Finite Group DatabasebetaFinite Group DatabasebetaOrder 1944.
458 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 458 conjugacy classes of subgroups — too many to draw. Only its 28 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 | A4×AΣL(1,𝔽27) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C3⋊(C3⋊(C3⋊S4)) | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | A4⋊((C3⋊D3)⋊C3) | — | 3 | 1 | no | maximal |
| 4 | #10619745 | — | 3 | 2 | no | maximal |
| 5 | #10619642 | — | 3 | 1 | no | maximal |
| 6 | C3⋊#4175 | — | 4 | 1 | no | maximal |
| 7 | C33×A4 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 8 | A4×3−1+2 | — | 2 | 1 | no | — |
| 9 | A4×H(3,𝔽3) | D3 | 1 | 1 | yes | characteristic, complemented |
| 10 | C22×AΣL(1,𝔽27) | D3 | 1 | 1 | yes | characteristic, complemented |
| 11 | C2×#4175 | — | 3 | 1 | no | — |
| 12 | #66368 | — | 2 | 1 | no | — |
| 13 | #66366 | — | 2 | 2 | no | — |
| 14 | #66363 | D3 | 1 | 2 | yes | complemented |
| 15 | #66322 | — | 3 | 1 | no | — |
| 16 | C3×AΣL(1,𝔽27) | — | 4 | 1 | no | Sylow |
| 17 | C3⋊(C3⋊S4) | — | 3 | 1 | no | — |
| 18 | C3⋊(C3⋊S4) | — | 3 | 2 | no | — |
| 19 | C3⋊(C3⋊S4) | — | 9 | 1 | no | — |
| 20 | C3⋊(C3⋊S4) | — | 9 | 2 | no | — |
| 21 | C3⋊(C3⋊S4) | — | 9 | 9 | no | — |
| 22 | C3×C3⋊S4 | — | 9 | 1 | no | — |
| 23 | #6312 | — | 3 | 1 | no | — |
| 24 | (C3⋊S4)⋊C3 | — | 9 | 2 | no | — |
| 25 | #638⋊C3 | — | 9 | 1 | no | — |
| 26 | C3⋊(C3⋊(C3⋊D3)) | — | 4 | 1 | no | — |
| 27 | C3⋊((C3⋊D3)⋊C3) | — | 12 | 1 | no | — |
| 28 | C2×AΣL(1,𝔽27) | — | 3 | 1 | no | — |
| 29 | #4175 | — | 6 | 1 | no | — |
| 30 | #4175 | — | 12 | 2 | no | — |
| 31 | C22×C33 | C3×D3 | 1 | 1 | yes | characteristic, complemented |
| 32 | C2×C3⋊(C3⋊D3) | — | 3 | 1 | no | — |
| 33 | C32×A4 | — | 3 | 2 | no | — |
| 34 | C32×A4 | — | 3 | 1 | no | — |
| 35 | C32×A4 | — | 3 | 9 | no | — |
| 36 | C32×A4 | — | 3 | 1 | no | — |
| 37 | C32×A4 | C3×D3 | 1 | 1 | yes | characteristic |
| 38 | C32×A4 | C3×D3 | 1 | 2 | yes | complemented |
| 39 | C3⋊(C3⋊Dic3) | — | 3 | 1 | no | — |
| 40 | C22×3−1+2 | — | 2 | 1 | no | — |
| 41 | C22×H(3,𝔽3) | C3⋊D3 | 1 | 1 | yes | characteristic, complemented |
| 42 | C2×(C3⋊D3)⋊C3 | — | 9 | 1 | no | — |
| 43 | C22⋊H(3,𝔽3) | — | 2 | 1 | no | — |
| 44 | C22⋊H(3,𝔽3) | — | 3 | 2 | no | — |
