Finite Group DatabasebetaFinite Group DatabasebetaOrder 1296.
548 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 548 conjugacy classes of subgroups — too many to draw. Only its 53 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×((C3⋊D3)⋊C3)⋊D3 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #10619902 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #10619890 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C3×#6198 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #6292⋊C3 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #88955 | — | 3 | 2 | no | maximal |
| 7 | #88655 | — | 3 | 1 | no | maximal |
| 8 | C2×C3⋊#272 | C22 | 1 | 1 | yes | characteristic |
| 9 | C2×C3⋊((C3⋊D3)⋊C3) | C22 | 1 | 2 | yes | complemented |
| 10 | C2×C3×(C3⋊D3)⋊C3 | C22 | 1 | 2 | yes | complemented |
| 11 | ((C3⋊D3)⋊C3)⋊D3 | — | 2 | 4 | no | — |
| 12 | C3×#1084 | C22 | 1 | 1 | yes | characteristic, complemented |
| 13 | Dic3×H(3,𝔽3) | C22 | 1 | 1 | yes | characteristic, complemented |
| 14 | C2×#1113 | — | 3 | 2 | no | — |
| 15 | C2×D3×C3⋊D3 | — | 3 | 2 | no | — |
| 16 | #6292 | — | 3 | 2 | no | — |
| 17 | #6292 | D3 | 1 | 2 | yes | complemented |
| 18 | C3×#626 | — | 3 | 2 | no | — |
| 19 | C3×#626 | — | 3 | 2 | no | — |
| 20 | C3×#624 | — | 3 | 2 | no | — |
| 21 | C3×#624 | — | 6 | 1 | no | — |
| 22 | C2×((C3⋊D3)⋊C3)⋊C2 | — | 3 | 2 | no | — |
| 23 | C2×((C3⋊D3)⋊C3)⋊C2 | — | 6 | 2 | no | — |
| 24 | C4×#272 | — | 3 | 1 | no | — |
| 25 | C4⋊((C3⋊D3)⋊C3) | — | 3 | 2 | no | — |
| 26 | #6198 | — | 3 | 1 | no | — |
| 27 | #6198 | D3 | 1 | 1 | yes | characteristic, complemented |
| 28 | C2×C3×H(3,𝔽3) | C23 | 1 | 1 | yes | characteristic |
| 29 | C3⋊#272 | D4 | 1 | 2 | yes | complemented |
| 30 | C3⋊((C3⋊D3)⋊C3) | — | 2 | 2 | no | — |
| 31 | C3×(C3⋊D3)⋊C3 | — | 2 | 2 | no | — |
| 32 | #3811 | — | 9 | 2 | no | — |
| 33 | D3×D12 | — | 9 | 1 | no | — |
| 34 | D3×D12 | — | 9 | 2 | no | — |
| 35 | C2×C3⋊(C3⋊D3) | C2×D3 | 1 | 2 | yes | complemented |
| 36 | C2×C3×C3⋊D3 | — | 3 | 2 | no | — |
| 37 | C2×C3×C3⋊D3 | — | 3 | 2 | no | — |
| 38 | C2×C3×C3⋊D3 | — | 6 | 1 | no | — |
| 39 | C2×C3×C3⋊D3 | C2×D3 | 1 | 2 | yes | complemented |
| 40 | C2×C32×D3 | — | 3 | 2 | no | — |
| 41 | #1113 | — | 6 | 4 | no | — |
| 42 | D3×C3⋊D3 | — | 6 | 4 | no | — |
| 43 | C32×Dic3 | — | 2 | 1 | no | — |
| 44 | C32×Dic3 | — | 3 | 2 | no | — |
| 45 | C32×Dic3 | — | 6 | 1 | no | — |
| 46 | C32×Dic3 | C2×D3 | 1 | 2 | yes | complemented |
| 47 | C2×#272 | — | 3 | 1 | no | — |
| 48 | C2×#272 | — | 6 | 2 | no | — |
| 49 | C2×#272 | — | 12 | 1 | no | — |
| 50 | C2×(C3⋊D3)⋊C3 | — | 2 | 2 | no | — |
