Finite Group DatabasebetaFinite Group DatabasebetaOrder 2187.
1193 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 1193 conjugacy classes of subgroups — too many to draw. Only its 98 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×#6912 | C3 | 1 | 2 | yes | maximal, complemented |
| 2 | C3×#6911 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C3×#6910 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C3×#6910 | C3 | 1 | 2 | yes | maximal, complemented |
| 5 | C3×#6909 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C3×#6909 | C3 | 1 | 2 | yes | maximal, complemented |
| 7 | C32×#731 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C32×#731 | C3 | 1 | 2 | yes | maximal, complemented |
| 9 | C3×#6893 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 10 | C32×3−1+2 | C32 | 1 | 2 | yes | — |
| 11 | C32×3−1+2 | C32 | 1 | 2 | yes | — |
| 12 | C32×3−1+2 | C32 | 1 | 2 | yes | — |
| 13 | C32×3−1+2 | C32 | 1 | 1 | yes | characteristic |
| 14 | C32×3−1+2 | C32 | 1 | 1 | yes | characteristic |
| 15 | C32×H(3,𝔽3) | C32 | 1 | 1 | yes | characteristic |
| 16 | C32×H(3,𝔽3) | C32 | 1 | 1 | yes | characteristic |
| 17 | C32×H(3,𝔽3) | C32 | 1 | 2 | yes | — |
| 18 | C33×C9 | C32 | 1 | 1 | yes | characteristic |
| 19 | #6912 | — | 3 | 6 | no | — |
| 20 | #6912 | — | 3 | 6 | no | — |
| 21 | #6912 | — | 3 | 6 | no | — |
| 22 | #6911 | — | 3 | 3 | no | — |
| 23 | #6911 | — | 3 | 6 | no | — |
| 24 | #6910 | — | 3 | 6 | no | — |
| 25 | #6910 | — | 3 | 6 | no | — |
| 26 | #6910 | — | 3 | 6 | no | — |
| 27 | #6910 | — | 3 | 3 | no | — |
| 28 | #6910 | — | 3 | 6 | no | — |
| 29 | #6909 | — | 3 | 6 | no | — |
| 30 | #6909 | — | 3 | 6 | no | — |
| 31 | #6909 | — | 3 | 3 | no | — |
| 32 | #6909 | — | 3 | 6 | no | — |
| 33 | #6909 | — | 3 | 6 | no | — |
| 34 | C3×#733 | — | 3 | 6 | no | — |
| 35 | C3×#732 | — | 3 | 3 | no | — |
| 36 | C3×#731 | — | 3 | 3 | no | — |
| 37 | C3×#731 | — | 3 | 6 | no | — |
| 38 | C3×#731 | — | 3 | 6 | no | — |
| 39 | C3×#731 | — | 3 | 3 | no | — |
| 40 | C3×#731 | — | 3 | 6 | no | — |
| 41 | C3×#731 | — | 3 | 3 | no | — |
| 42 | C3×#731 | — | 3 | 6 | no | — |
| 43 | C3×#731 | — | 3 | 3 | no | — |
| 44 | C3×#731 | — | 3 | 3 | no | — |
| 45 | C3×#731 | — | 3 | 6 | no | — |
| 46 | C3×AΣL(1,𝔽27) | — | 3 | 6 | no | — |
