Finite Group DatabasebetaFinite Group DatabasebetaOrder 1152.
701 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 701 conjugacy classes of subgroups — too many to draw. Only its 63 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 | C2×#64267 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×C3×#986 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | Dic3×#41⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #10593215 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | #85346 | — | 3 | 1 | no | maximal |
| 6 | #83932 | — | 3 | 1 | no | maximal |
| 7 | C2×C3×#41⋊C3 | C22 | 1 | 1 | yes | characteristic |
| 8 | #64267 | — | 2 | 2 | no | — |
| 9 | C2×C3⋊GL(2,𝔽3) | C22 | 1 | 2 | yes | complemented |
| 10 | C3×#986 | — | 2 | 2 | no | — |
| 11 | C2×C3×GL(2,𝔽3) | C22 | 1 | 2 | yes | complemented |
| 12 | #63911 | — | 4 | 1 | no | maximal |
| 13 | Dic3×SL(2,𝔽3) | C22 | 1 | 2 | yes | complemented |
| 14 | C2×#986 | — | 3 | 1 | no | — |
| 15 | C2×#986 | — | 6 | 1 | no | — |
| 16 | C2×#986 | D3 | 1 | 1 | yes | characteristic, complemented |
| 17 | C2×C3×AGL(1,ℤ/8) | — | 3 | 1 | no | — |
| 18 | Dic3×#41 | D3 | 1 | 1 | yes | characteristic, complemented |
| 19 | C2×#949 | — | 3 | 1 | no | — |
| 20 | C2×#940 | — | 3 | 1 | no | — |
| 21 | C4×#41⋊C3 | — | 3 | 1 | no | — |
| 22 | C4⋊GL(2,𝔽3) | — | 3 | 4 | no | — |
| 23 | #5215 | — | 3 | 1 | no | — |
| 24 | #5163 | — | 3 | 4 | no | — |
| 25 | #5150 | — | 3 | 4 | no | — |
| 26 | #5131 | — | 3 | 1 | no | — |
| 27 | #5118 | — | 3 | 1 | no | — |
| 28 | C2×C4⋊(C3⋊D3) | — | 4 | 1 | no | — |
| 29 | C2×C3×D12 | — | 4 | 1 | no | — |
| 30 | C3×#41⋊C3 | D4 | 1 | 2 | yes | complemented |
| 31 | C2×C3×SL(2,𝔽3) | C23 | 1 | 1 | yes | characteristic |
| 32 | C2×#626 | — | 4 | 4 | no | — |
| 33 | C3⋊GL(2,𝔽3) | — | 2 | 2 | no | — |
| 34 | C3×GL(2,𝔽3) | — | 2 | 2 | no | — |
| 35 | C3×C4×Dic3 | — | 4 | 1 | no | — |
| 36 | #3038 | — | 9 | 1 | no | Sylow |
| 37 | C2×C3×#41 | C2×D3 | 1 | 1 | yes | characteristic |
| 38 | C22×C3×D4 | — | 3 | 1 | no | — |
| 39 | C22×#67 | — | 6 | 2 | no | — |
| 40 | C22×D12 | — | 3 | 1 | no | — |
| 41 | C2×#41⋊C3 | — | 2 | 1 | no | — |
| 42 | C2×#41⋊C3 | C2×D3 | 1 | 1 | yes | characteristic |
| 43 | #986 | — | 2 | 2 | no | — |
| 44 | #986 | — | 6 | 4 | no | — |
| 45 | #986 | — | 6 | 2 | no | — |
| 46 | C2×GL(2,𝔽3) | — | 3 | 2 | no | — |
| 47 | C2×GL(2,𝔽3) | — | 6 | 2 | no | — |
