Finite Group DatabasebetaFinite Group DatabasebetaOrder 1152.
573 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 573 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, complemented |
| 2 | C2×#63692 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #10589529 | C2 | 1 | 4 | yes | maximal, complemented |
| 4 | #10589526 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #84101 | — | 3 | 1 | no | maximal |
| 6 | #63700 | — | 2 | 2 | no | — |
| 7 | C2×#41⋊C9 | C22 | 1 | 1 | yes | characteristic |
| 8 | C2×#41⋊C9 | C22 | 1 | 2 | yes | complemented |
| 9 | #63692 | — | 2 | 2 | no | — |
| 10 | C2×#3689 | C22 | 1 | 2 | yes | complemented |
| 11 | #63509 | — | 4 | 1 | no | maximal |
| 12 | #63421 | C22 | 1 | 2 | yes | complemented |
| 13 | C2×C3×H(5,𝔽2) | D3 | 1 | 1 | yes | characteristic |
| 14 | #5830 | — | 3 | 1 | no | — |
| 15 | C2×#949 | — | 3 | 1 | no | — |
| 16 | C22×#213 | — | 3 | 1 | no | — |
| 17 | #5238 | — | 3 | 4 | no | — |
| 18 | #5234 | — | 3 | 1 | no | — |
| 19 | #5233 | — | 3 | 1 | no | — |
| 20 | #5217 | — | 3 | 4 | no | — |
| 21 | #5215 | — | 3 | 1 | no | — |
| 22 | C2×Q8×C9 | — | 4 | 1 | no | — |
| 23 | C2×D36 | — | 4 | 1 | no | — |
| 24 | #41⋊C9 | — | 2 | 2 | no | — |
| 25 | #41⋊C9 | D4 | 1 | 2 | yes | complemented |
| 26 | C2×Q8⋊C9 | C23 | 1 | 1 | yes | characteristic |
| 27 | #3689 | — | 2 | 2 | no | — |
| 28 | (C2×D9)⋊C4 | — | 4 | 4 | no | — |
| 29 | C4×Dic9 | — | 4 | 1 | no | — |
| 30 | #3008 | — | 9 | 1 | no | Sylow |
| 31 | C3×H(5,𝔽2) | — | 2 | 2 | no | — |
| 32 | C3×H(5,𝔽2) | — | 6 | 2 | no | — |
| 33 | C2×C3×#41 | — | 3 | 1 | no | — |
| 34 | C2×C3×#41 | C2×D3 | 1 | 2 | yes | — |
| 35 | C2×C3×#41 | C2×D3 | 1 | 1 | yes | characteristic |
| 36 | C22×C3×D4 | — | 3 | 1 | no | — |
| 37 | C22×C3×D4 | — | 3 | 2 | no | — |
| 38 | C22×D12 | — | 3 | 1 | no | — |
| 39 | #949 | — | 6 | 2 | no | — |
| 40 | #949 | — | 6 | 2 | no | — |
| 41 | #947 | — | 6 | 1 | no | — |
| 42 | C2×#215 | — | 3 | 2 | no | — |
| 43 | #940 | — | 6 | 1 | no | — |
| 44 | #937 | — | 6 | 2 | no | — |
| 45 | C2×#213 | — | 3 | 2 | no | — |
| 46 | C2×#213 | — | 3 | 4 | no | — |
| 47 | C2×#213 | — | 6 | 4 | no | — |
| 48 | #930 | — | 3 | 4 | no | — |
| 49 | #926 | — | 3 | 1 | no | — |
| 50 | C2×#208 | — | 3 | 1 | no | — |
| 51 | C22×C3⋊C8 | — | 3 | 1 | no | — |
| 52 | #835 | — | 3 | 2 | no | — |
