Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
626 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 626 conjugacy classes of subgroups — too many to draw. Only its 90 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 | C2×#6295 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×#6292 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | C2×#6291 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C2×C3×#626 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | C3×Dic32 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #63979 | — | 3 | 1 | no | maximal |
| 7 | #63911 | — | 3 | 2 | no | maximal |
| 8 | C22×C3⋊(C3⋊D3) | C22 | 1 | 1 | yes | characteristic |
| 9 | C22×C3×C3⋊D3 | C22 | 1 | 2 | yes | — |
| 10 | C22×C32×D3 | C22 | 1 | 1 | yes | characteristic |
| 11 | C2×C3×C3⋊Dic3 | C22 | 1 | 1 | yes | characteristic |
| 12 | C2×C32×Dic3 | C22 | 1 | 2 | yes | — |
| 13 | #6295 | — | 2 | 2 | no | — |
| 14 | #6292 | — | 2 | 4 | no | — |
| 15 | #6291 | — | 2 | 2 | no | — |
| 16 | C3×#626 | — | 2 | 4 | no | — |
| 17 | C2×#638 | — | 3 | 1 | no | — |
| 18 | C2×C4⋊(C3⋊D3) | — | 3 | 2 | no | — |
| 19 | C2×C3×#67 | — | 3 | 2 | no | — |
| 20 | C2×C3×D12 | — | 3 | 2 | no | — |
| 21 | C2×#626 | — | 3 | 2 | no | — |
| 22 | C2×#626 | — | 3 | 2 | no | — |
| 23 | C2×#626 | — | 3 | 2 | no | — |
| 24 | C2×#626 | — | 3 | 2 | no | — |
| 25 | C2×#626 | — | 6 | 2 | no | — |
| 26 | C2×#626 | — | 6 | 1 | no | — |
| 27 | C2×#626 | D3 | 1 | 2 | yes | complemented |
| 28 | C2×#625 | — | 3 | 1 | no | — |
| 29 | C3×C4×Dic3 | — | 3 | 2 | no | — |
| 30 | Dic32 | D3 | 1 | 1 | yes | characteristic, complemented |
| 31 | C22×C33 | C23 | 1 | 1 | yes | characteristic |
| 32 | C2×C3⋊(C3⋊D3) | — | 2 | 1 | no | — |
| 33 | C2×C3⋊(C3⋊D3) | — | 2 | 2 | no | — |
| 34 | C2×C3×C3⋊D3 | — | 2 | 2 | no | — |
| 35 | C2×C3×C3⋊D3 | — | 2 | 2 | no | — |
| 36 | C2×C3×C3⋊D3 | — | 2 | 2 | no | — |
| 37 | C2×C32×D3 | — | 2 | 1 | no | — |
| 38 | C2×C32×D3 | — | 2 | 2 | no | — |
| 39 | C3×C3⋊Dic3 | D4 | 1 | 2 | yes | complemented |
| 40 | C32×Dic3 | D4 | 1 | 4 | yes | complemented |
| 41 | #940 | — | 9 | 2 | no | — |
| 42 | #874 | — | 9 | 1 | no | — |
| 43 | C23×C32 | — | 3 | 1 | no | — |
| 44 | C22×C3⋊D3 | — | 3 | 2 | no | — |
| 45 | C22×C3⋊D3 | — | 3 | 1 | no | — |
| 46 | C22×C3⋊D3 | — | 6 | 1 | no | — |
| 47 | C22×C3⋊D3 | — | 6 | 2 | no | — |
| 48 | C22×C3⋊D3 | — | 12 | 1 | no | — |
| 49 | C22×C3⋊D3 | C2×D3 | 1 | 2 | yes | — |
| 50 | C22×C3×D3 | — | 2 | 1 | no | — |
| 51 | C22×C3×D3 | — | 3 | 2 | no | — |
| 52 | C22×C3×D3 | — | 3 | 2 | no | — |
| 53 | C22×C3×D3 | — | 6 | 2 | no | — |
| 54 | C22×C3×D3 | C2×D3 | 1 | 2 | yes | — |
