Finite Group DatabasebetaFinite Group DatabasebetaOrder 1152.
541 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 541 conjugacy classes of subgroups — too many to draw. Only its 57 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×#63700 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×#63690 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #10589530 | C2 | 1 | 4 | yes | maximal, direct product factor |
| 4 | C2×#63422 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | C2×#5242 | — | 3 | 1 | no | maximal |
| 6 | #63700 | — | 2 | 1 | no | — |
| 7 | #63700 | C22 | 1 | 2 | yes | — |
| 8 | C2×#41⋊C9 | — | 2 | 1 | no | — |
| 9 | C2×#41⋊C9 | C22 | 1 | 1 | yes | characteristic |
| 10 | #63690 | — | 2 | 1 | no | — |
| 11 | #63690 | C22 | 1 | 2 | yes | complemented |
| 12 | C2×#3688 | — | 2 | 1 | no | — |
| 13 | C2×#3674 | — | 4 | 1 | no | maximal |
| 14 | #63422 | C22 | 1 | 2 | yes | complemented |
| 15 | C2×C3×H(5,𝔽2) | D3 | 1 | 1 | yes | characteristic |
| 16 | C2×#951 | — | 3 | 1 | no | — |
| 17 | C2×#951 | — | 3 | 1 | no | — |
| 18 | C2×#936 | — | 3 | 1 | no | — |
| 19 | #5242 | — | 3 | 4 | no | — |
| 20 | #5242 | — | 3 | 4 | no | — |
| 21 | C2×#837 | — | 3 | 1 | no | — |
| 22 | C2×#837 | — | 3 | 1 | no | — |
| 23 | C2×#833 | — | 3 | 1 | no | — |
| 24 | C2×Q8×C9 | — | 4 | 1 | no | — |
| 25 | C2×Dic18 | — | 4 | 1 | no | — |
| 26 | #41⋊C9 | — | 2 | 2 | no | — |
| 27 | #41⋊C9 | D4 | 1 | 1 | yes | characteristic |
| 28 | #41⋊C9 | C23 | 1 | 1 | yes | characteristic |
| 29 | C2×Q8⋊C9 | D4 | 1 | 1 | yes | characteristic |
| 30 | #3688 | — | 2 | 2 | no | — |
| 31 | #3674 | — | 4 | 4 | no | — |
| 32 | C2×C9⋊C8 | — | 4 | 1 | no | — |
| 33 | C2×#455 | — | 9 | 1 | no | Sylow |
| 34 | C3×H(5,𝔽2) | — | 2 | 1 | no | — |
| 35 | C3×H(5,𝔽2) | — | 3 | 2 | no | — |
| 36 | C3×H(5,𝔽2) | — | 6 | 1 | no | — |
| 37 | C3×H(5,𝔽2) | C2×D3 | 1 | 2 | yes | — |
| 38 | C2×C3×#41 | — | 2 | 1 | no | — |
| 39 | C2×C3×#41 | — | 3 | 1 | no | — |
| 40 | C2×C3×#41 | C2×D3 | 1 | 1 | yes | characteristic |
| 41 | C22×C3×D4 | — | 3 | 1 | no | — |
| 42 | C22×C3×D4 | — | 6 | 1 | no | — |
| 43 | C22×Dic6 | — | 3 | 1 | no | — |
| 44 | C2×#217 | — | 6 | 2 | no | — |
| 45 | #951 | — | 3 | 2 | no | — |
| 46 | #951 | — | 3 | 4 | no | — |
| 47 | #951 | — | 3 | 2 | no | — |
| 48 | #951 | — | 6 | 1 | no | — |
