Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
1216 conjugacy classes of subgroups; only the 164 normal subgroups are shown.
This group has 1216 conjugacy classes of subgroups — too many to draw. Only its 164 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 | C32×#1004 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #11723977 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #11723379 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | C3×#63660 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #11722043 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #10592941 | — | 3 | 1 | no | maximal |
| 7 | #10591513 | — | 3 | 4 | no | maximal |
| 8 | C2×C33×D4 | C22 | 1 | 1 | yes | characteristic |
| 9 | C2×#6308 | C22 | 1 | 1 | yes | characteristic, complemented |
| 10 | C2×C32×#67 | — | 2 | 2 | no | — |
| 11 | C32×#237 | — | 2 | 2 | no | — |
| 12 | C32×D3×D4 | — | 2 | 2 | no | — |
| 13 | C32×#235 | C22 | 1 | 2 | yes | — |
| 14 | #88864 | — | 2 | 2 | no | — |
| 15 | C4×C3⋊(C3⋊Dic3) | C22 | 1 | 1 | yes | characteristic, complemented |
| 16 | C3×#3748 | C22 | 1 | 2 | yes | complemented |
| 17 | #88799 | — | 2 | 2 | no | — |
| 18 | #88794 | — | 2 | 2 | no | — |
| 19 | C32×H(5,𝔽2) | — | 3 | 1 | no | — |
| 20 | C3×#1004 | D3 | 1 | 4 | yes | — |
| 21 | #64160 | — | 3 | 2 | no | — |
| 22 | #64145 | — | 3 | 1 | no | — |
| 23 | #64145 | — | 3 | 4 | no | — |
| 24 | #64145 | — | 6 | 4 | no | — |
| 25 | #63831 | — | 3 | 8 | no | — |
| 26 | #63664 | — | 3 | 2 | no | — |
| 27 | #63660 | D3 | 1 | 1 | yes | characteristic, complemented |
| 28 | C3×#833 | — | 3 | 4 | no | — |
| 29 | #63568 | — | 3 | 8 | no | — |
| 30 | C23×C33 | D4 | 1 | 2 | yes | — |
| 31 | C22×C32×D3 | — | 2 | 2 | no | — |
| 32 | C33×D4 | — | 2 | 2 | no | — |
| 33 | C2×C33×C4 | C23 | 1 | 1 | yes | characteristic |
| 34 | C2×C3⋊(C3⋊Dic3) | — | 2 | 1 | no | — |
| 35 | C2×C3⋊(C3⋊Dic3) | — | 2 | 1 | no | — |
| 36 | #6308 | — | 2 | 2 | no | — |
| 37 | C32×#67 | — | 2 | 4 | no | — |
| 38 | C32×#67 | — | 2 | 2 | no | — |
| 39 | C2×C32×Dic3 | — | 2 | 2 | no | — |
| 40 | C32×D12 | D4 | 1 | 2 | yes | — |
| 41 | C32×C4×D3 | — | 2 | 2 | no | — |
| 42 | C32×Dic6 | D4 | 1 | 2 | yes | — |
| 43 | C3×C3⋊(C3⋊C8) | — | 2 | 2 | no | — |
| 44 | #5242 | — | 9 | 4 | no | — |
| 45 | #4746 | — | 9 | 1 | no | — |
| 46 | C32×#41 | — | 3 | 2 | no | — |
| 47 | C32×#41 | — | 6 | 2 | no | — |
| 48 | C2×C32×D4 | — | 2 | 4 | no | — |
| 49 | C2×C32×D4 | — | 6 | 2 | no | — |
| 50 | C2×C32×D4 | — | 6 | 2 | no | — |
| 51 | C2×C32×D4 | C2×D3 | 1 | 1 | yes | characteristic |
| 52 | C2×C32×D4 | C2×D3 | 1 | 4 | yes | — |
| 53 | C2×#634 | — | 3 | 1 | no | — |
| 54 | C2×#634 | — | 3 | 4 | no | — |
| 55 | C2×#634 | — | 6 | 4 | no | — |
