Finite Group DatabasebetaFinite Group DatabasebetaOrder 1296.
892 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 892 conjugacy classes of subgroups — too many to draw. Only its 64 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 | C3×A4×C3⋊D3 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C3⋊(C3⋊(C3⋊S4)) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C32×C3⋊S4 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | D3×C3⋊S4 | — | 3 | 4 | no | maximal |
| 5 | C3⋊D3×S4 | — | 3 | 3 | no | maximal |
| 6 | C3⋊D3×#67 | — | 3 | 1 | no | maximal |
| 7 | C33×A4 | C22 | 1 | 1 | yes | characteristic, complemented |
| 8 | C3⋊D32 | — | 4 | 1 | no | maximal |
| 9 | C22×C3×C3⋊D3 | D3 | 1 | 1 | yes | characteristic, complemented |
| 10 | C2×D3×C3⋊D3 | — | 3 | 1 | no | — |
| 11 | A4×C3⋊D3 | D3 | 1 | 3 | yes | complemented |
| 12 | C3×D3×A4 | — | 3 | 4 | no | — |
| 13 | C3⋊(C3⋊S4) | — | 3 | 3 | no | — |
| 14 | C3⋊(C3⋊S4) | — | 3 | 4 | no | — |
| 15 | C3⋊(C3⋊S4) | — | 6 | 4 | no | — |
| 16 | C3⋊(C3⋊S4) | — | 6 | 12 | no | — |
| 17 | C3×C3⋊S4 | D3 | 1 | 4 | yes | complemented |
| 18 | C32×S4 | — | 3 | 3 | no | — |
| 19 | #6312 | — | 3 | 1 | no | — |
| 20 | C32×#67 | — | 3 | 1 | no | — |
| 21 | #6292 | — | 3 | 1 | no | — |
| 22 | #6290 | — | 3 | 1 | no | — |
| 23 | Dic3×C3⋊D3 | — | 3 | 1 | no | — |
| 24 | C3⋊(C3⋊(C3⋊D3)) | — | 4 | 1 | no | — |
| 25 | C32×C3⋊D3 | — | 4 | 1 | no | — |
| 26 | C32×C3⋊D3 | — | 4 | 1 | no | — |
| 27 | C2×C3⋊S4 | — | 9 | 1 | no | — |
| 28 | D3×S4 | — | 9 | 12 | no | — |
| 29 | D4×C3⋊D3 | — | 9 | 1 | no | — |
| 30 | D3×#67 | — | 9 | 4 | no | — |
| 31 | C22×C33 | C2×D3 | 1 | 1 | yes | characteristic, complemented |
| 32 | C2×C3⋊(C3⋊D3) | — | 3 | 1 | no | — |
| 33 | C2×C3×C3⋊D3 | — | 3 | 1 | no | — |
| 34 | C2×C3×C3⋊D3 | — | 3 | 1 | no | — |
| 35 | C2×C32×D3 | — | 3 | 1 | no | — |
| 36 | C32×A4 | — | 2 | 4 | no | — |
| 37 | C32×A4 | — | 2 | 12 | no | — |
| 38 | C32×A4 | C2×D3 | 1 | 3 | yes | complemented |
| 39 | C32×A4 | C2×D3 | 1 | 4 | yes | complemented |
| 40 | D3×C3⋊D3 | — | 6 | 1 | no | — |
| 41 | D3×C3⋊D3 | — | 6 | 1 | no | — |
| 42 | D3×C3⋊D3 | — | 12 | 4 | no | — |
| 43 | D3×C3⋊D3 | — | 12 | 3 | no | — |
| 44 | C3⋊(C3⋊Dic3) | — | 3 | 1 | no | — |
| 45 | C32×Dic3 | — | 3 | 1 | no | — |