| 51 | C2×(C3⋊D3)⋊C3 | — | 3 | 2 | no | — |
| 52 | C2×(C3⋊D3)⋊C3 | — | 6 | 2 | no | — |
| 53 | C2×(C3⋊D3)⋊C3 | C2×D3 | 1 | 2 | yes | complemented |
| 54 | ((C3⋊D3)⋊C3)⋊C2 | — | 6 | 4 | no | — |
| 55 | ((C3⋊D3)⋊C3)⋊C2 | — | 6 | 8 | no | — |
| 56 | C4×H(3,𝔽3) | — | 3 | 1 | no | — |
| 57 | #1084 | C2×D3 | 1 | 1 | yes | characteristic, complemented |
| 58 | C3×H(3,𝔽3) | C2×D4 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 59 | C22×C3⋊D3 | — | 9 | 2 | no | — |
| 60 | C2×D32 | — | 9 | 2 | no | — |
| 61 | C2×D32 | — | 9 | 2 | no | — |
| 62 | C2×D32 | — | 9 | 2 | no | — |
| 63 | C2×D32 | — | 9 | 2 | no | — |
| 64 | C2×D32 | — | 18 | 2 | no | — |
| 65 | C4⋊(C3⋊D3) | — | 3 | 2 | no | — |
| 66 | C4⋊(C3⋊D3) | — | 9 | 1 | no | — |
| 67 | C3×#67 | — | 9 | 2 | no | — |
| 68 | C3×#67 | — | 9 | 2 | no | — |
| 69 | C3×D12 | — | 9 | 2 | no | — |
| 70 | C3×D12 | — | 9 | 1 | no | — |
| 71 | C3×C4×D3 | — | 9 | 2 | no | — |
| 72 | C3×C4×D3 | — | 9 | 1 | no | — |
| 73 | C3×C4×D3 | — | 18 | 1 | no | — |
| 74 | #626 | — | 3 | 2 | no | — |
| 75 | #626 | — | 3 | 2 | no | — |
| 76 | #626 | — | 9 | 2 | no | — |
| 77 | #626 | — | 9 | 2 | no | — |
| 78 | #626 | — | 9 | 2 | no | — |
| 79 | #626 | — | 18 | 2 | no | — |
| 80 | #624 | — | 3 | 2 | no | — |
| 81 | #624 | — | 6 | 1 | no | — |
| 82 | C2×C33 | — | 2 | 1 | no | — |
| 83 | C2×C33 | C22×D3 | 1 | 2 | yes | — |
| 84 | C3⋊(C3⋊D3) | — | 2 | 2 | no | — |
| 85 | C3×C3⋊D3 | — | 2 | 2 | no | — |
| 86 | C3×C3⋊D3 | — | 3 | 4 | no | — |
| 87 | C3×C3⋊D3 | — | 6 | 2 | no | — |
| 88 | C3×C3⋊D3 | — | 6 | 2 | no | — |
| 89 | C32×D3 | — | 6 | 2 | no | — |
| 90 | C2×H(3,𝔽3) | — | 2 | 2 | no | — |
| 91 | C2×H(3,𝔽3) | — | 4 | 1 | no | — |
| 92 | C2×H(3,𝔽3) | C22×D3 | 1 | 1 | yes | characteristic |
| 93 | #272 | — | 3 | 2 | no | — |
| 94 | #272 | — | 6 | 4 | no | — |
| 95 | #272 | — | 12 | 2 | no | — |
| 96 | (C3⋊D3)⋊C3 | — | 2 | 4 | no | — |
| 97 | (C3⋊D3)⋊C3 | — | 2 | 2 | no | — |
| 98 | (C3⋊D3)⋊C3 | — | 6 | 4 | no | — |
| 99 | (C3⋊D3)⋊C3 | — | 6 | 2 | no | — |
| 100 | D3×D4 | — | 27 | 1 | no | — |
| 101 | D3×D4 | — | 27 | 2 | no | — |
| 102 | C2×D12 | — | 27 | 1 | no | — |
| 103 | C22×C32 | — | 9 | 2 | no | — |
| 104 | C2×C3⋊D3 | — | 3 | 2 | no | — |
| 105 | C2×C3⋊D3 | — | 3 | 2 | no | — |
| 106 | C2×C3⋊D3 | — | 3 | 2 | no | — |
| 107 | C2×C3⋊D3 | — | 6 | 1 | no | — |
| 108 | C2×C3⋊D3 | — | 6 | 2 | no | — |
| 109 | C2×C3⋊D3 | — | 9 | 2 | no | — |
| 110 | C2×C3⋊D3 | — | 18 | 2 | no | — |
| 111 | C2×C3⋊D3 | — | 18 | 4 | no | — |
| 112 | C2×C3⋊D3 | D32 | 1 | 2 | yes | complemented |
| 113 | C2×C3×D3 | — | 3 | 2 | no | — |
| 114 | C2×C3×D3 | — | 3 | 2 | no | — |
| 115 | C2×C3×D3 | — | 6 | 2 | no | — |
