Finite Group DatabasebetaFinite Group DatabasebetaOrder 1152.
421 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 421 conjugacy classes of subgroups — too many to draw. Only its 49 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×#1010⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #10593660 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C2×C3×GU(2,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #73155 | — | 3 | 1 | no | maximal |
| 5 | #70387 | — | 3 | 1 | no | maximal |
| 6 | C2×C3×#41⋊C3 | C22 | 1 | 1 | yes | characteristic |
| 7 | #1010⋊C3 | — | 2 | 1 | no | — |
| 8 | #1010⋊C3 | C4 | 1 | 2 | yes | complemented |
| 9 | C2×D3×SL(2,𝔽3) | — | 2 | 1 | no | — |
| 10 | C3⋊#859 | — | 2 | 1 | no | — |
| 11 | C3×GU(2,𝔽3) | — | 2 | 1 | no | — |
| 12 | #63563 | — | 4 | 1 | no | maximal |
| 13 | C2×#1010 | D3 | 1 | 1 | yes | characteristic |
| 14 | C2×2−1+4⋊C3 | — | 3 | 1 | no | — |
| 15 | #5426 | — | 3 | 1 | no | — |
| 16 | #5426 | — | 6 | 1 | no | — |
| 17 | C2×GU(2,𝔽3) | D3 | 1 | 1 | yes | characteristic, complemented |
| 18 | C2×C3×#104 | — | 3 | 1 | no | — |
| 19 | #5232 | — | 3 | 1 | no | — |
| 20 | C2×#824 | — | 3 | 1 | no | — |
| 21 | #5133 | — | 3 | 1 | no | — |
| 22 | #4929 | — | 3 | 1 | no | — |
| 23 | C2×#805 | — | 3 | 1 | no | — |
| 24 | C2×C3×D12 | — | 4 | 1 | no | — |
| 25 | C3×#41⋊C3 | C2×C4 | 1 | 1 | yes | characteristic |
| 26 | C3×#41⋊C3 | D4 | 1 | 1 | yes | characteristic |
| 27 | C2×C3×SL(2,𝔽3) | D4 | 1 | 1 | yes | characteristic |
| 28 | D3×SL(2,𝔽3) | — | 2 | 2 | no | — |
| 29 | C4⋊(C3⋊Dic3) | — | 4 | 1 | no | — |
| 30 | C2×C3×C3⋊C8 | — | 4 | 1 | no | — |
| 31 | #1793 | — | 9 | 1 | no | Sylow |
| 32 | C2×C3×#41 | C2×D3 | 1 | 1 | yes | characteristic |
| 33 | #1010 | — | 2 | 1 | no | — |
| 34 | #1010 | — | 3 | 2 | no | — |
| 35 | #1010 | — | 6 | 1 | no | — |
| 36 | #1010 | Dic3 | 1 | 2 | yes | complemented |
| 37 | C2×D3×Q8 | — | 2 | 1 | no | — |
| 38 | C2×#237 | — | 6 | 1 | no | — |
| 39 | C2×#235 | — | 3 | 1 | no | — |
| 40 | C22×Dic6 | — | 3 | 1 | no | — |
| 41 | 2−1+4⋊C3 | — | 3 | 2 | no | — |
| 42 | 2−1+4⋊C3 | — | 6 | 1 | no | — |
| 43 | C2×#41⋊C3 | — | 2 | 1 | no | — |
| 44 | C2×#41⋊C3 | C2×D3 | 1 | 1 | yes | characteristic |
| 45 | C22×SL(2,𝔽3) | — | 6 | 1 | no | — |
| 46 | C2×C3×M16 | — | 3 | 1 | no | — |
| 47 | C2×C3×C42 | — | 3 | 1 | no | — |
