Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
2268 conjugacy classes of subgroups; only the 217 normal subgroups are shown.
This group has 2268 conjugacy classes of subgroups — too many to draw. Only its 217 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 | C22×C3⋊(C3⋊S4) | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | C32×D4×A4 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #11724473 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | C4⋊(C3⋊(C3⋊S4)) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | C4×C3⋊(C3⋊S4) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | D4×C3⋊S4 | — | 3 | 12 | no | maximal |
| 7 | D4×#638 | — | 3 | 1 | no | maximal |
| 8 | C22×C32×A4 | C22 | 1 | 2 | yes | complemented |
| 9 | C2×C3⋊(C3⋊S4) | — | 2 | 4 | no | — |
| 10 | C2×C3⋊(C3⋊S4) | C22 | 1 | 1 | yes | characteristic |
| 11 | C2×C3⋊(C3⋊S4) | C22 | 1 | 2 | yes | complemented |
| 12 | D4×C3⋊(C3⋊D3) | — | 4 | 1 | no | maximal |
| 13 | C32×C4×A4 | C22 | 1 | 1 | yes | characteristic, complemented |
| 14 | C3⋊(C3⋊SL(2,ℤ/4)) | C22 | 1 | 1 | yes | characteristic, complemented |
| 15 | C22×C3⋊S4 | — | 3 | 24 | no | — |
| 16 | C22×C32×D4 | D3 | 1 | 1 | yes | characteristic, complemented |
| 17 | C22×#638 | — | 3 | 2 | no | — |
| 18 | C2×D4×C3⋊D3 | — | 3 | 1 | no | — |
| 19 | C3×D4×A4 | D3 | 1 | 12 | yes | complemented |
| 20 | #64269 | — | 3 | 24 | no | — |
| 21 | C4⋊(C3⋊S4) | — | 3 | 12 | no | — |
| 22 | C4×C3⋊S4 | — | 3 | 12 | no | — |
| 23 | #64162 | — | 3 | 2 | no | — |
| 24 | #64149 | — | 3 | 1 | no | — |
| 25 | #64148 | — | 3 | 2 | no | — |
| 26 | #64147 | — | 3 | 1 | no | — |
| 27 | #64146 | — | 3 | 2 | no | — |
| 28 | D4×C3⋊Dic3 | — | 3 | 1 | no | — |
| 29 | #64139 | — | 3 | 1 | no | — |
| 30 | C4×#638 | — | 3 | 1 | no | — |
| 31 | C22×C3⋊(C3⋊D3) | — | 4 | 2 | no | — |
| 32 | C2×C32×A4 | — | 2 | 2 | no | — |
| 33 | C2×C32×A4 | C23 | 1 | 1 | yes | characteristic |
| 34 | C3⋊(C3⋊S4) | — | 2 | 2 | no | — |
| 35 | C3⋊(C3⋊S4) | D4 | 1 | 2 | yes | direct product factor |
| 36 | C33×D4 | — | 4 | 1 | no | — |
| 37 | #6312 | — | 4 | 2 | no | — |
| 38 | C4⋊(C3⋊(C3⋊D3)) | — | 4 | 1 | no | — |
| 39 | C4×C3⋊(C3⋊D3) | — | 4 | 1 | no | — |
| 40 | D4×S4 | — | 9 | 9 | no | — |
| 41 | D4×#67 | — | 9 | 4 | no | — |
| 42 | C24×C32 | C2×D3 | 1 | 2 | yes | complemented |
| 43 | C23×C3⋊D3 | — | 3 | 2 | no | — |
| 44 | C22×C3×A4 | C2×D3 | 1 | 24 | yes | complemented |
| 45 | C2×C3⋊S4 | — | 3 | 12 | no | — |