| 45 | #1094 | — | 2 | 2 | no | — |
| 46 | C9⋊A4 | — | 6 | 2 | no | — |
| 47 | C9×A4 | — | 6 | 1 | no | — |
| 48 | (C3⋊Dic3)⋊C3 | — | 9 | 1 | no | — |
| 49 | C34 | — | 4 | 1 | no | — |
| 50 | C3×3−1+2 | — | 8 | 1 | no | — |
| 51 | C3×H(3,𝔽3) | — | 4 | 1 | no | — |
| 52 | AΣL(1,𝔽27) | — | 4 | 2 | no | — |
| 53 | AΣL(1,𝔽27) | — | 8 | 2 | no | — |
| 54 | AΣL(1,𝔽27) | — | 8 | 1 | no | — |
| 55 | AΣL(1,𝔽27) | S4 | 1 | 1 | yes | characteristic, complemented |
| 56 | C3⋊S4 | — | 9 | 1 | no | — |
| 57 | C3⋊S4 | — | 9 | 2 | no | — |
| 58 | C3⋊S4 | — | 27 | 2 | no | — |
| 59 | C3⋊S4 | — | 27 | 3 | no | — |
| 60 | C3⋊S4 | — | 27 | 9 | no | — |
| 61 | C3⋊S4 | — | 27 | 18 | no | — |
| 62 | C3⋊S4 | — | 27 | 6 | no | — |
| 63 | C3×S4 | — | 27 | 3 | no | — |
| 64 | #638 | — | 9 | 1 | no | — |
| 65 | #638 | — | 27 | 1 | no | — |
| 66 | #638 | — | 27 | 3 | no | — |
| 67 | C3×#67 | — | 27 | 1 | no | — |
| 68 | C2×C33 | — | 3 | 1 | no | — |
| 69 | C3⋊(C3⋊D3) | — | 6 | 1 | no | — |
| 70 | C3⋊(C3⋊D3) | — | 12 | 1 | no | — |
| 71 | C3⋊(C3⋊D3) | — | 12 | 2 | no | — |
| 72 | C3⋊(C3⋊D3) | — | 36 | 2 | no | — |
| 73 | C3⋊(C3⋊D3) | — | 36 | 1 | no | — |
| 74 | C3⋊(C3⋊D3) | — | 36 | 9 | no | — |
| 75 | C3×C3⋊D3 | — | 36 | 1 | no | — |
| 76 | C2×3−1+2 | — | 6 | 1 | no | — |
| 77 | C2×H(3,𝔽3) | — | 3 | 1 | no | — |
| 78 | (C3⋊D3)⋊C3 | — | 18 | 1 | no | — |
| 79 | (C3⋊D3)⋊C3 | — | 36 | 2 | no | — |
| 80 | C22×C32 | — | 3 | 1 | no | — |
| 81 | C22×C32 | — | 3 | 3 | no | — |
| 82 | C22×C32 | — | 3 | 1 | no | — |
| 83 | C22×C32 | C3×C3⋊D3 | 1 | 1 | yes | characteristic |
| 84 | C2×C3⋊D3 | — | 9 | 1 | no | — |
| 85 | C2×C3⋊D3 | — | 27 | 1 | no | — |
| 86 | C2×C3⋊D3 | — | 27 | 3 | no | — |
| 87 | C2×C3×D3 | — | 27 | 1 | no | — |
| 88 | C3×A4 | — | 3 | 18 | no | — |
| 89 | C3×A4 | — | 3 | 2 | no | — |
| 90 | C3×A4 | — | 3 | 3 | no | — |
| 91 | C3×A4 | — | 3 | 9 | no | — |
| 92 | C3×A4 | — | 3 | 6 | no | — |
| 93 | C3×A4 | — | 6 | 1 | no | — |
| 94 | C3×A4 | — | 9 | 3 | no | — |
| 95 | C3×A4 | (C3⋊D3)⋊C3 | 1 | 2 | yes | — |
| 96 | C3×A4 | (C3⋊D3)⋊C3 | 1 | 1 | yes | characteristic |
| 97 | C3⋊Dic3 | — | 9 | 1 | no | — |
| 98 | C3⋊Dic3 | — | 27 | 1 | no | — |
| 99 | C3⋊Dic3 | — | 27 | 3 | no | — |
| 100 | C3×Dic3 | — | 27 | 1 | no | — |
| 101 | C22×C9 | — | 6 | 1 | no | — |