| 48 | C2×C4⋊Dic3 | — | 3 | 1 | no | — |
| 49 | C2×#208 | — | 3 | 1 | no | — |
| 50 | GU(2,𝔽3) | — | 2 | 1 | no | — |
| 51 | #859 | — | 6 | 2 | no | — |
| 52 | #859 | — | 6 | 1 | no | — |
| 53 | C3×#104 | — | 6 | 1 | no | — |
| 54 | C3×#104 | — | 6 | 1 | no | — |
| 55 | #835 | — | 6 | 1 | no | — |
| 56 | #832 | — | 6 | 1 | no | — |
| 57 | #832 | — | 12 | 1 | no | — |
| 58 | C3⋊#95 | — | 6 | 2 | no | — |
| 59 | #824 | — | 6 | 2 | no | — |
| 60 | #821 | — | 6 | 1 | no | — |
| 61 | #816 | — | 6 | 1 | no | — |
| 62 | #805 | — | 6 | 2 | no | — |
| 63 | C22×C3×D3 | — | 8 | 1 | no | — |
| 64 | C2×C32×C4 | — | 4 | 1 | no | — |
| 65 | C2×C3⋊Dic3 | — | 8 | 1 | no | — |
| 66 | C3×D12 | — | 4 | 2 | no | — |
| 67 | C3×D12 | — | 8 | 1 | no | — |
| 68 | C3×SL(2,𝔽3) | C22⋊C4 | 1 | 1 | yes | characteristic |
| 69 | C3×C3⋊C8 | — | 8 | 1 | no | — |
| 70 | C2×2−1+4 | — | 3 | 1 | no | — |
| 71 | C2×#104 | — | 3 | 1 | no | — |
| 72 | C2×#104 | — | 9 | 1 | no | — |
| 73 | #417 | — | 9 | 1 | no | — |
| 74 | #417 | — | 9 | 1 | no | — |
| 75 | C2×#101 | — | 9 | 1 | no | — |
| 76 | #391 | — | 9 | 1 | no | — |
| 77 | C3×#41 | — | 3 | 1 | no | — |
| 78 | C3×#41 | — | 3 | 1 | no | — |
| 79 | C3×#41 | C2×Dic3 | 1 | 1 | yes | characteristic |
| 80 | C3×#41 | #67 | 1 | 1 | yes | characteristic |
| 81 | C2×C3×Q8 | #67 | 1 | 1 | yes | characteristic |
| 82 | C2×C3×D4 | — | 3 | 1 | no | — |
| 83 | C2×C3×D4 | — | 12 | 1 | no | — |
| 84 | C22×C3×C4 | — | 3 | 1 | no | — |
| 85 | C22×C3×C4 | — | 6 | 1 | no | — |
| 86 | C2×#67 | — | 6 | 1 | no | — |
| 87 | C22×Dic3 | — | 3 | 2 | no | — |
| 88 | C22×Dic3 | — | 6 | 1 | no | — |
| 89 | D3×Q8 | — | 2 | 2 | no | — |
| 90 | D3×Q8 | — | 6 | 2 | no | — |
| 91 | #237 | — | 6 | 2 | no | — |
| 92 | #237 | — | 6 | 4 | no | — |
| 93 | #237 | — | 6 | 2 | no | — |
| 94 | #235 | — | 3 | 2 | no | — |
| 95 | #235 | — | 3 | 2 | no | — |
| 96 | #235 | — | 6 | 1 | no | — |
| 97 | #235 | — | 6 | 1 | no | — |
| 98 | C2×D12 | S4 | 1 | 1 | yes | characteristic |
| 99 | C2×C4×D3 | — | 6 | 1 | no | — |
| 100 | C2×Dic6 | — | 3 | 1 | no | — |
| 101 | C2×Dic6 | — | 3 | 2 | no | — |
| 102 | C2×Dic6 | — | 3 | 2 | no | — |
| 103 | C2×Dic6 | — | 6 | 1 | no | — |
| 104 | C2×Dic6 | — | 6 | 1 | no | — |
| 105 | C2×Dic6 | — | 6 | 1 | no | — |
| 106 | #41⋊C3 | — | 2 | 1 | no | — |
| 107 | #41⋊C3 | — | 2 | 1 | no | — |
