Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
1036 conjugacy classes of subgroups; only the 182 normal subgroups are shown.
This group has 1036 conjugacy classes of subgroups — too many to draw. Only its 182 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 | C4×#6296 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #11723390 | C2 | 1 | 3 | yes | maximal |
| 3 | C3×#63870 | C2 | 1 | 3 | yes | maximal |
| 4 | #11723037 | C2 | 1 | 3 | yes | maximal |
| 5 | #11723015 | C2 | 1 | 3 | yes | maximal |
| 6 | #11722945 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | #11722945 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | #10594597 | — | 3 | 3 | no | maximal |
| 9 | C2×#6296 | C22 | 1 | 1 | yes | characteristic |
| 10 | C2×#6293 | C22 | 1 | 3 | yes | — |
| 11 | #88857 | C22 | 1 | 1 | yes | characteristic |
| 12 | C3×C4×C3⋊Dic3 | C22 | 1 | 3 | yes | — |
| 13 | C32×Dic3⋊C4 | C22 | 1 | 3 | yes | — |
| 14 | #88815 | C22 | 1 | 3 | yes | — |
| 15 | #88807 | C22 | 1 | 3 | yes | — |
| 16 | Dic3×C3⋊Dic3 | C22 | 1 | 3 | yes | — |
| 17 | Dic3×C3⋊Dic3 | C22 | 1 | 3 | yes | — |
| 18 | C3×#3724 | C22 | 1 | 3 | yes | — |
| 19 | C3×#3724 | C22 | 1 | 3 | yes | — |
| 20 | C3×Dic3⋊Dic3 | C22 | 1 | 6 | yes | — |
| 21 | #64096 | — | 3 | 3 | no | — |
| 22 | C3×#889 | — | 3 | 6 | no | — |
| 23 | #63919 | — | 3 | 3 | no | — |
| 24 | C4×#627 | — | 3 | 3 | no | — |
| 25 | #63870 | — | 3 | 3 | no | — |
| 26 | #63870 | D3 | 1 | 3 | yes | — |
| 27 | #63846 | — | 3 | 6 | no | — |
| 28 | #63840 | — | 3 | 6 | no | — |
| 29 | #63838 | — | 3 | 3 | no | — |
| 30 | #63838 | — | 3 | 3 | no | — |
| 31 | #63837 | — | 3 | 6 | no | — |
| 32 | #63837 | — | 6 | 3 | no | — |
| 33 | C2×C33×C4 | C23 | 1 | 1 | yes | characteristic |
| 34 | C2×C3⋊(C3⋊Dic3) | C23 | 1 | 1 | yes | characteristic |
| 35 | C2×C3⋊(C3⋊Dic3) | C23 | 1 | 1 | yes | characteristic |
| 36 | C2×C3×C3⋊Dic3 | C23 | 1 | 3 | yes | — |
| 37 | C2×C3×C3⋊Dic3 | C23 | 1 | 3 | yes | — |
| 38 | C2×C32×Dic3 | C23 | 1 | 3 | yes | — |
| 39 | C2×C32×Dic3 | C23 | 1 | 3 | yes | — |
| 40 | #6296 | Q8 | 1 | 4 | yes | complemented |
| 41 | #6293 | — | 2 | 6 | no | — |
| 42 | #5685 | — | 9 | 3 | no | — |
| 43 | C2×#634 | — | 3 | 3 | no | — |
| 44 | C2×#627 | — | 3 | 3 | no | — |
| 45 | C2×#627 | — | 3 | 6 | no | — |
| 46 | C2×#627 | — | 6 | 3 | no | — |
| 47 | C32×C4⋊C4 | — | 3 | 3 | no | — |
| 48 | #3750 | — | 3 | 3 | no | — |
| 49 | #3750 | — | 3 | 3 | no | — |
| 50 | #3750 | — | 6 | 3 | no | — |
| 51 | #3750 | — | 12 | 1 | no | — |
