← #13136243
Subgroups — #13136243
Order 1728.
588 conjugacy classes of subgroups.
The subgroup lattice has 240 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×#88399 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×C4⋊C4×C27 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C4×#6171 | C2 | 1 | 4 | yes | maximal, complemented |
| 4 | #11720175 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #11720142 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #11720141 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | C4⋊C4×D27 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #11720136 | C2 | 1 | 2 | yes | maximal, complemented |
| 9 | #10589606 | — | 3 | 1 | no | maximal |
| 10 | C22×C4×C27 | C22 | 1 | 2 | yes | — |
| 11 | C22×C4×C27 | C22 | 1 | 1 | yes | characteristic |
| 12 | C2×#6171 | C22 | 1 | 2 | yes | — |
| 13 | #88399 | C4 | 1 | 8 | yes | complemented |
| 14 | C2×D108 | C22 | 1 | 1 | yes | characteristic |
| 15 | C2×C4×D27 | C22 | 1 | 4 | yes | — |
| 16 | C2×C4×D27 | C22 | 1 | 2 | yes | — |
| 17 | C2×Dic54 | C22 | 1 | 1 | yes | characteristic |
| 18 | C4⋊C4×C27 | C22 | 1 | 2 | yes | — |
| 19 | C4⋊C4×C27 | C22 | 1 | 2 | yes | — |
| 20 | #88372 | C22 | 1 | 2 | yes | — |
| 21 | (C2×D27)⋊C4 | C22 | 1 | 4 | yes | — |
| 22 | C4⋊Dic27 | C22 | 1 | 1 | yes | characteristic |
| 23 | C4⋊Dic27 | C22 | 1 | 1 | yes | characteristic |
| 24 | Dic27⋊C4 | C22 | 1 | 4 | yes | — |
| 25 | C4×Dic27 | C22 | 1 | 4 | yes | — |
| 26 | C4×Dic27 | C22 | 1 | 2 | yes | — |
| 27 | C2×#3697 | — | 3 | 1 | no | — |
| 28 | C2×C9×C4⋊C4 | D3 | 1 | 1 | yes | characteristic |
| 29 | C4×#611 | — | 3 | 4 | no | — |
| 30 | #63488 | — | 3 | 1 | no | — |
| 31 | #63455 | — | 3 | 2 | no | — |
| 32 | #63454 | — | 3 | 2 | no | — |
| 33 | C4⋊C4×D9 | — | 3 | 2 | no | — |
| 34 | #63449 | — | 3 | 2 | no | — |
| 35 | C23×C27 | C23 | 1 | 1 | yes | characteristic |
| 36 | C22×D27 | C23 | 1 | 2 | yes | — |
| 37 | C2×C4×C27 | — | 4 | 2 | no | — |
| 38 | C2×C4×C27 | C2×C4 | 1 | 4 | yes | complemented |
| 39 | C2×C4×C27 | C23 | 1 | 4 | yes | — |
| 40 | C2×C4×C27 | C23 | 1 | 1 | yes | characteristic |
| 41 | C2×C4×C27 | C23 | 1 | 1 | yes | characteristic |
| 42 | #6171 | C2×C4 | 1 | 8 | yes | complemented |
| 43 | C2×Dic27 | C23 | 1 | 2 | yes | — |
| 44 | C2×Dic27 | C23 | 1 | 2 | yes | — |
| 45 | C2×Dic27 | C23 | 1 | 2 | yes | — |
| 46 | D108 | C2×C4 | 1 | 4 | yes | complemented |
| 47 | C4×D27 | — | 4 | 4 | no | — |
| 48 | C4×D27 | C2×C4 | 1 | 8 | yes | complemented |
| 49 | Dic54 | C2×C4 | 1 | 4 | yes | complemented |
| 50 | #5504 | — | 9 | 1 | no | — |
| 51 | C22×C4×C9 | C2×D3 | 1 | 2 | yes | — |
| 52 | C22×C4×C9 | C2×D3 | 1 | 1 | yes | characteristic |
