Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
730 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 730 conjugacy classes of subgroups — too many to draw. Only its 135 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 | #11722667 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #11721857 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #11721783 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | C3×C9×#101 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #11720558 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #10592452 | — | 3 | 1 | no | maximal |
| 7 | #10588433 | — | 3 | 3 | no | maximal |
| 8 | C2×C3×Q8×C9 | C22 | 1 | 1 | yes | characteristic |
| 9 | #88746 | — | 2 | 2 | no | — |
| 10 | Q8×C3⋊D9 | — | 2 | 2 | no | — |
| 11 | #88736 | — | 2 | 2 | no | — |
| 12 | #88736 | C22 | 1 | 2 | yes | — |
| 13 | C2×C4⋊(C3⋊D9) | C22 | 1 | 1 | yes | characteristic |
| 14 | C3×C9×M16 | C22 | 1 | 2 | yes | — |
| 15 | #88536 | — | 2 | 2 | no | — |
| 16 | C4×C3⋊Dic9 | C22 | 1 | 1 | yes | characteristic |
| 17 | #88525 | — | 2 | 2 | no | — |
| 18 | #88524 | — | 2 | 2 | no | — |
| 19 | #64364 | — | 3 | 1 | no | — |
| 20 | #64156 | — | 3 | 1 | no | — |
| 21 | #64117 | — | 3 | 2 | no | — |
| 22 | #63713 | — | 3 | 3 | no | — |
| 23 | C32×#101 | D3 | 1 | 1 | yes | characteristic |
| 24 | #63652 | — | 3 | 2 | no | — |
| 25 | #63509 | — | 3 | 3 | no | — |
| 26 | #63470 | — | 3 | 6 | no | — |
| 27 | C9×#101 | D3 | 1 | 3 | yes | complemented |
| 28 | #63384 | — | 3 | 6 | no | — |
| 29 | C22×C3⋊D9 | — | 2 | 1 | no | — |
| 30 | C3×Q8×C9 | — | 2 | 2 | no | — |
| 31 | C2×C3×C4×C9 | D4 | 1 | 2 | yes | — |
| 32 | C2×C3×C4×C9 | C23 | 1 | 1 | yes | characteristic |
| 33 | #6233 | — | 2 | 2 | no | — |
| 34 | C2×C3⋊Dic9 | — | 2 | 1 | no | — |
| 35 | C4⋊(C3⋊D9) | — | 2 | 2 | no | — |
| 36 | C4⋊(C3⋊D9) | — | 2 | 2 | no | — |
| 37 | C4⋊(C3⋊D9) | D4 | 1 | 2 | yes | — |
| 38 | C4×C3⋊D9 | — | 2 | 4 | no | — |
| 39 | C4×C3⋊D9 | — | 2 | 2 | no | — |
| 40 | #6226 | — | 2 | 2 | no | — |
| 41 | #6226 | D4 | 1 | 2 | yes | — |
| 42 | C3×C8×C9 | — | 2 | 2 | no | — |
| 43 | #4753 | — | 9 | 3 | no | — |
| 44 | #4753 | — | 9 | 1 | no | — |
| 45 | C2×C32×Q8 | C2×D3 | 1 | 1 | yes | characteristic |
| 46 | #3832 | — | 6 | 2 | no | — |
| 47 | Q8×C3⋊D3 | — | 6 | 2 | no | — |