| 52 | C4×C3⋊Dic3 | — | 3 | 3 | no | — |
| 53 | C4×C3⋊Dic3 | — | 3 | 3 | no | — |
| 54 | C4×C3⋊Dic3 | C2×D3 | 1 | 3 | yes | — |
| 55 | C3×C4⋊Dic3 | — | 3 | 6 | no | — |
| 56 | C3×Dic3⋊C4 | — | 2 | 3 | no | — |
| 57 | C3×Dic3⋊C4 | — | 3 | 6 | no | — |
| 58 | C3×Dic3⋊C4 | — | 3 | 6 | no | — |
| 59 | C3×Dic3⋊C4 | — | 3 | 6 | no | — |
| 60 | C3×Dic3⋊C4 | C2×D3 | 1 | 6 | yes | — |
| 61 | C3×C4×Dic3 | — | 3 | 6 | no | — |
| 62 | C3×C4×Dic3 | — | 6 | 3 | no | — |
| 63 | #3724 | — | 3 | 3 | no | — |
| 64 | #3724 | C2×D3 | 1 | 3 | yes | — |
| 65 | #3724 | C2×D3 | 1 | 3 | yes | — |
| 66 | Dic3⋊Dic3 | — | 3 | 6 | no | — |
| 67 | Dic3⋊Dic3 | — | 3 | 6 | no | — |
| 68 | Dic3⋊Dic3 | — | 6 | 3 | no | — |
| 69 | Dic3⋊Dic3 | C2×D3 | 1 | 6 | yes | — |
| 70 | Dic32 | — | 3 | 6 | no | — |
| 71 | Dic32 | — | 3 | 6 | no | — |
| 72 | Dic32 | — | 6 | 3 | no | — |
| 73 | Dic32 | — | 6 | 3 | no | — |
| 74 | C22×C33 | C24 | 1 | 1 | yes | characteristic |
| 75 | C33×C4 | — | 2 | 1 | no | — |
| 76 | C3⋊(C3⋊Dic3) | — | 2 | 1 | no | — |
| 77 | C3⋊(C3⋊Dic3) | #41 | 1 | 2 | yes | — |
| 78 | C3×C3⋊Dic3 | — | 2 | 3 | no | — |
| 79 | C3×C3⋊Dic3 | C2×Q8 | 1 | 6 | yes | — |
| 80 | C32×Dic3 | — | 2 | 3 | no | — |
| 81 | C32×Dic3 | — | 2 | 3 | no | — |
| 82 | C3×#125 | — | 9 | 3 | no | — |
| 83 | #890 | — | 9 | 3 | no | — |
| 84 | #889 | — | 3 | 6 | no | — |
| 85 | #889 | — | 9 | 6 | no | — |
| 86 | #888 | — | 9 | 3 | no | — |
| 87 | #888 | — | 9 | 6 | no | — |
| 88 | #887 | — | 9 | 6 | no | — |
| 89 | #887 | — | 9 | 3 | no | — |
| 90 | #887 | — | 9 | 3 | no | — |
| 91 | #887 | — | 18 | 3 | no | — |
| 92 | C4×Dic6 | — | 9 | 3 | no | — |
| 93 | C4×Dic6 | — | 9 | 3 | no | — |
| 94 | C2×C32×C4 | — | 2 | 3 | no | — |
| 95 | C2×C32×C4 | — | 3 | 3 | no | — |
| 96 | C2×C32×C4 | — | 3 | 3 | no | — |
| 97 | C2×C32×C4 | — | 4 | 1 | no | — |
| 98 | C2×C32×C4 | C22×D3 | 1 | 3 | yes | — |
| 99 | C2×C3⋊Dic3 | — | 3 | 3 | no | — |
| 100 | C2×C3⋊Dic3 | — | 3 | 3 | no | — |
| 101 | C2×C3⋊Dic3 | — | 6 | 3 | no | — |
| 102 | C2×C3⋊Dic3 | — | 6 | 3 | no | — |
| 103 | C2×C3⋊Dic3 | — | 12 | 1 | no | — |
| 104 | C2×C3⋊Dic3 | — | 12 | 1 | no | — |
| 105 | C2×C3⋊Dic3 | C22×D3 | 1 | 3 | yes | — |
| 106 | C2×C3⋊Dic3 | C22×D3 | 1 | 3 | yes | — |
| 107 | #634 | — | 6 | 6 | no | — |
| 108 | C2×C3×Dic3 | — | 2 | 3 | no | — |
| 109 | C2×C3×Dic3 | — | 2 | 3 | no | — |
| 110 | C2×C3×Dic3 | — | 3 | 6 | no | — |
| 111 | C2×C3×Dic3 | — | 3 | 6 | no | — |
| 112 | C2×C3×Dic3 | — | 6 | 3 | no | — |
| 113 | C2×C3×Dic3 | — | 6 | 3 | no | — |