| 116 | C2×C3×D3 | — | 9 | 2 | no | — |
| 117 | C2×C3×D3 | — | 9 | 2 | no | — |
| 118 | C2×C3×D3 | — | 9 | 1 | no | — |
| 119 | C2×C3×D3 | — | 9 | 2 | no | — |
| 120 | C2×C3×D3 | — | 9 | 2 | no | — |
| 121 | C2×C3×D3 | — | 18 | 1 | no | — |
| 122 | C2×C3×D3 | — | 18 | 2 | no | — |
| 123 | C2×C3×D3 | — | 18 | 2 | no | — |
| 124 | C2×C3×D3 | — | 36 | 1 | no | — |
| 125 | D32 | — | 18 | 4 | no | — |
| 126 | D32 | — | 18 | 4 | no | — |
| 127 | D32 | — | 18 | 8 | no | — |
| 128 | D32 | — | 18 | 4 | no | — |
| 129 | D32 | — | 18 | 4 | no | — |
| 130 | C32×C4 | — | 3 | 2 | no | — |
| 131 | C32×C4 | — | 6 | 1 | no | — |
| 132 | C32×C4 | — | 9 | 1 | no | — |
| 133 | C3×Dic3 | — | 3 | 2 | no | — |
| 134 | C3×Dic3 | — | 3 | 2 | no | — |
| 135 | C3×Dic3 | — | 3 | 2 | no | — |
| 136 | C3×Dic3 | — | 6 | 2 | no | — |
| 137 | C3×Dic3 | — | 6 | 1 | no | — |
| 138 | C3×Dic3 | — | 6 | 1 | no | — |
| 139 | C3×Dic3 | — | 6 | 1 | no | — |
| 140 | C3×Dic3 | — | 12 | 1 | no | — |
| 141 | C3×Dic3 | D32 | 1 | 1 | yes | characteristic |
| 142 | C33 | — | 2 | 1 | no | — |
| 143 | C33 | D3×D4 | 1 | 2 | yes | complemented |
| 144 | H(3,𝔽3) | — | 2 | 2 | no | — |
| 145 | H(3,𝔽3) | — | 4 | 1 | no | — |
| 146 | H(3,𝔽3) | D3×D4 | 1 | 1 | yes | characteristic, complemented |
| 147 | C22×D3 | — | 27 | 2 | no | — |
| 148 | C22×D3 | — | 27 | 2 | no | — |
| 149 | C22×D3 | — | 27 | 2 | no | — |
| 150 | C22×D3 | — | 27 | 2 | no | — |
| 151 | C22×D3 | — | 54 | 2 | no | — |
| 152 | C3×D4 | — | 27 | 1 | no | — |
| 153 | C3×D4 | — | 27 | 2 | no | — |
| 154 | C2×C3×C4 | — | 27 | 1 | no | — |
| 155 | #67 | — | 9 | 2 | no | — |
| 156 | #67 | — | 9 | 2 | no | — |
| 157 | #67 | — | 27 | 2 | no | — |
| 158 | D12 | — | 9 | 2 | no | — |
| 159 | D12 | — | 9 | 1 | no | — |
| 160 | D12 | — | 27 | 2 | no | — |
| 161 | D12 | — | 27 | 1 | no | — |
| 162 | D12 | — | 27 | 1 | no | — |
| 163 | D12 | — | 54 | 1 | no | — |
| 164 | C4×D3 | — | 9 | 2 | no | — |
| 165 | C4×D3 | — | 9 | 1 | no | — |
| 166 | C4×D3 | — | 18 | 1 | no | — |
| 167 | C2×C32 | — | 2 | 2 | no | — |
| 168 | C2×C32 | — | 2 | 1 | no | — |
| 169 | C2×C32 | — | 3 | 2 | no | — |
| 170 | C2×C32 | — | 4 | 1 | no | — |
| 171 | C2×C32 | — | 6 | 2 | no | — |
| 172 | C2×C32 | — | 6 | 1 | no | — |
| 173 | C2×C32 | — | 12 | 1 | no | — |
| 174 | C2×C32 | — | 18 | 2 | no | — |
| 175 | C2×C32 | C2×D32 | 1 | 1 | yes | characteristic |
| 176 | C2×C32 | C2×D32 | 1 | 2 | yes | — |
| 177 | C3⋊D3 | — | 2 | 2 | no | — |
| 178 | C3⋊D3 | — | 3 | 4 | no | — |
| 179 | C3⋊D3 | — | 6 | 2 | no | — |
| 180 | C3⋊D3 | — | 6 | 2 | no | — |
| 181 | C3⋊D3 | — | 6 | 4 | no | — |