| 48 | C2×GL(2,𝔽3) | C2×D3 | 1 | 2 | yes | complemented |
| 49 | C3×AGL(1,ℤ/8) | — | 6 | 2 | no | — |
| 50 | C3×AGL(1,ℤ/8) | — | 6 | 2 | no | — |
| 51 | C2×C3×SD16 | — | 3 | 2 | no | — |
| 52 | C2×C3×D8 | — | 3 | 2 | no | — |
| 53 | C2×C3×M16 | — | 3 | 1 | no | — |
| 54 | #949 | — | 6 | 2 | no | — |
| 55 | #949 | — | 6 | 2 | no | — |
| 56 | Q8×Dic3 | C2×D3 | 1 | 2 | yes | — |
| 57 | C2×#215 | — | 3 | 2 | no | — |
| 58 | #940 | — | 3 | 4 | no | — |
| 59 | #940 | — | 6 | 1 | no | — |
| 60 | #940 | — | 6 | 1 | no | — |
| 61 | D4×Dic3 | — | 3 | 2 | no | — |
| 62 | C2×#213 | — | 3 | 2 | no | — |
| 63 | #926 | — | 3 | 1 | no | — |
| 64 | C2×C4×Dic3 | — | 3 | 1 | no | — |
| 65 | C2×#208 | — | 3 | 1 | no | — |
| 66 | #874 | — | 12 | 1 | no | — |
| 67 | C4×SL(2,𝔽3) | — | 3 | 2 | no | — |
| 68 | #832 | — | 3 | 4 | no | — |
| 69 | #821 | — | 3 | 4 | no | — |
| 70 | Dic3⋊C8 | — | 3 | 4 | no | — |
| 71 | C22×C3⋊D3 | — | 4 | 2 | no | — |
| 72 | C22×C3×D3 | — | 4 | 2 | no | — |
| 73 | C2×C32×C4 | — | 4 | 1 | no | — |
| 74 | C4⋊(C3⋊D3) | — | 8 | 2 | no | — |
| 75 | C2×C3×Dic3 | — | 4 | 2 | no | — |
| 76 | C3×D12 | — | 8 | 2 | no | — |
| 77 | C3×SL(2,𝔽3) | C2×D4 | 1 | 1 | yes | characteristic, complemented |
| 78 | #626 | — | 8 | 8 | no | — |
| 79 | C2×AGL(1,ℤ/8) | — | 3 | 1 | no | — |
| 80 | C2×AGL(1,ℤ/8) | — | 9 | 1 | no | — |
| 81 | C2×#127 | — | 9 | 1 | no | — |
| 82 | C4×#41 | — | 3 | 1 | no | — |
| 83 | #460 | — | 9 | 4 | no | — |
| 84 | #459 | — | 9 | 4 | no | — |
| 85 | #423 | — | 9 | 1 | no | — |
| 86 | #418 | — | 9 | 1 | no | — |
| 87 | #418 | — | 9 | 1 | no | — |
| 88 | C24×C3 | — | 6 | 1 | no | — |
| 89 | C23×D3 | — | 6 | 1 | no | — |
| 90 | C3×#41 | — | 3 | 2 | no | — |
| 91 | C3×#41 | D12 | 1 | 2 | yes | complemented |
| 92 | C2×C3×Q8 | C22×D3 | 1 | 1 | yes | characteristic |
| 93 | C2×C3×D4 | — | 3 | 1 | no | — |
| 94 | C2×C3×D4 | — | 3 | 2 | no | — |
| 95 | C2×C3×D4 | — | 6 | 2 | no | — |
| 96 | C2×C3×D4 | — | 6 | 2 | no | — |
| 97 | C2×C3×D4 | — | 6 | 1 | no | — |
| 98 | C22×C3×C4 | — | 3 | 1 | no | — |
| 99 | C2×#67 | — | 6 | 4 | no | — |
| 100 | C2×#67 | — | 6 | 4 | no | — |
| 101 | C2×#67 | — | 12 | 8 | no | — |
| 102 | C22×Dic3 | — | 3 | 2 | no | — |
| 103 | C2×D12 | — | 3 | 2 | no | — |
| 104 | C2×D12 | — | 4 | 1 | no | — |
| 105 | C2×D12 | — | 6 | 2 | no | — |
| 106 | C2×D12 | — | 6 | 2 | no | — |
| 107 | C2×D12 | — | 6 | 1 | no | — |