| 46 | C2×C3⋊S4 | — | 3 | 24 | no | — |
| 47 | C2×C3⋊S4 | — | 6 | 48 | no | — |
| 48 | C2×C32×D4 | — | 3 | 1 | no | — |
| 49 | C2×C32×D4 | — | 3 | 2 | no | — |
| 50 | C2×C32×D4 | — | 3 | 1 | no | — |
| 51 | C22×C32×C4 | C2×D3 | 1 | 1 | yes | characteristic, complemented |
| 52 | C2×#638 | — | 3 | 1 | no | — |
| 53 | C2×#638 | — | 3 | 2 | no | — |
| 54 | C2×#638 | — | 3 | 2 | no | — |
| 55 | C2×#638 | — | 3 | 2 | no | — |
| 56 | C2×#638 | — | 3 | 2 | no | — |
| 57 | C2×#638 | — | 3 | 2 | no | — |
| 58 | C2×#638 | — | 6 | 4 | no | — |
| 59 | C2×#638 | — | 6 | 4 | no | — |
| 60 | C22×C3⋊Dic3 | — | 3 | 2 | no | — |
| 61 | D4×C3⋊D3 | — | 6 | 2 | no | — |
| 62 | D4×C3⋊D3 | — | 6 | 1 | no | — |
| 63 | D4×C3⋊D3 | — | 6 | 1 | no | — |
| 64 | D4×C3⋊D3 | — | 12 | 12 | no | — |
| 65 | C2×C4⋊(C3⋊D3) | — | 3 | 1 | no | — |
| 66 | C2×C4×C3⋊D3 | — | 3 | 1 | no | — |
| 67 | C3×C4×A4 | C2×D3 | 1 | 12 | yes | complemented |
| 68 | C3⋊SL(2,ℤ/4) | — | 3 | 12 | no | — |
| 69 | #3757 | — | 3 | 2 | no | — |
| 70 | #3757 | — | 3 | 2 | no | — |
| 71 | #3757 | — | 3 | 1 | no | — |
| 72 | #3752 | — | 3 | 1 | no | — |
| 73 | #3752 | — | 3 | 2 | no | — |
| 74 | C4⋊(C3⋊Dic3) | — | 3 | 1 | no | — |
| 75 | #3750 | — | 3 | 1 | no | — |
| 76 | C4×C3⋊Dic3 | — | 3 | 1 | no | — |
| 77 | C22×C33 | — | 4 | 2 | no | — |
| 78 | C2×C3⋊(C3⋊D3) | — | 4 | 1 | no | — |
| 79 | C2×C3⋊(C3⋊D3) | — | 4 | 2 | no | — |
| 80 | C2×C3⋊(C3⋊D3) | — | 8 | 4 | no | — |
| 81 | C32×A4 | C2×D4 | 1 | 1 | yes | characteristic, complemented |
| 82 | C33×C4 | — | 4 | 1 | no | — |
| 83 | C3⋊(C3⋊Dic3) | — | 4 | 1 | no | — |
| 84 | C22×S4 | — | 9 | 18 | no | — |
| 85 | C22×C3×D4 | C3⋊D3 | 1 | 4 | yes | complemented |
| 86 | C22×#67 | — | 9 | 8 | no | — |
| 87 | C2×D3×D4 | — | 9 | 4 | no | — |
| 88 | D4×A4 | C3⋊D3 | 1 | 9 | yes | complemented |
| 89 | GL(2,ℤ/4) | — | 9 | 18 | no | — |
| 90 | C4⋊S4 | — | 9 | 9 | no | — |
| 91 | C4×S4 | — | 9 | 9 | no | — |
| 92 | #953 | — | 9 | 8 | no | — |
| 93 | #940 | — | 9 | 4 | no | — |
| 94 | #939 | — | 9 | 8 | no | — |
| 95 | #938 | — | 9 | 4 | no | — |
| 96 | #937 | — | 9 | 8 | no | — |
| 97 | D4×Dic3 | — | 9 | 4 | no | — |
| 98 | #930 | — | 9 | 4 | no | — |
| 99 | C4×#67 | — | 9 | 4 | no | — |
| 100 | C23×C32 | — | 2 | 2 | no | — |
| 101 | C23×C32 | — | 3 | 2 | no | — |
| 102 | C23×C32 | — | 3 | 2 | no | — |
| 103 | C23×C32 | — | 6 | 2 | no | — |
| 104 | C23×C32 | C22×D3 | 1 | 1 | yes | characteristic |
| 105 | C22×C3⋊D3 | — | 3 | 1 | no | — |