| 48 | #3828 | — | 3 | 2 | no | — |
| 49 | #3828 | — | 6 | 2 | no | — |
| 50 | C2×C4⋊(C3⋊D3) | — | 3 | 1 | no | — |
| 51 | C32×M16 | C2×D3 | 1 | 2 | yes | — |
| 52 | #3752 | — | 6 | 2 | no | — |
| 53 | C4×C3⋊Dic3 | — | 3 | 1 | no | — |
| 54 | #3745 | — | 6 | 2 | no | — |
| 55 | #3744 | — | 6 | 2 | no | — |
| 56 | C2×Q8×C9 | C2×D3 | 1 | 3 | yes | — |
| 57 | #3701 | — | 6 | 6 | no | — |
| 58 | Q8×D9 | — | 6 | 6 | no | — |
| 59 | #3697 | — | 3 | 6 | no | — |
| 60 | #3697 | — | 6 | 6 | no | — |
| 61 | C2×D36 | — | 3 | 3 | no | — |
| 62 | C9×M16 | C2×D3 | 1 | 6 | yes | — |
| 63 | (C2×D9)⋊C4 | — | 6 | 6 | no | — |
| 64 | C4×Dic9 | — | 3 | 3 | no | — |
| 65 | D72 | — | 6 | 6 | no | — |
| 66 | #3664 | — | 6 | 6 | no | — |
| 67 | C22×C3×C9 | C2×D4 | 1 | 1 | yes | characteristic |
| 68 | C2×C3⋊D9 | — | 2 | 2 | no | — |
| 69 | C2×C3⋊D9 | — | 4 | 2 | no | — |
| 70 | C2×C3⋊D9 | — | 4 | 1 | no | — |
| 71 | C3×C4×C9 | — | 2 | 2 | no | — |
| 72 | C3×C4×C9 | C2×D4 | 1 | 2 | yes | — |
| 73 | C3⋊Dic9 | — | 2 | 2 | no | — |
| 74 | C3⋊Dic9 | — | 4 | 1 | no | — |
| 75 | #1007 | — | 9 | 3 | no | — |
| 76 | #1007 | — | 9 | 1 | no | — |
| 77 | #947 | — | 9 | 3 | no | — |
| 78 | #947 | — | 9 | 1 | no | — |
| 79 | #908 | — | 9 | 6 | no | — |
| 80 | #908 | — | 9 | 2 | no | — |
| 81 | C3×#101 | D9 | 1 | 3 | yes | complemented |
| 82 | C3×#101 | C3⋊D3 | 1 | 1 | yes | characteristic |
| 83 | #825 | — | 9 | 6 | no | — |
| 84 | #825 | — | 9 | 2 | no | — |
| 85 | C22×C3⋊D3 | — | 6 | 1 | no | — |
| 86 | C32×Q8 | — | 2 | 2 | no | — |
| 87 | C2×C32×C4 | D12 | 1 | 2 | yes | — |
| 88 | C2×C32×C4 | C22×D3 | 1 | 1 | yes | characteristic |
| 89 | #638 | — | 6 | 2 | no | — |
| 90 | C2×C3⋊Dic3 | — | 6 | 1 | no | — |
| 91 | C4⋊(C3⋊D3) | — | 3 | 2 | no | — |
| 92 | C4⋊(C3⋊D3) | — | 6 | 2 | no | — |
| 93 | C4⋊(C3⋊D3) | — | 6 | 2 | no | — |
| 94 | C4×C3⋊D3 | — | 6 | 4 | no | — |
| 95 | C4×C3⋊D3 | — | 6 | 2 | no | — |
| 96 | #634 | — | 3 | 2 | no | — |
| 97 | #634 | — | 6 | 2 | no | — |
| 98 | C22×D9 | — | 6 | 3 | no | — |
| 99 | C32×C8 | — | 2 | 2 | no | — |
| 100 | Q8×C9 | — | 2 | 6 | no | — |
| 101 | C2×C4×C9 | D12 | 1 | 6 | yes | — |
| 102 | C2×C4×C9 | C22×D3 | 1 | 3 | yes | — |
| 103 | #611 | — | 6 | 6 | no | — |
| 104 | C2×Dic9 | — | 6 | 3 | no | — |
| 105 | D36 | — | 3 | 6 | no | — |
