Finite Group DatabasebetaFinite Group DatabasebetaOrder 1152.
782 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 782 conjugacy classes of subgroups — too many to draw. Only its 144 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 | #10594771 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #10592618 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C3×#4919 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #10591040 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #10591038 | C2 | 1 | 4 | yes | maximal, complemented |
| 6 | #10590234 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | D3×D48 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #10590227 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | #82031 | — | 3 | 2 | no | maximal |
| 10 | #64320 | — | 2 | 2 | no | — |
| 11 | #64128 | C22 | 1 | 2 | yes | complemented |
| 12 | C3×#919 | C22 | 1 | 4 | yes | complemented |
| 13 | #63941 | — | 2 | 2 | no | — |
| 14 | #63939 | — | 2 | 4 | no | — |
| 15 | #63803 | C22 | 1 | 1 | yes | characteristic |
| 16 | #63797 | C22 | 1 | 2 | yes | complemented |
| 17 | D3×D24 | C22 | 1 | 4 | yes | complemented |
| 18 | C32×Q32 | C22 | 1 | 1 | yes | characteristic, complemented |
| 19 | #63655 | C22 | 1 | 2 | yes | complemented |
| 20 | #63626 | C22 | 1 | 1 | yes | characteristic, complemented |
| 21 | C16×C3⋊D3 | C22 | 1 | 1 | yes | characteristic, complemented |
| 22 | C3×#828 | C22 | 1 | 4 | yes | complemented |
| 23 | C3×D48 | C22 | 1 | 2 | yes | complemented |
| 24 | C3×D3×C16 | C22 | 1 | 2 | yes | complemented |
| 25 | #63546 | C22 | 1 | 4 | yes | complemented |
| 26 | #63545 | C22 | 1 | 2 | yes | complemented |
| 27 | #63544 | C22 | 1 | 1 | yes | characteristic, complemented |
| 28 | #63543 | C22 | 1 | 2 | yes | complemented |
| 29 | #5769 | — | 3 | 4 | no | — |
| 30 | C3×#508 | — | 3 | 2 | no | — |
| 31 | #5193 | — | 3 | 4 | no | — |
| 32 | #4919 | — | 3 | 2 | no | — |
| 33 | #4919 | — | 6 | 1 | no | — |
| 34 | #4919 | D3 | 1 | 2 | yes | complemented |
| 35 | #4914 | — | 3 | 4 | no | — |
| 36 | #4914 | — | 3 | 4 | no | — |
| 37 | D3×D16 | — | 3 | 2 | no | — |
| 38 | D3×D16 | — | 3 | 2 | no | — |
| 39 | C2×D48 | — | 3 | 2 | no | — |
| 40 | #4901 | — | 3 | 2 | no | — |
| 41 | #3832 | — | 2 | 2 | no | — |
| 42 | C3×#239 | — | 2 | 4 | no | — |
| 43 | #3802 | — | 2 | 2 | no | — |
| 44 | #3802 | — | 4 | 2 | no | — |