| 106 | C22×C3⋊D3 | — | 3 | 2 | no | — |
| 107 | C22×C3⋊D3 | — | 6 | 4 | no | — |
| 108 | C22×C3⋊D3 | — | 6 | 2 | no | — |
| 109 | C22×C3⋊D3 | — | 6 | 2 | no | — |
| 110 | C22×C3⋊D3 | — | 12 | 24 | no | — |
| 111 | C22×C3⋊D3 | — | 12 | 2 | no | — |
| 112 | C2×C3×A4 | — | 2 | 24 | no | — |
| 113 | C2×C3×A4 | C22×D3 | 1 | 12 | yes | — |
| 114 | C3⋊S4 | — | 3 | 24 | no | — |
| 115 | C3⋊S4 | — | 6 | 24 | no | — |
| 116 | C32×D4 | — | 3 | 2 | no | — |
| 117 | C32×D4 | — | 3 | 1 | no | — |
| 118 | C32×D4 | — | 4 | 12 | no | — |
| 119 | C32×D4 | — | 6 | 1 | no | — |
| 120 | C32×D4 | S4 | 1 | 1 | yes | characteristic, complemented |
| 121 | C2×C32×C4 | — | 3 | 1 | no | — |
| 122 | C2×C32×C4 | — | 3 | 1 | no | — |
| 123 | #638 | — | 3 | 2 | no | — |
| 124 | #638 | — | 3 | 2 | no | — |
| 125 | #638 | — | 6 | 4 | no | — |
| 126 | #638 | — | 6 | 4 | no | — |
| 127 | #638 | — | 6 | 2 | no | — |
| 128 | #638 | — | 6 | 2 | no | — |
| 129 | #638 | — | 6 | 2 | no | — |
| 130 | #638 | — | 6 | 2 | no | — |
| 131 | #638 | — | 12 | 24 | no | — |
| 132 | #638 | — | 12 | 2 | no | — |
| 133 | C2×C3⋊Dic3 | — | 3 | 1 | no | — |
| 134 | C2×C3⋊Dic3 | — | 3 | 1 | no | — |
| 135 | C2×C3⋊Dic3 | — | 3 | 1 | no | — |
| 136 | C2×C3⋊Dic3 | — | 3 | 2 | no | — |
| 137 | C2×C3⋊Dic3 | — | 6 | 4 | no | — |
| 138 | C4⋊(C3⋊D3) | — | 6 | 1 | no | — |
| 139 | C4⋊(C3⋊D3) | — | 6 | 1 | no | — |
| 140 | C4⋊(C3⋊D3) | — | 12 | 12 | no | — |
| 141 | C4×C3⋊D3 | — | 6 | 1 | no | — |
| 142 | C4×C3⋊D3 | — | 6 | 1 | no | — |
| 143 | C4×C3⋊D3 | — | 12 | 12 | no | — |
| 144 | D42 | — | 27 | 1 | no | Sylow |
| 145 | C2×C33 | — | 4 | 1 | no | — |
| 146 | C2×C33 | — | 8 | 2 | no | — |
| 147 | C3⋊(C3⋊D3) | — | 4 | 2 | no | — |
| 148 | C3⋊(C3⋊D3) | — | 8 | 2 | no | — |
| 149 | C24×C3 | C2×C3⋊D3 | 1 | 8 | yes | complemented |
| 150 | C23×D3 | — | 9 | 8 | no | — |
| 151 | C22×A4 | C2×C3⋊D3 | 1 | 18 | yes | complemented |
| 152 | C2×S4 | — | 9 | 9 | no | — |
| 153 | C2×S4 | — | 9 | 18 | no | — |
| 154 | C2×S4 | — | 18 | 36 | no | — |
| 155 | C2×C3×D4 | — | 3 | 4 | no | — |
| 156 | C2×C3×D4 | — | 3 | 8 | no | — |
| 157 | C2×C3×D4 | — | 3 | 4 | no | — |
| 158 | C22×C3×C4 | C2×C3⋊D3 | 1 | 4 | yes | complemented |
| 159 | C2×#67 | — | 9 | 4 | no | — |
| 160 | C2×#67 | — | 9 | 8 | no | — |
| 161 | C2×#67 | — | 9 | 8 | no | — |
| 162 | C2×#67 | — | 9 | 8 | no | — |
| 163 | C2×#67 | — | 9 | 8 | no | — |
| 164 | C2×#67 | — | 9 | 8 | no | — |
