Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
298 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 298 conjugacy classes of subgroups — too many to draw. Only its 44 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 | D3×#3777 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #11724116 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #11724104 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #11724101 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #11724100 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C3×AGL(1,𝔽9)⋊C4 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | (C2×D3)⋊AGL(1,𝔽9) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #10596412 | — | 3 | 1 | no | maximal |
| 9 | C2×#157⋊D3 | C22 | 1 | 1 | yes | characteristic |
| 10 | C2×D3×#157 | — | 2 | 1 | no | — |
| 11 | C2×D3×#157 | C22 | 1 | 1 | yes | characteristic |
| 12 | C2×C3⋊AGL(1,𝔽9) | C22 | 1 | 1 | yes | characteristic |
| 13 | C2×C3×AGL(1,𝔽9) | C22 | 1 | 1 | yes | characteristic |
| 14 | #88955 | — | 2 | 1 | no | — |
| 15 | #88946 | C22 | 1 | 1 | yes | characteristic |
| 16 | #88945 | — | 2 | 1 | no | — |
| 17 | C3×#3777 | C22 | 1 | 1 | yes | characteristic |
| 18 | #88932 | — | 2 | 1 | no | — |
| 19 | C3×#3773 | C22 | 1 | 1 | yes | characteristic |
| 20 | C2×#3777 | — | 3 | 1 | no | — |
| 21 | #64242 | — | 3 | 1 | no | — |
| 22 | C22⋊AGL(1,𝔽9) | — | 3 | 1 | no | — |
| 23 | AGL(1,𝔽9)⋊C4 | — | 3 | 1 | no | — |
| 24 | AGL(1,𝔽9)⋊C4 | D3 | 1 | 1 | yes | characteristic, complemented |
| 25 | GO+(4,𝔽2)⋊C4 | — | 3 | 1 | no | — |
| 26 | GO+(4,𝔽2)⋊C4 | — | 3 | 1 | no | — |
| 27 | C2×#1113 | — | 2 | 1 | no | — |
| 28 | C2×D3×C3⋊D3 | D4 | 1 | 1 | yes | characteristic |
| 29 | C2×C3⋊#157 | — | 2 | 1 | no | — |
| 30 | C2×C3⋊#157 | D4 | 1 | 1 | yes | characteristic |
| 31 | C2×C3×#157 | — | 2 | 1 | no | — |
| 32 | C2×C3×#157 | C23 | 1 | 1 | yes | characteristic |
| 33 | #157⋊D3 | — | 2 | 1 | no | — |
| 34 | #157⋊D3 | — | 2 | 1 | no | — |
| 35 | D3×#157 | — | 2 | 1 | no | — |
| 36 | D3×#157 | — | 2 | 1 | no | — |
| 37 | D3×#157 | — | 4 | 2 | no | — |
| 38 | C3⋊AGL(1,𝔽9) | — | 2 | 1 | no | — |
| 39 | C3×AGL(1,𝔽9) | — | 2 | 1 | no | — |
| 40 | #6292 | — | 4 | 1 | no | — |
| 41 | C3×#626 | — | 4 | 1 | no | — |
| 42 | C3×#624 | — | 2 | 1 | no | — |
| 43 | #4863 | — | 9 | 1 | no | maximal |
| 44 | C22×#157 | — | 3 | 1 | no | — |
| 45 | C22×#157 | — | 6 | 1 | no | — |
| 46 | C2×GO+(4,𝔽2) | — | 3 | 1 | no | — |
| 47 | C2×AGL(1,𝔽9) | — | 3 | 1 | no | — |
| 48 | C2×AGL(1,𝔽9) | C2×D3 | 1 | 1 | yes | characteristic |
| 49 | #3811 | — | 6 | 1 | no | — |
| 50 | D3×D12 | — | 12 | 1 | no | — |