| 46 | C34 | — | 4 | 1 | no | Sylow |
| 47 | C22×C3⋊D3 | — | 9 | 1 | no | — |
| 48 | C22×C3⋊D3 | C3⋊D3 | 1 | 1 | yes | characteristic, complemented |
| 49 | C22×C3×D3 | — | 3 | 4 | no | — |
| 50 | C2×C3×A4 | — | 9 | 1 | no | — |
| 51 | C2×D32 | — | 9 | 4 | no | — |
| 52 | D3×A4 | — | 3 | 12 | no | — |
| 53 | C3⋊S4 | — | 9 | 1 | no | — |
| 54 | C3⋊S4 | — | 9 | 12 | no | — |
| 55 | C3⋊S4 | — | 18 | 4 | no | — |
| 56 | C3⋊S4 | — | 18 | 12 | no | — |
| 57 | C3⋊S4 | — | 18 | 12 | no | — |
| 58 | C3⋊S4 | — | 18 | 24 | no | — |
| 59 | C3⋊S4 | C3⋊D3 | 1 | 1 | yes | characteristic, direct product factor |
| 60 | C3×S4 | — | 3 | 12 | no | — |
| 61 | C32×D4 | — | 9 | 1 | no | — |
| 62 | #638 | — | 9 | 1 | no | — |
| 63 | #638 | — | 9 | 1 | no | — |
| 64 | #638 | — | 9 | 4 | no | — |
| 65 | #638 | — | 18 | 4 | no | — |
| 66 | C4⋊(C3⋊D3) | — | 9 | 1 | no | — |
| 67 | C4×C3⋊D3 | — | 9 | 1 | no | — |
| 68 | C3×#67 | — | 3 | 4 | no | — |
| 69 | #626 | — | 9 | 4 | no | — |
| 70 | #625 | — | 9 | 4 | no | — |
| 71 | D3×Dic3 | — | 9 | 4 | no | — |
| 72 | C2×C33 | — | 3 | 1 | no | — |
| 73 | C3⋊(C3⋊D3) | — | 6 | 1 | no | — |
| 74 | C3⋊(C3⋊D3) | — | 12 | 3 | no | — |
| 75 | C3⋊(C3⋊D3) | — | 12 | 4 | no | — |
| 76 | C3⋊(C3⋊D3) | — | 24 | 4 | no | — |
| 77 | C3⋊(C3⋊D3) | — | 24 | 12 | no | — |
| 78 | C3×C3⋊D3 | — | 3 | 1 | no | — |
| 79 | C3×C3⋊D3 | — | 4 | 4 | no | — |
| 80 | C3×C3⋊D3 | — | 4 | 3 | no | — |
| 81 | C3×C3⋊D3 | S4 | 1 | 1 | yes | characteristic, complemented |
| 82 | C32×D3 | — | 6 | 1 | no | — |
| 83 | C32×D3 | — | 12 | 4 | no | — |
| 84 | C32×D3 | — | 12 | 3 | no | — |
| 85 | C2×S4 | — | 27 | 3 | no | — |
| 86 | C2×#67 | — | 27 | 1 | no | — |
| 87 | D3×D4 | — | 27 | 4 | no | — |
| 88 | C22×C32 | — | 2 | 4 | no | — |
| 89 | C22×C32 | — | 9 | 1 | no | — |
| 90 | C22×C32 | D32 | 1 | 4 | yes | complemented |
| 91 | C22×C32 | C2×C3⋊D3 | 1 | 1 | yes | characteristic, complemented |
| 92 | C2×C3⋊D3 | — | 3 | 1 | no | — |
| 93 | C2×C3⋊D3 | — | 3 | 1 | no | — |
| 94 | C2×C3⋊D3 | — | 9 | 1 | no | — |
| 95 | C2×C3⋊D3 | — | 9 | 4 | no | — |
| 96 | C2×C3⋊D3 | — | 18 | 4 | no | — |
| 97 | C2×C3⋊D3 | — | 18 | 1 | no | — |
| 98 | C2×C3⋊D3 | — | 18 | 1 | no | — |
| 99 | C2×C3⋊D3 | — | 36 | 1 | no | — |