| 49 | #951 | — | 6 | 2 | no | — |
| 50 | #951 | — | 6 | 1 | no | — |
| 51 | C2×#216 | — | 6 | 1 | no | — |
| 52 | C2×#216 | — | 6 | 1 | no | — |
| 53 | #936 | — | 3 | 4 | no | — |
| 54 | #936 | — | 6 | 2 | no | — |
| 55 | C2×#214 | — | 6 | 1 | no | — |
| 56 | C2×#214 | — | 6 | 1 | no | — |
| 57 | C2×C4×Dic3 | — | 3 | 1 | no | — |
| 58 | C2×#208 | — | 3 | 1 | no | — |
| 59 | C2×#208 | — | 3 | 1 | no | — |
| 60 | #837 | — | 3 | 2 | no | — |
| 61 | #837 | — | 3 | 4 | no | — |
| 62 | #837 | — | 3 | 2 | no | — |
| 63 | #833 | — | 3 | 4 | no | — |
| 64 | Q8×C9 | — | 4 | 2 | no | — |
| 65 | Q8×C9 | — | 8 | 1 | no | — |
| 66 | C2×C4×C9 | — | 4 | 1 | no | — |
| 67 | C2×C4×C9 | — | 8 | 1 | no | — |
| 68 | C2×Dic9 | — | 8 | 1 | no | — |
| 69 | Dic18 | — | 4 | 2 | no | — |
| 70 | Dic18 | — | 8 | 1 | no | — |
| 71 | Q8⋊C9 | C2×D4 | 1 | 1 | yes | characteristic |
| 72 | C9⋊C8 | — | 4 | 2 | no | — |
| 73 | C2×H(5,𝔽2) | D9 | 1 | 1 | yes | characteristic |
| 74 | C2×#137 | — | 9 | 1 | no | — |
| 75 | C2×#137 | — | 9 | 1 | no | — |
| 76 | C2×#124 | — | 9 | 1 | no | — |
| 77 | #455 | — | 9 | 4 | no | — |
| 78 | #455 | — | 9 | 4 | no | — |
| 79 | C2×#104 | — | 9 | 1 | no | — |
| 80 | C2×#104 | — | 9 | 1 | no | — |
| 81 | C2×#100 | — | 9 | 1 | no | — |
| 82 | C24×C3 | — | 3 | 2 | no | — |
| 83 | C3×#41 | — | 2 | 2 | no | — |
| 84 | C3×#41 | — | 3 | 1 | no | — |
| 85 | C3×#41 | — | 3 | 1 | no | — |
| 86 | C3×#41 | — | 3 | 2 | no | — |
| 87 | C3×#41 | — | 3 | 2 | no | — |
| 88 | C3×#41 | — | 6 | 1 | no | — |
| 89 | C3×#41 | — | 6 | 1 | no | — |
| 90 | C3×#41 | — | 6 | 2 | no | — |
| 91 | C3×#41 | #67 | 1 | 1 | yes | characteristic |
| 92 | C3×#41 | C22×D3 | 1 | 1 | yes | characteristic |
| 93 | C2×C3×Q8 | #67 | 1 | 1 | yes | characteristic |
| 94 | C2×C3×Q8 | S4 | 1 | 1 | yes | characteristic |
| 95 | C2×C3×D4 | — | 3 | 1 | no | — |
| 96 | C2×C3×D4 | — | 3 | 2 | no | — |
| 97 | C2×C3×D4 | — | 3 | 2 | no | — |
| 98 | C2×C3×D4 | — | 3 | 1 | no | — |
| 99 | C2×C3×D4 | — | 6 | 1 | no | — |
| 100 | C2×C3×D4 | — | 6 | 1 | no | — |
| 101 | C2×C3×D4 | — | 6 | 1 | no | — |
| 102 | C2×C3×D4 | — | 6 | 1 | no | — |
| 103 | C2×C3×D4 | — | 6 | 2 | no | — |
| 104 | C2×C3×D4 | — | 6 | 2 | no | — |
| 105 | C2×C3×D4 | — | 6 | 4 | no | — |
| 106 | C22×C3×C4 | — | 3 | 1 | no | — |