| 51 | #3777 | — | 3 | 1 | no | — |
| 52 | #3777 | — | 6 | 1 | no | — |
| 53 | #3777 | C2×D3 | 1 | 1 | yes | characteristic |
| 54 | #3773 | C2×D3 | 1 | 1 | yes | characteristic |
| 55 | D32⋊C4 | — | 6 | 1 | no | — |
| 56 | C2×C3⋊(C3⋊D3) | SD16 | 1 | 1 | yes | characteristic |
| 57 | C2×C3×C3⋊D3 | — | 4 | 1 | no | — |
| 58 | C2×C3×C3⋊D3 | C2×D4 | 1 | 1 | yes | characteristic |
| 59 | C2×C32×D3 | SD16 | 1 | 1 | yes | characteristic |
| 60 | #1113 | — | 4 | 1 | no | — |
| 61 | #1113 | — | 4 | 1 | no | — |
| 62 | D3×C3⋊D3 | — | 2 | 1 | no | — |
| 63 | D3×C3⋊D3 | — | 2 | 1 | no | — |
| 64 | C3⋊#157 | — | 2 | 1 | no | — |
| 65 | C3⋊#157 | — | 4 | 1 | no | — |
| 66 | C3×#157 | — | 4 | 1 | no | — |
| 67 | C3×#157 | #41 | 1 | 1 | yes | characteristic |
| 68 | C3×#157 | #41 | 1 | 1 | yes | characteristic |
| 69 | C32×Dic3 | — | 4 | 1 | no | — |
| 70 | #895 | — | 9 | 1 | no | — |
| 71 | D3×C4⋊C4 | — | 9 | 1 | no | — |
| 72 | C3×#106 | — | 9 | 1 | no | — |
| 73 | #821 | — | 9 | 1 | no | — |
| 74 | (C2×D3)⋊C8 | — | 9 | 1 | no | — |
| 75 | #809 | — | 9 | 1 | no | — |
| 76 | #808 | — | 9 | 1 | no | — |
| 77 | C22×C3⋊D3 | — | 3 | 1 | no | — |
| 78 | C2×D32 | — | 6 | 1 | no | — |
| 79 | C2×D32 | — | 12 | 1 | no | — |
| 80 | C2×D32 | — | 12 | 1 | no | — |
| 81 | C2×#157 | — | 2 | 1 | no | — |
| 82 | C2×#157 | — | 3 | 1 | no | — |
| 83 | C2×#157 | — | 6 | 1 | no | — |
| 84 | C2×#157 | — | 6 | 1 | no | — |
| 85 | C2×#157 | — | 6 | 1 | no | — |
| 86 | C2×#157 | — | 12 | 2 | no | — |
| 87 | C2×#157 | C22×D3 | 1 | 1 | yes | characteristic |
| 88 | GO+(4,𝔽2) | — | 6 | 1 | no | — |
| 89 | GO+(4,𝔽2) | — | 6 | 1 | no | — |
| 90 | AGL(1,𝔽9) | — | 2 | 1 | no | — |
| 91 | AGL(1,𝔽9) | — | 6 | 1 | no | — |
| 92 | C4⋊(C3⋊D3) | — | 12 | 1 | no | — |
| 93 | C3×#67 | — | 12 | 1 | no | — |
| 94 | C3×D12 | — | 12 | 1 | no | — |
| 95 | C3×C4×D3 | — | 12 | 1 | no | — |
| 96 | #626 | — | 4 | 1 | no | — |
| 97 | #626 | — | 12 | 1 | no | — |
| 98 | #626 | — | 12 | 1 | no | — |
| 99 | #626 | — | 24 | 1 | no | — |
| 100 | #624 | — | 2 | 1 | no | — |
| 101 | #481 | — | 27 | 1 | no | Sylow |
| 102 | C2×C33 | C2×SD16 | 1 | 1 | yes | characteristic |
| 103 | C3⋊(C3⋊D3) | — | 2 | 1 | no | — |
| 104 | C3×C3⋊D3 | — | 8 | 1 | no | — |
| 105 | C3×C3⋊D3 | #123 | 1 | 1 | yes | characteristic |
| 106 | C3×C3⋊D3 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 107 | C32×D3 | — | 2 | 1 | no | — |
| 108 | D3×D4 | — | 36 | 1 | no | — |
| 109 | C2×D12 | — | 9 | 1 | no | — |
| 110 | C2×D12 | — | 18 | 1 | no | — |
| 111 | C2×C4×D3 | — | 9 | 1 | no | — |
| 112 | C2×C4×D3 | — | 18 | 1 | no | — |