| 102 | C22⋊C9 | — | 6 | 2 | no | — |
| 103 | C33 | — | 4 | 1 | no | — |
| 104 | C33 | — | 4 | 2 | no | — |
| 105 | C33 | — | 12 | 1 | no | — |
| 106 | C33 | — | 12 | 2 | no | — |
| 107 | C33 | — | 12 | 1 | no | — |
| 108 | C33 | — | 12 | 9 | no | — |
| 109 | C33 | C3×S4 | 1 | 1 | yes | characteristic, complemented |
| 110 | 3−1+2 | — | 2 | 1 | no | — |
| 111 | 3−1+2 | — | 8 | 2 | no | — |
| 112 | 3−1+2 | — | 24 | 2 | no | — |
| 113 | H(3,𝔽3) | — | 8 | 1 | no | — |
| 114 | H(3,𝔽3) | — | 12 | 2 | no | — |
| 115 | H(3,𝔽3) | C3⋊S4 | 1 | 1 | yes | characteristic, complemented |
| 116 | C3×C9 | — | 24 | 1 | no | — |
| 117 | S4 | — | 27 | 3 | no | — |
| 118 | S4 | — | 81 | 2 | no | — |
| 119 | S4 | — | 81 | 6 | no | — |
| 120 | C3×D4 | — | 81 | 1 | no | — |
| 121 | #67 | — | 27 | 1 | no | — |
| 122 | #67 | — | 81 | 1 | no | — |
| 123 | #67 | — | 81 | 3 | no | — |
| 124 | C2×C32 | — | 3 | 1 | no | — |
| 125 | C2×C32 | — | 9 | 1 | no | — |
| 126 | C2×C32 | — | 9 | 1 | no | — |
| 127 | C2×C32 | — | 9 | 3 | no | — |
| 128 | C3⋊D3 | — | 18 | 1 | no | — |
| 129 | C3⋊D3 | — | 36 | 2 | no | — |
| 130 | C3⋊D3 | — | 36 | 1 | no | — |
| 131 | C3⋊D3 | — | 54 | 1 | no | — |
| 132 | C3⋊D3 | — | 54 | 3 | no | — |
| 133 | C3⋊D3 | — | 108 | 2 | no | — |
| 134 | C3⋊D3 | — | 108 | 18 | no | — |
| 135 | C3⋊D3 | — | 108 | 3 | no | — |
| 136 | C3⋊D3 | — | 108 | 9 | no | — |
| 137 | C3⋊D3 | — | 108 | 6 | no | — |
| 138 | C3×D3 | — | 54 | 1 | no | — |
| 139 | C3×D3 | — | 108 | 3 | no | — |
| 140 | C2×C9 | — | 18 | 1 | no | — |
| 141 | C22×C3 | — | 3 | 3 | no | — |
| 142 | C22×C3 | — | 3 | 1 | no | — |
| 143 | C22×C3 | — | 9 | 1 | no | — |
| 144 | C22×C3 | — | 81 | 1 | no | — |
| 145 | C22×C3 | C3⋊((C3⋊D3)⋊C3) | 1 | 1 | yes | characteristic |
| 146 | C2×D3 | — | 27 | 1 | no | — |
| 147 | C2×D3 | — | 81 | 1 | no | — |
| 148 | C2×D3 | — | 81 | 3 | no | — |
| 149 | A4 | — | 3 | 6 | no | — |
| 150 | A4 | — | 3 | 2 | no | — |
| 151 | A4 | — | 18 | 1 | no | — |
| 152 | A4 | #4175 | 1 | 3 | yes | complemented |
| 153 | C3×C4 | — | 81 | 1 | no | — |
| 154 | Dic3 | — | 27 | 1 | no | — |
| 155 | Dic3 | — | 81 | 1 | no | — |
| 156 | Dic3 | — | 81 | 3 | no | — |
| 157 | C32 | — | 3 | 3 | no | — |
| 158 | C32 | — | 3 | 1 | no | — |
| 159 | C32 | — | 3 | 1 | no | — |
| 160 | C32 | — | 4 | 1 | no | — |