| 100 | C2×C3×D3 | — | 3 | 4 | no | — |
| 101 | C2×C3×D3 | — | 9 | 4 | no | — |
| 102 | C2×C3×D3 | — | 9 | 4 | no | — |
| 103 | C3×A4 | — | 2 | 24 | no | — |
| 104 | C3×A4 | — | 2 | 4 | no | — |
| 105 | C3×A4 | — | 2 | 12 | no | — |
| 106 | C3×A4 | — | 2 | 12 | no | — |
| 107 | C3×A4 | D32 | 1 | 12 | yes | complemented |
| 108 | C3×A4 | C2×C3⋊D3 | 1 | 1 | yes | characteristic, complemented |
| 109 | D32 | — | 18 | 4 | no | — |
| 110 | D32 | — | 18 | 4 | no | — |
| 111 | D32 | — | 36 | 12 | no | — |
| 112 | C32×C4 | — | 9 | 1 | no | — |
| 113 | C3⋊Dic3 | — | 9 | 1 | no | — |
| 114 | C3⋊Dic3 | — | 9 | 4 | no | — |
| 115 | C3⋊Dic3 | — | 18 | 4 | no | — |
| 116 | C3×Dic3 | — | 3 | 4 | no | — |
| 117 | C33 | — | 4 | 4 | no | — |
| 118 | C33 | — | 4 | 3 | no | — |
| 119 | C33 | — | 8 | 12 | no | — |
| 120 | C33 | — | 8 | 4 | no | — |
| 121 | C33 | C2×S4 | 1 | 1 | yes | characteristic, complemented |
| 122 | C23×C3 | — | 9 | 1 | no | — |
| 123 | C22×D3 | — | 3 | 4 | no | — |
| 124 | C22×D3 | — | 27 | 1 | no | — |
| 125 | C22×D3 | — | 27 | 4 | no | — |
| 126 | C2×A4 | — | 9 | 3 | no | — |
| 127 | S4 | — | 3 | 3 | no | — |
| 128 | S4 | — | 27 | 3 | no | — |
| 129 | S4 | — | 54 | 12 | no | — |
| 130 | C3×D4 | — | 9 | 4 | no | — |
| 131 | #67 | — | 3 | 1 | no | — |
| 132 | #67 | — | 27 | 1 | no | — |
| 133 | #67 | — | 27 | 1 | no | — |
| 134 | #67 | — | 27 | 4 | no | — |
| 135 | #67 | — | 27 | 1 | no | — |
| 136 | #67 | — | 27 | 4 | no | — |
| 137 | #67 | — | 54 | 4 | no | — |
| 138 | C2×Dic3 | — | 27 | 1 | no | — |
| 139 | D12 | — | 27 | 4 | no | — |
| 140 | C4×D3 | — | 27 | 4 | no | — |
| 141 | C2×C32 | — | 3 | 1 | no | — |
| 142 | C2×C32 | — | 3 | 4 | no | — |
| 143 | C2×C32 | — | 6 | 4 | no | — |
| 144 | C2×C32 | — | 18 | 1 | no | — |
| 145 | C2×C32 | — | 36 | 1 | no | — |
| 146 | C3⋊D3 | — | 3 | 1 | no | — |
| 147 | C3⋊D3 | — | 4 | 1 | no | — |
| 148 | C3⋊D3 | — | 18 | 1 | no | — |
| 149 | C3⋊D3 | — | 18 | 4 | no | — |
| 150 | C3⋊D3 | — | 36 | 4 | no | — |
| 151 | C3⋊D3 | — | 36 | 1 | no | — |
| 152 | C3⋊D3 | — | 36 | 12 | no | — |
| 153 | C3⋊D3 | — | 72 | 4 | no | — |
| 154 | C3⋊D3 | — | 72 | 12 | no | — |
| 155 | C3⋊D3 | — | 72 | 12 | no | — |
| 156 | C3⋊D3 | — | 72 | 24 | no | — |
| 157 | C3⋊D3 | C3⋊S4 | 1 | 1 | yes | characteristic, direct product factor |