| 113 | C2×C3×C8 | — | 9 | 1 | no | — |
| 114 | C3×C4⋊C4 | — | 9 | 1 | no | — |
| 115 | C3×C4⋊C4 | — | 9 | 1 | no | — |
| 116 | (C2×D3)⋊C4 | — | 18 | 1 | no | — |
| 117 | C4⋊Dic3 | — | 9 | 1 | no | — |
| 118 | Dic3⋊C4 | — | 18 | 1 | no | — |
| 119 | C2×C3⋊C8 | — | 9 | 1 | no | — |
| 120 | C22×C32 | — | 3 | 1 | no | — |
| 121 | C2×C3⋊D3 | — | 3 | 1 | no | — |
| 122 | C2×C3⋊D3 | — | 4 | 1 | no | — |
| 123 | C2×C3⋊D3 | — | 6 | 1 | no | — |
| 124 | C2×C3⋊D3 | — | 6 | 1 | no | — |
| 125 | C2×C3⋊D3 | — | 12 | 1 | no | — |
| 126 | C2×C3⋊D3 | — | 24 | 1 | no | — |
| 127 | C2×C3⋊D3 | D3×D4 | 1 | 1 | yes | characteristic |
| 128 | C2×C3×D3 | — | 4 | 1 | no | — |
| 129 | C2×C3×D3 | — | 12 | 1 | no | — |
| 130 | C2×C3×D3 | — | 12 | 1 | no | — |
| 131 | C2×C3×D3 | — | 12 | 1 | no | — |
| 132 | C2×C3×D3 | — | 24 | 1 | no | — |
| 133 | D32 | — | 12 | 1 | no | — |
| 134 | D32 | — | 12 | 1 | no | — |
| 135 | D32 | — | 24 | 1 | no | — |
| 136 | D32 | — | 24 | 1 | no | — |
| 137 | D32 | — | 24 | 1 | no | — |
| 138 | D32 | — | 24 | 1 | no | — |
| 139 | #157 | — | 4 | 1 | no | — |
| 140 | #157 | — | 6 | 1 | no | — |
| 141 | #157 | — | 12 | 1 | no | — |
| 142 | #157 | #235 | 1 | 1 | yes | characteristic |
| 143 | #157 | #239 | 1 | 1 | yes | characteristic |
| 144 | C32×C4 | — | 12 | 1 | no | — |
| 145 | C3×Dic3 | — | 4 | 1 | no | — |
| 146 | C3×Dic3 | — | 4 | 1 | no | — |
| 147 | C3×Dic3 | — | 8 | 1 | no | — |
| 148 | #121 | — | 27 | 1 | no | — |
| 149 | C2×C4⋊C4 | — | 27 | 1 | no | — |
| 150 | #106 | — | 9 | 1 | no | — |
| 151 | #106 | — | 27 | 1 | no | — |
| 152 | #102 | — | 27 | 1 | no | — |
| 153 | #102 | — | 27 | 1 | no | — |
| 154 | C22⋊C8 | — | 27 | 1 | no | — |
| 155 | C33 | #481 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 156 | C22×D3 | — | 9 | 1 | no | — |
| 157 | C22×D3 | — | 18 | 1 | no | — |
| 158 | C22×D3 | — | 36 | 1 | no | — |
| 159 | C22×D3 | — | 36 | 1 | no | — |
| 160 | C3×D4 | — | 36 | 1 | no | — |
| 161 | C2×C3×C4 | — | 9 | 1 | no | — |
| 162 | C2×C3×C4 | — | 18 | 1 | no | — |
| 163 | C2×C3×C4 | — | 18 | 1 | no | — |
| 164 | #67 | — | 12 | 1 | no | — |
| 165 | #67 | — | 36 | 1 | no | — |
| 166 | C2×Dic3 | — | 9 | 1 | no | — |
| 167 | C2×Dic3 | — | 18 | 1 | no | — |
| 168 | D12 | — | 12 | 1 | no | — |
| 169 | D12 | — | 18 | 1 | no | — |
| 170 | D12 | — | 18 | 1 | no | — |
| 171 | D12 | — | 36 | 1 | no | — |
| 172 | D12 | — | 36 | 1 | no | — |
| 173 | D12 | — | 72 | 1 | no | — |
| 174 | C4×D3 | — | 12 | 1 | no | — |
| 175 | C4×D3 | — | 18 | 1 | no | — |
| 176 | C4×D3 | — | 18 | 1 | no | — |
| 177 | C4×D3 | — | 36 | 2 | no | — |
| 178 | C3×C8 | — | 18 | 1 | no | — |