| 182 | C3⋊D3 | — | 6 | 2 | no | — |
| 183 | C3⋊D3 | — | 18 | 2 | no | — |
| 184 | C3⋊D3 | — | 36 | 2 | no | — |
| 185 | C3×D3 | — | 6 | 4 | no | — |
| 186 | C3×D3 | — | 6 | 2 | no | — |
| 187 | C3×D3 | — | 6 | 2 | no | — |
| 188 | C3×D3 | — | 9 | 2 | no | — |
| 189 | C3×D3 | — | 9 | 4 | no | — |
| 190 | C3×D3 | — | 18 | 2 | no | — |
| 191 | C3×D3 | — | 18 | 2 | no | — |
| 192 | C3×D3 | — | 18 | 4 | no | — |
| 193 | C3×D3 | — | 18 | 2 | no | — |
| 194 | C3×D3 | — | 18 | 2 | no | — |
| 195 | C3×D3 | — | 36 | 2 | no | — |
| 196 | C3×D3 | — | 36 | 2 | no | — |
| 197 | C2×D4 | — | 81 | 1 | no | Sylow |
| 198 | C22×C3 | — | 9 | 2 | no | — |
| 199 | C22×C3 | — | 9 | 2 | no | — |
| 200 | C22×C3 | — | 18 | 2 | no | — |
| 201 | C22×C3 | — | 27 | 2 | no | — |
| 202 | C22×C3 | — | 27 | 1 | no | — |
| 203 | C2×D3 | — | 3 | 2 | no | — |
| 204 | C2×D3 | — | 9 | 2 | no | — |
| 205 | C2×D3 | — | 9 | 2 | no | — |
| 206 | C2×D3 | — | 9 | 1 | no | — |
| 207 | C2×D3 | — | 9 | 2 | no | — |
| 208 | C2×D3 | — | 9 | 2 | no | — |
| 209 | C2×D3 | — | 18 | 1 | no | — |
| 210 | C2×D3 | — | 18 | 2 | no | — |
| 211 | C2×D3 | — | 18 | 2 | no | — |
| 212 | C2×D3 | — | 27 | 2 | no | — |
| 213 | C2×D3 | — | 36 | 1 | no | — |
| 214 | C2×D3 | — | 54 | 2 | no | — |
| 215 | C2×D3 | — | 54 | 4 | no | — |
| 216 | C2×D3 | — | 54 | 4 | no | — |
| 217 | C2×D3 | — | 54 | 4 | no | — |
| 218 | C2×D3 | — | 54 | 4 | no | — |
| 219 | C2×D3 | — | 54 | 8 | no | — |
| 220 | C3×C4 | — | 3 | 1 | no | — |
| 221 | C3×C4 | — | 9 | 2 | no | — |
| 222 | C3×C4 | — | 9 | 1 | no | — |
| 223 | C3×C4 | — | 9 | 1 | no | — |
| 224 | C3×C4 | — | 18 | 1 | no | — |
| 225 | C3×C4 | — | 18 | 1 | no | — |
| 226 | Dic3 | — | 3 | 2 | no | — |
| 227 | Dic3 | — | 6 | 1 | no | — |
| 228 | Dic3 | ((C3⋊D3)⋊C3)⋊C2 | 1 | 1 | yes | characteristic, complemented |
| 229 | C32 | — | 2 | 2 | no | — |
| 230 | C32 | — | 2 | 1 | no | — |
| 231 | C32 | — | 3 | 2 | no | — |
| 232 | C32 | — | 4 | 1 | no | — |
| 233 | C32 | — | 6 | 2 | no | — |
| 234 | C32 | — | 6 | 1 | no | — |
| 235 | C32 | — | 12 | 1 | no | — |
| 236 | C32 | #3802 | 1 | 1 | yes | characteristic |
| 237 | C32 | #3811 | 1 | 2 | yes | complemented |
| 238 | C23 | — | 81 | 2 | no | — |
| 239 | D4 | — | 27 | 2 | no | — |
| 240 | D4 | — | 27 | 1 | no | — |
| 241 | D4 | — | 81 | 1 | no | — |
| 242 | C2×C4 | — | 27 | 1 | no | — |
| 243 | C2×C3 | — | 2 | 1 | no | — |
| 244 | C2×C3 | — | 3 | 2 | no | — |
| 245 | C2×C3 | — | 6 | 1 | no | — |
| 246 | C2×C3 | — | 6 | 2 | no | — |
| 247 | C2×C3 | — | 12 | 1 | no | — |
| 248 | C2×C3 | — | 18 | 2 | no | — |
| 249 | C2×C3 | — | 18 | 4 | no | — |
| 250 | C2×C3 | — | 18 | 2 | no | — |
