Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
760 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 760 conjugacy classes of subgroups — too many to draw. Only its 73 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 | Q8×C22⋊H(3,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C4×#6258 | C2 | 1 | 3 | yes | maximal |
| 3 | #11722380 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | C3×Q8×S4 | — | 3 | 3 | no | maximal |
| 5 | C3×Q8×#67 | — | 3 | 1 | no | maximal |
| 6 | C2×#6258 | C22 | 1 | 1 | yes | characteristic |
| 7 | Q8×#272 | — | 4 | 1 | no | maximal |
| 8 | C4×C22⋊H(3,𝔽3) | C22 | 1 | 3 | yes | — |
| 9 | #88613 | C22 | 1 | 3 | yes | — |
| 10 | C22×C32×Q8 | D3 | 1 | 1 | yes | characteristic, complemented |
| 11 | C2×C3×D3×Q8 | — | 3 | 1 | no | — |
| 12 | C3×Q8×A4 | D3 | 1 | 3 | yes | complemented |
| 13 | C3×C4×S4 | — | 3 | 9 | no | — |
| 14 | C3×A4⋊Q8 | — | 3 | 9 | no | — |
| 15 | C3×#946 | — | 3 | 3 | no | — |
| 16 | C3×Q8×Dic3 | — | 3 | 1 | no | — |
| 17 | C3×#944 | — | 3 | 3 | no | — |
| 18 | C3×C4×#67 | — | 3 | 3 | no | — |
| 19 | C3×#924 | — | 3 | 3 | no | — |
| 20 | C2×C22⋊H(3,𝔽3) | C23 | 1 | 1 | yes | characteristic |
| 21 | #6258 | Q8 | 1 | 2 | yes | direct product factor |
| 22 | Q8×H(3,𝔽3) | — | 4 | 1 | no | — |
| 23 | C4×#272 | — | 4 | 3 | no | — |
| 24 | #6229 | — | 4 | 3 | no | — |
| 25 | Q8×S4 | — | 3 | 3 | no | — |
| 26 | C3×D4×Q8 | — | 9 | 1 | no | — |
| 27 | Q8×#67 | — | 3 | 1 | no | — |
| 28 | C2×C3×S4 | — | 3 | 3 | no | — |
| 29 | C2×C32×Q8 | — | 3 | 1 | no | — |
| 30 | C2×C32×Q8 | — | 3 | 3 | no | — |
| 31 | C22×C32×C4 | C2×D3 | 1 | 3 | yes | — |
| 32 | C2×C3×#67 | — | 3 | 1 | no | — |
| 33 | C3×D3×Q8 | — | 6 | 3 | no | — |
| 34 | C3×D3×Q8 | — | 6 | 1 | no | — |
| 35 | C3×D3×Q8 | — | 12 | 3 | no | — |
| 36 | C2×C3×C4×D3 | — | 3 | 3 | no | — |
| 37 | C2×C3×Dic6 | — | 3 | 3 | no | — |
| 38 | C3×C4×A4 | C2×D3 | 1 | 9 | yes | — |
| 39 | C3×SL(2,ℤ/4) | — | 3 | 9 | no | — |
| 40 | C3×#217 | — | 3 | 3 | no | — |
| 41 | C3×(C2×D3)⋊C4 | — | 3 | 3 | no | — |
| 42 | C3×C4⋊Dic3 | — | 3 | 3 | no | — |
| 43 | C3×Dic3⋊C4 | — | 3 | 6 | no | — |
| 44 | C3×Dic3⋊C4 | — | 3 | 3 | no | — |
| 45 | C3×C4×Dic3 | — | 3 | 3 | no | — |
| 46 | C2×#272 | — | 4 | 1 | no | — |
| 47 | C22⋊H(3,𝔽3) | C2×Q8 | 1 | 1 | yes | characteristic, complemented |
| 48 | C4×H(3,𝔽3) | — | 4 | 3 | no | — |
| 49 | #1084 | — | 4 | 3 | no | — |
| 50 | C22×C3×Q8 | — | 3 | 1 | no | — |
| 51 | C22×C3×Q8 | — | 9 | 1 | no | — |