| 108 | C2×D12 | — | 12 | 1 | no | — |
| 109 | C2×D12 | — | 12 | 4 | no | — |
| 110 | C2×D12 | — | 24 | 1 | no | — |
| 111 | #41⋊C3 | — | 2 | 2 | no | — |
| 112 | #41⋊C3 | #67 | 1 | 2 | yes | complemented |
| 113 | C2×SL(2,𝔽3) | — | 2 | 1 | no | — |
| 114 | C2×SL(2,𝔽3) | C22×D3 | 1 | 1 | yes | characteristic |
| 115 | GL(2,𝔽3) | — | 2 | 2 | no | — |
| 116 | GL(2,𝔽3) | — | 6 | 4 | no | — |
| 117 | GL(2,𝔽3) | — | 6 | 2 | no | — |
| 118 | C3×SD16 | — | 6 | 2 | no | — |
| 119 | C3×SD16 | — | 6 | 2 | no | — |
| 120 | C3×D8 | — | 6 | 4 | no | — |
| 121 | C3×M16 | — | 6 | 2 | no | — |
| 122 | C2×C3×C8 | — | 3 | 2 | no | — |
| 123 | C3×C42 | — | 12 | 1 | no | — |
| 124 | #217 | — | 6 | 1 | no | — |
| 125 | #215 | — | 6 | 2 | no | — |
| 126 | #215 | — | 6 | 2 | no | — |
| 127 | #213 | — | 6 | 4 | no | — |
| 128 | C4⋊Dic3 | — | 3 | 2 | no | — |
| 129 | C4×Dic3 | — | 3 | 1 | no | — |
| 130 | C4×Dic3 | — | 3 | 2 | no | — |
| 131 | C4×Dic3 | S4 | 1 | 1 | yes | characteristic |
| 132 | #208 | — | 6 | 2 | no | — |
| 133 | C2×C3⋊C8 | — | 3 | 2 | no | — |
| 134 | C22×C32 | — | 4 | 1 | no | — |
| 135 | C2×C3⋊D3 | — | 8 | 2 | no | — |
| 136 | C2×C3⋊D3 | — | 8 | 4 | no | — |
| 137 | C2×C3×D3 | — | 8 | 4 | no | — |
| 138 | C2×C3×D3 | — | 8 | 2 | no | — |
| 139 | C32×C4 | — | 4 | 2 | no | — |
| 140 | C3×Dic3 | — | 4 | 4 | no | — |
| 141 | C2×#41 | D32 | 1 | 1 | yes | characteristic |
| 142 | C22×D4 | — | 3 | 1 | no | — |
| 143 | C22×D4 | — | 9 | 1 | no | — |
| 144 | C22×D4 | — | 18 | 2 | no | — |
| 145 | AGL(1,ℤ/8) | — | 6 | 2 | no | — |
| 146 | AGL(1,ℤ/8) | — | 6 | 2 | no | — |
| 147 | AGL(1,ℤ/8) | — | 18 | 2 | no | — |
| 148 | AGL(1,ℤ/8) | — | 18 | 2 | no | — |
| 149 | C2×SD16 | — | 3 | 2 | no | — |
| 150 | C2×SD16 | — | 9 | 2 | no | — |
| 151 | C2×D8 | — | 3 | 2 | no | — |
| 152 | C2×D8 | — | 9 | 2 | no | — |
| 153 | C2×M16 | — | 3 | 1 | no | — |
| 154 | C2×M16 | — | 9 | 1 | no | — |
| 155 | #127 | — | 9 | 4 | no | — |
| 156 | #127 | — | 18 | 1 | no | — |
| 157 | #127 | — | 18 | 1 | no | — |
| 158 | C4×Q8 | — | 3 | 2 | no | — |
| 159 | C4×D4 | — | 9 | 2 | no | — |
| 160 | #117 | — | 9 | 1 | no | — |
| 161 | C2×C42 | — | 9 | 1 | no | — |
| 162 | C4⋊C8 | — | 9 | 4 | no | — |
| 163 | #102 | — | 9 | 4 | no | — |
| 164 | #102 | — | 9 | 4 | no | — |
| 165 | C23×C3 | — | 3 | 1 | no | — |
| 166 | C23×C3 | — | 6 | 2 | no | — |
| 167 | C23×C3 | — | 12 | 2 | no | — |