| 179 | C3⋊C8 | — | 18 | 1 | no | — |
| 180 | C2×C32 | — | 4 | 1 | no | — |
| 181 | C2×C32 | — | 6 | 1 | no | — |
| 182 | C2×C32 | — | 8 | 1 | no | — |
| 183 | C2×C32 | D3×SD16 | 1 | 1 | yes | characteristic |
| 184 | C3⋊D3 | — | 6 | 1 | no | — |
| 185 | C3⋊D3 | — | 8 | 1 | no | — |
| 186 | C3⋊D3 | — | 24 | 1 | no | — |
| 187 | C3⋊D3 | — | 48 | 1 | no | — |
| 188 | C3⋊D3 | #894 | 1 | 1 | yes | characteristic |
| 189 | C3⋊D3 | #914 | 1 | 1 | yes | characteristic |
| 190 | C3×D3 | — | 8 | 1 | no | — |
| 191 | C3×D3 | — | 12 | 1 | no | — |
| 192 | C3×D3 | — | 12 | 1 | no | — |
| 193 | C3×D3 | — | 24 | 1 | no | — |
| 194 | C3×D3 | — | 24 | 1 | no | — |
| 195 | C3×D3 | — | 48 | 1 | no | — |
| 196 | C2×D4 | — | 27 | 1 | no | — |
| 197 | C2×D4 | — | 54 | 1 | no | — |
| 198 | C22×C4 | — | 27 | 1 | no | — |
| 199 | C22×C4 | — | 54 | 1 | no | — |
| 200 | C2×C8 | — | 9 | 1 | no | — |
| 201 | C2×C8 | — | 27 | 1 | no | — |
| 202 | C4⋊C4 | — | 9 | 1 | no | — |
| 203 | C4⋊C4 | — | 9 | 1 | no | — |
| 204 | C4⋊C4 | — | 27 | 1 | no | — |
| 205 | C4⋊C4 | — | 54 | 1 | no | — |
| 206 | C22⋊C4 | — | 54 | 1 | no | — |
| 207 | C22×C3 | — | 9 | 1 | no | — |
| 208 | C22×C3 | — | 12 | 1 | no | — |
| 209 | C22×C3 | — | 36 | 1 | no | — |
| 210 | C2×D3 | — | 9 | 1 | no | — |
| 211 | C2×D3 | — | 12 | 1 | no | — |
| 212 | C2×D3 | — | 12 | 1 | no | — |
| 213 | C2×D3 | — | 12 | 1 | no | — |
| 214 | C2×D3 | — | 18 | 1 | no | — |
| 215 | C2×D3 | — | 18 | 1 | no | — |
| 216 | C2×D3 | — | 24 | 1 | no | — |
| 217 | C2×D3 | — | 36 | 1 | no | — |
| 218 | C2×D3 | — | 36 | 1 | no | — |
| 219 | C2×D3 | — | 36 | 1 | no | — |
| 220 | C2×D3 | — | 72 | 1 | no | — |
| 221 | C2×D3 | — | 72 | 1 | no | — |
| 222 | C2×D3 | — | 72 | 1 | no | — |
| 223 | C2×D3 | — | 72 | 1 | no | — |
| 224 | C2×D3 | — | 72 | 1 | no | — |
| 225 | C2×D3 | AΓL(1,𝔽9) | 1 | 1 | yes | characteristic |
| 226 | C3×C4 | — | 9 | 1 | no | — |
| 227 | C3×C4 | — | 9 | 1 | no | — |
| 228 | C3×C4 | — | 12 | 1 | no | — |
| 229 | C3×C4 | — | 12 | 1 | no | — |
| 230 | C3×C4 | — | 24 | 1 | no | — |
| 231 | C3×C4 | — | 36 | 1 | no | — |
| 232 | Dic3 | — | 4 | 1 | no | — |
| 233 | Dic3 | — | 18 | 1 | no | — |
| 234 | Dic3 | — | 36 | 1 | no | — |
| 235 | C32 | — | 4 | 1 | no | — |
| 236 | C32 | — | 8 | 1 | no | — |
| 237 | C32 | #4863 | 1 | 1 | yes | characteristic, complemented |
| 238 | C23 | — | 27 | 1 | no | — |
| 239 | C23 | — | 54 | 1 | no | — |
| 240 | D4 | — | 36 | 1 | no | — |
| 241 | D4 | — | 54 | 1 | no | — |
| 242 | D4 | — | 54 | 1 | no | — |
| 243 | D4 | — | 108 | 1 | no | — |
| 244 | C2×C4 | — | 9 | 1 | no | — |
| 245 | C2×C4 | — | 18 | 1 | no | — |