| 56 | C2×C3×#67 | — | 2 | 8 | no | — |
| 57 | C3×#237 | — | 2 | 8 | no | — |
| 58 | C3×D3×D4 | — | 2 | 8 | no | — |
| 59 | C3×#235 | C2×D3 | 1 | 8 | yes | — |
| 60 | #3757 | — | 6 | 16 | no | — |
| 61 | #3757 | — | 6 | 2 | no | — |
| 62 | #3757 | — | 6 | 8 | no | — |
| 63 | #3756 | — | 6 | 2 | no | — |
| 64 | #3754 | — | 6 | 2 | no | — |
| 65 | C4×C3⋊Dic3 | — | 3 | 1 | no | — |
| 66 | C4×C3⋊Dic3 | — | 3 | 4 | no | — |
| 67 | C4×C3⋊Dic3 | — | 6 | 4 | no | — |
| 68 | #3748 | C2×D3 | 1 | 2 | yes | complemented |
| 69 | C3×#208 | — | 3 | 8 | no | — |
| 70 | #3719 | — | 6 | 8 | no | — |
| 71 | #3716 | — | 6 | 8 | no | — |
| 72 | C22×C33 | — | 2 | 2 | no | — |
| 73 | C22×C33 | — | 4 | 2 | no | — |
| 74 | C22×C33 | C2×D4 | 1 | 1 | yes | characteristic |
| 75 | C2×C32×D3 | — | 2 | 2 | no | — |
| 76 | C2×C32×D3 | — | 8 | 2 | no | — |
| 77 | C33×C4 | C2×D4 | 1 | 2 | yes | — |
| 78 | C3⋊(C3⋊Dic3) | — | 4 | 1 | no | — |
| 79 | C3⋊(C3⋊Dic3) | — | 4 | 1 | no | — |
| 80 | C32×Dic3 | — | 2 | 2 | no | — |
| 81 | C3×H(5,𝔽2) | — | 3 | 4 | no | — |
| 82 | #1004 | C3⋊D3 | 1 | 1 | yes | characteristic |
| 83 | #951 | — | 9 | 8 | no | — |
| 84 | #936 | — | 9 | 4 | no | — |
| 85 | #936 | — | 9 | 1 | no | — |
| 86 | #936 | — | 18 | 4 | no | — |
| 87 | #909 | — | 9 | 2 | no | — |
| 88 | C3×#100 | — | 9 | 1 | no | — |
| 89 | #837 | — | 9 | 8 | no | — |
| 90 | #833 | — | 3 | 4 | no | — |
| 91 | #825 | — | 9 | 2 | no | — |
| 92 | C23×C32 | — | 2 | 8 | no | — |
| 93 | C23×C32 | — | 6 | 2 | no | — |
| 94 | C23×C32 | D12 | 1 | 2 | yes | — |
| 95 | C23×C32 | #67 | 1 | 8 | yes | — |
| 96 | C22×C3×D3 | — | 2 | 8 | no | — |
| 97 | C32×Q8 | — | 3 | 2 | no | — |
| 98 | C32×D4 | — | 2 | 2 | no | — |
| 99 | C32×D4 | — | 2 | 8 | no | — |
| 100 | C32×D4 | — | 3 | 2 | no | — |
| 101 | C32×D4 | — | 4 | 8 | no | — |
| 102 | C32×D4 | — | 6 | 4 | no | — |
| 103 | C32×D4 | — | 6 | 2 | no | — |
| 104 | C2×C32×C4 | — | 2 | 4 | no | — |
| 105 | C2×C32×C4 | — | 6 | 2 | no | — |
| 106 | C2×C32×C4 | — | 6 | 2 | no | — |
| 107 | C2×C32×C4 | C22×D3 | 1 | 4 | yes | — |
| 108 | C2×C32×C4 | C22×D3 | 1 | 1 | yes | characteristic |
| 109 | C2×C3⋊Dic3 | — | 6 | 8 | no | — |
| 110 | C2×C3⋊Dic3 | — | 6 | 8 | no | — |
| 111 | C2×C3⋊Dic3 | — | 6 | 1 | no | — |
| 112 | C2×C3⋊Dic3 | — | 6 | 1 | no | — |
| 113 | C2×C3⋊Dic3 | — | 6 | 4 | no | — |
| 114 | C2×C3⋊Dic3 | — | 6 | 4 | no | — |
| 115 | #634 | — | 6 | 2 | no | — |
| 116 | #634 | — | 6 | 8 | no | — |
| 117 | #634 | — | 12 | 8 | no | — |
| 118 | C3×#67 | — | 2 | 16 | no | — |
| 119 | C3×#67 | — | 2 | 8 | no | — |
| 120 | C2×C3×Dic3 | — | 2 | 8 | no | — |
| 121 | C3×D12 | #67 | 1 | 8 | yes | — |
| 122 | C3×C4×D3 | — | 2 | 8 | no | — |
| 123 | C3×Dic6 | #67 | 1 | 8 | yes | — |
