← #13164954
Subgroups — #13164954
Order 1728.
2513 conjugacy classes of subgroups.
The subgroup lattice has 259 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | 3−1+2×H(5,𝔽2) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #11722642 | C2 | 1 | 6 | yes | maximal, complemented |
| 3 | #41×AGL(1,ℤ/9) | C2 | 1 | 6 | yes | maximal, complemented |
| 4 | #63710⋊C3 | C2 | 1 | 9 | yes | maximal, complemented |
| 5 | C2×#3699⋊C3 | C2 | 1 | 9 | yes | maximal, complemented |
| 6 | C3×#5966 | — | 3 | 1 | no | maximal |
| 7 | #10593141 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #41×3−1+2 | C22 | 1 | 6 | yes | complemented |
| 9 | C2×D4×3−1+2 | C22 | 1 | 9 | yes | complemented |
| 10 | C2×#611⋊C3 | C22 | 1 | 18 | yes | complemented |
| 11 | C22×3−1+2⋊C4 | C22 | 1 | 6 | yes | complemented |
| 12 | #88724 | C22 | 1 | 2 | yes | complemented |
| 13 | Q8×AGL(1,ℤ/9) | C22 | 1 | 6 | yes | complemented |
| 14 | #3699⋊C3 | C22 | 1 | 18 | yes | complemented |
| 15 | #3699⋊C3 | C22 | 1 | 36 | yes | complemented |
| 16 | D4×AGL(1,ℤ/9) | C22 | 1 | 18 | yes | complemented |
| 17 | #3697⋊C3 | C22 | 1 | 18 | yes | complemented |
| 18 | C2×C4×AGL(1,ℤ/9) | C22 | 1 | 9 | yes | complemented |
| 19 | C2×3−1+2⋊Q8 | C22 | 1 | 9 | yes | complemented |
| 20 | C32×H(5,𝔽2) | D3 | 1 | 1 | yes | characteristic |
| 21 | C3×#1010 | — | 3 | 6 | no | — |
| 22 | C3×D3×#41 | — | 3 | 6 | no | — |
| 23 | C3×#1004 | — | 3 | 9 | no | — |
| 24 | C2×C3×#237 | — | 3 | 9 | no | — |
| 25 | C9×H(5,𝔽2) | — | 2 | 1 | no | — |
| 26 | C9×H(5,𝔽2) | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 27 | #63716 | C2×C3 | 1 | 6 | yes | complemented |
| 28 | #41×D9 | C2×C3 | 1 | 6 | yes | complemented |
| 29 | #63710 | C2×C3 | 1 | 9 | yes | complemented |
| 30 | C2×#3699 | C2×C3 | 1 | 9 | yes | complemented |
| 31 | C23×3−1+2 | C23 | 1 | 6 | yes | — |
| 32 | C22×AGL(1,ℤ/9) | C23 | 1 | 9 | yes | — |
| 33 | Q8×3−1+2 | C23 | 1 | 2 | yes | — |
| 34 | D4×3−1+2 | C23 | 1 | 18 | yes | — |
| 35 | C2×C4×3−1+2 | C23 | 1 | 9 | yes | — |
| 36 | #611⋊C3 | C23 | 1 | 36 | yes | — |
| 37 | C2×3−1+2⋊C4 | C23 | 1 | 18 | yes | — |
| 38 | C2×3−1+2⋊C4 | C23 | 1 | 9 | yes | — |
| 39 | C4⋊AGL(1,ℤ/9) | C23 | 1 | 6 | yes | — |
| 40 | C4×AGL(1,ℤ/9) | C23 | 1 | 6 | yes | — |
| 41 | C4×AGL(1,ℤ/9) | C23 | 1 | 18 | yes | — |
| 42 | 3−1+2⋊Q8 | C23 | 1 | 18 | yes | — |
| 43 | C3×#585 | — | 9 | 1 | no | — |
| 44 | #5966 | — | 3 | 1 | no | — |
| 45 | C32×#41 | C2×D3 | 1 | 6 | yes | — |
| 46 | C2×C32×D4 | C2×D3 | 1 | 9 | yes | — |
| 47 | C2×C3×#67 | — | 3 | 18 | no | — |
| 48 | C22×C3×Dic3 | — | 3 | 6 | no | — |
| 49 | C3×#239 | — | 3 | 2 | no | — |