| 52 | C22×C3×Q8 | C3⋊D3 | 1 | 1 | yes | characteristic |
| 53 | C2×D3×Q8 | — | 3 | 1 | no | — |
| 54 | Q8×A4 | — | 3 | 3 | no | — |
| 55 | C4×S4 | — | 3 | 9 | no | — |
| 56 | A4⋊Q8 | — | 3 | 9 | no | — |
| 57 | C3×C4⋊Q8 | — | 9 | 3 | no | — |
| 58 | C3×C22⋊Q8 | — | 9 | 3 | no | — |
| 59 | C3×C22⋊Q8 | — | 9 | 3 | no | — |
| 60 | C3×C4×Q8 | — | 9 | 1 | no | — |
| 61 | C3×C4×D4 | — | 9 | 3 | no | — |
| 62 | #946 | — | 3 | 3 | no | — |
| 63 | Q8×Dic3 | — | 3 | 1 | no | — |
| 64 | #944 | — | 3 | 3 | no | — |
| 65 | C4×#67 | — | 3 | 3 | no | — |
| 66 | #924 | — | 3 | 3 | no | — |
| 67 | C23×C32 | C22×D3 | 1 | 1 | yes | characteristic |
| 68 | C22×C3×D3 | — | 3 | 1 | no | — |
| 69 | C2×C3×A4 | C22×D3 | 1 | 3 | yes | — |
| 70 | C3×S4 | — | 3 | 6 | no | — |
| 71 | C32×Q8 | — | 3 | 3 | no | — |
| 72 | C32×Q8 | — | 4 | 3 | no | — |
| 73 | C32×Q8 | — | 6 | 1 | no | — |
| 74 | C32×Q8 | S4 | 1 | 1 | yes | characteristic, complemented |
| 75 | C2×C32×C4 | — | 3 | 3 | no | — |
| 76 | C2×C32×C4 | — | 3 | 3 | no | — |
| 77 | C3×#67 | — | 3 | 2 | no | — |
| 78 | C3×#67 | — | 3 | 2 | no | — |
| 79 | C2×C3×Dic3 | — | 3 | 1 | no | — |
| 80 | C2×C3×Dic3 | — | 3 | 3 | no | — |
| 81 | C2×C3×Dic3 | — | 3 | 3 | no | — |
| 82 | C3×C4×D3 | — | 6 | 3 | no | — |
| 83 | C3×C4×D3 | — | 6 | 3 | no | — |
| 84 | C3×C4×D3 | — | 12 | 9 | no | — |
| 85 | C3×Dic6 | — | 6 | 3 | no | — |
| 86 | C3×Dic6 | — | 6 | 3 | no | — |
| 87 | C3×Dic6 | — | 12 | 9 | no | — |
| 88 | D4×Q8 | — | 9 | 1 | no | Sylow |
| 89 | C2×H(3,𝔽3) | — | 4 | 1 | no | — |
| 90 | #272 | — | 4 | 2 | no | — |
| 91 | C2×S4 | — | 3 | 3 | no | — |
| 92 | C2×C3×Q8 | — | 3 | 1 | no | — |
| 93 | C2×C3×Q8 | — | 3 | 1 | no | — |
| 94 | C2×C3×Q8 | — | 3 | 3 | no | — |
| 95 | C2×C3×Q8 | — | 3 | 3 | no | — |
| 96 | C2×C3×Q8 | — | 6 | 1 | no | — |
| 97 | C2×C3×Q8 | — | 6 | 3 | no | — |
| 98 | C2×C3×Q8 | — | 9 | 3 | no | — |
| 99 | C2×C3×Q8 | — | 18 | 3 | no | — |
| 100 | C2×C3×Q8 | — | 18 | 1 | no | — |
| 101 | C2×C3×D4 | — | 9 | 1 | no | — |
| 102 | C22×C3×C4 | — | 3 | 3 | no | — |
| 103 | C22×C3×C4 | — | 9 | 3 | no | — |
| 104 | C22×C3×C4 | C2×C3⋊D3 | 1 | 3 | yes | — |
| 105 | C2×#67 | — | 3 | 1 | no | — |
| 106 | D3×Q8 | — | 6 | 3 | no | — |
| 107 | D3×Q8 | — | 6 | 1 | no | — |
| 108 | D3×Q8 | — | 12 | 3 | no | — |
| 109 | C2×C4×D3 | — | 3 | 3 | no | — |
| 110 | C2×Dic6 | — | 3 | 3 | no | — |
| 111 | C4×A4 | — | 3 | 9 | no | — |
| 112 | SL(2,ℤ/4) | — | 3 | 9 | no | — |
| 113 | C3×C4⋊C4 | — | 9 | 3 | no | — |