| 165 | C2×#67 | — | 18 | 16 | no | — |
| 166 | C2×#67 | — | 18 | 16 | no | — |
| 167 | C22×Dic3 | — | 9 | 8 | no | — |
| 168 | D3×D4 | — | 18 | 8 | no | — |
| 169 | D3×D4 | — | 18 | 4 | no | — |
| 170 | D3×D4 | — | 18 | 4 | no | — |
| 171 | D3×D4 | — | 36 | 9 | no | — |
| 172 | C2×D12 | — | 9 | 4 | no | — |
| 173 | C2×C4×D3 | — | 9 | 4 | no | — |
| 174 | C4×A4 | C2×C3⋊D3 | 1 | 9 | yes | complemented |
| 175 | SL(2,ℤ/4) | — | 9 | 9 | no | — |
| 176 | #217 | — | 9 | 8 | no | — |
| 177 | #217 | — | 9 | 8 | no | — |
| 178 | #217 | — | 9 | 4 | no | — |
| 179 | (C2×D3)⋊C4 | — | 9 | 4 | no | — |
| 180 | (C2×D3)⋊C4 | — | 9 | 8 | no | — |
| 181 | C4⋊Dic3 | — | 9 | 4 | no | — |
| 182 | Dic3⋊C4 | — | 9 | 4 | no | — |
| 183 | C4×Dic3 | — | 9 | 4 | no | — |
| 184 | C22×C32 | — | 3 | 1 | no | — |
| 185 | C22×C32 | — | 3 | 1 | no | — |
| 186 | C22×C32 | — | 3 | 2 | no | — |
| 187 | C22×C32 | — | 4 | 24 | no | — |
| 188 | C22×C32 | — | 6 | 2 | no | — |
| 189 | C22×C32 | — | 6 | 2 | no | — |
| 190 | C22×C32 | — | 6 | 2 | no | — |
| 191 | C22×C32 | — | 6 | 2 | no | — |
| 192 | C22×C32 | D3×D4 | 1 | 1 | yes | characteristic, complemented |
| 193 | C22×C32 | C2×S4 | 1 | 2 | yes | complemented |
| 194 | C2×C3⋊D3 | — | 3 | 2 | no | — |
| 195 | C2×C3⋊D3 | — | 6 | 1 | no | — |
| 196 | C2×C3⋊D3 | — | 6 | 1 | no | — |
| 197 | C2×C3⋊D3 | — | 6 | 2 | no | — |
| 198 | C2×C3⋊D3 | — | 6 | 2 | no | — |
| 199 | C2×C3⋊D3 | — | 6 | 2 | no | — |
| 200 | C2×C3⋊D3 | — | 12 | 12 | no | — |
| 201 | C2×C3⋊D3 | — | 12 | 24 | no | — |
| 202 | C2×C3⋊D3 | — | 12 | 4 | no | — |
| 203 | C2×C3⋊D3 | — | 12 | 4 | no | — |
| 204 | C2×C3⋊D3 | — | 24 | 48 | no | — |
| 205 | C3×A4 | D3×D4 | 1 | 12 | yes | complemented |
| 206 | C32×C4 | — | 3 | 1 | no | — |
| 207 | C32×C4 | — | 4 | 12 | no | — |
| 208 | C32×C4 | C2×S4 | 1 | 1 | yes | characteristic, complemented |
| 209 | C3⋊Dic3 | — | 3 | 2 | no | — |
| 210 | C3⋊Dic3 | — | 6 | 2 | no | — |
| 211 | C3⋊Dic3 | — | 6 | 1 | no | — |
| 212 | C3⋊Dic3 | — | 6 | 1 | no | — |
| 213 | C3⋊Dic3 | — | 12 | 12 | no | — |
| 214 | C22×D4 | — | 27 | 1 | no | — |
| 215 | C22×D4 | — | 27 | 2 | no | — |
| 216 | C22×D4 | C3⋊(C3⋊D3) | 1 | 1 | yes | characteristic, complemented |
| 217 | #127 | — | 27 | 1 | no | — |
| 218 | #121 | — | 27 | 1 | no | — |
| 219 | #121 | — | 27 | 2 | no | — |
| 220 | #121 | — | 27 | 1 | no | — |
| 221 | #120 | — | 27 | 2 | no | — |
| 222 | #120 | — | 27 | 2 | no | — |
| 223 | C4×D4 | — | 27 | 1 | no | — |