| 45 | D3×D12 | — | 2 | 4 | no | — |
| 46 | D3×D12 | — | 4 | 4 | no | — |
| 47 | #3798 | — | 4 | 2 | no | — |
| 48 | #3798 | D4 | 1 | 1 | yes | characteristic |
| 49 | C32×Q16 | C23 | 1 | 2 | yes | — |
| 50 | #3755 | — | 2 | 2 | no | — |
| 51 | #3745 | C23 | 1 | 2 | yes | — |
| 52 | C8×C3⋊D3 | C23 | 1 | 1 | yes | characteristic |
| 53 | C3×#215 | — | 2 | 4 | no | — |
| 54 | C3×D24 | C23 | 1 | 4 | yes | — |
| 55 | C3×D3×C8 | C23 | 1 | 2 | yes | — |
| 56 | #3714 | — | 2 | 4 | no | — |
| 57 | #3713 | — | 2 | 2 | no | — |
| 58 | #3712 | D4 | 1 | 1 | yes | characteristic |
| 59 | #3711 | D4 | 1 | 2 | yes | — |
| 60 | C32×C16 | C23 | 1 | 1 | yes | characteristic |
| 61 | C3⋊(C3⋊C16) | C23 | 1 | 1 | yes | characteristic |
| 62 | C3×C3⋊C16 | C23 | 1 | 2 | yes | — |
| 63 | #3415 | — | 9 | 1 | no | Sylow |
| 64 | #1009 | — | 6 | 4 | no | — |
| 65 | C3×#135 | — | 3 | 4 | no | — |
| 66 | #949 | — | 6 | 4 | no | — |
| 67 | #919 | — | 3 | 4 | no | — |
| 68 | #919 | — | 6 | 2 | no | — |
| 69 | #919 | C2×D3 | 1 | 4 | yes | complemented |
| 70 | #914 | — | 6 | 4 | no | — |
| 71 | #914 | — | 6 | 4 | no | — |
| 72 | D3×D8 | — | 3 | 4 | no | — |
| 73 | D3×D8 | — | 3 | 4 | no | — |
| 74 | C2×D24 | — | 3 | 4 | no | — |
| 75 | #901 | — | 3 | 2 | no | — |
| 76 | C3×Q32 | — | 2 | 1 | no | — |
| 77 | C3×Q32 | C2×D3 | 1 | 2 | yes | complemented |
| 78 | C3×SD32 | — | 3 | 4 | no | — |
| 79 | C3×D16 | — | 3 | 2 | no | — |
| 80 | C2×C3×C16 | — | 3 | 2 | no | — |
| 81 | #828 | — | 3 | 4 | no | — |
| 82 | #828 | — | 6 | 2 | no | — |
| 83 | #828 | C2×D3 | 1 | 4 | yes | complemented |
| 84 | #826 | — | 3 | 4 | no | — |
| 85 | #826 | — | 3 | 4 | no | — |
| 86 | #812 | — | 3 | 2 | no | — |
| 87 | D48 | — | 3 | 4 | no | — |
| 88 | D48 | — | 3 | 2 | no | — |
| 89 | D48 | — | 6 | 1 | no | — |
| 90 | D48 | C2×D3 | 1 | 2 | yes | complemented |
| 91 | #798 | — | 3 | 2 | no | — |
| 92 | #798 | — | 3 | 2 | no | — |
| 93 | D3×C16 | — | 3 | 2 | no | — |
| 94 | D3×C16 | — | 6 | 1 | no | — |
| 95 | D3×C16 | C2×D3 | 1 | 2 | yes | complemented |
| 96 | C2×D32 | — | 4 | 2 | no | — |
| 97 | C2×D32 | — | 4 | 4 | no | — |
| 98 | C32×Q8 | — | 2 | 2 | no | — |
| 99 | C4⋊(C3⋊D3) | — | 2 | 2 | no | — |
| 100 | C4⋊(C3⋊D3) | — | 4 | 2 | no | — |
| 101 | C4×C3⋊D3 | — | 4 | 2 | no | — |
| 102 | C4×C3⋊D3 | C2×D4 | 1 | 1 | yes | characteristic |
| 103 | C3×D12 | — | 2 | 4 | no | — |