| 55 | C2×C32×C4 | — | 3 | 2 | no | — |
| 56 | #638 | — | 6 | 2 | no | — |
| 57 | C2×C3⋊Dic3 | C2×D3 | 1 | 1 | yes | characteristic |
| 58 | C4⋊(C3⋊D3) | — | 6 | 4 | no | — |
| 59 | C3×#67 | — | 6 | 4 | no | — |
| 60 | C2×C3×Dic3 | — | 2 | 2 | no | — |
| 61 | C2×C3×Dic3 | — | 3 | 2 | no | — |
| 62 | C2×C3×Dic3 | — | 6 | 1 | no | — |
| 63 | C2×C3×Dic3 | C2×D3 | 1 | 2 | yes | — |
| 64 | C2×C3×Dic3 | C2×D3 | 1 | 2 | yes | — |
| 65 | C3×D12 | — | 6 | 4 | no | — |
| 66 | #626 | — | 2 | 4 | no | — |
| 67 | #626 | — | 6 | 8 | no | — |
| 68 | #626 | — | 6 | 4 | no | — |
| 69 | #626 | — | 6 | 4 | no | — |
| 70 | #626 | — | 6 | 4 | no | — |
| 71 | #626 | — | 6 | 4 | no | — |
| 72 | #626 | — | 6 | 4 | no | — |
| 73 | #625 | — | 6 | 2 | no | — |
| 74 | C2×C33 | C2×D4 | 1 | 1 | yes | characteristic |
| 75 | C2×C33 | C2×D4 | 1 | 2 | yes | — |
| 76 | C3⋊(C3⋊D3) | — | 4 | 1 | no | — |
| 77 | C3×C3⋊D3 | — | 4 | 2 | no | — |
| 78 | C32×D3 | — | 4 | 1 | no | — |
| 79 | C2×C3×D4 | — | 9 | 2 | no | — |
| 80 | C2×#67 | — | 3 | 2 | no | — |
| 81 | C2×#67 | — | 9 | 2 | no | — |
| 82 | C2×#67 | — | 9 | 2 | no | — |
| 83 | C2×#67 | — | 9 | 2 | no | — |
| 84 | C2×#67 | — | 18 | 1 | no | — |
| 85 | C2×D12 | — | 3 | 2 | no | — |
| 86 | C2×D12 | — | 9 | 2 | no | — |
| 87 | C2×D12 | — | 9 | 2 | no | — |
| 88 | C2×D12 | — | 9 | 1 | no | — |
| 89 | C2×D12 | — | 9 | 1 | no | — |
| 90 | C2×D12 | — | 18 | 2 | no | — |
| 91 | C3×C42 | — | 9 | 1 | no | — |
| 92 | C4×Dic3 | — | 3 | 2 | no | — |
| 93 | C22×C32 | — | 2 | 2 | no | — |
| 94 | C22×C32 | — | 2 | 1 | no | — |
| 95 | C22×C32 | — | 4 | 1 | no | — |
| 96 | C22×C32 | — | 6 | 1 | no | — |
| 97 | C22×C32 | — | 6 | 2 | no | — |
| 98 | C22×C32 | C22×D3 | 1 | 1 | yes | characteristic |
| 99 | C22×C32 | C22×D3 | 1 | 2 | yes | — |
| 100 | C2×C3⋊D3 | — | 2 | 2 | no | — |
| 101 | C2×C3⋊D3 | — | 2 | 2 | no | — |
| 102 | C2×C3⋊D3 | — | 2 | 2 | no | — |
| 103 | C2×C3⋊D3 | — | 6 | 1 | no | — |
| 104 | C2×C3⋊D3 | — | 6 | 2 | no | — |
| 105 | C2×C3⋊D3 | — | 6 | 4 | no | — |
| 106 | C2×C3⋊D3 | — | 6 | 2 | no | — |
| 107 | C2×C3⋊D3 | — | 6 | 2 | no | — |
| 108 | C2×C3⋊D3 | — | 6 | 2 | no | — |
| 109 | C2×C3⋊D3 | — | 6 | 2 | no | — |
| 110 | C2×C3⋊D3 | — | 12 | 2 | no | — |
| 111 | C2×C3⋊D3 | — | 12 | 2 | no | — |
| 112 | C2×C3⋊D3 | — | 12 | 2 | no | — |
| 113 | C2×C3⋊D3 | — | 12 | 4 | no | — |
| 114 | C2×C3⋊D3 | — | 12 | 2 | no | — |
| 115 | C2×C3×D3 | — | 2 | 2 | no | — |
| 116 | C2×C3×D3 | — | 2 | 2 | no | — |
| 117 | C2×C3×D3 | — | 2 | 2 | no | — |
| 118 | C2×C3×D3 | — | 2 | 2 | no | — |
| 119 | C2×C3×D3 | — | 4 | 2 | no | — |
| 120 | C2×C3×D3 | — | 6 | 4 | no | — |
| 121 | C2×C3×D3 | — | 6 | 2 | no | — |