| 107 | C22×C3×C4 | — | 6 | 1 | no | — |
| 108 | C22×Dic3 | — | 6 | 1 | no | — |
| 109 | C22×Dic3 | — | 6 | 1 | no | — |
| 110 | C2×Dic6 | — | 3 | 2 | no | — |
| 111 | C2×Dic6 | — | 6 | 1 | no | — |
| 112 | C2×Dic6 | — | 6 | 1 | no | — |
| 113 | C2×Dic6 | — | 6 | 1 | no | — |
| 114 | C2×Dic6 | — | 6 | 1 | no | — |
| 115 | C2×Dic6 | — | 6 | 1 | no | — |
| 116 | #217 | — | 6 | 8 | no | — |
| 117 | #216 | — | 6 | 2 | no | — |
| 118 | #216 | — | 6 | 2 | no | — |
| 119 | #216 | — | 6 | 4 | no | — |
| 120 | #214 | — | 6 | 2 | no | — |
| 121 | #214 | — | 6 | 2 | no | — |
| 122 | #214 | — | 6 | 4 | no | — |
| 123 | C4×Dic3 | — | 3 | 2 | no | — |
| 124 | C4×Dic3 | — | 6 | 1 | no | — |
| 125 | #208 | — | 3 | 2 | no | — |
| 126 | #208 | — | 3 | 2 | no | — |
| 127 | #208 | — | 6 | 1 | no | — |
| 128 | #208 | — | 6 | 1 | no | — |
| 129 | C2×C3⋊C8 | — | 6 | 1 | no | — |
| 130 | C2×C3⋊C8 | — | 6 | 1 | no | — |
| 131 | C22×C9 | — | 4 | 1 | no | — |
| 132 | C4×C9 | — | 4 | 1 | no | — |
| 133 | C4×C9 | — | 4 | 1 | no | — |
| 134 | C4×C9 | — | 8 | 2 | no | — |
| 135 | Dic9 | — | 8 | 2 | no | — |
| 136 | H(5,𝔽2) | — | 2 | 1 | no | — |
| 137 | H(5,𝔽2) | — | 3 | 2 | no | — |
| 138 | H(5,𝔽2) | — | 6 | 1 | no | — |
| 139 | H(5,𝔽2) | C2×D9 | 1 | 2 | yes | — |
| 140 | C2×#41 | — | 2 | 1 | no | — |
| 141 | C2×#41 | — | 3 | 1 | no | — |
| 142 | C2×#41 | C2×D9 | 1 | 1 | yes | characteristic |
| 143 | C22×Q8 | — | 9 | 1 | no | — |
| 144 | C22×D4 | — | 3 | 1 | no | — |
| 145 | C22×D4 | — | 6 | 1 | no | — |
| 146 | #137 | — | 9 | 2 | no | — |
| 147 | #137 | — | 9 | 4 | no | — |
| 148 | #137 | — | 9 | 2 | no | — |
| 149 | #137 | — | 18 | 1 | no | — |
| 150 | #137 | — | 18 | 2 | no | — |
| 151 | #137 | — | 18 | 1 | no | — |
| 152 | C2×Q16 | — | 18 | 1 | no | — |
| 153 | C2×Q16 | — | 18 | 1 | no | — |
| 154 | C2×SD16 | — | 18 | 1 | no | — |
| 155 | C2×SD16 | — | 18 | 1 | no | — |
| 156 | C2×M16 | — | 9 | 1 | no | — |
| 157 | C2×M16 | — | 9 | 1 | no | — |
| 158 | #124 | — | 9 | 4 | no | — |
| 159 | #124 | — | 18 | 2 | no | — |
| 160 | C2×C22⋊C4 | — | 18 | 2 | no | — |
| 161 | C2×C42 | — | 9 | 1 | no | — |
| 162 | #104 | — | 9 | 2 | no | — |
| 163 | #104 | — | 9 | 4 | no | — |
| 164 | #104 | — | 9 | 2 | no | — |
| 165 | #100 | — | 9 | 4 | no | — |
| 166 | C23×C3 | — | 3 | 1 | no | — |