| 114 | C3×C4⋊C4 | — | 9 | 6 | no | — |
| 115 | C3×C4⋊C4 | — | 9 | 3 | no | — |
| 116 | C3×C22⋊C4 | — | 9 | 3 | no | — |
| 117 | C3×C22⋊C4 | — | 9 | 3 | no | — |
| 118 | C3×C42 | — | 9 | 3 | no | — |
| 119 | #217 | — | 3 | 3 | no | — |
| 120 | (C2×D3)⋊C4 | — | 3 | 3 | no | — |
| 121 | C4⋊Dic3 | — | 3 | 3 | no | — |
| 122 | Dic3⋊C4 | — | 3 | 6 | no | — |
| 123 | Dic3⋊C4 | — | 3 | 3 | no | — |
| 124 | C4×Dic3 | — | 3 | 3 | no | — |
| 125 | C22×C32 | — | 3 | 1 | no | — |
| 126 | C22×C32 | — | 3 | 1 | no | — |
| 127 | C22×C32 | D3×Q8 | 1 | 1 | yes | characteristic, complemented |
| 128 | C2×C3×D3 | — | 3 | 2 | no | — |
| 129 | C2×C3×D3 | — | 6 | 1 | no | — |
| 130 | C2×C3×D3 | — | 6 | 1 | no | — |
| 131 | C2×C3×D3 | — | 12 | 3 | no | — |
| 132 | C3×A4 | D3×Q8 | 1 | 3 | yes | complemented |
| 133 | C32×C4 | — | 3 | 3 | no | — |
| 134 | C32×C4 | — | 4 | 9 | no | — |
| 135 | C32×C4 | C2×S4 | 1 | 3 | yes | — |
| 136 | C3×Dic3 | — | 3 | 2 | no | — |
| 137 | C3×Dic3 | — | 6 | 3 | no | — |
| 138 | C3×Dic3 | — | 6 | 3 | no | — |
| 139 | C3×Dic3 | — | 12 | 9 | no | — |
| 140 | C22×Q8 | — | 9 | 1 | no | — |
| 141 | C22×Q8 | #272 | 1 | 1 | yes | characteristic, complemented |
| 142 | C4⋊Q8 | — | 9 | 3 | no | — |
| 143 | C22⋊Q8 | — | 9 | 3 | no | — |
| 144 | C22⋊Q8 | — | 9 | 3 | no | — |
| 145 | C4×Q8 | — | 9 | 1 | no | — |
| 146 | C4×D4 | — | 9 | 3 | no | — |
| 147 | H(3,𝔽3) | — | 4 | 1 | no | Sylow |
| 148 | C23×C3 | — | 3 | 1 | no | — |
| 149 | C23×C3 | — | 9 | 1 | no | — |
| 150 | C23×C3 | C22×C3⋊D3 | 1 | 1 | yes | characteristic |
| 151 | C22×D3 | — | 3 | 1 | no | — |
| 152 | C2×A4 | — | 3 | 3 | no | — |
| 153 | S4 | — | 3 | 6 | no | — |
| 154 | C3×Q8 | — | 3 | 3 | no | — |
| 155 | C3×Q8 | — | 3 | 3 | no | — |
| 156 | C3×Q8 | — | 3 | 1 | no | — |
| 157 | C3×Q8 | — | 6 | 3 | no | — |
| 158 | C3×Q8 | — | 6 | 1 | no | — |
| 159 | C3×Q8 | — | 6 | 3 | no | — |
| 160 | C3×Q8 | — | 12 | 3 | no | — |
| 161 | C3×Q8 | — | 18 | 3 | no | — |
| 162 | C3×Q8 | — | 18 | 3 | no | — |
| 163 | C3×Q8 | C3⋊S4 | 1 | 1 | yes | characteristic |
| 164 | C3×D4 | — | 9 | 2 | no | — |
| 165 | C3×D4 | — | 9 | 2 | no | — |
| 166 | C2×C3×C4 | — | 3 | 3 | no | — |
| 167 | C2×C3×C4 | — | 3 | 3 | no | — |
| 168 | C2×C3×C4 | — | 3 | 3 | no | — |
| 169 | C2×C3×C4 | — | 3 | 3 | no | — |
| 170 | C2×C3×C4 | — | 6 | 3 | no | — |
| 171 | C2×C3×C4 | — | 6 | 3 | no | — |
| 172 | C2×C3×C4 | — | 9 | 1 | no | — |
| 173 | C2×C3×C4 | — | 9 | 3 | no | — |
| 174 | C2×C3×C4 | — | 9 | 3 | no | — |