| 47 | C3×AΣL(1,𝔽27) | — | 3 | 3 | no | — |
| 48 | #6893 | — | 3 | 6 | no | — |
| 49 | #6893 | — | 3 | 3 | no | — |
| 50 | C3×C32⋊C9 | — | 3 | 3 | no | — |
| 51 | C34 | — | 3 | 1 | no | — |
| 52 | C34 | — | 3 | 2 | no | — |
| 53 | C34 | — | 3 | 1 | no | — |
| 54 | C34 | C33 | 1 | 1 | yes | characteristic |
| 55 | C3×3−1+2 | — | 3 | 2 | no | — |
| 56 | C3×3−1+2 | — | 3 | 2 | no | — |
| 57 | C3×3−1+2 | — | 3 | 2 | no | — |
| 58 | C3×3−1+2 | — | 3 | 2 | no | — |
| 59 | C3×3−1+2 | — | 3 | 2 | no | — |
| 60 | C3×3−1+2 | — | 3 | 2 | no | — |
| 61 | C3×3−1+2 | — | 3 | 2 | no | — |
| 62 | C3×3−1+2 | — | 3 | 2 | no | — |
| 63 | C3×3−1+2 | — | 3 | 2 | no | — |
| 64 | C3×3−1+2 | — | 3 | 2 | no | — |
| 65 | C3×3−1+2 | — | 3 | 2 | no | — |
| 66 | C3×3−1+2 | — | 3 | 2 | no | — |
| 67 | C3×3−1+2 | — | 3 | 2 | no | — |
| 68 | C3×3−1+2 | — | 3 | 2 | no | — |
| 69 | C3×3−1+2 | — | 3 | 2 | no | — |
| 70 | C3×3−1+2 | — | 3 | 2 | no | — |
| 71 | C3×3−1+2 | — | 3 | 2 | no | — |
| 72 | C3×3−1+2 | — | 3 | 2 | no | — |
| 73 | C3×3−1+2 | — | 3 | 2 | no | — |
| 74 | C3×3−1+2 | — | 3 | 1 | no | — |
| 75 | C3×3−1+2 | — | 3 | 1 | no | — |
| 76 | C3×3−1+2 | — | 3 | 2 | no | — |
| 77 | C3×3−1+2 | — | 3 | 2 | no | — |
| 78 | C3×3−1+2 | — | 3 | 1 | no | — |
| 79 | C3×3−1+2 | — | 3 | 1 | no | — |
| 80 | C3×3−1+2 | — | 3 | 2 | no | — |
| 81 | C3×3−1+2 | — | 3 | 2 | no | — |
| 82 | C3×3−1+2 | — | 3 | 2 | no | — |
| 83 | C3×3−1+2 | — | 3 | 2 | no | — |
| 84 | C3×3−1+2 | — | 3 | 1 | no | — |
| 85 | C3×3−1+2 | — | 3 | 1 | no | — |
| 86 | C3×3−1+2 | — | 3 | 2 | no | — |
| 87 | C3×3−1+2 | — | 3 | 2 | no | — |
| 88 | C3×3−1+2 | — | 3 | 2 | no | — |
| 89 | C3×3−1+2 | — | 3 | 2 | no | — |
| 90 | C3×3−1+2 | — | 3 | 2 | no | — |
| 91 | C3×3−1+2 | — | 3 | 2 | no | — |
| 92 | C3×3−1+2 | — | 3 | 2 | no | — |
| 93 | C3×3−1+2 | — | 3 | 2 | no | — |
| 94 | C3×3−1+2 | — | 3 | 2 | no | — |
| 95 | C3×3−1+2 | — | 3 | 2 | no | — |
| 96 | C3×3−1+2 | — | 3 | 2 | no | — |
| 97 | C3×3−1+2 | — | 3 | 2 | no | — |
| 98 | C3×3−1+2 | — | 3 | 2 | no | — |
| 99 | C3×3−1+2 | — | 3 | 2 | no | — |
| 100 | C3×3−1+2 | — | 3 | 2 | no | — |
| 101 | C3×3−1+2 | — | 3 | 2 | no | — |
| 102 | C3×3−1+2 | H(3,𝔽3) | 1 | 3 | yes | — |
| 103 | C3×3−1+2 | H(3,𝔽3) | 1 | 6 | yes | — |
| 104 | C3×3−1+2 | H(3,𝔽3) | 1 | 6 | yes | — |