| 167 | C23×C3 | — | 3 | 4 | no | — |
| 168 | C23×C3 | — | 6 | 2 | no | — |
| 169 | C23×C3 | — | 6 | 1 | no | — |
| 170 | C23×C3 | — | 12 | 4 | no | — |
| 171 | C3×Q8 | — | 2 | 1 | no | — |
| 172 | C3×Q8 | — | 3 | 1 | no | — |
| 173 | C3×Q8 | C2×#67 | 1 | 1 | yes | characteristic |
| 174 | C3×Q8 | C2×S4 | 1 | 2 | yes | — |
| 175 | C3×D4 | — | 3 | 1 | no | — |
| 176 | C3×D4 | — | 3 | 1 | no | — |
| 177 | C3×D4 | — | 3 | 2 | no | — |
| 178 | C3×D4 | — | 6 | 2 | no | — |
| 179 | C3×D4 | — | 6 | 2 | no | — |
| 180 | C3×D4 | — | 6 | 1 | no | — |
| 181 | C3×D4 | — | 6 | 2 | no | — |
| 182 | C3×D4 | — | 6 | 2 | no | — |
| 183 | C3×D4 | — | 6 | 1 | no | — |
| 184 | C2×C3×C4 | — | 2 | 1 | no | — |
| 185 | C2×C3×C4 | — | 3 | 1 | no | — |
| 186 | C2×C3×C4 | — | 3 | 1 | no | — |
| 187 | C2×C3×C4 | — | 3 | 1 | no | — |
| 188 | C2×C3×C4 | — | 3 | 1 | no | — |
| 189 | C2×C3×C4 | — | 3 | 1 | no | — |
| 190 | C2×C3×C4 | — | 6 | 2 | no | — |
| 191 | C2×C3×C4 | — | 6 | 2 | no | — |
| 192 | C2×C3×C4 | C2×S4 | 1 | 1 | yes | characteristic |
| 193 | C2×Dic3 | — | 6 | 2 | no | — |
| 194 | C2×Dic3 | — | 6 | 2 | no | — |
| 195 | C2×Dic3 | — | 12 | 1 | no | — |
| 196 | C2×Dic3 | — | 12 | 2 | no | — |
| 197 | C2×Dic3 | — | 12 | 1 | no | — |
| 198 | Dic6 | — | 6 | 2 | no | — |
| 199 | Dic6 | — | 6 | 2 | no | — |
| 200 | Dic6 | — | 12 | 1 | no | — |
| 201 | Dic6 | — | 12 | 1 | no | — |
| 202 | C3⋊C8 | — | 6 | 2 | no | — |
| 203 | C3⋊C8 | — | 6 | 2 | no | — |
| 204 | C2×C9 | — | 4 | 1 | no | — |
| 205 | C2×C9 | — | 4 | 2 | no | — |
| 206 | C24 | — | 3 | 2 | no | — |
| 207 | #41 | — | 2 | 2 | no | — |
| 208 | #41 | — | 3 | 1 | no | — |
| 209 | #41 | — | 3 | 1 | no | — |
| 210 | #41 | — | 3 | 2 | no | — |
| 211 | #41 | — | 3 | 2 | no | — |
| 212 | #41 | — | 6 | 1 | no | — |
| 213 | #41 | — | 6 | 1 | no | — |
| 214 | #41 | — | 6 | 2 | no | — |
| 215 | #41 | #611 | 1 | 1 | yes | characteristic |
| 216 | #41 | C22×D9 | 1 | 1 | yes | characteristic |
| 217 | C2×Q8 | — | 9 | 2 | no | — |
| 218 | C2×Q8 | — | 18 | 1 | no | — |
| 219 | C2×Q8 | — | 18 | 1 | no | — |
| 220 | C2×Q8 | — | 18 | 1 | no | — |
| 221 | C2×Q8 | — | 18 | 1 | no | — |
| 222 | C2×Q8 | — | 18 | 1 | no | — |
| 223 | C2×Q8 | #611 | 1 | 1 | yes | characteristic |
| 224 | C2×Q8 | (C22⋊C9)⋊C2 | 1 | 1 | yes | characteristic |
| 225 | C2×D4 | — | 3 | 1 | no | — |