| 50 | C3×D3×Q8 | — | 3 | 6 | no | — |
| 51 | C3×#237 | — | 3 | 18 | no | — |
| 52 | C3×#237 | — | 3 | 36 | no | — |
| 53 | C3×D3×D4 | — | 3 | 18 | no | — |
| 54 | C3×#235 | — | 3 | 18 | no | — |
| 55 | C2×C3×C4×D3 | — | 3 | 9 | no | — |
| 56 | C2×C3×Dic6 | — | 3 | 9 | no | — |
| 57 | C9×#41 | — | 2 | 6 | no | — |
| 58 | C9×#41 | C22×C3 | 1 | 6 | yes | complemented |
| 59 | C2×D4×C9 | — | 2 | 9 | no | — |
| 60 | C2×D4×C9 | C22×C3 | 1 | 9 | yes | complemented |
| 61 | C2×#611 | C22×C3 | 1 | 18 | yes | complemented |
| 62 | C22×Dic9 | C22×C3 | 1 | 6 | yes | complemented |
| 63 | #3701 | C22×C3 | 1 | 2 | yes | complemented |
| 64 | Q8×D9 | C22×C3 | 1 | 6 | yes | complemented |
| 65 | #3699 | C22×C3 | 1 | 18 | yes | complemented |
| 66 | #3699 | C22×C3 | 1 | 36 | yes | complemented |
| 67 | D4×D9 | C22×C3 | 1 | 18 | yes | complemented |
| 68 | #3697 | C22×C3 | 1 | 18 | yes | complemented |
| 69 | C2×C4×D9 | C22×C3 | 1 | 9 | yes | complemented |
| 70 | C2×Dic18 | C22×C3 | 1 | 9 | yes | complemented |
| 71 | C22×3−1+2 | — | 4 | 6 | no | — |
| 72 | C22×3−1+2 | C24 | 1 | 9 | yes | — |
| 73 | C2×AGL(1,ℤ/9) | — | 4 | 9 | no | — |
| 74 | C2×AGL(1,ℤ/9) | C24 | 1 | 6 | yes | — |
| 75 | C4×3−1+2 | C24 | 1 | 6 | yes | — |
| 76 | 3−1+2⋊C4 | C24 | 1 | 9 | yes | — |
| 77 | 3−1+2⋊C4 | C24 | 1 | 1 | yes | characteristic |
| 78 | C3×2−1+4 | — | 9 | 6 | no | — |
| 79 | C3×H(5,𝔽2) | — | 3 | 1 | no | — |
| 80 | C3×H(5,𝔽2) | — | 9 | 9 | no | — |
| 81 | C3×H(5,𝔽2) | C3×D3 | 1 | 1 | yes | characteristic |
| 82 | C2×C3×#41 | — | 9 | 9 | no | — |
| 83 | C2×C3×#41 | — | 9 | 6 | no | — |
| 84 | #1010 | — | 3 | 6 | no | — |
| 85 | D3×#41 | — | 3 | 6 | no | — |
| 86 | #1004 | — | 3 | 9 | no | — |
| 87 | C2×#237 | — | 3 | 9 | no | — |
| 88 | C23×C32 | C22×D3 | 1 | 6 | yes | — |
| 89 | C22×C3×D3 | — | 3 | 9 | no | — |
| 90 | C32×Q8 | C22×D3 | 1 | 2 | yes | — |
| 91 | C32×D4 | C22×D3 | 1 | 18 | yes | — |
| 92 | C2×C32×C4 | C22×D3 | 1 | 9 | yes | — |
| 93 | C3×#67 | — | 3 | 36 | no | — |
| 94 | C2×C3×Dic3 | — | 3 | 18 | no | — |
| 95 | C2×C3×Dic3 | — | 3 | 9 | no | — |
| 96 | C3×D12 | — | 3 | 6 | no | — |
| 97 | C3×C4×D3 | — | 3 | 6 | no | — |
| 98 | C3×C4×D3 | — | 3 | 18 | no | — |
| 99 | C3×Dic6 | — | 3 | 18 | no | — |
| 100 | C23×C9 | — | 2 | 6 | no | — |
| 101 | C23×C9 | C23×C3 | 1 | 6 | yes | — |
| 102 | C22×D9 | C23×C3 | 1 | 9 | yes | — |
| 103 | Q8×C9 | — | 2 | 2 | no | — |
| 104 | Q8×C9 | C23×C3 | 1 | 2 | yes | — |
| 105 | D4×C9 | — | 2 | 18 | no | — |
| 106 | D4×C9 | C23×C3 | 1 | 18 | yes | — |
| 107 | C2×C4×C9 | — | 2 | 9 | no | — |
| 108 | C2×C4×C9 | C23×C3 | 1 | 9 | yes | — |