| 53 | #832 | — | 3 | 4 | no | — |
| 54 | #832 | — | 3 | 2 | no | — |
| 55 | #830 | — | 3 | 4 | no | — |
| 56 | C22×D9 | — | 4 | 2 | no | — |
| 57 | Q8×C9 | — | 8 | 2 | no | — |
| 58 | C2×C4×C9 | — | 4 | 1 | no | — |
| 59 | C2×C4×C9 | — | 4 | 2 | no | — |
| 60 | C2×Dic9 | — | 4 | 2 | no | — |
| 61 | D36 | — | 8 | 2 | no | — |
| 62 | Q8⋊C9 | C2×D4 | 1 | 1 | yes | characteristic |
| 63 | C2×H(5,𝔽2) | D9 | 1 | 1 | yes | characteristic, complemented |
| 64 | C2×AGL(1,ℤ/8) | — | 9 | 1 | no | — |
| 65 | C22×D8 | — | 9 | 1 | no | — |
| 66 | #535 | — | 9 | 1 | no | — |
| 67 | #449 | — | 9 | 4 | no | — |
| 68 | #447 | — | 9 | 4 | no | — |
| 69 | #418 | — | 9 | 1 | no | — |
| 70 | #416 | — | 9 | 1 | no | — |
| 71 | #408 | — | 9 | 1 | no | — |
| 72 | C24×C3 | — | 6 | 1 | no | — |
| 73 | C23×D3 | — | 6 | 1 | no | — |
| 74 | C3×#41 | — | 2 | 2 | no | — |
| 75 | C3×#41 | — | 3 | 2 | no | — |
| 76 | C3×#41 | — | 6 | 2 | no | — |
| 77 | C3×#41 | — | 6 | 2 | no | — |
| 78 | C3×#41 | — | 6 | 2 | no | — |
| 79 | C3×#41 | #67 | 1 | 2 | yes | — |
| 80 | C2×C3×Q8 | S4 | 1 | 1 | yes | characteristic |
| 81 | C2×C3×Q8 | C22×D3 | 1 | 1 | yes | characteristic |
| 82 | C2×C3×D4 | — | 3 | 1 | no | — |
| 83 | C2×C3×D4 | — | 3 | 2 | no | — |
| 84 | C2×C3×D4 | — | 3 | 4 | no | — |
| 85 | C2×C3×D4 | — | 3 | 2 | no | — |
| 86 | C2×C3×D4 | — | 3 | 1 | no | — |
| 87 | C2×C3×D4 | — | 6 | 2 | no | — |
| 88 | C2×C3×D4 | — | 6 | 4 | no | — |
| 89 | C2×C3×D4 | — | 6 | 4 | no | — |
| 90 | C22×C3×C4 | — | 3 | 1 | no | — |
| 91 | C22×C3×C4 | — | 3 | 2 | no | — |
| 92 | C2×#67 | — | 6 | 4 | no | — |
| 93 | C2×D12 | — | 3 | 2 | no | — |
| 94 | C2×D12 | — | 6 | 2 | no | — |
| 95 | C2×D12 | — | 6 | 2 | no | — |
| 96 | C2×D12 | — | 6 | 1 | no | — |
| 97 | #217 | — | 6 | 1 | no | — |
| 98 | #215 | — | 6 | 2 | no | — |
| 99 | #215 | — | 6 | 2 | no | — |
| 100 | #213 | — | 6 | 8 | no | — |
| 101 | #213 | — | 6 | 4 | no | — |
| 102 | (C2×D3)⋊C4 | — | 6 | 4 | no | — |
| 103 | C4⋊Dic3 | — | 3 | 2 | no | — |
| 104 | C4×Dic3 | — | 6 | 1 | no | — |
| 105 | #208 | — | 6 | 2 | no | — |
| 106 | C2×C3⋊C8 | — | 3 | 2 | no | — |
| 107 | C2×C3⋊C8 | — | 3 | 2 | no | — |
| 108 | C2×C3⋊C8 | — | 6 | 2 | no | — |
| 109 | C22×C9 | — | 4 | 1 | no | — |
| 110 | C2×D9 | — | 8 | 2 | no | — |
| 111 | C2×D9 | — | 8 | 4 | no | — |