| 105 | C3×3−1+2 | H(3,𝔽3) | 1 | 6 | yes | — |
| 106 | C3×3−1+2 | H(3,𝔽3) | 1 | 3 | yes | — |
| 107 | C3×H(3,𝔽3) | — | 3 | 2 | no | — |
| 108 | C3×H(3,𝔽3) | — | 3 | 1 | no | — |
| 109 | C3×H(3,𝔽3) | — | 3 | 1 | no | — |
| 110 | C3×H(3,𝔽3) | — | 3 | 2 | no | — |
| 111 | C3×H(3,𝔽3) | — | 3 | 2 | no | — |
| 112 | C3×H(3,𝔽3) | — | 3 | 2 | no | — |
| 113 | C3×H(3,𝔽3) | — | 3 | 1 | no | — |
| 114 | C3×H(3,𝔽3) | — | 3 | 2 | no | — |
| 115 | C3×H(3,𝔽3) | — | 3 | 2 | no | — |
| 116 | C3×H(3,𝔽3) | — | 3 | 2 | no | — |
| 117 | C3×H(3,𝔽3) | — | 3 | 2 | no | — |
| 118 | C3×H(3,𝔽3) | — | 3 | 1 | no | — |
| 119 | C3×H(3,𝔽3) | — | 3 | 2 | no | — |
| 120 | C3×H(3,𝔽3) | — | 3 | 1 | no | — |
| 121 | C3×H(3,𝔽3) | — | 3 | 2 | no | — |
| 122 | C3×H(3,𝔽3) | — | 3 | 2 | no | — |
| 123 | C3×H(3,𝔽3) | — | 3 | 2 | no | — |
| 124 | C3×H(3,𝔽3) | — | 3 | 1 | no | — |
| 125 | C3×H(3,𝔽3) | — | 3 | 2 | no | — |
| 126 | C3×H(3,𝔽3) | — | 3 | 2 | no | — |
| 127 | C3×H(3,𝔽3) | — | 3 | 2 | no | — |
| 128 | C3×H(3,𝔽3) | — | 3 | 2 | no | — |
| 129 | C3×H(3,𝔽3) | — | 3 | 2 | no | — |
| 130 | C3×H(3,𝔽3) | — | 3 | 2 | no | — |
| 131 | C3×H(3,𝔽3) | — | 3 | 2 | no | — |
| 132 | C3×H(3,𝔽3) | H(3,𝔽3) | 1 | 3 | yes | — |
| 133 | C3×H(3,𝔽3) | H(3,𝔽3) | 1 | 3 | yes | — |
| 134 | C3×H(3,𝔽3) | H(3,𝔽3) | 1 | 6 | yes | — |
| 135 | C32×C9 | — | 3 | 2 | no | — |
| 136 | C32×C9 | — | 3 | 1 | no | — |
| 137 | C32×C9 | — | 3 | 1 | no | — |
| 138 | C32×C9 | — | 3 | 1 | no | — |
| 139 | C32×C9 | — | 3 | 2 | no | — |
| 140 | C32×C9 | — | 3 | 2 | no | — |
| 141 | C32×C9 | — | 3 | 2 | no | — |
| 142 | C32×C9 | — | 3 | 1 | no | — |
| 143 | C32×C9 | — | 3 | 2 | no | — |
| 144 | C32×C9 | — | 3 | 2 | no | — |
| 145 | C32×C9 | — | 3 | 1 | no | — |
| 146 | C32×C9 | — | 3 | 1 | no | — |
| 147 | C32×C9 | — | 3 | 2 | no | — |
| 148 | C32×C9 | H(3,𝔽3) | 1 | 3 | yes | — |
| 149 | #733 | — | 3 | 54 | no | — |
| 150 | #732 | — | 3 | 27 | no | — |
| 151 | #731 | — | 3 | 27 | no | — |
| 152 | #731 | — | 3 | 54 | no | — |
| 153 | #731 | — | 3 | 54 | no | — |
| 154 | #731 | — | 3 | 27 | no | — |
| 155 | AΣL(1,𝔽27) | — | 3 | 54 | no | — |
| 156 | AΣL(1,𝔽27) | — | 3 | 27 | no | — |
| 157 | C32⋊C9 | — | 3 | 27 | no | — |
| 158 | C33 | — | 3 | 3 | no | — |
| 159 | C33 | — | 3 | 3 | no | — |
| 160 | C33 | — | 3 | 6 | no | — |
| 161 | C33 | — | 9 | 3 | no | — |