| 108 | #41⋊C3 | C4×D3 | 1 | 1 | yes | characteristic |
| 109 | #41⋊C3 | #67 | 1 | 1 | yes | characteristic |
| 110 | C2×SL(2,𝔽3) | — | 2 | 1 | no | — |
| 111 | C2×SL(2,𝔽3) | — | 6 | 2 | no | — |
| 112 | C2×SL(2,𝔽3) | D12 | 1 | 1 | yes | characteristic |
| 113 | C3×M16 | — | 6 | 1 | no | — |
| 114 | C3×M16 | — | 6 | 1 | no | — |
| 115 | C2×C3×C8 | — | 6 | 1 | no | — |
| 116 | C3×C42 | — | 6 | 1 | no | — |
| 117 | C3×C42 | — | 6 | 1 | no | — |
| 118 | C4⋊Dic3 | — | 6 | 1 | no | — |
| 119 | C4⋊Dic3 | — | 6 | 1 | no | — |
| 120 | C4⋊Dic3 | — | 12 | 1 | no | — |
| 121 | C4⋊Dic3 | — | 24 | 1 | no | — |
| 122 | #208 | — | 6 | 1 | no | — |
| 123 | #208 | — | 6 | 1 | no | — |
| 124 | C2×C3⋊C8 | — | 4 | 1 | no | — |
| 125 | C2×C3⋊C8 | — | 6 | 1 | no | — |
| 126 | C22×C32 | — | 4 | 1 | no | — |
| 127 | C2×C3×D3 | — | 8 | 2 | no | — |
| 128 | C2×C3×D3 | — | 16 | 1 | no | — |
| 129 | C2×C3×D3 | — | 16 | 1 | no | — |
| 130 | C32×C4 | — | 4 | 1 | no | — |
| 131 | C32×C4 | — | 4 | 1 | no | — |
| 132 | C3⋊Dic3 | — | 16 | 1 | no | — |
| 133 | 2−1+4 | — | 3 | 2 | no | — |
| 134 | 2−1+4 | — | 6 | 1 | no | — |
| 135 | 2−1+4 | — | 9 | 2 | no | — |
| 136 | 2−1+4 | — | 18 | 1 | no | — |
| 137 | C2×#41 | — | 9 | 1 | no | — |
| 138 | C2×#41 | — | 18 | 1 | no | — |
| 139 | C2×#41 | D32 | 1 | 1 | yes | characteristic |
| 140 | C22×Q8 | — | 6 | 1 | no | — |
| 141 | C22×Q8 | — | 9 | 1 | no | — |
| 142 | C2×M16 | — | 3 | 1 | no | — |
| 143 | C2×M16 | — | 9 | 1 | no | — |
| 144 | C2×C4⋊C4 | — | 9 | 1 | no | — |
| 145 | C2×C42 | — | 3 | 1 | no | — |
| 146 | #104 | — | 6 | 1 | no | — |
| 147 | #104 | — | 6 | 1 | no | — |
| 148 | #104 | — | 18 | 2 | no | — |
| 149 | #103 | — | 18 | 1 | no | — |
| 150 | #103 | — | 18 | 1 | no | — |
| 151 | #102 | — | 18 | 1 | no | — |
| 152 | #102 | — | 18 | 1 | no | — |
| 153 | #101 | — | 18 | 2 | no | — |
| 154 | #95 | — | 18 | 2 | no | — |
| 155 | C23×C3 | — | 3 | 1 | no | — |
| 156 | C23×C3 | — | 24 | 1 | no | — |
| 157 | C22×D3 | — | 2 | 1 | no | — |
| 158 | C3×Q8 | — | 3 | 1 | no | — |
| 159 | C3×Q8 | #217 | 1 | 1 | yes | characteristic |
| 160 | C3×D4 | — | 3 | 1 | no | — |
| 161 | C3×D4 | — | 3 | 1 | no | — |
| 162 | C3×D4 | — | 6 | 1 | no | — |
| 163 | C3×D4 | — | 12 | 2 | no | — |
| 164 | C3×D4 | — | 24 | 1 | no | — |
| 165 | C2×C3×C4 | — | 3 | 1 | no | — |
| 166 | C2×C3×C4 | — | 3 | 1 | no | — |
| 167 | C2×C3×C4 | — | 3 | 1 | no | — |