| 53 | C2×#611 | — | 3 | 2 | no | — |
| 54 | #3697 | — | 3 | 8 | no | — |
| 55 | C2×D36 | — | 3 | 1 | no | — |
| 56 | C2×C4×D9 | — | 3 | 2 | no | — |
| 57 | C2×C4×D9 | — | 3 | 4 | no | — |
| 58 | C2×Dic18 | — | 3 | 1 | no | — |
| 59 | C9×C4⋊C4 | C2×D3 | 1 | 2 | yes | — |
| 60 | C9×C4⋊C4 | C2×D3 | 1 | 2 | yes | — |
| 61 | #3676 | — | 3 | 2 | no | — |
| 62 | (C2×D9)⋊C4 | — | 3 | 4 | no | — |
| 63 | C4⋊Dic9 | — | 3 | 1 | no | — |
| 64 | C4⋊Dic9 | — | 3 | 1 | no | — |
| 65 | Dic9⋊C4 | — | 3 | 4 | no | — |
| 66 | C4×Dic9 | — | 3 | 2 | no | — |
| 67 | C4×Dic9 | — | 3 | 4 | no | — |
| 68 | C22×C27 | — | 2 | 1 | no | — |
| 69 | C22×C27 | — | 2 | 1 | no | — |
| 70 | C22×C27 | C22×C4 | 1 | 2 | yes | — |
| 71 | C22×C27 | C24 | 1 | 1 | yes | characteristic |
| 72 | C2×D27 | — | 2 | 2 | no | — |
| 73 | C2×D27 | — | 2 | 2 | no | — |
| 74 | C2×D27 | C22×C4 | 1 | 4 | yes | — |
| 75 | C4×C27 | — | 2 | 4 | no | — |
| 76 | C4×C27 | C22×C4 | 1 | 2 | yes | — |
| 77 | C4×C27 | C22×C4 | 1 | 2 | yes | — |
| 78 | Dic27 | — | 2 | 2 | no | — |
| 79 | Dic27 | — | 2 | 2 | no | — |
| 80 | Dic27 | C22×C4 | 1 | 4 | yes | — |
| 81 | C2×#235 | — | 9 | 1 | no | — |
| 82 | C2×C3×C4⋊C4 | D9 | 1 | 1 | yes | characteristic |
| 83 | C4×#67 | — | 9 | 4 | no | — |
| 84 | #926 | — | 9 | 1 | no | — |
| 85 | #893 | — | 9 | 2 | no | — |
| 86 | #892 | — | 9 | 2 | no | — |
| 87 | D3×C4⋊C4 | — | 9 | 2 | no | — |
| 88 | #887 | — | 9 | 2 | no | — |
| 89 | C23×C9 | C22×D3 | 1 | 1 | yes | characteristic |
| 90 | C22×D9 | — | 3 | 2 | no | — |
| 91 | C2×C4×C9 | — | 4 | 2 | no | — |
| 92 | C2×C4×C9 | C4×D3 | 1 | 4 | yes | — |
| 93 | C2×C4×C9 | C22×D3 | 1 | 4 | yes | — |
| 94 | C2×C4×C9 | C22×D3 | 1 | 1 | yes | characteristic |
| 95 | C2×C4×C9 | C22×D3 | 1 | 1 | yes | characteristic |
| 96 | #611 | — | 3 | 8 | no | — |
| 97 | C2×Dic9 | — | 3 | 2 | no | — |
| 98 | C2×Dic9 | — | 3 | 2 | no | — |
| 99 | C2×Dic9 | — | 3 | 2 | no | — |
| 100 | D36 | — | 3 | 4 | no | — |
| 101 | C4×D9 | — | 3 | 8 | no | — |
| 102 | C4×D9 | — | 12 | 4 | no | — |
| 103 | Dic18 | — | 3 | 4 | no | — |
| 104 | #520 | — | 27 | 1 | no | Sylow |
| 105 | C2×C27 | — | 2 | 2 | no | — |
| 106 | C2×C27 | C23×C4 | 1 | 1 | yes | characteristic |
| 107 | C2×C27 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 108 | C2×C27 | 2−1+4 | 1 | 1 | yes | characteristic |
| 109 | D27 | — | 2 | 4 | no | — |
| 110 | C22×C3×C4 | C2×D9 | 1 | 2 | yes | — |
| 111 | C22×C3×C4 | C2×D9 | 1 | 1 | yes | characteristic |
| 112 | C2×#67 | — | 9 | 2 | no | — |
| 113 | #235 | — | 9 | 8 | no | — |
| 114 | C2×D12 | — | 9 | 1 | no | — |
| 115 | C2×C4×D3 | — | 9 | 2 | no | — |
| 116 | C2×C4×D3 | — | 9 | 4 | no | — |