| 122 | C2×C3×D3 | — | 6 | 2 | no | — |
| 123 | C2×C3×D3 | — | 6 | 2 | no | — |
| 124 | C2×C3×D3 | — | 6 | 2 | no | — |
| 125 | C2×C3×D3 | — | 6 | 2 | no | — |
| 126 | C2×C3×D3 | — | 6 | 2 | no | — |
| 127 | C2×C3×D3 | — | 12 | 2 | no | — |
| 128 | C2×C3×D3 | — | 12 | 2 | no | — |
| 129 | C32×C4 | — | 3 | 4 | no | — |
| 130 | C3⋊Dic3 | D12 | 1 | 2 | yes | complemented |
| 131 | C3×Dic3 | — | 2 | 4 | no | — |
| 132 | C3×Dic3 | — | 3 | 4 | no | — |
| 133 | C3×Dic3 | — | 6 | 2 | no | — |
| 134 | C3×Dic3 | D12 | 1 | 4 | yes | complemented |
| 135 | C3×Dic3 | #67 | 1 | 4 | yes | complemented |
| 136 | #127 | — | 27 | 1 | no | Sylow |
| 137 | C33 | #127 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 138 | C23×C3 | — | 3 | 2 | no | — |
| 139 | C23×C3 | — | 6 | 1 | no | — |
| 140 | C23×C3 | — | 9 | 2 | no | — |
| 141 | C22×D3 | — | 3 | 2 | no | — |
| 142 | C22×D3 | — | 3 | 2 | no | — |
| 143 | C22×D3 | — | 6 | 2 | no | — |
| 144 | C22×D3 | — | 9 | 1 | no | — |
| 145 | C22×D3 | — | 9 | 2 | no | — |
| 146 | C22×D3 | — | 18 | 2 | no | — |
| 147 | C22×D3 | — | 18 | 1 | no | — |
| 148 | C22×D3 | — | 36 | 1 | no | — |
| 149 | C22×D3 | D32 | 1 | 1 | yes | characteristic |
| 150 | C3×D4 | — | 18 | 4 | no | — |
| 151 | C2×C3×C4 | — | 3 | 2 | no | — |
| 152 | C2×C3×C4 | — | 3 | 2 | no | — |
| 153 | C2×C3×C4 | — | 6 | 2 | no | — |
| 154 | C2×C3×C4 | — | 9 | 1 | no | — |
| 155 | #67 | — | 6 | 4 | no | — |
| 156 | #67 | — | 18 | 4 | no | — |
| 157 | #67 | — | 18 | 4 | no | — |
| 158 | #67 | — | 18 | 4 | no | — |
| 159 | #67 | — | 18 | 4 | no | — |
| 160 | C2×Dic3 | — | 3 | 2 | no | — |
| 161 | C2×Dic3 | — | 6 | 1 | no | — |
| 162 | C2×Dic3 | D32 | 1 | 2 | yes | — |
| 163 | D12 | — | 6 | 4 | no | — |
| 164 | D12 | — | 18 | 4 | no | — |
| 165 | D12 | — | 18 | 4 | no | — |
| 166 | D12 | — | 18 | 8 | no | — |
| 167 | D12 | — | 18 | 2 | no | — |
| 168 | D12 | — | 18 | 2 | no | — |
| 169 | C2×C32 | — | 2 | 2 | no | — |
| 170 | C2×C32 | — | 2 | 2 | no | — |
| 171 | C2×C32 | — | 2 | 2 | no | — |
| 172 | C2×C32 | — | 2 | 2 | no | — |
| 173 | C2×C32 | — | 2 | 1 | no | — |
| 174 | C2×C32 | — | 4 | 2 | no | — |
| 175 | C2×C32 | — | 4 | 1 | no | — |
| 176 | C2×C32 | — | 12 | 1 | no | — |
| 177 | C2×C32 | C2×D12 | 1 | 1 | yes | characteristic |
| 178 | C2×C32 | C2×D12 | 1 | 2 | yes | — |
| 179 | C2×C32 | D3×D4 | 1 | 2 | yes | — |
| 180 | C2×C32 | D3×D4 | 1 | 2 | yes | — |
| 181 | C2×C32 | C2×#67 | 1 | 2 | yes | — |
| 182 | C3⋊D3 | — | 4 | 2 | no | — |
| 183 | C3⋊D3 | — | 12 | 4 | no | — |
| 184 | C3⋊D3 | — | 12 | 2 | no | — |
| 185 | C3⋊D3 | — | 12 | 4 | no | — |
| 186 | C3⋊D3 | — | 12 | 1 | no | — |