| 224 | C4×D4 | — | 27 | 1 | no | — |
| 225 | C33 | — | 4 | 1 | no | Sylow |
| 226 | C23×C3 | — | 2 | 8 | no | — |
| 227 | C23×C3 | — | 3 | 8 | no | — |
| 228 | C23×C3 | — | 3 | 8 | no | — |
| 229 | C23×C3 | — | 6 | 8 | no | — |
| 230 | C23×C3 | C22×C3⋊D3 | 1 | 4 | yes | — |
| 231 | C22×D3 | — | 9 | 4 | no | — |
| 232 | C22×D3 | — | 9 | 8 | no | — |
| 233 | C22×D3 | — | 18 | 16 | no | — |
| 234 | C22×D3 | — | 18 | 8 | no | — |
| 235 | C22×D3 | — | 18 | 8 | no | — |
| 236 | C22×D3 | — | 36 | 18 | no | — |
| 237 | C22×D3 | — | 36 | 8 | no | — |
| 238 | C2×A4 | — | 2 | 18 | no | — |
| 239 | C2×A4 | C22×C3⋊D3 | 1 | 9 | yes | — |
| 240 | S4 | — | 9 | 18 | no | — |
| 241 | S4 | — | 18 | 18 | no | — |
| 242 | C3×D4 | — | 3 | 8 | no | — |
| 243 | C3×D4 | — | 3 | 4 | no | — |
| 244 | C3×D4 | — | 4 | 9 | no | — |
| 245 | C3×D4 | — | 6 | 4 | no | — |
| 246 | C3×D4 | C3⋊S4 | 1 | 4 | yes | complemented |
| 247 | C2×C3×C4 | — | 3 | 4 | no | — |
| 248 | C2×C3×C4 | — | 3 | 4 | no | — |
| 249 | #67 | — | 9 | 8 | no | — |
| 250 | #67 | — | 9 | 8 | no | — |
| 251 | #67 | — | 18 | 16 | no | — |
| 252 | #67 | — | 18 | 16 | no | — |
| 253 | #67 | — | 18 | 8 | no | — |
| 254 | #67 | — | 18 | 8 | no | — |
| 255 | #67 | — | 18 | 8 | no | — |
| 256 | #67 | — | 18 | 8 | no | — |
| 257 | #67 | — | 36 | 18 | no | — |
| 258 | #67 | — | 36 | 8 | no | — |
| 259 | C2×Dic3 | — | 9 | 4 | no | — |
| 260 | C2×Dic3 | — | 9 | 4 | no | — |
| 261 | C2×Dic3 | — | 9 | 4 | no | — |
| 262 | C2×Dic3 | — | 9 | 8 | no | — |
| 263 | C2×Dic3 | — | 18 | 16 | no | — |
| 264 | D12 | — | 18 | 4 | no | — |
| 265 | D12 | — | 18 | 4 | no | — |
| 266 | D12 | — | 36 | 9 | no | — |
| 267 | C4×D3 | — | 18 | 4 | no | — |
| 268 | C4×D3 | — | 18 | 4 | no | — |
| 269 | C4×D3 | — | 36 | 9 | no | — |
| 270 | C2×C32 | — | 2 | 2 | no | — |
| 271 | C2×C32 | — | 3 | 1 | no | — |
| 272 | C2×C32 | — | 3 | 1 | no | — |
| 273 | C2×C32 | — | 4 | 12 | no | — |
| 274 | C2×C32 | — | 6 | 2 | no | — |
| 275 | C2×C32 | — | 8 | 24 | no | — |
| 276 | C2×C32 | C22×S4 | 1 | 1 | yes | characteristic |
| 277 | C3⋊D3 | — | 6 | 2 | no | — |
| 278 | C3⋊D3 | — | 12 | 24 | no | — |
| 279 | C3⋊D3 | — | 12 | 2 | no | — |
| 280 | C3⋊D3 | — | 24 | 24 | no | — |
| 281 | C24 | — | 27 | 2 | no | — |
| 282 | C24 | C2×C3⋊(C3⋊D3) | 1 | 2 | yes | complemented |
| 283 | C2×D4 | — | 3 | 1 | no | — |
| 284 | C2×D4 | — | 3 | 2 | no | — |
| 285 | C2×D4 | — | 3 | 1 | no | — |
| 286 | C2×D4 | — | 27 | 1 | no | — |