| 175 | C2×C3×C4 | — | 18 | 3 | no | — |
| 176 | C2×C3×C4 | — | 18 | 3 | no | — |
| 177 | #67 | — | 3 | 2 | no | — |
| 178 | #67 | — | 3 | 2 | no | — |
| 179 | C2×Dic3 | — | 3 | 1 | no | — |
| 180 | C2×Dic3 | — | 3 | 3 | no | — |
| 181 | C2×Dic3 | — | 3 | 3 | no | — |
| 182 | C4×D3 | — | 6 | 3 | no | — |
| 183 | C4×D3 | — | 6 | 3 | no | — |
| 184 | C4×D3 | — | 12 | 9 | no | — |
| 185 | Dic6 | — | 6 | 3 | no | — |
| 186 | Dic6 | — | 6 | 3 | no | — |
| 187 | Dic6 | — | 12 | 9 | no | — |
| 188 | C2×C32 | — | 3 | 1 | no | — |
| 189 | C2×C32 | — | 3 | 1 | no | — |
| 190 | C2×C32 | — | 4 | 3 | no | — |
| 191 | C2×C32 | C22×S4 | 1 | 1 | yes | characteristic |
| 192 | C3×D3 | — | 6 | 2 | no | — |
| 193 | C3×D3 | — | 12 | 6 | no | — |
| 194 | C2×Q8 | — | 3 | 1 | no | — |
| 195 | C2×Q8 | — | 3 | 3 | no | — |
| 196 | C2×Q8 | — | 9 | 3 | no | — |
| 197 | C2×Q8 | — | 18 | 3 | no | — |
| 198 | C2×Q8 | — | 18 | 1 | no | — |
| 199 | C2×D4 | — | 9 | 1 | no | — |
| 200 | C22×C4 | — | 9 | 3 | no | — |
| 201 | C22×C4 | C2×#272 | 1 | 3 | yes | — |
| 202 | C4⋊C4 | — | 9 | 3 | no | — |
| 203 | C4⋊C4 | — | 9 | 3 | no | — |
| 204 | C4⋊C4 | — | 9 | 6 | no | — |
| 205 | C22⋊C4 | — | 9 | 3 | no | — |
| 206 | C22⋊C4 | — | 9 | 3 | no | — |
| 207 | C42 | — | 9 | 3 | no | — |
| 208 | C22×C3 | — | 3 | 1 | no | — |
| 209 | C22×C3 | — | 3 | 1 | no | — |
| 210 | C22×C3 | — | 3 | 1 | no | — |
| 211 | C22×C3 | — | 3 | 1 | no | — |
| 212 | C22×C3 | — | 3 | 1 | no | — |
| 213 | C22×C3 | — | 6 | 1 | no | — |
| 214 | C22×C3 | — | 6 | 1 | no | — |
| 215 | C22×C3 | — | 9 | 2 | no | — |
| 216 | C22×C3 | — | 18 | 1 | no | — |
| 217 | C22×C3 | — | 18 | 1 | no | — |
| 218 | C22×C3 | Q8×C3⋊D3 | 1 | 1 | yes | characteristic |
| 219 | C2×D3 | — | 3 | 2 | no | — |
| 220 | C2×D3 | — | 6 | 1 | no | — |
| 221 | C2×D3 | — | 6 | 1 | no | — |
| 222 | C2×D3 | — | 12 | 3 | no | — |
| 223 | A4 | — | 3 | 3 | no | — |
| 224 | C3×C4 | — | 3 | 3 | no | — |
| 225 | C3×C4 | — | 3 | 3 | no | — |
| 226 | C3×C4 | — | 3 | 3 | no | — |
| 227 | C3×C4 | — | 6 | 3 | no | — |
| 228 | C3×C4 | — | 9 | 2 | no | — |
| 229 | C3×C4 | — | 12 | 9 | no | — |
| 230 | C3×C4 | — | 18 | 3 | no | — |
| 231 | C3×C4 | — | 18 | 3 | no | — |
| 232 | C3×C4 | C2×C3⋊S4 | 1 | 3 | yes | — |
| 233 | Dic3 | — | 3 | 2 | no | — |
| 234 | Dic3 | — | 6 | 3 | no | — |
| 235 | Dic3 | — | 6 | 3 | no | — |
| 236 | Dic3 | — | 12 | 9 | no | — |
| 237 | C32 | — | 4 | 3 | no | — |
| 238 | C32 | Q8×S4 | 1 | 1 | yes | characteristic, complemented |