| 112 | C4×C9 | — | 4 | 2 | no | — |
| 113 | C4×C9 | — | 8 | 2 | no | — |
| 114 | Dic9 | — | 8 | 2 | no | — |
| 115 | H(5,𝔽2) | — | 2 | 2 | no | — |
| 116 | H(5,𝔽2) | — | 6 | 2 | no | — |
| 117 | C2×#41 | — | 3 | 1 | no | — |
| 118 | C2×#41 | C2×D9 | 1 | 2 | yes | — |
| 119 | C2×#41 | C2×D9 | 1 | 1 | yes | characteristic |
| 120 | C22×D4 | — | 3 | 1 | no | — |
| 121 | C22×D4 | — | 3 | 2 | no | — |
| 122 | C22×D4 | — | 9 | 1 | no | — |
| 123 | AGL(1,ℤ/8) | — | 18 | 2 | no | — |
| 124 | AGL(1,ℤ/8) | — | 18 | 2 | no | — |
| 125 | C2×SD16 | — | 9 | 2 | no | — |
| 126 | C2×D8 | — | 9 | 4 | no | — |
| 127 | C2×D8 | — | 9 | 2 | no | — |
| 128 | C2×D8 | — | 18 | 4 | no | — |
| 129 | C2×M16 | — | 9 | 1 | no | — |
| 130 | C22×C8 | — | 9 | 1 | no | — |
| 131 | #127 | — | 18 | 1 | no | — |
| 132 | #124 | — | 18 | 1 | no | — |
| 133 | #121 | — | 9 | 4 | no | — |
| 134 | #120 | — | 18 | 2 | no | — |
| 135 | #117 | — | 9 | 1 | no | — |
| 136 | #103 | — | 9 | 2 | no | — |
| 137 | #102 | — | 9 | 2 | no | — |
| 138 | #102 | — | 9 | 4 | no | — |
| 139 | C22⋊C8 | — | 9 | 4 | no | — |
| 140 | C23×C3 | — | 3 | 1 | no | — |
| 141 | C23×C3 | — | 3 | 2 | no | — |
| 142 | C23×C3 | — | 6 | 4 | no | — |
| 143 | C23×C3 | — | 24 | 2 | no | — |
| 144 | C22×D3 | — | 6 | 2 | no | — |
| 145 | C22×D3 | — | 12 | 2 | no | — |
| 146 | C22×D3 | — | 24 | 2 | no | — |
| 147 | C3×Q8 | — | 2 | 2 | no | — |
| 148 | C3×Q8 | — | 3 | 1 | no | — |
| 149 | C3×Q8 | C2×#67 | 1 | 1 | yes | characteristic |
| 150 | C3×D4 | — | 3 | 2 | no | — |
| 151 | C3×D4 | — | 3 | 4 | no | — |
| 152 | C3×D4 | — | 6 | 4 | no | — |
| 153 | C3×D4 | — | 6 | 1 | no | — |
| 154 | C3×D4 | — | 6 | 2 | no | — |
| 155 | C3×D4 | — | 6 | 2 | no | — |
| 156 | C2×C3×C4 | — | 3 | 1 | no | — |
| 157 | C2×C3×C4 | — | 3 | 2 | no | — |
| 158 | C2×C3×C4 | — | 3 | 2 | no | — |
| 159 | C2×C3×C4 | — | 6 | 2 | no | — |
| 160 | C2×C3×C4 | — | 6 | 2 | no | — |
| 161 | C2×C3×C4 | C2×S4 | 1 | 2 | yes | — |
| 162 | C2×C3×C4 | C2×S4 | 1 | 1 | yes | characteristic |
| 163 | #67 | — | 24 | 4 | no | — |
| 164 | C2×Dic3 | — | 6 | 2 | no | — |
| 165 | D12 | — | 6 | 2 | no | — |
| 166 | D12 | — | 6 | 2 | no | — |
| 167 | D12 | — | 12 | 2 | no | — |
| 168 | C3⋊C8 | — | 6 | 2 | no | — |
| 169 | C3⋊C8 | — | 6 | 2 | no | — |
| 170 | C2×C9 | — | 4 | 2 | no | — |
| 171 | C2×C9 | — | 4 | 1 | no | — |