| 287 | C2×D4 | — | 27 | 1 | no | — |
| 288 | C2×D4 | — | 27 | 2 | no | — |
| 289 | C2×D4 | — | 27 | 2 | no | — |
| 290 | C2×D4 | — | 27 | 2 | no | — |
| 291 | C2×D4 | — | 27 | 2 | no | — |
| 292 | C2×D4 | — | 27 | 2 | no | — |
| 293 | C2×D4 | — | 54 | 2 | no | — |
| 294 | C2×D4 | — | 54 | 1 | no | — |
| 295 | C2×D4 | — | 54 | 4 | no | — |
| 296 | C2×D4 | — | 54 | 4 | no | — |
| 297 | C2×D4 | — | 54 | 1 | no | — |
| 298 | C22×C4 | — | 27 | 1 | no | — |
| 299 | C22×C4 | — | 27 | 2 | no | — |
| 300 | C22×C4 | C2×C3⋊(C3⋊D3) | 1 | 1 | yes | characteristic, complemented |
| 301 | C4⋊C4 | — | 27 | 1 | no | — |
| 302 | C4⋊C4 | — | 27 | 1 | no | — |
| 303 | C22⋊C4 | — | 27 | 1 | no | — |
| 304 | C22⋊C4 | — | 27 | 2 | no | — |
| 305 | C22⋊C4 | — | 27 | 2 | no | — |
| 306 | C22⋊C4 | — | 27 | 2 | no | — |
| 307 | C22⋊C4 | — | 27 | 1 | no | — |
| 308 | C42 | — | 27 | 1 | no | — |
| 309 | C22×C3 | — | 3 | 4 | no | — |
| 310 | C22×C3 | — | 3 | 4 | no | — |
| 311 | C22×C3 | — | 3 | 8 | no | — |
| 312 | C22×C3 | — | 4 | 18 | no | — |
| 313 | C22×C3 | — | 6 | 8 | no | — |
| 314 | C22×C3 | — | 6 | 8 | no | — |
| 315 | C22×C3 | — | 6 | 8 | no | — |
| 316 | C22×C3 | — | 6 | 8 | no | — |
| 317 | C22×C3 | D4×C3⋊D3 | 1 | 4 | yes | complemented |
| 318 | C22×C3 | C2×C3⋊S4 | 1 | 8 | yes | complemented |
| 319 | C2×D3 | — | 9 | 8 | no | — |
| 320 | C2×D3 | — | 18 | 4 | no | — |
| 321 | C2×D3 | — | 18 | 4 | no | — |
| 322 | C2×D3 | — | 18 | 8 | no | — |
| 323 | C2×D3 | — | 18 | 8 | no | — |
| 324 | C2×D3 | — | 18 | 8 | no | — |
| 325 | C2×D3 | — | 36 | 9 | no | — |
| 326 | C2×D3 | — | 36 | 18 | no | — |
| 327 | C2×D3 | — | 36 | 16 | no | — |
| 328 | C2×D3 | — | 36 | 16 | no | — |
| 329 | C2×D3 | — | 72 | 36 | no | — |
| 330 | A4 | D4×C3⋊D3 | 1 | 9 | yes | complemented |
| 331 | C3×C4 | — | 3 | 4 | no | — |
| 332 | C3×C4 | — | 4 | 9 | no | — |
| 333 | C3×C4 | C2×C3⋊S4 | 1 | 4 | yes | complemented |
| 334 | Dic3 | — | 9 | 8 | no | — |
| 335 | Dic3 | — | 18 | 8 | no | — |
| 336 | Dic3 | — | 18 | 4 | no | — |
| 337 | Dic3 | — | 18 | 4 | no | — |
| 338 | Dic3 | — | 36 | 9 | no | — |
| 339 | C32 | — | 4 | 12 | no | — |
| 340 | C32 | D4×S4 | 1 | 1 | yes | characteristic, complemented |
| 341 | C23 | — | 2 | 2 | no | — |
| 342 | C23 | — | 3 | 2 | no | — |
| 343 | C23 | — | 3 | 2 | no | — |
| 344 | C23 | — | 6 | 2 | no | — |
| 345 | C23 | — | 27 | 1 | no | — |
| 346 | C23 | — | 27 | 2 | no | — |
| 347 | C23 | — | 54 | 2 | no | — |