| 246 | C2×C4 | — | 18 | 1 | no | — |
| 247 | C2×C4 | — | 27 | 1 | no | — |
| 248 | C2×C4 | — | 54 | 1 | no | — |
| 249 | C2×C4 | — | 54 | 1 | no | — |
| 250 | C2×C4 | — | 54 | 1 | no | — |
| 251 | C2×C4 | — | 108 | 2 | no | — |
| 252 | C8 | — | 18 | 1 | no | — |
| 253 | C8 | — | 54 | 1 | no | — |
| 254 | C2×C3 | — | 4 | 1 | no | — |
| 255 | C2×C3 | — | 8 | 1 | no | — |
| 256 | C2×C3 | — | 9 | 1 | no | — |
| 257 | C2×C3 | — | 9 | 1 | no | — |
| 258 | C2×C3 | — | 24 | 1 | no | — |
| 259 | C2×C3 | — | 72 | 1 | no | — |
| 260 | C2×C3 | C2×AΓL(1,𝔽9) | 1 | 1 | yes | characteristic |
| 261 | D3 | — | 2 | 1 | no | — |
| 262 | D3 | — | 12 | 1 | no | — |
| 263 | D3 | — | 12 | 1 | no | — |
| 264 | D3 | — | 18 | 1 | no | — |
| 265 | D3 | — | 24 | 1 | no | — |
| 266 | D3 | — | 24 | 1 | no | — |
| 267 | D3 | — | 48 | 1 | no | — |
| 268 | D3 | — | 72 | 1 | no | — |
| 269 | D3 | — | 144 | 1 | no | — |
| 270 | C22 | — | 3 | 1 | no | — |
| 271 | C22 | — | 9 | 1 | no | — |
| 272 | C22 | — | 27 | 1 | no | — |
| 273 | C22 | — | 36 | 1 | no | — |
| 274 | C22 | — | 54 | 1 | no | — |
| 275 | C22 | — | 54 | 1 | no | — |
| 276 | C22 | — | 108 | 1 | no | — |
| 277 | C22 | — | 108 | 1 | no | — |
| 278 | C4 | — | 9 | 1 | no | — |
| 279 | C4 | — | 9 | 1 | no | — |
| 280 | C4 | — | 12 | 1 | no | — |
| 281 | C4 | — | 36 | 1 | no | — |
| 282 | C4 | — | 54 | 1 | no | — |
| 283 | C4 | — | 108 | 1 | no | — |
| 284 | C3 | — | 4 | 1 | no | — |
| 285 | C3 | — | 8 | 1 | no | — |
| 286 | C3 | #10596412 | 1 | 1 | yes | characteristic, complemented |
| 287 | C2 | — | 6 | 1 | no | — |
| 288 | C2 | — | 9 | 1 | no | — |
| 289 | C2 | — | 9 | 1 | no | — |
| 290 | C2 | — | 54 | 1 | no | — |
| 291 | C2 | — | 72 | 1 | no | — |
| 292 | C2 | D3×AΓL(1,𝔽9) | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D3×AΓL(1,𝔽9) |
| commutator, cosocle | C2×C3×#157 | C23 |
| Fitting, socle | C2×C33 | C2×SD16 |
| prime radical | #11724116 | C2 |
| hypocenter | C33 | #481 |
| LCS | G ▷ C2×C3×#157 ▷ C3×C3⋊D3 ▷ C33 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. D3×#3777 | C2 | semidirect |
| 3. #11724104 | C2 | semidirect |
| 4. #11724101 | C2 | semidirect |
| 5. #11724100 | C2 | semidirect |
| 6. C3×AGL(1,𝔽9)⋊C4 | C2 | semidirect |
| 7. (C2×D3)⋊AGL(1,𝔽9) | C2 | semidirect |
| 24. AGL(1,𝔽9)⋊C4 | D3 | semidirect |
| 155. C33 | #481 | semidirect |
| 237. C32 | #4863 | semidirect |
| 286. C3 | #10596412 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| D3×#3777 | 2 |
| #11724116 | 2 |
| #11724104 | 2 |
| #11724101 | 2 |
| #11724100 | 2 |
| C3×AGL(1,𝔽9)⋊C4 | 2 |
| (C2×D3)⋊AGL(1,𝔽9) | 2 |
| #10596412 | 3 |
| #4863 | 9 |