| 114 | C2×C3×Dic3 | C22×D3 | 1 | 6 | yes | — |
| 115 | C2×C3×Dic3 | C22×D3 | 1 | 6 | yes | — |
| 116 | #627 | — | 3 | 12 | no | — |
| 117 | #627 | — | 6 | 12 | no | — |
| 118 | #627 | — | 6 | 12 | no | — |
| 119 | #557 | — | 27 | 1 | no | Sylow |
| 120 | C2×C33 | C22×Q8 | 1 | 1 | yes | characteristic |
| 121 | C2×C33 | C2×#41 | 1 | 1 | yes | characteristic |
| 122 | C2×C33 | 2−1+4 | 1 | 1 | yes | characteristic |
| 123 | C2×Dic6 | — | 9 | 3 | no | — |
| 124 | C2×Dic6 | — | 9 | 6 | no | — |
| 125 | C2×Dic6 | — | 9 | 3 | no | — |
| 126 | C2×Dic6 | — | 18 | 3 | no | — |
| 127 | C3×C4⋊C4 | — | 3 | 6 | no | — |
| 128 | C3×C4⋊C4 | — | 6 | 3 | no | — |
| 129 | C3×C4⋊C4 | — | 9 | 3 | no | — |
| 130 | C3×C4⋊C4 | — | 9 | 3 | no | — |
| 131 | C3×C4⋊C4 | — | 9 | 6 | no | — |
| 132 | C3×C42 | — | 9 | 3 | no | — |
| 133 | C4⋊Dic3 | — | 3 | 6 | no | — |
| 134 | C4⋊Dic3 | — | 9 | 3 | no | — |
| 135 | C4⋊Dic3 | — | 9 | 3 | no | — |
| 136 | Dic3⋊C4 | — | 3 | 6 | no | — |
| 137 | Dic3⋊C4 | — | 3 | 6 | no | — |
| 138 | Dic3⋊C4 | — | 3 | 6 | no | — |
| 139 | Dic3⋊C4 | — | 9 | 6 | no | — |
| 140 | Dic3⋊C4 | — | 9 | 6 | no | — |
| 141 | Dic3⋊C4 | — | 9 | 3 | no | — |
| 142 | Dic3⋊C4 | — | 18 | 3 | no | — |
| 143 | Dic3⋊C4 | — | 18 | 3 | no | — |
| 144 | Dic3⋊C4 | — | 36 | 1 | no | — |
| 145 | Dic3⋊C4 | D32 | 1 | 3 | yes | — |
| 146 | C4×Dic3 | — | 3 | 6 | no | — |
| 147 | C4×Dic3 | — | 6 | 3 | no | — |
| 148 | C4×Dic3 | — | 9 | 6 | no | — |
| 149 | C4×Dic3 | — | 9 | 6 | no | — |
| 150 | C4×Dic3 | — | 9 | 3 | no | — |
| 151 | C4×Dic3 | — | 9 | 3 | no | — |
| 152 | C4×Dic3 | — | 18 | 3 | no | — |
| 153 | C4×Dic3 | — | 18 | 3 | no | — |
| 154 | C22×C32 | — | 2 | 3 | no | — |
| 155 | C22×C32 | — | 4 | 1 | no | — |
| 156 | C22×C32 | C23×D3 | 1 | 3 | yes | — |
| 157 | C32×C4 | — | 2 | 3 | no | — |
| 158 | C32×C4 | — | 4 | 3 | no | — |
| 159 | C32×C4 | — | 6 | 3 | no | — |
| 160 | C32×C4 | — | 6 | 3 | no | — |
| 161 | C32×C4 | — | 8 | 1 | no | — |
| 162 | C3⋊Dic3 | — | 2 | 3 | no | — |
| 163 | C3⋊Dic3 | — | 3 | 6 | no | — |
| 164 | C3⋊Dic3 | — | 6 | 6 | no | — |
| 165 | C3⋊Dic3 | — | 6 | 3 | no | — |
| 166 | C3⋊Dic3 | — | 12 | 2 | no | — |
| 167 | C3⋊Dic3 | — | 12 | 3 | no | — |
| 168 | C3⋊Dic3 | — | 24 | 1 | no | — |
| 169 | C3⋊Dic3 | D3×Q8 | 1 | 6 | yes | — |
| 170 | C3×Dic3 | — | 2 | 6 | no | — |
| 171 | C3×Dic3 | — | 2 | 6 | no | — |
| 172 | C3×Dic3 | — | 2 | 6 | no | — |
| 173 | C3×Dic3 | — | 3 | 12 | no | — |
| 174 | C3×Dic3 | — | 4 | 3 | no | — |
| 175 | C3×Dic3 | — | 6 | 6 | no | — |
| 176 | C3×Dic3 | — | 6 | 6 | no | — |