| 168 | C23×C3 | — | 12 | 4 | no | — |
| 169 | C22×D3 | — | 4 | 2 | no | — |
| 170 | C22×D3 | — | 6 | 2 | no | — |
| 171 | C22×D3 | — | 12 | 2 | no | — |
| 172 | C22×D3 | — | 12 | 2 | no | — |
| 173 | C22×D3 | — | 12 | 4 | no | — |
| 174 | C22×D3 | — | 24 | 2 | no | — |
| 175 | C3×Q8 | — | 3 | 1 | no | — |
| 176 | C3×Q8 | C2×D12 | 1 | 1 | yes | characteristic |
| 177 | C3×D4 | — | 3 | 2 | no | — |
| 178 | C3×D4 | — | 6 | 1 | no | — |
| 179 | C3×D4 | — | 6 | 2 | no | — |
| 180 | C3×D4 | — | 6 | 2 | no | — |
| 181 | C3×D4 | — | 12 | 2 | no | — |
| 182 | C2×C3×C4 | — | 3 | 1 | no | — |
| 183 | C2×C3×C4 | — | 3 | 2 | no | — |
| 184 | C2×C3×C4 | — | 3 | 2 | no | — |
| 185 | C2×C3×C4 | — | 4 | 1 | no | — |
| 186 | C2×C3×C4 | — | 8 | 1 | no | — |
| 187 | C2×C3×C4 | — | 12 | 2 | no | — |
| 188 | C2×C3×C4 | C2×S4 | 1 | 1 | yes | characteristic |
| 189 | #67 | — | 12 | 8 | no | — |
| 190 | #67 | — | 12 | 8 | no | — |
| 191 | C2×Dic3 | — | 3 | 2 | no | — |
| 192 | C2×Dic3 | — | 6 | 4 | no | — |
| 193 | C2×Dic3 | C2×S4 | 1 | 2 | yes | — |
| 194 | D12 | — | 6 | 2 | no | — |
| 195 | D12 | — | 6 | 2 | no | — |
| 196 | D12 | — | 8 | 2 | no | — |
| 197 | D12 | — | 12 | 2 | no | — |
| 198 | D12 | — | 24 | 2 | no | — |
| 199 | D12 | — | 24 | 4 | no | — |
| 200 | D12 | — | 24 | 8 | no | — |
| 201 | SL(2,𝔽3) | — | 2 | 1 | no | — |
| 202 | SL(2,𝔽3) | C2×#67 | 1 | 1 | yes | characteristic, complemented |
| 203 | C3×C8 | — | 6 | 2 | no | — |
| 204 | C3⋊C8 | — | 6 | 2 | no | — |
| 205 | C2×C32 | — | 4 | 1 | no | — |
| 206 | C2×C32 | — | 4 | 2 | no | — |
| 207 | C3⋊D3 | — | 16 | 2 | no | — |
| 208 | C3×D3 | — | 16 | 2 | no | — |
| 209 | C24 | — | 6 | 1 | no | — |
| 210 | C24 | — | 18 | 1 | no | — |
| 211 | #41 | — | 3 | 2 | no | — |
| 212 | #41 | #626 | 1 | 2 | yes | complemented |
| 213 | C2×Q8 | C2×D32 | 1 | 1 | yes | characteristic |
| 214 | C2×D4 | — | 3 | 1 | no | — |
| 215 | C2×D4 | — | 3 | 2 | no | — |
| 216 | C2×D4 | — | 6 | 2 | no | — |
| 217 | C2×D4 | — | 6 | 2 | no | — |
| 218 | C2×D4 | — | 6 | 1 | no | — |
| 219 | C2×D4 | — | 9 | 2 | no | — |
| 220 | C2×D4 | — | 18 | 2 | no | — |
| 221 | C2×D4 | — | 18 | 2 | no | — |
| 222 | C2×D4 | — | 18 | 4 | no | — |
| 223 | C2×D4 | — | 18 | 4 | no | — |
| 224 | C2×D4 | — | 18 | 1 | no | — |
| 225 | C2×D4 | — | 36 | 8 | no | — |
| 226 | C22×C4 | — | 3 | 1 | no | — |
| 227 | C22×C4 | — | 9 | 2 | no | — |