| 187 | C3⋊D3 | — | 12 | 2 | no | — |
| 188 | C3×D3 | — | 4 | 2 | no | — |
| 189 | C3×D3 | — | 4 | 2 | no | — |
| 190 | C3×D3 | — | 12 | 4 | no | — |
| 191 | C3×D3 | — | 12 | 2 | no | — |
| 192 | C3×D3 | — | 12 | 2 | no | — |
| 193 | C2×D4 | — | 9 | 2 | no | — |
| 194 | C2×D4 | — | 27 | 2 | no | — |
| 195 | C2×D4 | — | 27 | 1 | no | — |
| 196 | C2×D4 | — | 27 | 1 | no | — |
| 197 | C42 | — | 9 | 1 | no | — |
| 198 | C22×C3 | — | 2 | 1 | no | — |
| 199 | C22×C3 | — | 2 | 2 | no | — |
| 200 | C22×C3 | — | 4 | 1 | no | — |
| 201 | C22×C3 | — | 6 | 2 | no | — |
| 202 | C22×C3 | — | 6 | 2 | no | — |
| 203 | C22×C3 | — | 6 | 2 | no | — |
| 204 | C22×C3 | — | 6 | 2 | no | — |
| 205 | C22×C3 | — | 12 | 2 | no | — |
| 206 | C22×C3 | — | 18 | 2 | no | — |
| 207 | C22×C3 | — | 18 | 2 | no | — |
| 208 | C22×C3 | — | 18 | 2 | no | — |
| 209 | C22×C3 | C2×D32 | 1 | 2 | yes | — |
| 210 | C22×C3 | C2×D32 | 1 | 1 | yes | characteristic |
| 211 | C2×D3 | — | 2 | 2 | no | — |
| 212 | C2×D3 | — | 2 | 1 | no | — |
| 213 | C2×D3 | — | 6 | 4 | no | — |
| 214 | C2×D3 | — | 6 | 2 | no | — |
| 215 | C2×D3 | — | 6 | 2 | no | — |
| 216 | C2×D3 | — | 6 | 2 | no | — |
| 217 | C2×D3 | — | 6 | 2 | no | — |
| 218 | C2×D3 | — | 6 | 2 | no | — |
| 219 | C2×D3 | — | 6 | 2 | no | — |
| 220 | C2×D3 | — | 12 | 2 | no | — |
| 221 | C2×D3 | — | 12 | 2 | no | — |
| 222 | C2×D3 | — | 18 | 4 | no | — |
| 223 | C2×D3 | — | 18 | 2 | no | — |
| 224 | C2×D3 | — | 18 | 2 | no | — |
| 225 | C2×D3 | — | 18 | 2 | no | — |
| 226 | C2×D3 | — | 18 | 2 | no | — |
| 227 | C2×D3 | — | 18 | 2 | no | — |
| 228 | C2×D3 | — | 18 | 1 | no | — |
| 229 | C2×D3 | — | 36 | 2 | no | — |
| 230 | C2×D3 | — | 36 | 2 | no | — |
| 231 | C2×D3 | — | 36 | 2 | no | — |
| 232 | C2×D3 | — | 36 | 4 | no | — |
| 233 | C2×D3 | — | 36 | 2 | no | — |
| 234 | C3×C4 | — | 3 | 4 | no | — |
| 235 | C3×C4 | — | 3 | 4 | no | — |
| 236 | C3×C4 | — | 6 | 4 | no | — |
| 237 | C3×C4 | — | 9 | 2 | no | — |
| 238 | Dic3 | — | 3 | 4 | no | — |
| 239 | Dic3 | — | 6 | 2 | no | — |
| 240 | Dic3 | #626 | 1 | 4 | yes | complemented |
| 241 | C32 | — | 2 | 2 | no | — |
| 242 | C32 | — | 2 | 1 | no | — |
| 243 | C32 | — | 4 | 1 | no | — |
| 244 | C32 | #874 | 1 | 1 | yes | characteristic, complemented |
| 245 | C32 | #940 | 1 | 2 | yes | complemented |
| 246 | C23 | — | 3 | 1 | no | — |
| 247 | C23 | — | 9 | 2 | no | — |
| 248 | C23 | — | 27 | 1 | no | — |
| 249 | D4 | — | 18 | 4 | no | — |
| 250 | D4 | — | 54 | 4 | no | — |
| 251 | D4 | — | 54 | 2 | no | — |
| 252 | D4 | — | 54 | 2 | no | — |
| 253 | C2×C4 | — | 3 | 2 | no | — |
| 254 | C2×C4 | — | 9 | 1 | no | — |
| 255 | C2×C3 | — | 2 | 2 | no | — |