| 239 | C23 | — | 9 | 1 | no | — |
| 240 | C23 | C22×#272 | 1 | 1 | yes | characteristic |
| 241 | Q8 | — | 3 | 3 | no | — |
| 242 | Q8 | — | 6 | 1 | no | — |
| 243 | Q8 | — | 18 | 3 | no | — |
| 244 | Q8 | — | 18 | 3 | no | — |
| 245 | Q8 | #6258 | 1 | 1 | yes | characteristic, direct product factor |
| 246 | D4 | — | 9 | 2 | no | — |
| 247 | D4 | — | 9 | 2 | no | — |
| 248 | C2×C4 | — | 3 | 3 | no | — |
| 249 | C2×C4 | — | 3 | 3 | no | — |
| 250 | C2×C4 | — | 9 | 1 | no | — |
| 251 | C2×C4 | — | 9 | 3 | no | — |
| 252 | C2×C4 | — | 9 | 3 | no | — |
| 253 | C2×C4 | — | 18 | 3 | no | — |
| 254 | C2×C4 | — | 18 | 3 | no | — |
| 255 | C2×C3 | — | 3 | 1 | no | — |
| 256 | C2×C3 | — | 3 | 1 | no | — |
| 257 | C2×C3 | — | 3 | 1 | no | — |
| 258 | C2×C3 | — | 3 | 1 | no | — |
| 259 | C2×C3 | — | 3 | 1 | no | — |
| 260 | C2×C3 | — | 6 | 1 | no | — |
| 261 | C2×C3 | — | 6 | 1 | no | — |
| 262 | C2×C3 | — | 12 | 3 | no | — |
| 263 | C2×C3 | — | 18 | 2 | no | — |
| 264 | C2×C3 | C22×C3⋊S4 | 1 | 1 | yes | characteristic |
| 265 | D3 | — | 6 | 2 | no | — |
| 266 | D3 | — | 12 | 6 | no | — |
| 267 | C22 | — | 3 | 1 | no | — |
| 268 | C22 | — | 3 | 1 | no | — |
| 269 | C22 | — | 9 | 2 | no | — |
| 270 | C22 | — | 18 | 1 | no | — |
| 271 | C22 | — | 18 | 1 | no | — |
| 272 | C22 | Q8×#272 | 1 | 1 | yes | characteristic, complemented |
| 273 | C4 | — | 3 | 3 | no | — |
| 274 | C4 | — | 9 | 2 | no | — |
| 275 | C4 | — | 18 | 3 | no | — |
| 276 | C4 | — | 18 | 3 | no | — |
| 277 | C4 | C2×#6258 | 1 | 3 | yes | — |
| 278 | C3 | — | 3 | 1 | no | — |
| 279 | C3 | — | 12 | 3 | no | — |
| 280 | C3 | Q8×C3⋊S4 | 1 | 1 | yes | characteristic |
| 281 | C2 | — | 3 | 1 | no | — |
| 282 | C2 | — | 3 | 1 | no | — |
| 283 | C2 | — | 18 | 2 | no | — |
| 284 | C2 | C22×#6258 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. Q8×C22⋊H(3,𝔽3) | C2 | semidirect |
| 3. #11722380 | C2 | semidirect |
| 10. C22×C32×Q8 | D3 | semidirect |
| 12. C3×Q8×A4 | D3 | semidirect |
| 21. #6258 | Q8 | direct (N × H) |
| 47. C22⋊H(3,𝔽3) | C2×Q8 | semidirect |
| 74. C32×Q8 | S4 | semidirect |
| 127. C22×C32 | D3×Q8 | semidirect |
| 132. C3×A4 | D3×Q8 | semidirect |
| 141. C22×Q8 | #272 | semidirect |
| 238. C32 | Q8×S4 | semidirect |
| 245. Q8 | #6258 | direct (N × H) |
| 272. C22 | Q8×#272 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Q8×C22⋊H(3,𝔽3) | 2 |
| C4×#6258 | 2 |
| #11722380 | 2 |
| C3×Q8×S4 | 3 |
| C3×Q8×#67 | 3 |
| Q8×#272 | 4 |