| 228 | SD16 | — | 6 | 2 | no | — |
| 229 | SD16 | — | 6 | 2 | no | — |
| 230 | SD16 | — | 18 | 2 | no | — |
| 231 | SD16 | — | 18 | 2 | no | — |
| 232 | D8 | — | 6 | 4 | no | — |
| 233 | D8 | — | 18 | 4 | no | — |
| 234 | M16 | — | 6 | 2 | no | — |
| 235 | M16 | — | 18 | 2 | no | — |
| 236 | C2×C8 | — | 3 | 2 | no | — |
| 237 | C2×C8 | — | 9 | 2 | no | — |
| 238 | C4⋊C4 | — | 9 | 2 | no | — |
| 239 | C22⋊C4 | — | 18 | 1 | no | — |
| 240 | C42 | — | 3 | 1 | no | — |
| 241 | C42 | — | 9 | 2 | no | — |
| 242 | C42 | — | 9 | 1 | no | — |
| 243 | C22×C3 | — | 3 | 2 | no | — |
| 244 | C22×C3 | — | 4 | 1 | no | — |
| 245 | C22×C3 | — | 6 | 2 | no | — |
| 246 | C22×C3 | — | 8 | 1 | no | — |
| 247 | C22×C3 | — | 12 | 2 | no | — |
| 248 | C22×C3 | — | 12 | 4 | no | — |
| 249 | C22×C3 | — | 24 | 4 | no | — |
| 250 | C22×C3 | C22×S4 | 1 | 1 | yes | characteristic |
| 251 | C2×D3 | — | 8 | 2 | no | — |
| 252 | C2×D3 | — | 8 | 4 | no | — |
| 253 | C2×D3 | — | 12 | 2 | no | — |
| 254 | C2×D3 | — | 12 | 4 | no | — |
| 255 | C2×D3 | — | 24 | 4 | no | — |
| 256 | C2×D3 | — | 24 | 2 | no | — |
| 257 | C2×D3 | — | 24 | 4 | no | — |
| 258 | C2×D3 | — | 24 | 4 | no | — |
| 259 | C2×D3 | — | 48 | 4 | no | — |
| 260 | C3×C4 | — | 3 | 1 | no | — |
| 261 | C3×C4 | — | 3 | 1 | no | — |
| 262 | C3×C4 | — | 4 | 2 | no | — |
| 263 | C3×C4 | — | 8 | 2 | no | — |
| 264 | C3×C4 | — | 12 | 4 | no | — |
| 265 | C3×C4 | C4⋊S4 | 1 | 2 | yes | — |
| 266 | Dic3 | — | 6 | 2 | no | — |
| 267 | Dic3 | #986 | 1 | 4 | yes | complemented |
| 268 | C32 | — | 4 | 1 | no | Sylow |
| 269 | C23 | — | 3 | 1 | no | — |
| 270 | C23 | — | 6 | 2 | no | — |
| 271 | C23 | — | 12 | 2 | no | — |
| 272 | C23 | — | 12 | 4 | no | — |
| 273 | C23 | — | 18 | 2 | no | — |
| 274 | C23 | — | 36 | 2 | no | — |
| 275 | C23 | — | 36 | 4 | no | — |
| 276 | Q8 | — | 3 | 1 | no | — |
| 277 | Q8 | C2×#626 | 1 | 1 | yes | characteristic |
| 278 | D4 | — | 3 | 2 | no | — |
| 279 | D4 | — | 6 | 1 | no | — |
| 280 | D4 | — | 6 | 2 | no | — |
| 281 | D4 | — | 6 | 2 | no | — |
| 282 | D4 | — | 12 | 2 | no | — |
| 283 | D4 | — | 18 | 2 | no | — |
| 284 | D4 | — | 18 | 2 | no | — |
| 285 | D4 | — | 36 | 8 | no | — |
| 286 | D4 | — | 36 | 8 | no | — |
| 287 | D4 | — | 36 | 2 | no | — |
| 288 | C2×C4 | — | 3 | 2 | no | — |
| 289 | C2×C4 | — | 3 | 1 | no | — |
| 290 | C2×C4 | — | 3 | 2 | no | — |