| 106 | D36 | — | 6 | 6 | no | — |
| 107 | D36 | — | 6 | 6 | no | — |
| 108 | C4×D9 | — | 6 | 12 | no | — |
| 109 | C4×D9 | — | 6 | 6 | no | — |
| 110 | Dic18 | — | 3 | 6 | no | — |
| 111 | Dic18 | — | 6 | 6 | no | — |
| 112 | C8×C9 | — | 2 | 6 | no | — |
| 113 | #454 | — | 27 | 1 | no | Sylow |
| 114 | C2×C3×C9 | — | 2 | 1 | no | — |
| 115 | C2×C3×C9 | #120 | 1 | 1 | yes | characteristic |
| 116 | C3⋊D9 | — | 4 | 2 | no | — |
| 117 | C3⋊D9 | — | 8 | 1 | no | — |
| 118 | C2×C3×Q8 | C2×D9 | 1 | 3 | yes | — |
| 119 | C2×C3×Q8 | C2×C3⋊D3 | 1 | 1 | yes | characteristic |
| 120 | #239 | — | 18 | 2 | no | — |
| 121 | #239 | — | 18 | 6 | no | — |
| 122 | D3×Q8 | — | 18 | 2 | no | — |
| 123 | D3×Q8 | — | 18 | 6 | no | — |
| 124 | #235 | — | 9 | 6 | no | — |
| 125 | #235 | — | 9 | 2 | no | — |
| 126 | #235 | — | 18 | 6 | no | — |
| 127 | #235 | — | 18 | 2 | no | — |
| 128 | C2×D12 | — | 9 | 3 | no | — |
| 129 | C2×D12 | — | 9 | 1 | no | — |
| 130 | C3×M16 | C2×D9 | 1 | 6 | yes | — |
| 131 | C3×M16 | C2×C3⋊D3 | 1 | 2 | yes | — |
| 132 | (C2×D3)⋊C4 | — | 18 | 6 | no | — |
| 133 | (C2×D3)⋊C4 | — | 18 | 2 | no | — |
| 134 | C4×Dic3 | — | 9 | 3 | no | — |
| 135 | C4×Dic3 | — | 9 | 1 | no | — |
| 136 | D24 | — | 18 | 2 | no | — |
| 137 | D24 | — | 18 | 6 | no | — |
| 138 | #204 | — | 18 | 2 | no | — |
| 139 | #204 | — | 18 | 6 | no | — |
| 140 | C22×C32 | C2×D12 | 1 | 1 | yes | characteristic |
| 141 | C2×C3⋊D3 | — | 6 | 2 | no | — |
| 142 | C2×C3⋊D3 | — | 12 | 2 | no | — |
| 143 | C2×C3⋊D3 | — | 12 | 1 | no | — |
| 144 | C32×C4 | — | 2 | 2 | no | — |
| 145 | C32×C4 | D3×D4 | 1 | 2 | yes | — |
| 146 | C3⋊Dic3 | — | 6 | 2 | no | — |
| 147 | C3⋊Dic3 | — | 12 | 1 | no | — |
| 148 | C22×C9 | C2×D12 | 1 | 3 | yes | — |
| 149 | C2×D9 | — | 6 | 6 | no | — |
| 150 | C2×D9 | — | 12 | 6 | no | — |
| 151 | C2×D9 | — | 12 | 3 | no | — |
| 152 | C4×C9 | — | 2 | 6 | no | — |
| 153 | C4×C9 | D3×D4 | 1 | 6 | yes | — |
| 154 | Dic9 | — | 6 | 6 | no | — |
| 155 | Dic9 | — | 12 | 3 | no | — |
| 156 | 2−1+4 | — | 27 | 1 | no | — |
| 157 | AGL(1,ℤ/8) | — | 27 | 2 | no | — |
| 158 | #124 | — | 27 | 1 | no | — |
| 159 | #104 | — | 27 | 2 | no | — |
| 160 | #101 | C3⋊D9 | 1 | 1 | yes | characteristic, complemented |
| 161 | C3×C9 | #454 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 162 | C22×D3 | — | 18 | 3 | no | — |