| 348 | C23 | — | 54 | 4 | no | — |
| 349 | C23 | — | 54 | 2 | no | — |
| 350 | C23 | — | 108 | 2 | no | — |
| 351 | C23 | C22×C3⋊(C3⋊D3) | 1 | 1 | yes | characteristic |
| 352 | D4 | — | 3 | 2 | no | — |
| 353 | D4 | — | 3 | 1 | no | — |
| 354 | D4 | — | 6 | 1 | no | — |
| 355 | D4 | — | 27 | 2 | no | — |
| 356 | D4 | — | 27 | 2 | no | — |
| 357 | D4 | — | 54 | 1 | no | — |
| 358 | D4 | — | 54 | 4 | no | — |
| 359 | D4 | — | 54 | 2 | no | — |
| 360 | D4 | — | 54 | 2 | no | — |
| 361 | D4 | — | 54 | 4 | no | — |
| 362 | D4 | — | 54 | 2 | no | — |
| 363 | D4 | — | 54 | 2 | no | — |
| 364 | D4 | — | 54 | 1 | no | — |
| 365 | D4 | — | 108 | 2 | no | — |
| 366 | D4 | C3⋊(C3⋊S4) | 1 | 1 | yes | characteristic, direct product factor |
| 367 | C2×C4 | — | 3 | 1 | no | — |
| 368 | C2×C4 | — | 3 | 1 | no | — |
| 369 | C2×C4 | — | 27 | 1 | no | — |
| 370 | C2×C4 | — | 27 | 1 | no | — |
| 371 | C2×C4 | — | 27 | 2 | no | — |
| 372 | C2×C4 | — | 27 | 1 | no | — |
| 373 | C2×C4 | — | 54 | 1 | no | — |
| 374 | C2×C4 | — | 54 | 4 | no | — |
| 375 | C2×C4 | — | 54 | 1 | no | — |
| 376 | C2×C3 | — | 2 | 8 | no | — |
| 377 | C2×C3 | — | 3 | 4 | no | — |
| 378 | C2×C3 | — | 3 | 4 | no | — |
| 379 | C2×C3 | — | 4 | 9 | no | — |
| 380 | C2×C3 | — | 6 | 8 | no | — |
| 381 | C2×C3 | — | 8 | 18 | no | — |
| 382 | C2×C3 | C22×C3⋊S4 | 1 | 4 | yes | — |
| 383 | D3 | — | 18 | 8 | no | — |
| 384 | D3 | — | 36 | 18 | no | — |
| 385 | D3 | — | 36 | 8 | no | — |
| 386 | D3 | — | 72 | 18 | no | — |
| 387 | C22 | — | 3 | 1 | no | — |
| 388 | C22 | — | 3 | 1 | no | — |
| 389 | C22 | — | 3 | 2 | no | — |
| 390 | C22 | — | 6 | 2 | no | — |
| 391 | C22 | — | 6 | 2 | no | — |
| 392 | C22 | — | 6 | 2 | no | — |
| 393 | C22 | — | 6 | 2 | no | — |
| 394 | C22 | — | 27 | 2 | no | — |
| 395 | C22 | — | 54 | 1 | no | — |
| 396 | C22 | — | 54 | 2 | no | — |
| 397 | C22 | — | 54 | 2 | no | — |
| 398 | C22 | — | 54 | 1 | no | — |
| 399 | C22 | — | 54 | 2 | no | — |
| 400 | C22 | — | 108 | 4 | no | — |
| 401 | C22 | — | 108 | 4 | no | — |
| 402 | C22 | D4×C3⋊(C3⋊D3) | 1 | 1 | yes | characteristic, complemented |
| 403 | C22 | C2×C3⋊(C3⋊S4) | 1 | 2 | yes | complemented |
| 404 | C4 | — | 3 | 1 | no | — |
| 405 | C4 | — | 27 | 2 | no | — |
| 406 | C4 | — | 54 | 2 | no | — |
| 407 | C4 | — | 54 | 1 | no | — |
| 408 | C4 | — | 54 | 1 | no | — |
| 409 | C4 | C2×C3⋊(C3⋊S4) | 1 | 1 | yes | characteristic, complemented |
| 410 | C3 | — | 4 | 9 | no | — |