| 256 | C2×C3 | — | 2 | 2 | no | — |
| 257 | C2×C3 | — | 2 | 2 | no | — |
| 258 | C2×C3 | — | 2 | 1 | no | — |
| 259 | C2×C3 | — | 2 | 2 | no | — |
| 260 | C2×C3 | — | 4 | 2 | no | — |
| 261 | C2×C3 | — | 4 | 1 | no | — |
| 262 | C2×C3 | — | 12 | 2 | no | — |
| 263 | C2×C3 | — | 12 | 2 | no | — |
| 264 | C2×C3 | — | 36 | 2 | no | — |
| 265 | C2×C3 | D3×D12 | 1 | 2 | yes | — |
| 266 | C2×C3 | D3×D12 | 1 | 2 | yes | — |
| 267 | C2×C3 | C2×#626 | 1 | 2 | yes | — |
| 268 | C2×C3 | D3×#67 | 1 | 2 | yes | — |
| 269 | C2×C3 | #3811 | 1 | 1 | yes | characteristic |
| 270 | D3 | — | 4 | 1 | no | — |
| 271 | D3 | — | 12 | 4 | no | — |
| 272 | D3 | — | 12 | 2 | no | — |
| 273 | D3 | — | 12 | 2 | no | — |
| 274 | D3 | — | 36 | 4 | no | — |
| 275 | D3 | — | 36 | 2 | no | — |
| 276 | D3 | — | 36 | 4 | no | — |
| 277 | D3 | — | 36 | 2 | no | — |
| 278 | D3 | — | 36 | 1 | no | — |
| 279 | C22 | — | 6 | 2 | no | — |
| 280 | C22 | — | 6 | 1 | no | — |
| 281 | C22 | — | 18 | 2 | no | — |
| 282 | C22 | — | 18 | 2 | no | — |
| 283 | C22 | — | 18 | 2 | no | — |
| 284 | C22 | — | 54 | 2 | no | — |
| 285 | C22 | — | 54 | 1 | no | — |
| 286 | C22 | D33 | 1 | 1 | yes | characteristic |
| 287 | C4 | — | 3 | 4 | no | — |
| 288 | C4 | — | 9 | 2 | no | — |
| 289 | C3 | — | 2 | 2 | no | — |
| 290 | C3 | — | 2 | 1 | no | — |
| 291 | C3 | — | 4 | 1 | no | — |
| 292 | C3 | #63911 | 1 | 2 | yes | complemented |
| 293 | C3 | #63979 | 1 | 1 | yes | characteristic, complemented |
| 294 | C2 | — | 12 | 1 | no | — |
| 295 | C2 | — | 36 | 2 | no | — |
| 296 | C2 | — | 108 | 1 | no | — |
| 297 | C2 | D3×#626 | 1 | 2 | yes | — |
| 298 | C2 | #88955 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C22 | D33 |
| commutator, Fitting, socle, cosocle | C22×C33 | C23 |
| hypocenter | C33 | #127 |
| LCS | G ▷ C22×C33 ▷ C33 | |
| UCS | C1 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#6295 | C2 | semidirect |
| 2. C2×#6292 | C2 | semidirect |
| 3. C2×#6291 | C2 | semidirect |
| 4. C2×C3×#626 | C2 | semidirect |
| 5. C3×Dic32 | C2 | semidirect |
| 27. C2×#626 | D3 | semidirect |
| 30. Dic32 | D3 | semidirect |
| 39. C3×C3⋊Dic3 | D4 | semidirect |
| 40. C32×Dic3 | D4 | semidirect |
| 130. C3⋊Dic3 | D12 | semidirect |
| 134. C3×Dic3 | D12 | semidirect |
| 135. C3×Dic3 | #67 | semidirect |
| 137. C33 | #127 | semidirect |
| 240. Dic3 | #626 | semidirect |
| 244. C32 | #874 | semidirect |
| 245. C32 | #940 | semidirect |
| 292. C3 | #63911 | semidirect |
| 293. C3 | #63979 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#6295 | 2 |
| C2×#6292 | 2 |
| C2×#6291 | 2 |
| C2×C3×#626 | 2 |
| C3×Dic32 | 2 |
| #63979 | 3 |
| #63911 | 3 |