| 226 | C2×D4 | — | 3 | 2 | no | — |
| 227 | C2×D4 | — | 3 | 2 | no | — |
| 228 | C2×D4 | — | 3 | 1 | no | — |
| 229 | C2×D4 | — | 6 | 1 | no | — |
| 230 | C2×D4 | — | 6 | 1 | no | — |
| 231 | C2×D4 | — | 6 | 1 | no | — |
| 232 | C2×D4 | — | 6 | 1 | no | — |
| 233 | C2×D4 | — | 6 | 2 | no | — |
| 234 | C2×D4 | — | 6 | 2 | no | — |
| 235 | C2×D4 | — | 6 | 4 | no | — |
| 236 | C22×C4 | — | 3 | 1 | no | — |
| 237 | C22×C4 | — | 6 | 1 | no | — |
| 238 | C22×C4 | — | 18 | 1 | no | — |
| 239 | C22×C4 | — | 18 | 1 | no | — |
| 240 | Q16 | — | 18 | 2 | no | — |
| 241 | Q16 | — | 18 | 2 | no | — |
| 242 | Q16 | — | 18 | 4 | no | — |
| 243 | SD16 | — | 18 | 2 | no | — |
| 244 | SD16 | — | 18 | 2 | no | — |
| 245 | SD16 | — | 18 | 4 | no | — |
| 246 | M16 | — | 9 | 2 | no | — |
| 247 | M16 | — | 9 | 2 | no | — |
| 248 | M16 | — | 18 | 1 | no | — |
| 249 | M16 | — | 18 | 1 | no | — |
| 250 | C2×C8 | — | 18 | 1 | no | — |
| 251 | C2×C8 | — | 18 | 1 | no | — |
| 252 | C22⋊C4 | — | 18 | 8 | no | — |
| 253 | C42 | — | 9 | 2 | no | — |
| 254 | C42 | — | 18 | 1 | no | — |
| 255 | C22×C3 | — | 3 | 1 | no | — |
| 256 | C22×C3 | — | 3 | 1 | no | — |
| 257 | C22×C3 | — | 6 | 2 | no | — |
| 258 | C22×C3 | — | 6 | 2 | no | — |
| 259 | C22×C3 | — | 12 | 4 | no | — |
| 260 | C22×C3 | — | 12 | 2 | no | — |
| 261 | C22×C3 | — | 24 | 2 | no | — |
| 262 | C22×C3 | GL(2,ℤ/4) | 1 | 1 | yes | characteristic |
| 263 | C3×C4 | — | 2 | 2 | no | — |
| 264 | C3×C4 | — | 3 | 1 | no | — |
| 265 | C3×C4 | — | 3 | 1 | no | — |
| 266 | C3×C4 | GL(2,ℤ/4) | 1 | 1 | yes | characteristic |
| 267 | C3×C4 | C22×S4 | 1 | 1 | yes | characteristic |
| 268 | Dic3 | — | 12 | 2 | no | — |
| 269 | Dic3 | — | 12 | 2 | no | — |
| 270 | C9 | — | 4 | 1 | no | Sylow |
| 271 | C23 | — | 3 | 1 | no | — |
| 272 | C23 | — | 3 | 4 | no | — |
| 273 | C23 | — | 6 | 2 | no | — |
| 274 | C23 | — | 6 | 1 | no | — |
| 275 | C23 | — | 12 | 4 | no | — |
| 276 | Q8 | — | 2 | 1 | no | — |
| 277 | Q8 | — | 3 | 1 | no | — |
| 278 | Q8 | — | 18 | 2 | no | — |
| 279 | Q8 | — | 18 | 2 | no | — |
| 280 | Q8 | — | 36 | 1 | no | — |
| 281 | Q8 | — | 36 | 1 | no | — |
| 282 | Q8 | C2×#611 | 1 | 1 | yes | characteristic |
| 283 | Q8 | C2×(C22⋊C9)⋊C2 | 1 | 2 | yes | — |
| 284 | D4 | — | 3 | 1 | no | — |
| 285 | D4 | — | 3 | 1 | no | — |