| 162 | C33 | — | 9 | 9 | no | — |
| 163 | C33 | — | 9 | 2 | no | — |
| 164 | C33 | — | 9 | 1 | no | — |
| 165 | C33 | — | 9 | 1 | no | — |
| 166 | C33 | — | 9 | 1 | no | — |
| 167 | C33 | — | 9 | 2 | no | — |
| 168 | C33 | — | 9 | 2 | no | — |
| 169 | C33 | — | 9 | 1 | no | — |
| 170 | C33 | — | 9 | 2 | no | — |
| 171 | C33 | — | 27 | 6 | no | — |
| 172 | C33 | — | 27 | 3 | no | — |
| 173 | C33 | C3×H(3,𝔽3) | 1 | 2 | yes | — |
| 174 | C33 | C3×H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 175 | C33 | C3×H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 176 | C33 | C3×H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 177 | C33 | C3×H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 178 | C33 | C3×H(3,𝔽3) | 1 | 2 | yes | — |
| 179 | C33 | C3×H(3,𝔽3) | 1 | 2 | yes | — |
| 180 | C33 | C3×H(3,𝔽3) | 1 | 2 | yes | — |
| 181 | C33 | C3×H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 182 | 3−1+2 | — | 3 | 6 | no | — |
| 183 | 3−1+2 | — | 3 | 6 | no | — |
| 184 | 3−1+2 | — | 3 | 6 | no | — |
| 185 | 3−1+2 | — | 3 | 3 | no | — |
| 186 | 3−1+2 | — | 3 | 6 | no | — |
| 187 | 3−1+2 | — | 3 | 6 | no | — |
| 188 | 3−1+2 | — | 3 | 3 | no | — |
| 189 | 3−1+2 | — | 3 | 6 | no | — |
| 190 | 3−1+2 | — | 3 | 6 | no | — |
| 191 | 3−1+2 | — | 3 | 6 | no | — |
| 192 | 3−1+2 | — | 3 | 6 | no | — |
| 193 | 3−1+2 | — | 3 | 6 | no | — |
| 194 | 3−1+2 | — | 3 | 6 | no | — |
| 195 | 3−1+2 | — | 9 | 2 | no | — |
| 196 | 3−1+2 | — | 9 | 2 | no | — |
| 197 | 3−1+2 | — | 9 | 2 | no | — |
| 198 | 3−1+2 | — | 9 | 2 | no | — |
| 199 | 3−1+2 | — | 9 | 2 | no | — |
| 200 | 3−1+2 | — | 9 | 2 | no | — |
| 201 | 3−1+2 | — | 9 | 2 | no | — |
| 202 | 3−1+2 | — | 9 | 2 | no | — |
| 203 | 3−1+2 | — | 9 | 2 | no | — |
| 204 | 3−1+2 | — | 9 | 2 | no | — |
| 205 | 3−1+2 | — | 9 | 2 | no | — |
| 206 | 3−1+2 | — | 9 | 2 | no | — |
| 207 | 3−1+2 | — | 9 | 2 | no | — |
| 208 | 3−1+2 | — | 9 | 2 | no | — |
| 209 | 3−1+2 | — | 9 | 2 | no | — |
| 210 | 3−1+2 | — | 9 | 2 | no | — |
| 211 | 3−1+2 | — | 9 | 2 | no | — |
| 212 | 3−1+2 | — | 9 | 2 | no | — |
| 213 | 3−1+2 | — | 9 | 2 | no | — |
| 214 | 3−1+2 | — | 9 | 2 | no | — |
| 215 | 3−1+2 | — | 9 | 2 | no | — |
| 216 | 3−1+2 | — | 9 | 2 | no | — |
| 217 | 3−1+2 | — | 9 | 2 | no | — |
| 218 | 3−1+2 | — | 9 | 2 | no | — |