| 104 | C3×D12 | — | 4 | 4 | no | — |
| 105 | C3×C4×D3 | — | 4 | 4 | no | — |
| 106 | C3×C4×D3 | C2×D4 | 1 | 2 | yes | — |
| 107 | #627 | D8 | 1 | 1 | yes | characteristic |
| 108 | #626 | — | 4 | 4 | no | — |
| 109 | #626 | D8 | 1 | 2 | yes | — |
| 110 | #625 | — | 4 | 2 | no | — |
| 111 | #625 | D8 | 1 | 1 | yes | characteristic |
| 112 | C32×C8 | C24 | 1 | 1 | yes | characteristic |
| 113 | C3⋊(C3⋊C8) | C2×D4 | 1 | 1 | yes | characteristic |
| 114 | C3×C3⋊C8 | C2×D4 | 1 | 2 | yes | — |
| 115 | #576 | — | 9 | 2 | no | — |
| 116 | #509 | — | 9 | 2 | no | — |
| 117 | #509 | — | 9 | 4 | no | — |
| 118 | #508 | — | 3 | 2 | no | — |
| 119 | #508 | — | 9 | 1 | no | — |
| 120 | C2×D16 | — | 9 | 2 | no | — |
| 121 | C2×D16 | — | 9 | 1 | no | — |
| 122 | #504 | — | 9 | 1 | no | — |
| 123 | C3×#41 | — | 6 | 4 | no | — |
| 124 | #239 | — | 2 | 4 | no | — |
| 125 | #239 | — | 6 | 4 | no | — |
| 126 | #239 | — | 12 | 2 | no | — |
| 127 | D3×D4 | — | 6 | 4 | no | — |
| 128 | D3×D4 | — | 6 | 4 | no | — |
| 129 | D3×D4 | — | 12 | 4 | no | — |
| 130 | D3×D4 | — | 12 | 4 | no | — |
| 131 | #235 | — | 3 | 2 | no | — |
| 132 | #235 | — | 12 | 4 | no | — |
| 133 | C2×D12 | — | 6 | 4 | no | — |
| 134 | C2×D12 | — | 12 | 4 | no | — |
| 135 | C3×Q16 | — | 2 | 2 | no | — |
| 136 | C3×Q16 | C22×D3 | 1 | 4 | yes | — |
| 137 | C3×SD16 | — | 6 | 4 | no | — |
| 138 | C3×D8 | — | 3 | 4 | no | — |
| 139 | C2×C3×C8 | — | 3 | 2 | no | — |
| 140 | #215 | — | 2 | 4 | no | — |
| 141 | #215 | — | 6 | 4 | no | — |
| 142 | #215 | — | 12 | 2 | no | — |
| 143 | #213 | — | 6 | 4 | no | — |
| 144 | #213 | — | 6 | 4 | no | — |
| 145 | #208 | — | 3 | 2 | no | — |
| 146 | D24 | — | 3 | 4 | no | — |
| 147 | D24 | — | 6 | 2 | no | — |
| 148 | D24 | — | 6 | 4 | no | — |
| 149 | D24 | C22×D3 | 1 | 4 | yes | — |
| 150 | ΓL(1,𝔽25) | — | 3 | 2 | no | — |
| 151 | ΓL(1,𝔽25) | — | 3 | 2 | no | — |
| 152 | D3×C8 | — | 3 | 2 | no | — |
| 153 | D3×C8 | — | 6 | 1 | no | — |
| 154 | D3×C8 | C22×D3 | 1 | 2 | yes | — |
| 155 | C3×C16 | — | 2 | 1 | no | — |
| 156 | C3×C16 | — | 3 | 2 | no | — |
| 157 | C3×C16 | C22×D3 | 1 | 2 | yes | — |
| 158 | C3⋊C16 | — | 3 | 2 | no | — |
| 159 | C3⋊C16 | — | 6 | 1 | no | — |
| 160 | C3⋊C16 | C22×D3 | 1 | 2 | yes | — |
| 161 | C2×C3⋊D3 | — | 4 | 2 | no | — |
| 162 | C2×C3⋊D3 | C2×D8 | 1 | 1 | yes | characteristic |
| 163 | C2×C3×D3 | — | 4 | 4 | no | — |