| 124 | C3⋊(C3⋊C8) | — | 2 | 2 | no | — |
| 125 | C3×C3⋊C8 | — | 6 | 8 | no | — |
| 126 | #455 | — | 27 | 1 | no | Sylow |
| 127 | C2×C33 | — | 2 | 1 | no | — |
| 128 | C2×C33 | — | 4 | 2 | no | — |
| 129 | C2×C33 | #120 | 1 | 1 | yes | characteristic |
| 130 | C32×D3 | — | 4 | 2 | no | — |
| 131 | C3×#41 | — | 3 | 8 | no | — |
| 132 | C3×#41 | — | 6 | 8 | no | — |
| 133 | C2×C3×D4 | — | 2 | 4 | no | — |
| 134 | C2×C3×D4 | — | 6 | 8 | no | — |
| 135 | C2×C3×D4 | — | 6 | 8 | no | — |
| 136 | C2×C3×D4 | D32 | 1 | 4 | yes | — |
| 137 | C2×C3×D4 | C2×C3⋊D3 | 1 | 1 | yes | characteristic |
| 138 | C2×#67 | — | 2 | 2 | no | — |
| 139 | #237 | — | 2 | 2 | no | — |
| 140 | D3×D4 | — | 2 | 2 | no | — |
| 141 | #235 | C2×C3⋊D3 | 1 | 2 | yes | — |
| 142 | C2×Dic6 | — | 9 | 4 | no | — |
| 143 | C2×Dic6 | — | 9 | 1 | no | — |
| 144 | C2×Dic6 | — | 18 | 4 | no | — |
| 145 | C3×M16 | — | 9 | 2 | no | — |
| 146 | #217 | — | 18 | 16 | no | — |
| 147 | #217 | — | 18 | 8 | no | — |
| 148 | #217 | — | 18 | 2 | no | — |
| 149 | #216 | — | 18 | 8 | no | — |
| 150 | #214 | — | 18 | 8 | no | — |
| 151 | C4×Dic3 | — | 9 | 4 | no | — |
| 152 | C4×Dic3 | — | 9 | 1 | no | — |
| 153 | C4×Dic3 | — | 18 | 4 | no | — |
| 154 | #208 | — | 3 | 8 | no | — |
| 155 | Dic12 | — | 18 | 2 | no | — |
| 156 | #204 | — | 18 | 2 | no | — |
| 157 | C22×C32 | — | 2 | 2 | no | — |
| 158 | C22×C32 | — | 2 | 4 | no | — |
| 159 | C22×C32 | — | 2 | 8 | no | — |
| 160 | C22×C32 | — | 4 | 8 | no | — |
| 161 | C22×C32 | — | 4 | 2 | no | — |
| 162 | C22×C32 | — | 4 | 8 | no | — |
| 163 | C22×C32 | — | 4 | 16 | no | — |
| 164 | C22×C32 | — | 6 | 2 | no | — |
| 165 | C22×C32 | — | 24 | 2 | no | — |
| 166 | C22×C32 | C2×D12 | 1 | 1 | yes | characteristic |
| 167 | C22×C32 | C2×#67 | 1 | 4 | yes | — |
| 168 | C2×C3×D3 | — | 2 | 8 | no | — |
| 169 | C2×C3×D3 | — | 8 | 8 | no | — |
| 170 | C32×C4 | — | 2 | 8 | no | — |
| 171 | C32×C4 | — | 6 | 2 | no | — |
| 172 | C32×C4 | D3×D4 | 1 | 2 | yes | — |
| 173 | C32×C4 | C2×#67 | 1 | 8 | yes | — |
| 174 | C3⋊Dic3 | — | 12 | 8 | no | — |
| 175 | C3⋊Dic3 | — | 12 | 8 | no | — |
| 176 | C3⋊Dic3 | — | 12 | 4 | no | — |
| 177 | C3⋊Dic3 | — | 12 | 1 | no | — |
| 178 | C3⋊Dic3 | — | 12 | 1 | no | — |
| 179 | C3⋊Dic3 | — | 12 | 4 | no | — |
| 180 | C3×Dic3 | — | 2 | 8 | no | — |
| 181 | H(5,𝔽2) | — | 3 | 1 | no | — |
| 182 | #137 | — | 27 | 2 | no | — |
| 183 | #124 | — | 27 | 1 | no | — |
| 184 | #104 | — | 27 | 2 | no | — |
| 185 | #100 | — | 9 | 1 | no | — |
| 186 | C33 | #455 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 187 | C23×C3 | — | 2 | 8 | no | — |
| 188 | C23×C3 | — | 6 | 8 | no | — |
| 189 | C23×C3 | #626 | 1 | 8 | yes | — |
| 190 | C23×C3 | #638 | 1 | 2 | yes | — |