| 117 | C2×Dic6 | — | 9 | 1 | no | — |
| 118 | C3×C4⋊C4 | C2×D9 | 1 | 2 | yes | — |
| 119 | C3×C4⋊C4 | C2×D9 | 1 | 2 | yes | — |
| 120 | #217 | — | 9 | 2 | no | — |
| 121 | (C2×D3)⋊C4 | — | 9 | 4 | no | — |
| 122 | C4⋊Dic3 | — | 9 | 1 | no | — |
| 123 | C4⋊Dic3 | — | 9 | 1 | no | — |
| 124 | Dic3⋊C4 | — | 9 | 4 | no | — |
| 125 | C4×Dic3 | — | 9 | 4 | no | — |
| 126 | C4×Dic3 | — | 9 | 2 | no | — |
| 127 | C22×C9 | — | 2 | 1 | no | — |
| 128 | C22×C9 | — | 2 | 1 | no | — |
| 129 | C22×C9 | C2×C4×D3 | 1 | 2 | yes | — |
| 130 | C22×C9 | C23×D3 | 1 | 1 | yes | characteristic |
| 131 | C2×D9 | — | 3 | 4 | no | — |
| 132 | C2×D9 | — | 6 | 2 | no | — |
| 133 | C2×D9 | — | 6 | 2 | no | — |
| 134 | C4×C9 | — | 2 | 4 | no | — |
| 135 | C4×C9 | C2×C4×D3 | 1 | 2 | yes | — |
| 136 | C4×C9 | C2×C4×D3 | 1 | 2 | yes | — |
| 137 | Dic9 | — | 3 | 4 | no | — |
| 138 | Dic9 | — | 6 | 2 | no | — |
| 139 | Dic9 | — | 6 | 2 | no | — |
| 140 | C2×#41 | — | 27 | 1 | no | — |
| 141 | C4×Q8 | — | 27 | 2 | no | — |
| 142 | C4×D4 | — | 27 | 4 | no | — |
| 143 | C4×D4 | — | 27 | 2 | no | — |
| 144 | #117 | — | 27 | 1 | no | — |
| 145 | #117 | — | 27 | 2 | no | — |
| 146 | C2×C4⋊C4 | — | 27 | 2 | no | — |
| 147 | C2×C4⋊C4 | D27 | 1 | 1 | yes | characteristic, complemented |
| 148 | C27 | #520 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 149 | C23×C3 | C22×D9 | 1 | 1 | yes | characteristic |
| 150 | C22×D3 | — | 9 | 2 | no | — |
| 151 | C2×C3×C4 | — | 4 | 2 | no | — |
| 152 | C2×C3×C4 | C4×D9 | 1 | 4 | yes | — |
| 153 | C2×C3×C4 | C22×D9 | 1 | 4 | yes | — |
| 154 | C2×C3×C4 | C22×D9 | 1 | 1 | yes | characteristic |
| 155 | C2×C3×C4 | C22×D9 | 1 | 1 | yes | characteristic |
| 156 | #67 | — | 9 | 8 | no | — |
| 157 | C2×Dic3 | — | 9 | 2 | no | — |
| 158 | C2×Dic3 | — | 9 | 2 | no | — |
| 159 | C2×Dic3 | — | 9 | 2 | no | — |
| 160 | D12 | — | 9 | 4 | no | — |
| 161 | C4×D3 | — | 9 | 8 | no | — |
| 162 | C4×D3 | — | 36 | 4 | no | — |
| 163 | Dic6 | — | 9 | 4 | no | — |
| 164 | C2×C9 | — | 2 | 2 | no | — |
| 165 | C2×C9 | C22×C4×D3 | 1 | 1 | yes | characteristic |
| 166 | C2×C9 | #1004 | 1 | 1 | yes | characteristic |
| 167 | C2×C9 | #1007 | 1 | 1 | yes | characteristic |
| 168 | D9 | — | 6 | 4 | no | — |
| 169 | #41 | — | 27 | 8 | no | — |
| 170 | C2×Q8 | — | 27 | 1 | no | — |
| 171 | C2×D4 | — | 27 | 2 | no | — |
| 172 | C2×D4 | — | 27 | 1 | no | — |
| 173 | C22×C4 | — | 27 | 2 | no | — |
| 174 | C22×C4 | — | 27 | 4 | no | — |
| 175 | C22×C4 | C2×D27 | 1 | 2 | yes | — |
| 176 | C22×C4 | C2×D27 | 1 | 1 | yes | characteristic |
| 177 | C4⋊C4 | — | 27 | 1 | no | — |
| 178 | C4⋊C4 | — | 27 | 4 | no | — |
| 179 | C4⋊C4 | — | 27 | 1 | no | — |