| 164 | C2×C3×D3 | C2×D8 | 1 | 2 | yes | — |
| 165 | D32 | — | 16 | 4 | no | — |
| 166 | D32 | — | 16 | 2 | no | — |
| 167 | C32×C4 | — | 4 | 2 | no | — |
| 168 | C32×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 169 | C3⋊Dic3 | C2×D8 | 1 | 1 | yes | characteristic |
| 170 | C3×Dic3 | C2×D8 | 1 | 2 | yes | — |
| 171 | H(5,𝔽2) | — | 18 | 2 | no | — |
| 172 | AGL(1,ℤ/8) | — | 18 | 2 | no | — |
| 173 | AGL(1,ℤ/8) | — | 18 | 4 | no | — |
| 174 | #135 | — | 3 | 4 | no | — |
| 175 | #135 | — | 9 | 2 | no | — |
| 176 | C2×D8 | — | 9 | 4 | no | — |
| 177 | C2×D8 | — | 9 | 2 | no | — |
| 178 | #131 | — | 9 | 1 | no | — |
| 179 | Q32 | D32 | 1 | 1 | yes | characteristic, complemented |
| 180 | SD32 | — | 3 | 4 | no | — |
| 181 | SD32 | — | 9 | 2 | no | — |
| 182 | D16 | — | 3 | 2 | no | — |
| 183 | D16 | — | 9 | 4 | no | — |
| 184 | D16 | — | 9 | 2 | no | — |
| 185 | D16 | — | 9 | 1 | no | — |
| 186 | M32 | — | 9 | 1 | no | — |
| 187 | M32 | — | 9 | 2 | no | — |
| 188 | C2×C16 | — | 3 | 2 | no | — |
| 189 | C2×C16 | — | 9 | 1 | no | — |
| 190 | C22×D3 | — | 12 | 4 | no | — |
| 191 | C22×D3 | — | 12 | 4 | no | — |
| 192 | C22×D3 | — | 12 | 4 | no | — |
| 193 | C3×Q8 | — | 2 | 4 | no | — |
| 194 | C3×Q8 | — | 4 | 2 | no | — |
| 195 | C3×D4 | — | 6 | 4 | no | — |
| 196 | C3×D4 | — | 12 | 4 | no | — |
| 197 | C2×C3×C4 | — | 3 | 2 | no | — |
| 198 | C2×C3×C4 | — | 12 | 4 | no | — |
| 199 | #67 | — | 3 | 2 | no | — |
| 200 | #67 | — | 3 | 2 | no | — |
| 201 | #67 | — | 12 | 4 | no | — |
| 202 | #67 | — | 12 | 4 | no | — |
| 203 | D12 | — | 2 | 4 | no | — |
| 204 | D12 | — | 3 | 2 | no | — |
| 205 | D12 | — | 4 | 4 | no | — |
| 206 | D12 | — | 6 | 4 | no | — |
| 207 | D12 | — | 12 | 2 | no | — |
| 208 | D12 | — | 12 | 4 | no | — |
| 209 | D12 | — | 12 | 4 | no | — |
| 210 | D12 | — | 24 | 2 | no | — |
| 211 | C4×D3 | — | 3 | 2 | no | — |
| 212 | C4×D3 | — | 4 | 4 | no | — |
| 213 | C4×D3 | — | 6 | 1 | no | — |
| 214 | C4×D3 | — | 12 | 4 | no | — |
| 215 | C4×D3 | — | 24 | 2 | no | — |
| 216 | C4×D3 | D3×D4 | 1 | 2 | yes | — |
| 217 | Dic6 | — | 3 | 2 | no | — |
| 218 | C3×C8 | — | 2 | 1 | no | — |
| 219 | C3×C8 | — | 3 | 2 | no | — |
| 220 | C3×C8 | C23×D3 | 1 | 2 | yes | — |
| 221 | C3⋊C8 | — | 3 | 2 | no | — |
| 222 | C3⋊C8 | — | 6 | 1 | no | — |
| 223 | C3⋊C8 | D3×D4 | 1 | 2 | yes | — |
| 224 | C2×C32 | C22×D8 | 1 | 1 | yes | characteristic |