| 172 | D9 | — | 16 | 2 | no | — |
| 173 | C24 | — | 6 | 1 | no | — |
| 174 | C24 | — | 18 | 1 | no | — |
| 175 | #41 | — | 2 | 2 | no | — |
| 176 | #41 | — | 3 | 2 | no | — |
| 177 | #41 | — | 6 | 2 | no | — |
| 178 | #41 | — | 6 | 2 | no | — |
| 179 | #41 | — | 6 | 2 | no | — |
| 180 | #41 | #611 | 1 | 2 | yes | — |
| 181 | C2×Q8 | (C22⋊C9)⋊C2 | 1 | 1 | yes | characteristic |
| 182 | C2×Q8 | C22×D9 | 1 | 1 | yes | characteristic |
| 183 | C2×D4 | — | 3 | 1 | no | — |
| 184 | C2×D4 | — | 3 | 2 | no | — |
| 185 | C2×D4 | — | 3 | 4 | no | — |
| 186 | C2×D4 | — | 3 | 2 | no | — |
| 187 | C2×D4 | — | 3 | 1 | no | — |
| 188 | C2×D4 | — | 6 | 2 | no | — |
| 189 | C2×D4 | — | 6 | 4 | no | — |
| 190 | C2×D4 | — | 6 | 4 | no | — |
| 191 | C2×D4 | — | 9 | 2 | no | — |
| 192 | C2×D4 | — | 18 | 2 | no | — |
| 193 | C2×D4 | — | 18 | 2 | no | — |
| 194 | C2×D4 | — | 18 | 4 | no | — |
| 195 | C2×D4 | — | 18 | 1 | no | — |
| 196 | C22×C4 | — | 3 | 1 | no | — |
| 197 | C22×C4 | — | 3 | 2 | no | — |
| 198 | SD16 | — | 18 | 2 | no | — |
| 199 | SD16 | — | 18 | 2 | no | — |
| 200 | D8 | — | 18 | 8 | no | — |
| 201 | D8 | — | 18 | 4 | no | — |
| 202 | M16 | — | 18 | 2 | no | — |
| 203 | C2×C8 | — | 9 | 2 | no | — |
| 204 | C2×C8 | — | 9 | 2 | no | — |
| 205 | C2×C8 | — | 18 | 2 | no | — |
| 206 | C4⋊C4 | — | 9 | 2 | no | — |
| 207 | C22⋊C4 | — | 18 | 1 | no | — |
| 208 | C22⋊C4 | — | 18 | 4 | no | — |
| 209 | C42 | — | 18 | 1 | no | — |
| 210 | C22×C3 | — | 3 | 2 | no | — |
| 211 | C22×C3 | — | 6 | 2 | no | — |
| 212 | C22×C3 | — | 6 | 4 | no | — |
| 213 | C22×C3 | — | 6 | 2 | no | — |
| 214 | C22×C3 | — | 24 | 4 | no | — |
| 215 | C22×C3 | C22×S4 | 1 | 1 | yes | characteristic |
| 216 | C2×D3 | — | 12 | 4 | no | — |
| 217 | C2×D3 | — | 12 | 2 | no | — |
| 218 | C2×D3 | — | 24 | 4 | no | — |
| 219 | C3×C4 | — | 2 | 2 | no | — |
| 220 | C3×C4 | — | 3 | 1 | no | — |
| 221 | C3×C4 | — | 3 | 1 | no | — |
| 222 | C3×C4 | GL(2,ℤ/4) | 1 | 2 | yes | — |
| 223 | Dic3 | — | 12 | 2 | no | — |
| 224 | C9 | — | 4 | 1 | no | Sylow |
| 225 | C23 | — | 3 | 1 | no | — |
| 226 | C23 | — | 3 | 2 | no | — |
| 227 | C23 | — | 6 | 4 | no | — |
| 228 | C23 | — | 18 | 2 | no | — |
| 229 | C23 | — | 24 | 2 | no | — |
| 230 | C23 | — | 36 | 2 | no | — |
| 231 | C23 | — | 72 | 2 | no | — |