| 168 | C2×C3×C4 | — | 3 | 1 | no | — |
| 169 | C2×C3×C4 | — | 3 | 1 | no | — |
| 170 | C2×C3×C4 | — | 4 | 1 | no | — |
| 171 | C2×C3×C4 | — | 6 | 2 | no | — |
| 172 | C2×C3×C4 | — | 8 | 1 | no | — |
| 173 | C2×C3×C4 | — | 12 | 1 | no | — |
| 174 | C2×C3×C4 | — | 12 | 1 | no | — |
| 175 | C2×C3×C4 | C2×S4 | 1 | 1 | yes | characteristic |
| 176 | #67 | — | 6 | 2 | no | — |
| 177 | #67 | — | 6 | 2 | no | — |
| 178 | C2×Dic3 | — | 3 | 4 | no | — |
| 179 | C2×Dic3 | — | 6 | 2 | no | — |
| 180 | C2×Dic3 | — | 6 | 1 | no | — |
| 181 | C2×Dic3 | — | 12 | 2 | no | — |
| 182 | C2×Dic3 | — | 12 | 1 | no | — |
| 183 | C2×Dic3 | — | 24 | 1 | no | — |
| 184 | C2×Dic3 | — | 24 | 2 | no | — |
| 185 | D12 | — | 2 | 1 | no | — |
| 186 | D12 | SL(2,ℤ/4) | 1 | 2 | yes | — |
| 187 | C4×D3 | — | 6 | 2 | no | — |
| 188 | C4×D3 | — | 6 | 2 | no | — |
| 189 | Dic6 | — | 3 | 2 | no | — |
| 190 | Dic6 | — | 6 | 2 | no | — |
| 191 | Dic6 | — | 6 | 1 | no | — |
| 192 | Dic6 | — | 6 | 2 | no | — |
| 193 | SL(2,𝔽3) | — | 2 | 1 | no | — |
| 194 | SL(2,𝔽3) | (C2×D3)⋊C4 | 1 | 1 | yes | characteristic |
| 195 | C3×C8 | — | 12 | 1 | no | — |
| 196 | C3⋊C8 | — | 8 | 1 | no | — |
| 197 | C3⋊C8 | — | 12 | 1 | no | — |
| 198 | C2×C32 | — | 4 | 1 | no | — |
| 199 | C2×C32 | — | 4 | 1 | no | — |
| 200 | C2×C32 | — | 4 | 1 | no | — |
| 201 | C3×D3 | — | 16 | 2 | no | — |
| 202 | #41 | — | 3 | 1 | no | — |
| 203 | #41 | — | 3 | 1 | no | — |
| 204 | #41 | — | 9 | 2 | no | — |
| 205 | #41 | — | 9 | 2 | no | — |
| 206 | #41 | — | 18 | 1 | no | — |
| 207 | #41 | — | 18 | 1 | no | — |
| 208 | #41 | — | 18 | 2 | no | — |
| 209 | #41 | — | 18 | 4 | no | — |
| 210 | #41 | — | 18 | 2 | no | — |
| 211 | #41 | D3×Dic3 | 1 | 1 | yes | characteristic |
| 212 | #41 | #625 | 1 | 1 | yes | characteristic |
| 213 | C2×Q8 | — | 6 | 2 | no | — |
| 214 | C2×Q8 | — | 9 | 2 | no | — |
| 215 | C2×Q8 | — | 9 | 2 | no | — |
| 216 | C2×Q8 | — | 9 | 1 | no | — |
| 217 | C2×Q8 | — | 18 | 1 | no | — |
| 218 | C2×Q8 | — | 18 | 1 | no | — |
| 219 | C2×Q8 | — | 18 | 1 | no | — |
| 220 | C2×Q8 | — | 18 | 2 | no | — |
| 221 | C2×Q8 | #626 | 1 | 1 | yes | characteristic |
| 222 | C2×D4 | — | 3 | 1 | no | — |
| 223 | C2×D4 | — | 3 | 1 | no | — |
| 224 | C2×D4 | — | 18 | 1 | no | — |
| 225 | C22×C4 | — | 3 | 1 | no | — |
| 226 | C22×C4 | — | 6 | 1 | no | — |