| 411 | C3 | D4×C3⋊S4 | 1 | 4 | yes | complemented |
| 412 | C2 | — | 2 | 2 | no | — |
| 413 | C2 | — | 3 | 1 | no | — |
| 414 | C2 | — | 3 | 1 | no | — |
| 415 | C2 | — | 6 | 2 | no | — |
| 416 | C2 | — | 54 | 2 | no | — |
| 417 | C2 | — | 108 | 2 | no | — |
| 418 | C2 | C22×C3⋊(C3⋊S4) | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×C3⋊(C3⋊S4) |
| commutator, cosocle | C2×C32×A4 | C23 |
| Fitting | C22×C32×D4 | D3 |
| socle | C23×C32 | C22×D3 |
| hypercenter | D4 | C3⋊(C3⋊S4) |
| hypocenter | C32×A4 | C2×D4 |
| LCS | G ▷ C2×C32×A4 ▷ C32×A4 | |
| UCS | C1 ◁ C2 ◁ D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C3⋊(C3⋊S4) | C2 | semidirect |
| 2. C32×D4×A4 | C2 | semidirect |
| 3. #11724473 | C2 | semidirect |
| 4. C4⋊(C3⋊(C3⋊S4)) | C2 | semidirect |
| 5. C4×C3⋊(C3⋊S4) | C2 | semidirect |
| 8. C22×C32×A4 | C22 | semidirect |
| 11. C2×C3⋊(C3⋊S4) | C22 | semidirect |
| 13. C32×C4×A4 | C22 | semidirect |
| 14. C3⋊(C3⋊SL(2,ℤ/4)) | C22 | semidirect |
| 16. C22×C32×D4 | D3 | semidirect |
| 19. C3×D4×A4 | D3 | semidirect |
| 35. C3⋊(C3⋊S4) | D4 | direct (N × H) |
| 42. C24×C32 | C2×D3 | semidirect |
| 44. C22×C3×A4 | C2×D3 | semidirect |
| 51. C22×C32×C4 | C2×D3 | semidirect |
| 67. C3×C4×A4 | C2×D3 | semidirect |
| 81. C32×A4 | C2×D4 | semidirect |
| 85. C22×C3×D4 | C3⋊D3 | semidirect |
| 88. D4×A4 | C3⋊D3 | semidirect |
| 120. C32×D4 | S4 | semidirect |
| 149. C24×C3 | C2×C3⋊D3 | semidirect |
| 151. C22×A4 | C2×C3⋊D3 | semidirect |
| 158. C22×C3×C4 | C2×C3⋊D3 | semidirect |
| 174. C4×A4 | C2×C3⋊D3 | semidirect |
| 192. C22×C32 | D3×D4 | semidirect |
| 193. C22×C32 | C2×S4 | semidirect |
| 205. C3×A4 | D3×D4 | semidirect |
| 208. C32×C4 | C2×S4 | semidirect |
| 216. C22×D4 | C3⋊(C3⋊D3) | semidirect |
| 246. C3×D4 | C3⋊S4 | semidirect |
| 282. C24 | C2×C3⋊(C3⋊D3) | semidirect |
| 300. C22×C4 | C2×C3⋊(C3⋊D3) | semidirect |
| 317. C22×C3 | D4×C3⋊D3 | semidirect |
| 318. C22×C3 | C2×C3⋊S4 | semidirect |
| 330. A4 | D4×C3⋊D3 | semidirect |
| 333. C3×C4 | C2×C3⋊S4 | semidirect |
| 340. C32 | D4×S4 | semidirect |
| 366. D4 | C3⋊(C3⋊S4) | direct (N × H) |
| 402. C22 | D4×C3⋊(C3⋊D3) | semidirect |
| 403. C22 | C2×C3⋊(C3⋊S4) | semidirect |
| 409. C4 | C2×C3⋊(C3⋊S4) | semidirect |
| 411. C3 | D4×C3⋊S4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C3⋊(C3⋊S4) | 2 |
| C32×D4×A4 | 2 |
| #11724473 | 2 |
| C4⋊(C3⋊(C3⋊S4)) | 2 |
| C4×C3⋊(C3⋊S4) | 2 |
| D4×C3⋊S4 | 3 |
| D4×#638 | 3 |
| D4×C3⋊(C3⋊D3) | 4 |