| 163 | C22×D3 | — | 18 | 1 | no | — |
| 164 | C3×Q8 | — | 2 | 6 | no | — |
| 165 | C3×Q8 | — | 2 | 2 | no | — |
| 166 | C2×C3×C4 | D36 | 1 | 6 | yes | — |
| 167 | C2×C3×C4 | C22×D9 | 1 | 3 | yes | — |
| 168 | C2×C3×C4 | C4⋊(C3⋊D3) | 1 | 2 | yes | — |
| 169 | C2×C3×C4 | C22×C3⋊D3 | 1 | 1 | yes | characteristic |
| 170 | #67 | — | 18 | 6 | no | — |
| 171 | #67 | — | 18 | 2 | no | — |
| 172 | C2×Dic3 | — | 18 | 3 | no | — |
| 173 | C2×Dic3 | — | 18 | 1 | no | — |
| 174 | D12 | — | 9 | 6 | no | — |
| 175 | D12 | — | 9 | 2 | no | — |
| 176 | D12 | — | 18 | 2 | no | — |
| 177 | D12 | — | 18 | 6 | no | — |
| 178 | D12 | — | 18 | 6 | no | — |
| 179 | D12 | — | 18 | 2 | no | — |
| 180 | C4×D3 | — | 18 | 4 | no | — |
| 181 | C4×D3 | — | 18 | 12 | no | — |
| 182 | C4×D3 | — | 18 | 6 | no | — |
| 183 | C4×D3 | — | 18 | 2 | no | — |
| 184 | Dic6 | — | 9 | 6 | no | — |
| 185 | Dic6 | — | 9 | 2 | no | — |
| 186 | Dic6 | — | 18 | 2 | no | — |
| 187 | Dic6 | — | 18 | 6 | no | — |
| 188 | C3×C8 | — | 2 | 6 | no | — |
| 189 | C3×C8 | — | 2 | 2 | no | — |
| 190 | C2×C32 | — | 2 | 1 | no | — |
| 191 | C2×C32 | #882 | 1 | 1 | yes | characteristic |
| 192 | C3⋊D3 | — | 12 | 2 | no | — |
| 193 | C3⋊D3 | — | 24 | 1 | no | — |
| 194 | C2×C9 | — | 2 | 3 | no | — |
| 195 | C2×C9 | #882 | 1 | 3 | yes | — |
| 196 | D9 | — | 12 | 6 | no | — |
| 197 | D9 | — | 24 | 3 | no | — |
| 198 | #41 | — | 27 | 2 | no | — |
| 199 | #41 | — | 54 | 2 | no | — |
| 200 | #41 | — | 54 | 2 | no | — |
| 201 | C2×Q8 | — | 54 | 2 | no | — |
| 202 | C2×Q8 | C2×C3⋊D9 | 1 | 1 | yes | characteristic |
| 203 | C2×D4 | — | 27 | 1 | no | — |
| 204 | SD16 | — | 54 | 2 | no | — |
| 205 | D8 | — | 54 | 2 | no | — |
| 206 | M16 | C2×C3⋊D9 | 1 | 2 | yes | — |
| 207 | C22⋊C4 | — | 54 | 2 | no | — |
| 208 | C42 | — | 27 | 1 | no | — |
| 209 | C22×C3 | C2×D36 | 1 | 3 | yes | — |
| 210 | C22×C3 | C2×C4⋊(C3⋊D3) | 1 | 1 | yes | characteristic |
| 211 | C2×D3 | — | 18 | 2 | no | — |
| 212 | C2×D3 | — | 18 | 6 | no | — |
| 213 | C2×D3 | — | 36 | 2 | no | — |
| 214 | C2×D3 | — | 36 | 6 | no | — |
| 215 | C2×D3 | — | 36 | 1 | no | — |
| 216 | C2×D3 | — | 36 | 3 | no | — |
| 217 | C3×C4 | — | 2 | 2 | no | — |
| 218 | C3×C4 | — | 2 | 6 | no | — |
| 219 | C3×C4 | D4×D9 | 1 | 6 | yes | — |