| 232 | Q8 | — | 2 | 2 | no | — |
| 233 | Q8 | — | 3 | 1 | no | — |
| 234 | Q8 | C2×#611 | 1 | 1 | yes | characteristic |
| 235 | D4 | — | 3 | 2 | no | — |
| 236 | D4 | — | 3 | 4 | no | — |
| 237 | D4 | — | 6 | 4 | no | — |
| 238 | D4 | — | 6 | 1 | no | — |
| 239 | D4 | — | 6 | 2 | no | — |
| 240 | D4 | — | 6 | 2 | no | — |
| 241 | D4 | — | 18 | 2 | no | — |
| 242 | D4 | — | 18 | 2 | no | — |
| 243 | D4 | — | 36 | 2 | no | — |
| 244 | D4 | — | 72 | 4 | no | — |
| 245 | C2×C4 | — | 3 | 1 | no | — |
| 246 | C2×C4 | — | 3 | 2 | no | — |
| 247 | C2×C4 | — | 3 | 2 | no | — |
| 248 | C2×C4 | — | 6 | 2 | no | — |
| 249 | C2×C4 | — | 6 | 2 | no | — |
| 250 | C2×C4 | — | 18 | 2 | no | — |
| 251 | C2×C4 | C2×(C22⋊C9)⋊C2 | 1 | 2 | yes | — |
| 252 | C2×C4 | C2×(C22⋊C9)⋊C2 | 1 | 1 | yes | characteristic |
| 253 | C8 | — | 18 | 2 | no | — |
| 254 | C8 | — | 18 | 2 | no | — |
| 255 | C2×C3 | — | 6 | 2 | no | — |
| 256 | C2×C3 | — | 12 | 2 | no | — |
| 257 | C2×C3 | #5925 | 1 | 2 | yes | — |
| 258 | C2×C3 | C2×GL(2,ℤ/4) | 1 | 1 | yes | characteristic |
| 259 | D3 | — | 24 | 2 | no | — |
| 260 | C22 | — | 3 | 2 | no | — |
| 261 | C22 | — | 6 | 2 | no | — |
| 262 | C22 | — | 6 | 4 | no | — |
| 263 | C22 | — | 6 | 2 | no | — |
| 264 | C22 | — | 24 | 4 | no | — |
| 265 | C22 | — | 36 | 4 | no | — |
| 266 | C22 | — | 36 | 2 | no | — |
| 267 | C22 | — | 72 | 4 | no | — |
| 268 | C22 | C22×(C22⋊C9)⋊C2 | 1 | 1 | yes | characteristic |
| 269 | C4 | — | 2 | 2 | no | — |
| 270 | C4 | — | 3 | 1 | no | — |
| 271 | C4 | — | 3 | 1 | no | — |
| 272 | C4 | — | 36 | 2 | no | — |
| 273 | C4 | #63694 | 1 | 2 | yes | — |
| 274 | C3 | #85530 | 1 | 1 | yes | characteristic |
| 275 | C2 | — | 6 | 2 | no | — |
| 276 | C2 | — | 12 | 2 | no | — |
| 277 | C2 | — | 72 | 2 | no | — |
| 278 | C2 | #10593100 | 1 | 2 | yes | — |
| 279 | C2 | C2×#63694 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#63700 | C2 | semidirect |
| 2. C2×#63692 | C2 | semidirect |
| 3. #10589529 | C2 | semidirect |
| 4. #10589526 | C2 | semidirect |
| 8. C2×#41⋊C9 | C22 | semidirect |
| 10. C2×#3689 | C22 | semidirect |
| 12. #63421 | C22 | semidirect |
| 25. #41⋊C9 | D4 | semidirect |
| 63. C2×H(5,𝔽2) | D9 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#63700 | 2 |
| C2×#63692 | 2 |
| #10589529 | 2 |
| #10589526 | 2 |
| #84101 | 3 |
| #63509 | 4 |