| 177 | C3×Dic3 | — | 6 | 6 | no | — |
| 178 | C4⋊Q8 | — | 27 | 3 | no | — |
| 179 | #125 | — | 9 | 3 | no | — |
| 180 | #125 | — | 27 | 3 | no | — |
| 181 | C4×Q8 | — | 27 | 1 | no | — |
| 182 | C4×Q8 | — | 27 | 3 | no | — |
| 183 | C4×Q8 | — | 27 | 1 | no | — |
| 184 | C4×Q8 | — | 27 | 1 | no | — |
| 185 | C33 | #557 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 186 | C2×C3×C4 | — | 2 | 3 | no | — |
| 187 | C2×C3×C4 | — | 3 | 6 | no | — |
| 188 | C2×C3×C4 | — | 3 | 6 | no | — |
| 189 | C2×C3×C4 | — | 4 | 1 | no | — |
| 190 | C2×C3×C4 | — | 6 | 3 | no | — |
| 191 | C2×C3×C4 | — | 6 | 3 | no | — |
| 192 | C2×C3×C4 | — | 9 | 3 | no | — |
| 193 | C2×C3×C4 | — | 9 | 3 | no | — |
| 194 | C2×C3×C4 | C2×D32 | 1 | 3 | yes | — |
| 195 | C2×Dic3 | — | 3 | 6 | no | — |
| 196 | C2×Dic3 | — | 3 | 6 | no | — |
| 197 | C2×Dic3 | — | 6 | 3 | no | — |
| 198 | C2×Dic3 | — | 6 | 3 | no | — |
| 199 | C2×Dic3 | — | 9 | 3 | no | — |
| 200 | C2×Dic3 | — | 9 | 3 | no | — |
| 201 | C2×Dic3 | — | 18 | 3 | no | — |
| 202 | C2×Dic3 | — | 18 | 3 | no | — |
| 203 | C2×Dic3 | — | 36 | 1 | no | — |
| 204 | C2×Dic3 | — | 36 | 1 | no | — |
| 205 | C2×Dic3 | C2×D32 | 1 | 3 | yes | — |
| 206 | C2×Dic3 | C2×D32 | 1 | 3 | yes | — |
| 207 | Dic6 | — | 9 | 12 | no | — |
| 208 | Dic6 | — | 18 | 6 | no | — |
| 209 | Dic6 | — | 18 | 12 | no | — |
| 210 | Dic6 | — | 18 | 12 | no | — |
| 211 | C2×C32 | — | 2 | 3 | no | — |
| 212 | C2×C32 | — | 2 | 3 | no | — |
| 213 | C2×C32 | — | 2 | 3 | no | — |
| 214 | C2×C32 | — | 4 | 1 | no | — |
| 215 | C2×C32 | — | 4 | 1 | no | — |
| 216 | C2×C32 | — | 4 | 1 | no | — |
| 217 | C2×C32 | C2×D3×Q8 | 1 | 3 | yes | — |
| 218 | C2×C32 | D3×#41 | 1 | 3 | yes | — |
| 219 | C2×C32 | #1010 | 1 | 3 | yes | — |
| 220 | C2×Q8 | — | 27 | 3 | no | — |
| 221 | C2×Q8 | — | 27 | 1 | no | — |
| 222 | C4⋊C4 | — | 3 | 3 | no | — |
| 223 | C4⋊C4 | — | 9 | 3 | no | — |
| 224 | C4⋊C4 | — | 9 | 3 | no | — |
| 225 | C4⋊C4 | — | 9 | 6 | no | — |
| 226 | C4⋊C4 | — | 27 | 3 | no | — |
| 227 | C4⋊C4 | — | 27 | 3 | no | — |
| 228 | C4⋊C4 | — | 27 | 1 | no | — |
| 229 | C42 | — | 9 | 3 | no | — |
| 230 | C42 | — | 27 | 3 | no | — |
| 231 | C42 | — | 27 | 3 | no | — |
| 232 | C22×C3 | — | 2 | 3 | no | — |
| 233 | C22×C3 | — | 4 | 1 | no | — |
| 234 | C22×C3 | C22×D32 | 1 | 3 | yes | — |
| 235 | C3×C4 | — | 2 | 3 | no | — |
| 236 | C3×C4 | — | 4 | 3 | no | — |
| 237 | C3×C4 | — | 6 | 6 | no | — |
| 238 | C3×C4 | — | 6 | 6 | no | — |
| 239 | C3×C4 | — | 6 | 6 | no | — |
| 240 | C3×C4 | — | 8 | 1 | no | — |