| 219 | H(3,𝔽3) | — | 3 | 6 | no | — |
| 220 | H(3,𝔽3) | — | 3 | 3 | no | — |
| 221 | H(3,𝔽3) | — | 3 | 3 | no | — |
| 222 | H(3,𝔽3) | — | 3 | 6 | no | — |
| 223 | H(3,𝔽3) | — | 3 | 6 | no | — |
| 224 | H(3,𝔽3) | — | 3 | 3 | no | — |
| 225 | H(3,𝔽3) | — | 3 | 6 | no | — |
| 226 | H(3,𝔽3) | — | 3 | 3 | no | — |
| 227 | H(3,𝔽3) | — | 9 | 2 | no | — |
| 228 | H(3,𝔽3) | — | 9 | 2 | no | — |
| 229 | H(3,𝔽3) | — | 9 | 2 | no | — |
| 230 | H(3,𝔽3) | — | 9 | 2 | no | — |
| 231 | H(3,𝔽3) | — | 9 | 2 | no | — |
| 232 | H(3,𝔽3) | — | 9 | 2 | no | — |
| 233 | H(3,𝔽3) | — | 9 | 2 | no | — |
| 234 | H(3,𝔽3) | — | 9 | 2 | no | — |
| 235 | H(3,𝔽3) | — | 9 | 2 | no | — |
| 236 | H(3,𝔽3) | — | 9 | 2 | no | — |
| 237 | H(3,𝔽3) | — | 9 | 2 | no | — |
| 238 | H(3,𝔽3) | — | 9 | 2 | no | — |
| 239 | C3×C9 | — | 3 | 3 | no | — |
| 240 | C3×C9 | — | 3 | 6 | no | — |
| 241 | C3×C9 | — | 3 | 6 | no | — |
| 242 | C3×C9 | — | 3 | 3 | no | — |
| 243 | C3×C9 | — | 3 | 3 | no | — |
| 244 | C3×C9 | — | 3 | 6 | no | — |
| 245 | C3×C9 | — | 3 | 3 | no | — |
| 246 | C3×C9 | — | 3 | 6 | no | — |
| 247 | C3×C9 | — | 9 | 1 | no | — |
| 248 | C3×C9 | — | 9 | 2 | no | — |
| 249 | C3×C9 | — | 9 | 1 | no | — |
| 250 | C3×C9 | — | 9 | 2 | no | — |
| 251 | C3×C9 | — | 9 | 2 | no | — |
| 252 | C3×C9 | — | 9 | 1 | no | — |
| 253 | C3×C9 | — | 9 | 1 | no | — |
| 254 | C3×C9 | — | 9 | 2 | no | — |
| 255 | C3×C9 | — | 9 | 2 | no | — |
| 256 | C3×C9 | — | 9 | 2 | no | — |
| 257 | C3×C9 | — | 9 | 1 | no | — |
| 258 | C3×C9 | — | 9 | 2 | no | — |
| 259 | C3×C9 | — | 9 | 2 | no | — |
| 260 | C3×C9 | — | 9 | 2 | no | — |
| 261 | C3×C9 | — | 9 | 1 | no | — |
| 262 | C3×C9 | — | 9 | 1 | no | — |
| 263 | C3×C9 | — | 9 | 2 | no | — |
| 264 | C3×C9 | — | 9 | 2 | no | — |
| 265 | C3×C9 | — | 9 | 1 | no | — |
| 266 | C3×C9 | — | 9 | 1 | no | — |
| 267 | C3×C9 | — | 9 | 2 | no | — |
| 268 | C32 | — | 3 | 3 | no | — |
| 269 | C32 | — | 3 | 6 | no | — |
| 270 | C32 | — | 3 | 3 | no | — |
| 271 | C32 | — | 9 | 2 | no | — |
| 272 | C32 | — | 9 | 2 | no | — |
| 273 | C32 | — | 9 | 1 | no | — |
| 274 | C32 | — | 9 | 1 | no | — |
| 275 | C32 | — | 9 | 1 | no | — |
| 276 | C32 | — | 9 | 2 | no | — |
| 277 | C32 | — | 9 | 3 | no | — |
| 278 | C32 | — | 9 | 3 | no | — |
| 279 | C32 | — | 9 | 6 | no | — |