| 286 | D4 | — | 3 | 2 | no | — |
| 287 | D4 | — | 6 | 2 | no | — |
| 288 | D4 | — | 6 | 2 | no | — |
| 289 | D4 | — | 6 | 1 | no | — |
| 290 | D4 | — | 6 | 2 | no | — |
| 291 | D4 | — | 6 | 2 | no | — |
| 292 | D4 | — | 6 | 1 | no | — |
| 293 | C2×C4 | — | 2 | 1 | no | — |
| 294 | C2×C4 | — | 3 | 1 | no | — |
| 295 | C2×C4 | — | 3 | 1 | no | — |
| 296 | C2×C4 | — | 3 | 1 | no | — |
| 297 | C2×C4 | — | 3 | 1 | no | — |
| 298 | C2×C4 | — | 3 | 1 | no | — |
| 299 | C2×C4 | — | 6 | 2 | no | — |
| 300 | C2×C4 | — | 6 | 2 | no | — |
| 301 | C2×C4 | — | 18 | 2 | no | — |
| 302 | C2×C4 | — | 18 | 2 | no | — |
| 303 | C2×C4 | — | 36 | 1 | no | — |
| 304 | C2×C4 | — | 36 | 2 | no | — |
| 305 | C2×C4 | — | 36 | 1 | no | — |
| 306 | C2×C4 | C2×(C22⋊C9)⋊C2 | 1 | 1 | yes | characteristic |
| 307 | C8 | — | 18 | 2 | no | — |
| 308 | C8 | — | 18 | 2 | no | — |
| 309 | C2×C3 | — | 6 | 1 | no | — |
| 310 | C2×C3 | — | 6 | 1 | no | — |
| 311 | C2×C3 | — | 12 | 2 | no | — |
| 312 | C2×C3 | #5428 | 1 | 2 | yes | — |
| 313 | C2×C3 | C2×GL(2,ℤ/4) | 1 | 1 | yes | characteristic |
| 314 | C22 | — | 3 | 1 | no | — |
| 315 | C22 | — | 3 | 1 | no | — |
| 316 | C22 | — | 6 | 2 | no | — |
| 317 | C22 | — | 6 | 2 | no | — |
| 318 | C22 | — | 12 | 4 | no | — |
| 319 | C22 | — | 12 | 2 | no | — |
| 320 | C22 | — | 24 | 2 | no | — |
| 321 | C22 | #63694 | 1 | 1 | yes | characteristic |
| 322 | C4 | — | 2 | 2 | no | — |
| 323 | C4 | — | 3 | 1 | no | — |
| 324 | C4 | — | 3 | 1 | no | — |
| 325 | C4 | — | 36 | 2 | no | — |
| 326 | C4 | — | 36 | 2 | no | — |
| 327 | C4 | #63694 | 1 | 1 | yes | characteristic |
| 328 | C4 | C22×(C22⋊C9)⋊C2 | 1 | 1 | yes | characteristic |
| 329 | C3 | C2×#5428 | 1 | 1 | yes | characteristic |
| 330 | C2 | — | 6 | 1 | no | — |
| 331 | C2 | — | 6 | 1 | no | — |
| 332 | C2 | — | 12 | 2 | no | — |
| 333 | C2 | #10589530 | 1 | 2 | yes | direct product factor |
| 334 | C2 | C2×#63694 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#63690 | C2 | semidirect |
| 3. #10589530 | C2 | direct (N × H) |
| 4. C2×#63422 | C2 | semidirect |
| 11. #63690 | C22 | semidirect |
| 14. #63422 | C22 | semidirect |
| 333. C2 | #10589530 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#63700 | 2 |
| C2×#63690 | 2 |
| #10589530 | 2 |
| C2×#63422 | 2 |
| C2×#5242 | 3 |
| C2×#3674 | 4 |