| 109 | #611 | C23×C3 | 1 | 36 | yes | — |
| 110 | C2×Dic9 | C23×C3 | 1 | 18 | yes | — |
| 111 | C2×Dic9 | C23×C3 | 1 | 9 | yes | — |
| 112 | D36 | C23×C3 | 1 | 6 | yes | — |
| 113 | C4×D9 | C23×C3 | 1 | 6 | yes | — |
| 114 | C4×D9 | C23×C3 | 1 | 18 | yes | — |
| 115 | Dic18 | C23×C3 | 1 | 18 | yes | — |
| 116 | #585 | — | 9 | 1 | no | Sylow |
| 117 | C2×3−1+2 | — | 2 | 9 | no | — |
| 118 | C2×3−1+2 | C25 | 1 | 1 | yes | characteristic |
| 119 | AGL(1,ℤ/9) | — | 2 | 6 | no | — |
| 120 | C3×#41 | — | 3 | 6 | no | — |
| 121 | C3×#41 | — | 9 | 18 | no | — |
| 122 | C3×#41 | — | 9 | 18 | no | — |
| 123 | C3×#41 | — | 9 | 36 | no | — |
| 124 | C3×#41 | — | 9 | 2 | no | — |
| 125 | C3×#41 | C2×C3×D3 | 1 | 6 | yes | — |
| 126 | C2×C3×Q8 | — | 9 | 9 | no | — |
| 127 | C2×C3×Q8 | — | 9 | 6 | no | — |
| 128 | C2×C3×D4 | — | 3 | 9 | no | — |
| 129 | C2×C3×D4 | — | 9 | 18 | no | — |
| 130 | C2×C3×D4 | — | 9 | 18 | no | — |
| 131 | C2×C3×D4 | C2×C3×D3 | 1 | 9 | yes | — |
| 132 | C22×C3×C4 | — | 9 | 9 | no | — |
| 133 | C22×C3×C4 | — | 9 | 6 | no | — |
| 134 | C2×#67 | — | 3 | 18 | no | — |
| 135 | C22×Dic3 | — | 3 | 6 | no | — |
| 136 | #239 | — | 3 | 2 | no | — |
| 137 | D3×Q8 | — | 3 | 6 | no | — |
| 138 | #237 | — | 3 | 18 | no | — |
| 139 | #237 | — | 3 | 36 | no | — |
| 140 | D3×D4 | — | 3 | 18 | no | — |
| 141 | #235 | — | 3 | 18 | no | — |
| 142 | C2×C4×D3 | — | 3 | 9 | no | — |
| 143 | C2×Dic6 | — | 3 | 9 | no | — |
| 144 | C22×C32 | — | 4 | 6 | no | — |
| 145 | C22×C32 | C23×D3 | 1 | 9 | yes | — |
| 146 | C2×C3×D3 | — | 3 | 6 | no | — |
| 147 | C2×C3×D3 | — | 12 | 9 | no | — |
| 148 | C32×C4 | C23×D3 | 1 | 6 | yes | — |
| 149 | C3×Dic3 | — | 3 | 9 | no | — |
| 150 | C3×Dic3 | — | 3 | 1 | no | — |
| 151 | C22×C9 | — | 2 | 9 | no | — |
| 152 | C22×C9 | — | 4 | 6 | no | — |
| 153 | C22×C9 | — | 8 | 6 | no | — |
| 154 | C22×C9 | C24×C3 | 1 | 9 | yes | — |
| 155 | C2×D9 | — | 4 | 9 | no | — |
| 156 | C2×D9 | C24×C3 | 1 | 6 | yes | — |
| 157 | C4×C9 | — | 2 | 6 | no | — |
| 158 | C4×C9 | C24×C3 | 1 | 6 | yes | — |
| 159 | Dic9 | C24×C3 | 1 | 9 | yes | — |
| 160 | Dic9 | C24×C3 | 1 | 1 | yes | characteristic |
| 161 | 2−1+4 | — | 9 | 6 | no | — |
| 162 | H(5,𝔽2) | — | 9 | 9 | no | — |
| 163 | H(5,𝔽2) | AGL(1,ℤ/9) | 1 | 1 | yes | characteristic, complemented |
| 164 | C2×#41 | — | 9 | 6 | no | — |
| 165 | C2×#41 | — | 9 | 9 | no | — |
| 166 | 3−1+2 | #585 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 167 | C23×C3 | — | 3 | 6 | no | — |
| 168 | C23×C3 | — | 9 | 9 | no | — |
| 169 | C23×C3 | C22×C3×D3 | 1 | 6 | yes | — |
| 170 | C22×D3 | — | 3 | 9 | no | — |