| 161 | C32 | — | 4 | 2 | no | — |
| 162 | C32 | — | 12 | 3 | no | — |
| 163 | C32 | — | 12 | 9 | no | — |
| 164 | C32 | — | 12 | 6 | no | — |
| 165 | C32 | — | 12 | 18 | no | — |
| 166 | C32 | — | 12 | 2 | no | — |
| 167 | C32 | — | 24 | 1 | no | — |
| 168 | C32 | — | 36 | 3 | no | — |
| 169 | C32 | C3×C3⋊S4 | 1 | 1 | yes | characteristic |
| 170 | C9 | — | 6 | 1 | no | — |
| 171 | C9 | — | 24 | 2 | no | — |
| 172 | D4 | — | 81 | 1 | no | Sylow |
| 173 | C2×C3 | — | 3 | 1 | no | — |
| 174 | C2×C3 | — | 9 | 1 | no | — |
| 175 | C2×C3 | — | 9 | 3 | no | — |
| 176 | C2×C3 | — | 27 | 1 | no | — |
| 177 | C2×C3 | — | 162 | 1 | no | — |
| 178 | D3 | — | 54 | 1 | no | — |
| 179 | D3 | — | 324 | 2 | no | — |
| 180 | D3 | — | 324 | 6 | no | — |
| 181 | D3 | — | 108 | 3 | no | — |
| 182 | D3 | — | 162 | 1 | no | — |
| 183 | D3 | — | 162 | 3 | no | — |
| 184 | C22 | — | 81 | 1 | no | — |
| 185 | C22 | C3⋊#4175 | 1 | 1 | yes | characteristic, complemented |
| 186 | C4 | — | 81 | 1 | no | — |
| 187 | C3 | — | 3 | 3 | no | — |
| 188 | C3 | — | 3 | 1 | no | — |
| 189 | C3 | — | 4 | 3 | no | — |
| 190 | C3 | — | 9 | 1 | no | — |
| 191 | C3 | — | 12 | 2 | no | — |
| 192 | C3 | — | 12 | 6 | no | — |
| 193 | C3 | — | 72 | 1 | no | — |
| 194 | C3 | A4⋊((C3⋊D3)⋊C3) | 1 | 1 | yes | characteristic |
| 195 | C2 | — | 3 | 1 | no | — |
| 196 | C2 | — | 162 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C33×A4 | C2×C3 |
| Frattini | C32 | C3×C3⋊S4 |
| Fitting | C22×AΣL(1,𝔽27) | D3 |
| socle | C22×C3 | C3⋊((C3⋊D3)⋊C3) |
| LCS | G ▷ C33×A4 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. A4×AΣL(1,𝔽27) | C2 | semidirect |
| 2. C3⋊(C3⋊(C3⋊S4)) | C3 | semidirect |
| 7. C33×A4 | C2×C3 | semidirect |
| 9. A4×H(3,𝔽3) | D3 | semidirect |
| 10. C22×AΣL(1,𝔽27) | D3 | semidirect |
| 14. #66363 | D3 | semidirect |
| 31. C22×C33 | C3×D3 | semidirect |
| 38. C32×A4 | C3×D3 | semidirect |
| 41. C22×H(3,𝔽3) | C3⋊D3 | semidirect |
| 55. AΣL(1,𝔽27) | S4 | semidirect |
| 109. C33 | C3×S4 | semidirect |
| 115. H(3,𝔽3) | C3⋊S4 | semidirect |
| 152. A4 | #4175 | semidirect |
| 185. C22 | C3⋊#4175 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| A4×AΣL(1,𝔽27) | 2 |
| C3⋊(C3⋊(C3⋊S4)) | 3 |
| A4⋊((C3⋊D3)⋊C3) | 3 |
| #10619745 | 3 |
| #10619642 | 3 |
| C3⋊#4175 | 4 |