| 191 | C22×D3 | — | 2 | 2 | no | — |
| 192 | C3×Q8 | — | 3 | 8 | no | — |
| 193 | C3×D4 | — | 2 | 8 | no | — |
| 194 | C3×D4 | — | 2 | 2 | no | — |
| 195 | C3×D4 | — | 3 | 8 | no | — |
| 196 | C3×D4 | — | 4 | 8 | no | — |
| 197 | C3×D4 | — | 6 | 16 | no | — |
| 198 | C3×D4 | — | 6 | 8 | no | — |
| 199 | C2×C3×C4 | — | 2 | 4 | no | — |
| 200 | C2×C3×C4 | — | 6 | 8 | no | — |
| 201 | C2×C3×C4 | — | 6 | 8 | no | — |
| 202 | C2×C3×C4 | C2×D32 | 1 | 4 | yes | — |
| 203 | C2×C3×C4 | C22×C3⋊D3 | 1 | 1 | yes | characteristic |
| 204 | #67 | — | 2 | 4 | no | — |
| 205 | #67 | — | 2 | 2 | no | — |
| 206 | C2×Dic3 | — | 2 | 2 | no | — |
| 207 | C2×Dic3 | — | 18 | 8 | no | — |
| 208 | C2×Dic3 | — | 18 | 8 | no | — |
| 209 | C2×Dic3 | — | 18 | 4 | no | — |
| 210 | C2×Dic3 | — | 18 | 4 | no | — |
| 211 | C2×Dic3 | — | 18 | 1 | no | — |
| 212 | C2×Dic3 | — | 18 | 1 | no | — |
| 213 | D12 | #638 | 1 | 2 | yes | — |
| 214 | C4×D3 | — | 2 | 2 | no | — |
| 215 | Dic6 | — | 18 | 8 | no | — |
| 216 | Dic6 | — | 18 | 2 | no | — |
| 217 | Dic6 | — | 36 | 8 | no | — |
| 218 | Dic6 | #638 | 1 | 2 | yes | — |
| 219 | C3×C8 | — | 18 | 2 | no | — |
| 220 | C3⋊C8 | — | 6 | 8 | no | — |
| 221 | C2×C32 | — | 2 | 8 | no | — |
| 222 | C2×C32 | — | 2 | 4 | no | — |
| 223 | C2×C32 | — | 2 | 1 | no | — |
| 224 | C2×C32 | — | 2 | 4 | no | — |
| 225 | C2×C32 | — | 4 | 2 | no | — |
| 226 | C2×C32 | — | 4 | 8 | no | — |
| 227 | C2×C32 | — | 8 | 8 | no | — |
| 228 | C2×C32 | — | 12 | 2 | no | — |
| 229 | C2×C32 | #882 | 1 | 1 | yes | characteristic |
| 230 | C2×C32 | #953 | 1 | 4 | yes | — |
| 231 | C3×D3 | — | 4 | 8 | no | — |
| 232 | #41 | — | 3 | 2 | no | — |
| 233 | #41 | — | 6 | 2 | no | — |
| 234 | C2×Q8 | — | 27 | 1 | no | — |
| 235 | C2×D4 | — | 6 | 2 | no | — |
| 236 | C2×D4 | — | 6 | 2 | no | — |
| 237 | C2×D4 | D3×C3⋊D3 | 1 | 1 | yes | characteristic |
| 238 | Q16 | — | 54 | 2 | no | — |
| 239 | SD16 | — | 54 | 2 | no | — |
| 240 | M16 | — | 9 | 2 | no | — |
| 241 | C22⋊C4 | — | 54 | 2 | no | — |
| 242 | C42 | — | 27 | 1 | no | — |
| 243 | C22×C3 | — | 2 | 2 | no | — |
| 244 | C22×C3 | — | 2 | 8 | no | — |
| 245 | C22×C3 | — | 2 | 4 | no | — |
| 246 | C22×C3 | — | 4 | 8 | no | — |
| 247 | C22×C3 | — | 4 | 8 | no | — |
| 248 | C22×C3 | — | 4 | 2 | no | — |
| 249 | C22×C3 | — | 4 | 16 | no | — |
| 250 | C22×C3 | — | 6 | 8 | no | — |
| 251 | C22×C3 | — | 24 | 8 | no | — |
| 252 | C22×C3 | C2×#626 | 1 | 4 | yes | — |
| 253 | C22×C3 | C2×#638 | 1 | 1 | yes | characteristic |
| 254 | C2×D3 | — | 2 | 2 | no | — |
| 255 | C2×D3 | — | 8 | 2 | no | — |
| 256 | C3×C4 | — | 2 | 8 | no | — |
| 257 | C3×C4 | — | 6 | 8 | no | — |
| 258 | C3×C4 | D3×#67 | 1 | 8 | yes | — |
| 259 | C3×C4 | C2×#638 | 1 | 2 | yes | — |
