 | 1 | no | — |
| 46 | AGL(1,ℤ/8) | — | 9 | 1 | no | — |
| 47 | C2×SD16 | — | 9 | 1 | no | — |
| 48 | #131 | C3⋊D3 | 1 | 1 | yes | characteristic, complemented |
| 49 | #108 | — | 9 | 1 | no | — |
| 50 | #101 | — | 9 | 1 | no | — |
| 51 | #100 | — | 9 | 1 | no | — |
| 52 | C22×D3 | — | 6 | 1 | no | — |
| 53 | C3×Q8 | D12 | 1 | 1 | yes | characteristic |
| 54 | C3×D4 | — | 3 | 1 | no | — |
| 55 | C2×C3×C4 | — | 3 | 1 | no | — |
| 56 | C2×Dic3 | — | 6 | 1 | no | — |
| 57 | D12 | — | 6 | 1 | no | — |
| 58 | D12 | — | 6 | 1 | no | — |
| 59 | D12 | — | 12 | 3 | no | — |
| 60 | Dic6 | — | 6 | 1 | no | — |
| 61 | Dic6 | — | 6 | 1 | no | — |
| 62 | Dic6 | — | 12 | 3 | no | — |
| 63 | SL(2,𝔽3) | D12 | 1 | 3 | yes | — |
| 64 | C3×C8 | — | 3 | 1 | no | — |
| 65 | C3×C8 | — | 4 | 3 | no | — |
| 66 | C3×C8 | S4 | 1 | 1 | yes | characteristic |
| 67 | C3⋊C8 | — | 6 | 1 | no | — |
| 68 | C3⋊C8 | — | 6 | 1 | no | — |
| 69 | C2×C32 | — | 4 | 1 | no | — |
| 70 | C3⋊D3 | — | 16 | 1 | no | — |
| 71 | #41 | C2×C3⋊D3 | 1 | 1 | yes | characteristic |
| 72 | C2×Q8 | — | 9 | 1 | no | — |
| 73 | C2×D4 | — | 9 | 1 | no | — |
| 74 | Q16 | — | 18 | 1 | no | — |
| 75 | SD16 | — | 18 | 1 | no | — |
| 76 | SD16 | — | 18 | 1 | no | — |
| 77 | SD16 | — | 18 | 1 | no | — |
| 78 | SD16 | — | 18 | 1 | no | — |
| 79 | D8 | — | 18 | 1 | no | — |
| 80 | M16 | — | 3 | 1 | no | — |
| 81 | M16 | — | 9 | 1 | no | — |
| 82 | M16 | — | 9 | 1 | no | — |
| 83 | C2×C8 | — | 3 | 1 | no | — |
| 84 | C22×C3 | — | 3 | 1 | no | — |
| 85 | C2×D3 | — | 12 | 1 | no | — |
| 86 | C2×D3 | — | 24 | 3 | no | — |
| 87 | C2×D3 | — | 24 | 1 | no | — |
| 88 | C3×C4 | — | 3 | 1 | no | — |
| 89 | C3×C4 | — | 4 | 3 | no | — |
| 90 | C3×C4 | C2×S4 | 1 | 1 | yes | characteristic |
| 91 | Dic3 | — | 12 | 1 | no | — |
| 92 | Dic3 | — | 24 | 3 | no | — |
| 93 | C32 | — | 4 | 1 | no | Sylow |
| 94 | C23 | — | 18 | 1 | no | — |
| 95 | Q8 | — | 18 | 1 | no | — |
| 96 | Q8 | — | 18 | 1 | no | — |
| 97 | Q8 | C4⋊(C3⋊D3) | 1 | 1 | yes | characteristic |
| 98 | D4 | — | 3 | 1 | no | — |
| 99 | D4 | — | 18 | 1 | no | — |
| 100 | D4 | — | 18 | 1 | no | — |
| 101 | C2×C4 | — | 3 | 1 | no | — |
| 102 | C2×C4 | — | 18 | 1 | no | — |
| 103 | C8 | — | 3 | 1 | no | — |
| 104 | C8 | — | 18 | 1 | no | — |
| 105 | C8 | — | 18 | 1 | no | — |
| 106 | C8 | C3⋊S4 | 1 | 1 | yes | characteristic |
| 107 | C2×C3 | — | 4 | 3 | no | — |
| 108 | C2×C3 | — | 6 | 1 | no | — |
| 109 | C2×C3 | C4⋊S4 | 1 | 1 | yes | characteristic |
| 110 | D3 | — | 24 | 1 | no | — |
| 111 | D3 | — | 48 | 3 | no | — |
| 112 | C22 | — | 3 | 1 | no | — |
| 113 | C22 | — | 36 | 1 | no | — |
| 114 | C22 | — | 72 | 1 | no | — |
| 115 | C4 | — | 3 | 1 | no | — |
| 116 | C4 | — | 36 | 1 | no | — |
| 117 | C4 | C2×C3⋊S4 | 1 | 1 | yes | characteristic |
| 118 | C3 | — | 4 | 3 | no | — |
| 119 | C3 | #5406 | 1 | 1 | yes | characteristic, complemented |
| 120 | C2 | — | 6 | 1 | no | — |
| 121 | C2 | — | 72 | 1 | no | — |
| 122 | C2 | C4⋊(C3⋊S4) | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #64265 | C2 | semidirect |
| 3. C3×#131⋊C3 | C2 | semidirect |
| 12. C3×#131 | D3 | semidirect |
| 16. #131⋊C3 | D3 | semidirect |
| 48. #131 | C3⋊D3 | semidirect |
| 119. C3 | #5406 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #64267 | 2 |
| #64265 | 2 |
| C3×#131⋊C3 | 2 |
| #5406 | 3 |
| #5141 | 3 |
| #3744 | 4 |