| 171 | C3×Q8 | — | 3 | 2 | no | — |
| 172 | C3×Q8 | — | 9 | 18 | no | — |
| 173 | C3×Q8 | C22×C3×D3 | 1 | 2 | yes | — |
| 174 | C3×D4 | — | 3 | 18 | no | — |
| 175 | C3×D4 | — | 9 | 6 | no | — |
| 176 | C3×D4 | — | 9 | 36 | no | — |
| 177 | C3×D4 | C22×C3×D3 | 1 | 18 | yes | — |
| 178 | C2×C3×C4 | — | 3 | 9 | no | — |
| 179 | C2×C3×C4 | — | 9 | 6 | no | — |
| 180 | C2×C3×C4 | — | 9 | 18 | no | — |
| 181 | C2×C3×C4 | — | 9 | 9 | no | — |
| 182 | C2×C3×C4 | — | 9 | 18 | no | — |
| 183 | C2×C3×C4 | C22×C3×D3 | 1 | 9 | yes | — |
| 184 | #67 | — | 3 | 36 | no | — |
| 185 | C2×Dic3 | — | 3 | 18 | no | — |
| 186 | C2×Dic3 | — | 3 | 9 | no | — |
| 187 | D12 | — | 3 | 6 | no | — |
| 188 | C4×D3 | — | 3 | 6 | no | — |
| 189 | C4×D3 | — | 3 | 18 | no | — |
| 190 | Dic6 | — | 3 | 18 | no | — |
| 191 | C2×C32 | — | 2 | 9 | no | — |
| 192 | C2×C32 | C24×D3 | 1 | 1 | yes | characteristic |
| 193 | C3×D3 | — | 6 | 6 | no | — |
| 194 | C2×C9 | — | 2 | 9 | no | — |
| 195 | C2×C9 | — | 2 | 1 | no | — |
| 196 | C2×C9 | — | 4 | 9 | no | — |
| 197 | C2×C9 | C25×C3 | 1 | 1 | yes | characteristic |
| 198 | D9 | — | 2 | 6 | no | — |
| 199 | #41 | — | 9 | 2 | no | — |
| 200 | #41 | — | 9 | 18 | no | — |
| 201 | #41 | — | 9 | 36 | no | — |
| 202 | #41 | — | 9 | 18 | no | — |
| 203 | #41 | C2×AGL(1,ℤ/9) | 1 | 6 | yes | complemented |
| 204 | C2×Q8 | — | 9 | 6 | no | — |
| 205 | C2×Q8 | — | 9 | 9 | no | — |
| 206 | C2×D4 | — | 9 | 18 | no | — |
| 207 | C2×D4 | — | 9 | 18 | no | — |
| 208 | C2×D4 | C2×AGL(1,ℤ/9) | 1 | 9 | yes | complemented |
| 209 | C22×C4 | — | 9 | 6 | no | — |
| 210 | C22×C4 | — | 9 | 9 | no | — |
| 211 | C22×C3 | — | 3 | 9 | no | — |
| 212 | C22×C3 | — | 4 | 6 | no | — |
| 213 | C22×C3 | — | 9 | 6 | no | — |
| 214 | C22×C3 | — | 12 | 6 | no | — |
| 215 | C22×C3 | — | 36 | 9 | no | — |
| 216 | C22×C3 | C23×C3×D3 | 1 | 9 | yes | — |
| 217 | C2×D3 | — | 3 | 6 | no | — |
| 218 | C2×D3 | — | 12 | 9 | no | — |
| 219 | C3×C4 | — | 3 | 6 | no | — |
| 220 | C3×C4 | — | 9 | 1 | no | — |
| 221 | C3×C4 | — | 9 | 9 | no | — |
| 222 | C3×C4 | C23×C3×D3 | 1 | 6 | yes | — |
| 223 | Dic3 | — | 3 | 9 | no | — |
| 224 | Dic3 | — | 3 | 1 | no | — |
| 225 | C32 | #5966 | 1 | 1 | yes | characteristic |
| 226 | C9 | — | 2 | 1 | no | — |
| 227 | C9 | C3×#585 | 1 | 1 | yes | characteristic, complemented |
| 228 | C23 | — | 9 | 9 | no | — |
| 229 | C23 | C22×AGL(1,ℤ/9) | 1 | 6 | yes | — |
| 230 | Q8 | — | 9 | 18 | no | — |
| 231 | Q8 | C22×AGL(1,ℤ/9) | 1 | 2 | yes | — |
| 232 | D4 | — | 9 | 36 | no | — |
| 233 | D4 | — | 9 | 6 | no | — |
| 234 | D4 | C22×AGL(1,ℤ/9) | 1 | 18 | yes | — |