| 158 | C3×D3 | — | 3 | 4 | no | — |
| 159 | C3×D3 | — | 6 | 4 | no | — |
| 160 | C3×D3 | — | 9 | 4 | no | — |
| 161 | C3×D3 | — | 12 | 12 | no | — |
| 162 | C3×D3 | — | 12 | 12 | no | — |
| 163 | C2×D4 | — | 81 | 1 | no | Sylow |
| 164 | C22×C3 | — | 2 | 4 | no | — |
| 165 | C22×C3 | — | 9 | 4 | no | — |
| 166 | C22×C3 | — | 27 | 1 | no | — |
| 167 | C22×C3 | — | 27 | 1 | no | — |
| 168 | C22×C3 | D3×C3⋊D3 | 1 | 1 | yes | characteristic, complemented |
| 169 | C22×C3 | D3×C3⋊D3 | 1 | 4 | yes | complemented |
| 170 | C2×D3 | — | 3 | 1 | no | — |
| 171 | C2×D3 | — | 9 | 4 | no | — |
| 172 | C2×D3 | — | 9 | 4 | no | — |
| 173 | C2×D3 | — | 27 | 4 | no | — |
| 174 | C2×D3 | — | 27 | 1 | no | — |
| 175 | C2×D3 | — | 54 | 4 | no | — |
| 176 | C2×D3 | — | 54 | 4 | no | — |
| 177 | C2×D3 | — | 54 | 1 | no | — |
| 178 | C2×D3 | — | 54 | 1 | no | — |
| 179 | C2×D3 | — | 54 | 4 | no | — |
| 180 | C2×D3 | — | 108 | 3 | no | — |
| 181 | A4 | — | 2 | 12 | no | — |
| 182 | A4 | D3×C3⋊D3 | 1 | 3 | yes | complemented |
| 183 | C3×C4 | — | 9 | 4 | no | — |
| 184 | Dic3 | — | 3 | 1 | no | — |
| 185 | Dic3 | — | 27 | 4 | no | — |
| 186 | Dic3 | — | 27 | 1 | no | — |
| 187 | Dic3 | — | 54 | 4 | no | — |
| 188 | C32 | — | 2 | 4 | no | — |
| 189 | C32 | — | 4 | 1 | no | — |
| 190 | C32 | — | 4 | 12 | no | — |
| 191 | C32 | — | 8 | 12 | no | — |
| 192 | C32 | — | 8 | 24 | no | — |
| 193 | C32 | — | 8 | 4 | no | — |
| 194 | C32 | — | 8 | 12 | no | — |
| 195 | C32 | D3×S4 | 1 | 4 | yes | complemented |
| 196 | C32 | C2×C3⋊S4 | 1 | 1 | yes | characteristic, complemented |
| 197 | C23 | — | 9 | 1 | no | — |
| 198 | C23 | — | 81 | 1 | no | — |
| 199 | D4 | — | 9 | 1 | no | — |
| 200 | D4 | — | 81 | 1 | no | — |
| 201 | D4 | — | 81 | 1 | no | — |
| 202 | D4 | — | 81 | 1 | no | — |
| 203 | C2×C4 | — | 81 | 1 | no | — |
| 204 | C2×C3 | — | 3 | 1 | no | — |
| 205 | C2×C3 | — | 3 | 4 | no | — |
| 206 | C2×C3 | — | 6 | 4 | no | — |
| 207 | C2×C3 | — | 9 | 1 | no | — |
| 208 | C2×C3 | — | 18 | 4 | no | — |
| 209 | C2×C3 | — | 27 | 1 | no | — |
| 210 | C2×C3 | — | 36 | 3 | no | — |
| 211 | D3 | — | 3 | 4 | no | — |
| 212 | D3 | — | 6 | 1 | no | — |
| 213 | D3 | — | 9 | 4 | no | — |
| 214 | D3 | — | 12 | 3 | no | — |