| 241 | C3×C4 | — | 9 | 6 | no | — |
| 242 | C3×C4 | — | 12 | 3 | no | — |
| 243 | C3×C4 | — | 18 | 3 | no | — |
| 244 | Dic3 | — | 2 | 3 | no | — |
| 245 | Dic3 | — | 2 | 3 | no | — |
| 246 | Dic3 | — | 3 | 12 | no | — |
| 247 | Dic3 | — | 6 | 6 | no | — |
| 248 | Dic3 | — | 6 | 6 | no | — |
| 249 | Dic3 | — | 6 | 6 | no | — |
| 250 | Dic3 | — | 9 | 6 | no | — |
| 251 | Dic3 | — | 18 | 6 | no | — |
| 252 | Dic3 | — | 18 | 3 | no | — |
| 253 | Dic3 | — | 36 | 2 | no | — |
| 254 | Dic3 | — | 36 | 3 | no | — |
| 255 | Dic3 | — | 72 | 1 | no | — |
| 256 | C32 | — | 2 | 3 | no | — |
| 257 | C32 | — | 4 | 1 | no | — |
| 258 | C32 | #5685 | 1 | 3 | yes | complemented |
| 259 | Q8 | — | 27 | 4 | no | — |
| 260 | Q8 | — | 54 | 6 | no | — |
| 261 | C2×C4 | — | 3 | 3 | no | — |
| 262 | C2×C4 | — | 3 | 3 | no | — |
| 263 | C2×C4 | — | 9 | 3 | no | — |
| 264 | C2×C4 | — | 9 | 3 | no | — |
| 265 | C2×C4 | — | 27 | 1 | no | — |
| 266 | C2×C4 | — | 27 | 1 | no | — |
| 267 | C2×C4 | D33 | 1 | 1 | yes | characteristic |
| 268 | C2×C3 | — | 2 | 3 | no | — |
| 269 | C2×C3 | — | 2 | 3 | no | — |
| 270 | C2×C3 | — | 2 | 3 | no | — |
| 271 | C2×C3 | — | 4 | 1 | no | — |
| 272 | C2×C3 | — | 4 | 1 | no | — |
| 273 | C2×C3 | — | 4 | 1 | no | — |
| 274 | C2×C3 | #64306 | 1 | 3 | yes | — |
| 275 | C2×C3 | #64309 | 1 | 3 | yes | — |
| 276 | C2×C3 | D32×Q8 | 1 | 3 | yes | — |
| 277 | C22 | C2×D33 | 1 | 1 | yes | characteristic |
| 278 | C4 | — | 2 | 1 | no | — |
| 279 | C4 | — | 6 | 3 | no | — |
| 280 | C4 | — | 6 | 3 | no | — |
| 281 | C4 | — | 9 | 6 | no | — |
| 282 | C4 | — | 18 | 3 | no | — |
| 283 | C4 | — | 27 | 2 | no | — |
| 284 | C4 | — | 54 | 1 | no | — |
| 285 | C3 | — | 2 | 3 | no | — |
| 286 | C3 | — | 4 | 1 | no | — |
| 287 | C3 | #10594597 | 1 | 3 | yes | complemented |
| 288 | C2 | #11724378 | 1 | 1 | yes | characteristic |
| 289 | C2 | #11724398 | 1 | 1 | yes | characteristic |
| 290 | C2 | #11724428 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C2×D33 |
| commutator, socle, prime radical, cosocle | C22×C33 | C24 |
| Fitting | C2×C33×C4 | C23 |
| hypercenter | C2×C4 | D33 |
| hypocenter | C33 | #557 |
| LCS | G ▷ C22×C33 ▷ C33 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 40. #6296 | Q8 | semidirect |
| 185. C33 | #557 | semidirect |
| 258. C32 | #5685 | semidirect |
| 287. C3 | #10594597 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×#6296 | 2 |
| #11723390 | 2 |
| C3×#63870 | 2 |
| #11723037 | 2 |
| #11723015 | 2 |
| #11722945 | 2 |
| #10594597 | 3 |