| 291 | C2×C4 | — | 3 | 2 | no | — |
| 292 | C2×C4 | — | 9 | 2 | no | — |
| 293 | C2×C4 | — | 18 | 4 | no | — |
| 294 | C2×C4 | D3×S4 | 1 | 1 | yes | characteristic |
| 295 | C8 | — | 6 | 2 | no | — |
| 296 | C8 | — | 18 | 2 | no | — |
| 297 | C2×C3 | — | 4 | 2 | no | — |
| 298 | C2×C3 | — | 4 | 1 | no | — |
| 299 | C2×C3 | — | 6 | 2 | no | — |
| 300 | C2×C3 | — | 8 | 2 | no | — |
| 301 | C2×C3 | — | 8 | 1 | no | — |
| 302 | C2×C3 | — | 24 | 2 | no | — |
| 303 | C2×C3 | C2×C4⋊S4 | 1 | 1 | yes | characteristic |
| 304 | C2×C3 | C2×#986 | 1 | 2 | yes | — |
| 305 | D3 | — | 16 | 2 | no | — |
| 306 | D3 | — | 24 | 2 | no | — |
| 307 | D3 | — | 48 | 2 | no | — |
| 308 | D3 | — | 48 | 4 | no | — |
| 309 | C22 | — | 3 | 2 | no | — |
| 310 | C22 | — | 6 | 2 | no | — |
| 311 | C22 | — | 12 | 4 | no | — |
| 312 | C22 | — | 12 | 2 | no | — |
| 313 | C22 | — | 24 | 4 | no | — |
| 314 | C22 | — | 36 | 4 | no | — |
| 315 | C22 | — | 36 | 2 | no | — |
| 316 | C22 | — | 72 | 4 | no | — |
| 317 | C22 | C2×D3×S4 | 1 | 1 | yes | characteristic |
| 318 | C4 | — | 3 | 4 | no | — |
| 319 | C4 | — | 3 | 1 | no | — |
| 320 | C4 | — | 3 | 1 | no | — |
| 321 | C4 | — | 18 | 2 | no | — |
| 322 | C4 | #64207 | 1 | 2 | yes | — |
| 323 | C3 | — | 4 | 1 | no | — |
| 324 | C3 | — | 8 | 1 | no | — |
| 325 | C3 | #85346 | 1 | 1 | yes | characteristic, complemented |
| 326 | C2 | — | 6 | 2 | no | — |
| 327 | C2 | — | 24 | 2 | no | — |
| 328 | C2 | — | 72 | 2 | no | — |
| 329 | C2 | D3×#986 | 1 | 2 | yes | — |
| 330 | C2 | C2×#64207 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#64267 | C2 | semidirect |
| 2. C2×C3×#986 | C2 | semidirect |
| 3. Dic3×#41⋊C3 | C2 | semidirect |
| 4. #10593215 | C2 | semidirect |
| 9. C2×C3⋊GL(2,𝔽3) | C22 | semidirect |
| 11. C2×C3×GL(2,𝔽3) | C22 | semidirect |
| 13. Dic3×SL(2,𝔽3) | C22 | semidirect |
| 16. C2×#986 | D3 | semidirect |
| 18. Dic3×#41 | D3 | semidirect |
| 30. C3×#41⋊C3 | D4 | semidirect |
| 48. C2×GL(2,𝔽3) | C2×D3 | semidirect |
| 77. C3×SL(2,𝔽3) | C2×D4 | semidirect |
| 91. C3×#41 | D12 | semidirect |
| 112. #41⋊C3 | #67 | semidirect |
| 202. SL(2,𝔽3) | C2×#67 | semidirect |
| 212. #41 | #626 | semidirect |
| 267. Dic3 | #986 | semidirect |
| 325. C3 | #85346 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#64267 | 2 |
| C2×C3×#986 | 2 |
| Dic3×#41⋊C3 | 2 |
| #10593215 | 2 |
| #85346 | 3 |
| #83932 | 3 |
| #63911 | 4 |