| 220 | C3×C4 | D4×C3⋊D3 | 1 | 2 | yes | — |
| 221 | Dic3 | — | 18 | 2 | no | — |
| 222 | Dic3 | — | 18 | 6 | no | — |
| 223 | Dic3 | — | 36 | 1 | no | — |
| 224 | Dic3 | — | 36 | 3 | no | — |
| 225 | C32 | #4753 | 1 | 1 | yes | characteristic |
| 226 | C9 | #4753 | 1 | 3 | yes | complemented |
| 227 | C23 | — | 54 | 1 | no | — |
| 228 | Q8 | — | 2 | 2 | no | — |
| 229 | Q8 | — | 27 | 2 | no | — |
| 230 | Q8 | — | 54 | 2 | no | — |
| 231 | D4 | — | 27 | 2 | no | — |
| 232 | D4 | — | 54 | 2 | no | — |
| 233 | D4 | — | 54 | 2 | no | — |
| 234 | D4 | — | 54 | 2 | no | — |
| 235 | C2×C4 | — | 54 | 4 | no | — |
| 236 | C2×C4 | — | 54 | 2 | no | — |
| 237 | C2×C4 | — | 54 | 1 | no | — |
| 238 | C2×C4 | C4⋊(C3⋊D9) | 1 | 2 | yes | — |
| 239 | C2×C4 | C22×C3⋊D9 | 1 | 1 | yes | characteristic |
| 240 | C8 | — | 2 | 2 | no | — |
| 241 | C2×C3 | — | 2 | 3 | no | — |
| 242 | C2×C3 | — | 2 | 1 | no | — |
| 243 | C2×C3 | #63444 | 1 | 3 | yes | — |
| 244 | C2×C3 | #64091 | 1 | 1 | yes | characteristic |
| 245 | D3 | — | 36 | 6 | no | — |
| 246 | D3 | — | 36 | 2 | no | — |
| 247 | D3 | — | 72 | 3 | no | — |
| 248 | D3 | — | 72 | 1 | no | — |
| 249 | C22 | — | 54 | 2 | no | — |
| 250 | C22 | — | 108 | 2 | no | — |
| 251 | C22 | — | 108 | 1 | no | — |
| 252 | C22 | C2×C4⋊(C3⋊D9) | 1 | 1 | yes | characteristic |
| 253 | C4 | — | 2 | 2 | no | — |
| 254 | C4 | — | 54 | 2 | no | — |
| 255 | C4 | — | 108 | 1 | no | — |
| 256 | C4 | D4×C3⋊D9 | 1 | 2 | yes | — |
| 257 | C3 | #10588433 | 1 | 3 | yes | complemented |
| 258 | C3 | #10592452 | 1 | 1 | yes | characteristic |
| 259 | C2 | — | 2 | 1 | no | — |
| 260 | C2 | — | 108 | 2 | no | — |
| 261 | C2 | — | 216 | 1 | no | — |
| 262 | C2 | #11721728 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #11722667 | C2 | semidirect |
| 2. #11721857 | C2 | semidirect |
| 3. #11721783 | C2 | semidirect |
| 4. C3×C9×#101 | C2 | semidirect |
| 5. #11720558 | C2 | semidirect |
| 27. C9×#101 | D3 | semidirect |
| 81. C3×#101 | D9 | semidirect |
| 160. #101 | C3⋊D9 | semidirect |
| 161. C3×C9 | #454 | semidirect |
| 226. C9 | #4753 | semidirect |
| 257. C3 | #10588433 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11722667 | 2 |
| #11721857 | 2 |
| #11721783 | 2 |
| C3×C9×#101 | 2 |
| #11720558 | 2 |
| #10592452 | 3 |
| #10588433 | 3 |