| 235 | C2×C4 | — | 9 | 9 | no | — |
| 236 | C2×C4 | — | 9 | 18 | no | — |
| 237 | C2×C4 | — | 9 | 6 | no | — |
| 238 | C2×C4 | — | 9 | 18 | no | — |
| 239 | C2×C4 | C22×AGL(1,ℤ/9) | 1 | 9 | yes | — |
| 240 | C2×C3 | — | 2 | 9 | no | — |
| 241 | C2×C3 | — | 3 | 1 | no | — |
| 242 | C2×C3 | — | 6 | 9 | no | — |
| 243 | C2×C3 | — | 18 | 6 | no | — |
| 244 | C2×C3 | C24×C3×D3 | 1 | 1 | yes | characteristic |
| 245 | D3 | — | 6 | 6 | no | — |
| 246 | C22 | — | 4 | 6 | no | — |
| 247 | C22 | — | 9 | 6 | no | — |
| 248 | C22 | — | 36 | 9 | no | — |
| 249 | C22 | C23×AGL(1,ℤ/9) | 1 | 9 | yes | — |
| 250 | C4 | — | 9 | 1 | no | — |
| 251 | C4 | — | 9 | 9 | no | — |
| 252 | C4 | C23×AGL(1,ℤ/9) | 1 | 6 | yes | — |
| 253 | C3 | — | 3 | 1 | no | — |
| 254 | C3 | C3×#5966 | 1 | 1 | yes | characteristic |
| 255 | C2 | — | 2 | 9 | no | — |
| 256 | C2 | — | 18 | 6 | no | — |
| 257 | C2 | C24×AGL(1,ℤ/9) | 1 | 1 | yes | characteristic |
Complements
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|
| 1. 3−1+2×H(5,𝔽2) | C2 | semidirect |
| 2. #11722642 | C2 | semidirect |
| 3. #41×AGL(1,ℤ/9) | C2 | semidirect |
| 4. #63710⋊C3 | C2 | semidirect |
| 5. C2×#3699⋊C3 | C2 | semidirect |
| 7. #10593141 | C3 | semidirect |
| 8. #41×3−1+2 | C22 | semidirect |
| 9. C2×D4×3−1+2 | C22 | semidirect |
| 10. C2×#611⋊C3 | C22 | semidirect |
| 11. C22×3−1+2⋊C4 | C22 | semidirect |
| 12. #88724 | C22 | semidirect |
| 13. Q8×AGL(1,ℤ/9) | C22 | semidirect |
| 14. #3699⋊C3 | C22 | semidirect |
| 15. #3699⋊C3 | C22 | semidirect |
| 16. D4×AGL(1,ℤ/9) | C22 | semidirect |
| 17. #3697⋊C3 | C22 | semidirect |
| 18. C2×C4×AGL(1,ℤ/9) | C22 | semidirect |
| 19. C2×3−1+2⋊Q8 | C22 | semidirect |
| 26. C9×H(5,𝔽2) | C2×C3 | semidirect |
| 27. #63716 | C2×C3 | semidirect |
| 28. #41×D9 | C2×C3 | semidirect |
| 29. #63710 | C2×C3 | semidirect |
| 30. C2×#3699 | C2×C3 | semidirect |
| 58. C9×#41 | C22×C3 | semidirect |
| 60. C2×D4×C9 | C22×C3 | semidirect |
| 61. C2×#611 | C22×C3 | semidirect |
| 62. C22×Dic9 | C22×C3 | semidirect |
| 63. #3701 | C22×C3 | semidirect |
| 64. Q8×D9 | C22×C3 | semidirect |
| 65. #3699 | C22×C3 | semidirect |
| 66. #3699 | C22×C3 | semidirect |
| 67. D4×D9 | C22×C3 | semidirect |
| 68. #3697 | C22×C3 | semidirect |
| 69. C2×C4×D9 | C22×C3 | semidirect |
| 70. C2×Dic18 | C22×C3 | semidirect |
| 163. H(5,𝔽2) | AGL(1,ℤ/9) | semidirect |
| 166. 3−1+2 | #585 | semidirect |
| 203. #41 | C2×AGL(1,ℤ/9) | semidirect |
| 208. C2×D4 | C2×AGL(1,ℤ/9) | semidirect |
| 227. C9 | C3×#585 | semidirect |
Maximal subgroups
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
Finite Group Databasebeta