| 225 | C3⋊D3 | — | 2 | 1 | no | — |
| 226 | C3⋊D3 | — | 8 | 2 | no | — |
| 227 | C3×D3 | — | 2 | 2 | no | — |
| 228 | C3×D3 | — | 8 | 4 | no | — |
| 229 | #41 | — | 6 | 4 | no | — |
| 230 | #41 | — | 9 | 1 | no | — |
| 231 | #41 | — | 18 | 2 | no | — |
| 232 | #41 | — | 36 | 2 | no | — |
| 233 | C2×D4 | — | 18 | 4 | no | — |
| 234 | C2×D4 | — | 18 | 2 | no | — |
| 235 | C2×D4 | — | 36 | 4 | no | — |
| 236 | C2×D4 | — | 36 | 2 | no | — |
| 237 | Q16 | C2×D32 | 1 | 2 | yes | — |
| 238 | SD16 | — | 6 | 4 | no | — |
| 239 | SD16 | — | 18 | 2 | no | — |
| 240 | D8 | — | 3 | 4 | no | — |
| 241 | D8 | — | 9 | 2 | no | — |
| 242 | D8 | — | 18 | 4 | no | — |
| 243 | D8 | — | 18 | 2 | no | — |
| 244 | M16 | — | 9 | 1 | no | — |
| 245 | M16 | — | 9 | 2 | no | — |
| 246 | C2×C8 | — | 3 | 2 | no | — |
| 247 | C2×C8 | — | 9 | 1 | no | — |
| 248 | C16 | — | 3 | 2 | no | — |
| 249 | C16 | — | 9 | 1 | no | — |
| 250 | C16 | C2×D32 | 1 | 1 | yes | characteristic |
| 251 | C22×C3 | — | 3 | 2 | no | — |
| 252 | C22×C3 | — | 12 | 4 | no | — |
| 253 | C2×D3 | — | 3 | 2 | no | — |
| 254 | C2×D3 | — | 4 | 4 | no | — |
| 255 | C2×D3 | — | 6 | 1 | no | — |
| 256 | C2×D3 | — | 12 | 4 | no | — |
| 257 | C2×D3 | — | 24 | 2 | no | — |
| 258 | C2×D3 | — | 48 | 4 | no | — |
| 259 | C2×D3 | — | 48 | 4 | no | — |
| 260 | C2×D3 | — | 48 | 4 | no | — |
| 261 | C2×D3 | D3×D8 | 1 | 2 | yes | — |
| 262 | C3×C4 | — | 2 | 1 | no | — |
| 263 | C3×C4 | — | 3 | 2 | no | — |
| 264 | C3×C4 | — | 4 | 4 | no | — |
| 265 | C3×C4 | — | 8 | 2 | no | — |
| 266 | C3×C4 | C2×D3×D4 | 1 | 2 | yes | — |
| 267 | Dic3 | — | 3 | 2 | no | — |
| 268 | Dic3 | — | 6 | 1 | no | — |
| 269 | Dic3 | D3×D8 | 1 | 2 | yes | — |
| 270 | C32 | #3415 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 271 | C23 | — | 36 | 4 | no | — |
| 272 | C23 | — | 36 | 2 | no | — |
| 273 | Q8 | — | 2 | 2 | no | — |
| 274 | Q8 | — | 9 | 1 | no | — |
| 275 | D4 | — | 6 | 4 | no | — |
| 276 | D4 | — | 9 | 2 | no | — |
| 277 | D4 | — | 9 | 1 | no | — |
| 278 | D4 | — | 12 | 4 | no | — |
| 279 | D4 | — | 18 | 2 | no | — |
| 280 | D4 | — | 36 | 4 | no | — |
| 281 | D4 | — | 36 | 2 | no | — |
| 282 | D4 | — | 36 | 2 | no | — |
| 283 | C2×C4 | — | 3 | 2 | no | — |
| 284 | C2×C4 | — | 9 | 1 | no | — |
| 285 | C2×C4 | — | 12 | 4 | no | — |
| 286 | C2×C4 | — | 36 | 2 | no | — |
| 287 | C8 | — | 3 | 2 | no | — |
| 288 | C8 | — | 9 | 1 | no | — |