| 227 | C22×C4 | — | 9 | 2 | no | — |
| 228 | C22×C4 | — | 18 | 1 | no | — |
| 229 | C22×C4 | — | 18 | 1 | no | — |
| 230 | M16 | — | 6 | 1 | no | — |
| 231 | M16 | — | 6 | 1 | no | — |
| 232 | M16 | — | 18 | 1 | no | — |
| 233 | M16 | — | 18 | 1 | no | — |
| 234 | C2×C8 | — | 6 | 1 | no | — |
| 235 | C2×C8 | — | 18 | 1 | no | — |
| 236 | C4⋊C4 | — | 18 | 1 | no | — |
| 237 | C4⋊C4 | — | 18 | 1 | no | — |
| 238 | C42 | — | 6 | 1 | no | — |
| 239 | C42 | — | 6 | 1 | no | — |
| 240 | C22×C3 | — | 3 | 1 | no | — |
| 241 | C22×C3 | — | 3 | 1 | no | — |
| 242 | C22×C3 | — | 4 | 1 | no | — |
| 243 | C22×C3 | — | 6 | 1 | no | — |
| 244 | C22×C3 | — | 6 | 1 | no | — |
| 245 | C22×C3 | — | 8 | 1 | no | — |
| 246 | C22×C3 | — | 24 | 2 | no | — |
| 247 | C22×C3 | — | 48 | 1 | no | — |
| 248 | C22×C3 | — | 48 | 1 | no | — |
| 249 | C22×C3 | GL(2,ℤ/4) | 1 | 1 | yes | characteristic |
| 250 | C2×D3 | — | 2 | 2 | no | — |
| 251 | C2×D3 | — | 4 | 1 | no | — |
| 252 | C2×D3 | — | 4 | 1 | no | — |
| 253 | C3×C4 | — | 3 | 1 | no | — |
| 254 | C3×C4 | — | 3 | 1 | no | — |
| 255 | C3×C4 | — | 4 | 1 | no | — |
| 256 | C3×C4 | — | 4 | 1 | no | — |
| 257 | C3×C4 | — | 8 | 1 | no | — |
| 258 | C3×C4 | — | 8 | 1 | no | — |
| 259 | C3×C4 | — | 12 | 2 | no | — |
| 260 | C3×C4 | C2×SL(2,ℤ/4) | 1 | 1 | yes | characteristic |
| 261 | C3×C4 | GL(2,ℤ/4) | 1 | 1 | yes | characteristic |
| 262 | Dic3 | — | 6 | 2 | no | — |
| 263 | Dic3 | — | 24 | 1 | no | — |
| 264 | Dic3 | — | 48 | 1 | no | — |
| 265 | Dic3 | — | 48 | 2 | no | — |
| 266 | C32 | — | 4 | 1 | no | Sylow |
| 267 | C23 | — | 3 | 1 | no | — |
| 268 | C23 | — | 6 | 1 | no | — |
| 269 | Q8 | — | 3 | 1 | no | — |
| 270 | Q8 | — | 9 | 2 | no | — |
| 271 | Q8 | — | 18 | 2 | no | — |
| 272 | Q8 | — | 18 | 2 | no | — |
| 273 | Q8 | — | 18 | 1 | no | — |
| 274 | Q8 | (C2×D3)⋊Dic3 | 1 | 1 | yes | characteristic |
| 275 | D4 | — | 3 | 1 | no | — |
| 276 | D4 | — | 3 | 1 | no | — |
| 277 | D4 | — | 3 | 2 | no | — |
| 278 | D4 | — | 6 | 1 | no | — |
| 279 | D4 | — | 6 | 1 | no | — |
| 280 | D4 | — | 18 | 2 | no | — |
| 281 | D4 | — | 18 | 2 | no | — |
| 282 | C2×C4 | — | 3 | 1 | no | — |
| 283 | C2×C4 | — | 3 | 1 | no | — |
| 284 | C2×C4 | — | 3 | 1 | no | — |
| 285 | C2×C4 | — | 3 | 1 | no | — |
| 286 | C2×C4 | — | 3 | 1 | no | — |
| 287 | C2×C4 | — | 6 | 2 | no | — |
| 288 | C2×C4 | — | 9 | 4 | no | — |