| 215 | D3 | — | 54 | 1 | no | — |
| 216 | D3 | — | 54 | 4 | no | — |
| 217 | D3 | — | 108 | 4 | no | — |
| 218 | D3 | — | 108 | 3 | no | — |
| 219 | D3 | — | 216 | 12 | no | — |
| 220 | C22 | — | 9 | 1 | no | — |
| 221 | C22 | — | 27 | 1 | no | — |
| 222 | C22 | — | 27 | 1 | no | — |
| 223 | C22 | — | 81 | 1 | no | — |
| 224 | C22 | — | 162 | 1 | no | — |
| 225 | C22 | — | 162 | 1 | no | — |
| 226 | C22 | C3⋊D32 | 1 | 1 | yes | characteristic, complemented |
| 227 | C4 | — | 9 | 1 | no | — |
| 228 | C4 | — | 81 | 1 | no | — |
| 229 | C3 | — | 2 | 4 | no | — |
| 230 | C3 | — | 4 | 3 | no | — |
| 231 | C3 | — | 8 | 12 | no | — |
| 232 | C3 | C3⋊D3×S4 | 1 | 1 | yes | characteristic, complemented |
| 233 | C3 | D3×C3⋊S4 | 1 | 4 | yes | complemented |
| 234 | C2 | — | 3 | 1 | no | — |
| 235 | C2 | — | 9 | 1 | no | — |
| 236 | C2 | — | 18 | 1 | no | — |
| 237 | C2 | — | 27 | 1 | no | — |
| 238 | C2 | — | 162 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C33×A4 | C22 |
| Fitting, socle | C22×C33 | C2×D3 |
| LCS | G ▷ C33×A4 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×A4×C3⋊D3 | C2 | semidirect |
| 2. C3⋊(C3⋊(C3⋊S4)) | C2 | semidirect |
| 3. C32×C3⋊S4 | C2 | semidirect |
| 7. C33×A4 | C22 | semidirect |
| 9. C22×C3×C3⋊D3 | D3 | semidirect |
| 11. A4×C3⋊D3 | D3 | semidirect |
| 17. C3×C3⋊S4 | D3 | semidirect |
| 31. C22×C33 | C2×D3 | semidirect |
| 38. C32×A4 | C2×D3 | semidirect |
| 39. C32×A4 | C2×D3 | semidirect |
| 48. C22×C3⋊D3 | C3⋊D3 | semidirect |
| 59. C3⋊S4 | C3⋊D3 | direct (N × H) |
| 81. C3×C3⋊D3 | S4 | semidirect |
| 90. C22×C32 | D32 | semidirect |
| 91. C22×C32 | C2×C3⋊D3 | semidirect |
| 107. C3×A4 | D32 | semidirect |
| 108. C3×A4 | C2×C3⋊D3 | semidirect |
| 121. C33 | C2×S4 | semidirect |
| 157. C3⋊D3 | C3⋊S4 | direct (N × H) |
| 168. C22×C3 | D3×C3⋊D3 | semidirect |
| 169. C22×C3 | D3×C3⋊D3 | semidirect |
| 182. A4 | D3×C3⋊D3 | semidirect |
| 195. C32 | D3×S4 | semidirect |
| 196. C32 | C2×C3⋊S4 | semidirect |
| 226. C22 | C3⋊D32 | semidirect |
| 232. C3 | C3⋊D3×S4 | semidirect |
| 233. C3 | D3×C3⋊S4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×A4×C3⋊D3 | 2 |
| C3⋊(C3⋊(C3⋊S4)) | 2 |
| C32×C3⋊S4 | 2 |
| D3×C3⋊S4 | 3 |
| C3⋊D3×S4 | 3 |
| C3⋊D3×#67 | 3 |
| C3⋊D32 | 4 |