| 289 | C8 | C22×D32 | 1 | 1 | yes | characteristic |
| 290 | C2×C3 | — | 2 | 1 | no | — |
| 291 | C2×C3 | — | 6 | 2 | no | — |
| 292 | C2×C3 | — | 24 | 4 | no | — |
| 293 | C2×C3 | C2×D3×D8 | 1 | 2 | yes | — |
| 294 | D3 | — | 2 | 2 | no | — |
| 295 | D3 | — | 6 | 2 | no | — |
| 296 | D3 | — | 8 | 4 | no | — |
| 297 | D3 | — | 12 | 1 | no | — |
| 298 | D3 | — | 24 | 4 | no | — |
| 299 | D3 | — | 48 | 2 | no | — |
| 300 | C22 | — | 3 | 2 | no | — |
| 301 | C22 | — | 9 | 1 | no | — |
| 302 | C22 | — | 12 | 4 | no | — |
| 303 | C22 | — | 36 | 2 | no | — |
| 304 | C22 | — | 144 | 4 | no | — |
| 305 | C22 | — | 144 | 2 | no | — |
| 306 | C4 | — | 3 | 2 | no | — |
| 307 | C4 | — | 4 | 2 | no | — |
| 308 | C4 | — | 9 | 1 | no | — |
| 309 | C4 | D32×D4 | 1 | 1 | yes | characteristic |
| 310 | C3 | — | 2 | 1 | no | — |
| 311 | C3 | #82031 | 1 | 2 | yes | complemented |
| 312 | C2 | — | 6 | 2 | no | — |
| 313 | C2 | — | 18 | 1 | no | — |
| 314 | C2 | — | 24 | 4 | no | — |
| 315 | C2 | — | 72 | 2 | no | — |
| 316 | C2 | D32×D8 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #10594771 | C2 | semidirect |
| 2. #10592618 | C2 | semidirect |
| 3. C3×#4919 | C2 | semidirect |
| 4. #10591040 | C2 | semidirect |
| 5. #10591038 | C2 | semidirect |
| 6. #10590234 | C2 | semidirect |
| 7. D3×D48 | C2 | semidirect |
| 8. #10590227 | C2 | semidirect |
| 11. #64128 | C22 | semidirect |
| 12. C3×#919 | C22 | semidirect |
| 16. #63797 | C22 | semidirect |
| 17. D3×D24 | C22 | semidirect |
| 18. C32×Q32 | C22 | semidirect |
| 19. #63655 | C22 | semidirect |
| 20. #63626 | C22 | semidirect |
| 21. C16×C3⋊D3 | C22 | semidirect |
| 22. C3×#828 | C22 | semidirect |
| 23. C3×D48 | C22 | semidirect |
| 24. C3×D3×C16 | C22 | semidirect |
| 25. #63546 | C22 | semidirect |
| 26. #63545 | C22 | semidirect |
| 27. #63544 | C22 | semidirect |
| 28. #63543 | C22 | semidirect |
| 34. #4919 | D3 | semidirect |
| 69. #919 | C2×D3 | semidirect |
| 77. C3×Q32 | C2×D3 | semidirect |
| 83. #828 | C2×D3 | semidirect |
| 90. D48 | C2×D3 | semidirect |
| 95. D3×C16 | C2×D3 | semidirect |
| 179. Q32 | D32 | semidirect |
| 270. C32 | #3415 | semidirect |
| 311. C3 | #82031 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10594771 | 2 |
| #10592618 | 2 |
| C3×#4919 | 2 |
| #10591040 | 2 |
| #10591038 | 2 |
| #10590234 | 2 |
| D3×D48 | 2 |
| #10590227 | 2 |
| #82031 | 3 |