| 289 | C2×C4 | — | 12 | 1 | no | — |
| 290 | C2×C4 | — | 12 | 1 | no | — |
| 291 | C2×C4 | — | 18 | 2 | no | — |
| 292 | C2×C4 | — | 18 | 2 | no | — |
| 293 | C2×C4 | — | 18 | 2 | no | — |
| 294 | C2×C4 | — | 18 | 1 | no | — |
| 295 | C2×C4 | — | 36 | 2 | no | — |
| 296 | C2×C4 | — | 36 | 1 | no | — |
| 297 | C2×C4 | D3×S4 | 1 | 1 | yes | characteristic |
| 298 | C8 | — | 12 | 1 | no | — |
| 299 | C8 | — | 36 | 1 | no | — |
| 300 | C2×C3 | — | 4 | 1 | no | — |
| 301 | C2×C3 | — | 4 | 1 | no | — |
| 302 | C2×C3 | — | 4 | 1 | no | — |
| 303 | C2×C3 | — | 6 | 1 | no | — |
| 304 | C2×C3 | — | 6 | 1 | no | — |
| 305 | C2×C3 | — | 8 | 1 | no | — |
| 306 | C2×C3 | — | 8 | 1 | no | — |
| 307 | C2×C3 | — | 8 | 1 | no | — |
| 308 | C2×C3 | — | 48 | 2 | no | — |
| 309 | C2×C3 | #5430 | 1 | 1 | yes | characteristic |
| 310 | C2×C3 | #5431 | 1 | 1 | yes | characteristic |
| 311 | C2×C3 | #5432 | 1 | 1 | yes | characteristic |
| 312 | D3 | — | 4 | 2 | no | — |
| 313 | C22 | — | 3 | 1 | no | — |
| 314 | C22 | — | 3 | 1 | no | — |
| 315 | C22 | — | 6 | 1 | no | — |
| 316 | C22 | — | 6 | 1 | no | — |
| 317 | C22 | — | 6 | 2 | no | — |
| 318 | C22 | — | 12 | 1 | no | — |
| 319 | C22 | — | 12 | 1 | no | — |
| 320 | C22 | #64210 | 1 | 1 | yes | characteristic |
| 321 | C4 | — | 3 | 1 | no | — |
| 322 | C4 | — | 3 | 1 | no | — |
| 323 | C4 | — | 12 | 2 | no | — |
| 324 | C4 | — | 18 | 2 | no | — |
| 325 | C4 | — | 72 | 1 | no | — |
| 326 | C4 | D3×SL(2,ℤ/4) | 1 | 1 | yes | characteristic |
| 327 | C4 | #64209 | 1 | 1 | yes | characteristic |
| 328 | C3 | — | 4 | 1 | no | — |
| 329 | C3 | — | 8 | 1 | no | — |
| 330 | C3 | #73155 | 1 | 1 | yes | characteristic, complemented |
| 331 | C2 | — | 6 | 1 | no | — |
| 332 | C2 | — | 6 | 1 | no | — |
| 333 | C2 | — | 12 | 2 | no | — |
| 334 | C2 | #10593230 | 1 | 1 | yes | characteristic |
| 335 | C2 | #10593232 | 1 | 1 | yes | characteristic |
| 336 | C2 | (C2×D3)⋊SL(2,ℤ/4) | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #10593660 | C2 | semidirect |
| 3. C2×C3×GU(2,𝔽3) | C2 | semidirect |
| 8. #1010⋊C3 | C4 | semidirect |
| 17. C2×GU(2,𝔽3) | D3 | semidirect |
| 36. #1010 | Dic3 | semidirect |
| 330. C3 | #73155 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#1010⋊C3 | 2 |
| #10593660 | 2 |
| C2×C3×GU(2,𝔽3) | 2 |
| #73155 | 3 |
| #70387 | 3 |
| #63563 | 4 |