| 251 | C2×C3 | — | 27 | 2 | no | — |
| 252 | C2×C3 | — | 54 | 2 | no | — |
| 253 | C2×C3 | C2×((C3⋊D3)⋊C3)⋊C2 | 1 | 1 | yes | characteristic |
| 254 | C2×C3 | D33 | 1 | 1 | yes | characteristic |
| 255 | D3 | — | 6 | 2 | no | — |
| 256 | D3 | — | 9 | 4 | no | — |
| 257 | D3 | — | 9 | 2 | no | — |
| 258 | D3 | — | 18 | 2 | no | — |
| 259 | D3 | — | 18 | 4 | no | — |
| 260 | D3 | — | 18 | 2 | no | — |
| 261 | D3 | — | 18 | 2 | no | — |
| 262 | D3 | — | 18 | 2 | no | — |
| 263 | D3 | — | 36 | 2 | no | — |
| 264 | D3 | — | 36 | 2 | no | — |
| 265 | D3 | — | 54 | 2 | no | — |
| 266 | D3 | — | 54 | 4 | no | — |
| 267 | C22 | — | 9 | 2 | no | — |
| 268 | C22 | — | 27 | 1 | no | — |
| 269 | C22 | — | 27 | 2 | no | — |
| 270 | C22 | — | 162 | 4 | no | — |
| 271 | C4 | — | 3 | 1 | no | — |
| 272 | C4 | — | 9 | 1 | no | — |
| 273 | C3 | — | 2 | 1 | no | — |
| 274 | C3 | — | 3 | 2 | no | — |
| 275 | C3 | — | 6 | 2 | no | — |
| 276 | C3 | — | 6 | 1 | no | — |
| 277 | C3 | — | 12 | 1 | no | — |
| 278 | C3 | #88655 | 1 | 1 | yes | characteristic, complemented |
| 279 | C3 | #88954 | 1 | 1 | yes | characteristic |
| 280 | C2 | — | 18 | 2 | no | — |
| 281 | C2 | — | 27 | 2 | no | — |
| 282 | C2 | — | 54 | 2 | no | — |
| 283 | C2 | D3×((C3⋊D3)⋊C3)⋊C2 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D3×((C3⋊D3)⋊C3)⋊C2 |
| commutator, Fitting, cosocle | C2×C3×H(3,𝔽3) | C23 |
| Frattini | C2×C3 | D33 |
| socle | C2×C32 | C2×D32 |
| hypocenter | C3×H(3,𝔽3) | C2×D4 |
| LCS | G ▷ C2×C3×H(3,𝔽3) ▷ C3×H(3,𝔽3) | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×((C3⋊D3)⋊C3)⋊D3 | C2 | semidirect |
| 2. #10619902 | C2 | semidirect |
| 3. #10619890 | C2 | semidirect |
| 4. C3×#6198 | C2 | semidirect |
| 5. #6292⋊C3 | C2 | semidirect |
| 9. C2×C3⋊((C3⋊D3)⋊C3) | C22 | semidirect |
| 10. C2×C3×(C3⋊D3)⋊C3 | C22 | semidirect |
| 12. C3×#1084 | C22 | semidirect |
| 13. Dic3×H(3,𝔽3) | C22 | semidirect |
| 17. #6292 | D3 | semidirect |
| 27. #6198 | D3 | semidirect |
| 29. C3⋊#272 | D4 | semidirect |
| 35. C2×C3⋊(C3⋊D3) | C2×D3 | semidirect |
| 39. C2×C3×C3⋊D3 | C2×D3 | semidirect |
| 46. C32×Dic3 | C2×D3 | semidirect |
| 53. C2×(C3⋊D3)⋊C3 | C2×D3 | semidirect |
| 57. #1084 | C2×D3 | semidirect |
| 58. C3×H(3,𝔽3) | C2×D4 | semidirect |
| 112. C2×C3⋊D3 | D32 | semidirect |
| 143. C33 | D3×D4 | semidirect |
| 146. H(3,𝔽3) | D3×D4 | semidirect |
| 228. Dic3 | ((C3⋊D3)⋊C3)⋊C2 | semidirect |
| 237. C32 | #3811 | semidirect |
| 278. C3 | #88655 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×((C3⋊D3)⋊C3)⋊D3 | 2 |
| #10619902 | 2 |
| #10619890 | 2 |
| C3×#6198 | 2 |
| #6292⋊C3 | 2 |
| #88955 | 3 |
| #88655 | 3 |