| 260 | Dic3 | — | 2 | 2 | no | — |
| 261 | Dic3 | — | 36 | 8 | no | — |
| 262 | Dic3 | — | 36 | 8 | no | — |
| 263 | Dic3 | — | 36 | 4 | no | — |
| 264 | Dic3 | — | 36 | 1 | no | — |
| 265 | Dic3 | — | 36 | 1 | no | — |
| 266 | Dic3 | — | 36 | 4 | no | — |
| 267 | C32 | — | 2 | 4 | no | — |
| 268 | C32 | #4746 | 1 | 1 | yes | characteristic, complemented |
| 269 | C32 | #5242 | 1 | 4 | yes | complemented |
| 270 | C23 | — | 6 | 2 | no | — |
| 271 | C23 | #6291 | 1 | 2 | yes | — |
| 272 | Q8 | — | 3 | 2 | no | — |
| 273 | Q8 | — | 54 | 2 | no | — |
| 274 | D4 | — | 2 | 2 | no | — |
| 275 | D4 | — | 3 | 2 | no | — |
| 276 | D4 | — | 6 | 4 | no | — |
| 277 | D4 | — | 6 | 2 | no | — |
| 278 | C2×C4 | — | 6 | 2 | no | — |
| 279 | C2×C4 | — | 6 | 2 | no | — |
| 280 | C2×C4 | — | 54 | 1 | no | — |
| 281 | C2×C4 | — | 54 | 1 | no | — |
| 282 | C2×C4 | C2×D3×C3⋊D3 | 1 | 1 | yes | characteristic |
| 283 | C8 | — | 18 | 2 | no | — |
| 284 | C2×C3 | — | 2 | 8 | no | — |
| 285 | C2×C3 | — | 2 | 4 | no | — |
| 286 | C2×C3 | — | 2 | 1 | no | — |
| 287 | C2×C3 | — | 2 | 4 | no | — |
| 288 | C2×C3 | — | 4 | 8 | no | — |
| 289 | C2×C3 | — | 4 | 2 | no | — |
| 290 | C2×C3 | — | 8 | 8 | no | — |
| 291 | C2×C3 | — | 12 | 8 | no | — |
| 292 | C2×C3 | #63977 | 1 | 4 | yes | — |
| 293 | C2×C3 | #64162 | 1 | 1 | yes | characteristic |
| 294 | D3 | — | 4 | 2 | no | — |
| 295 | C22 | — | 2 | 2 | no | — |
| 296 | C22 | — | 4 | 2 | no | — |
| 297 | C22 | — | 6 | 2 | no | — |
| 298 | C22 | — | 24 | 2 | no | — |
| 299 | C22 | C2×#6291 | 1 | 1 | yes | characteristic |
| 300 | C4 | — | 6 | 2 | no | — |
| 301 | C4 | — | 108 | 1 | no | — |
| 302 | C4 | — | 108 | 1 | no | — |
| 303 | C4 | D3×#638 | 1 | 2 | yes | — |
| 304 | C3 | — | 2 | 4 | no | — |
| 305 | C3 | #10591513 | 1 | 4 | yes | complemented |
| 306 | C3 | #10592941 | 1 | 1 | yes | characteristic, complemented |
| 307 | C2 | — | 2 | 1 | no | — |
| 308 | C2 | — | 4 | 2 | no | — |
| 309 | C2 | — | 12 | 2 | no | — |
| 310 | C2 | #11723630 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #11723977 | C2 | semidirect |
| 3. #11723379 | C2 | semidirect |
| 4. C3×#63660 | C2 | semidirect |
| 5. #11722043 | C2 | semidirect |
| 9. C2×#6308 | C22 | semidirect |
| 15. C4×C3⋊(C3⋊Dic3) | C22 | semidirect |
| 16. C3×#3748 | C22 | semidirect |
| 27. #63660 | D3 | semidirect |
| 68. #3748 | C2×D3 | semidirect |
| 186. C33 | #455 | semidirect |
| 268. C32 | #4746 | semidirect |
| 269. C32 | #5242 | semidirect |
| 305. C3 | #10591513 | semidirect |
| 306. C3 | #10592941 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×#1004 | 2 |
| #11723977 | 2 |
| #11723379 | 2 |
| C3×#63660 | 2 |
| #11722043 | 2 |
| #10592941 | 3 |
| #10591513 | 3 |