| 280 | C32 | — | 9 | 9 | no | — |
| 281 | C32 | — | 27 | 3 | no | — |
| 282 | C32 | — | 27 | 6 | no | — |
| 283 | C32 | — | 27 | 6 | no | — |
| 284 | C32 | — | 27 | 3 | no | — |
| 285 | C32 | — | 27 | 3 | no | — |
| 286 | C32 | — | 27 | 3 | no | — |
| 287 | C32 | — | 27 | 6 | no | — |
| 288 | C32 | — | 27 | 6 | no | — |
| 289 | C32 | — | 27 | 3 | no | — |
| 290 | C32 | — | 27 | 6 | no | — |
| 291 | C32 | C32⋊H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 292 | C32 | C32⋊H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 293 | C32 | C32⋊H(3,𝔽3) | 1 | 2 | yes | — |
| 294 | C32 | C32⋊H(3,𝔽3) | 1 | 2 | yes | — |
| 295 | C32 | C32⋊H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 296 | C32 | C32⋊H(3,𝔽3) | 1 | 2 | yes | — |
| 297 | C32 | C32⋊H(3,𝔽3) | 1 | 2 | yes | — |
| 298 | C32 | C32⋊H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 299 | C32 | C32⋊H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 300 | C9 | — | 9 | 3 | no | — |
| 301 | C9 | — | 9 | 6 | no | — |
| 302 | C9 | — | 9 | 3 | no | — |
| 303 | C9 | — | 9 | 3 | no | — |
| 304 | C9 | — | 9 | 6 | no | — |
| 305 | C9 | — | 9 | 6 | no | — |
| 306 | C3 | — | 3 | 2 | no | — |
| 307 | C3 | — | 3 | 1 | no | — |
| 308 | C3 | — | 3 | 2 | no | — |
| 309 | C3 | — | 3 | 1 | no | — |
| 310 | C3 | — | 3 | 1 | no | — |
| 311 | C3 | — | 3 | 2 | no | — |
| 312 | C3 | — | 3 | 1 | no | — |
| 313 | C3 | — | 3 | 2 | no | — |
| 314 | C3 | — | 27 | 3 | no | — |
| 315 | C3 | — | 27 | 3 | no | — |
| 316 | C3 | — | 27 | 6 | no | — |
| 317 | C3 | #10624562 | 1 | 1 | yes | characteristic |
| 318 | C3 | #10624756 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C32 | C32⋊H(3,𝔽3) |
| commutator, Frattini, cosocle | C34 | C33 |
| LCS | G ▷ C34 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32 ◁ C34 ◁ G | |
| LECS | G ▷ C34 ▷ C3 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#6912 | C3 | semidirect |
| 2. C3×#6911 | C3 | semidirect |
| 3. C3×#6910 | C3 | semidirect |
| 4. C3×#6910 | C3 | semidirect |
| 5. C3×#6909 | C3 | semidirect |
| 6. C3×#6909 | C3 | semidirect |
| 7. C32×#731 | C3 | semidirect |
| 8. C32×#731 | C3 | semidirect |
| 9. C3×#6893 | C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#6912 | 3 |
| C3×#6911 | 3 |
| C3×#6910 | 3 |
| C3×#6909 | 3 |
| C32×#731 | 3 |
| C3×#6893 | 3 |