| 180 | C4⋊C4 | C2×D27 | 1 | 2 | yes | — |
| 181 | C4⋊C4 | C2×D27 | 1 | 2 | yes | — |
| 182 | C22⋊C4 | — | 27 | 2 | no | — |
| 183 | C22⋊C4 | — | 27 | 4 | no | — |
| 184 | C42 | — | 27 | 4 | no | — |
| 185 | C42 | — | 27 | 2 | no | — |
| 186 | C22×C3 | — | 2 | 1 | no | — |
| 187 | C22×C3 | — | 2 | 1 | no | — |
| 188 | C22×C3 | C2×C4×D9 | 1 | 2 | yes | — |
| 189 | C22×C3 | C23×D9 | 1 | 1 | yes | characteristic |
| 190 | C2×D3 | — | 9 | 4 | no | — |
| 191 | C2×D3 | — | 18 | 2 | no | — |
| 192 | C2×D3 | — | 18 | 2 | no | — |
| 193 | C3×C4 | — | 2 | 4 | no | — |
| 194 | C3×C4 | C2×C4×D9 | 1 | 2 | yes | — |
| 195 | C3×C4 | C2×C4×D9 | 1 | 2 | yes | — |
| 196 | Dic3 | — | 9 | 4 | no | — |
| 197 | Dic3 | — | 18 | 2 | no | — |
| 198 | Dic3 | — | 18 | 2 | no | — |
| 199 | C9 | #5504 | 1 | 1 | yes | characteristic |
| 200 | C23 | — | 27 | 2 | no | — |
| 201 | C23 | C22×D27 | 1 | 1 | yes | characteristic |
| 202 | Q8 | — | 27 | 4 | no | — |
| 203 | D4 | — | 27 | 8 | no | — |
| 204 | D4 | — | 27 | 4 | no | — |
| 205 | C2×C4 | — | 4 | 2 | no | — |
| 206 | C2×C4 | — | 27 | 2 | no | — |
| 207 | C2×C4 | — | 27 | 8 | no | — |
| 208 | C2×C4 | — | 27 | 2 | no | — |
| 209 | C2×C4 | — | 27 | 2 | no | — |
| 210 | C2×C4 | — | 108 | 4 | no | — |
| 211 | C2×C4 | C4×D27 | 1 | 4 | yes | complemented |
| 212 | C2×C4 | C22×D27 | 1 | 4 | yes | — |
| 213 | C2×C4 | C22×D27 | 1 | 1 | yes | characteristic |
| 214 | C2×C4 | C22×D27 | 1 | 1 | yes | characteristic |
| 215 | C2×C3 | — | 2 | 2 | no | — |
| 216 | C2×C3 | C22×C4×D9 | 1 | 1 | yes | characteristic |
| 217 | C2×C3 | #63710 | 1 | 1 | yes | characteristic |
| 218 | C2×C3 | #63713 | 1 | 1 | yes | characteristic |
| 219 | D3 | — | 18 | 4 | no | — |
| 220 | C22 | — | 2 | 1 | no | — |
| 221 | C22 | — | 2 | 1 | no | — |
| 222 | C22 | — | 27 | 4 | no | — |
| 223 | C22 | — | 54 | 2 | no | — |
| 224 | C22 | — | 54 | 2 | no | — |
| 225 | C22 | C2×C4×D27 | 1 | 2 | yes | — |
| 226 | C22 | C23×D27 | 1 | 1 | yes | characteristic |
| 227 | C4 | — | 2 | 4 | no | — |
| 228 | C4 | — | 27 | 4 | no | — |
| 229 | C4 | — | 54 | 2 | no | — |
| 230 | C4 | — | 54 | 2 | no | — |
| 231 | C4 | C2×C4×D27 | 1 | 2 | yes | — |
| 232 | C4 | C2×C4×D27 | 1 | 2 | yes | — |
| 233 | C3 | #10589606 | 1 | 1 | yes | characteristic |
| 234 | C2 | — | 2 | 2 | no | — |
| 235 | C2 | — | 54 | 4 | no | — |
| 236 | C2 | C22×C4×D27 | 1 | 1 | yes | characteristic |
| 237 | C2 | #11720670 | 1 | 1 | yes | characteristic |
| 238 | C2 | #11720673 | 1 | 1 | yes | characteristic |
Complements
Splittings G = N ⋊ H with N normal; H ≅ G/N.
Maximal subgroups
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
Finite Group Databasebeta