Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
1338 conjugacy classes of subgroups; only the 187 normal subgroups are shown.
This group has 1338 conjugacy classes of subgroups — too many to draw. Only its 187 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 | C22×#6296 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×C3×#3757 | C2 | 1 | 3 | yes | maximal |
| 3 | #11722082 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | #10591692 | — | 3 | 3 | no | maximal |
| 5 | C24×C33 | C22 | 1 | 1 | yes | characteristic |
| 6 | C22×C3×C3⋊Dic3 | C22 | 1 | 3 | yes | — |
| 7 | C22×C3×C3⋊Dic3 | C22 | 1 | 3 | yes | — |
| 8 | C2×#6296 | — | 2 | 6 | no | — |
| 9 | C3×#3757 | — | 2 | 6 | no | — |
| 10 | C22×#627 | — | 3 | 3 | no | — |
| 11 | C2×#3757 | D3 | 1 | 3 | yes | — |
| 12 | C2×C3×#217 | — | 3 | 6 | no | — |
| 13 | C2×C3×#217 | — | 6 | 3 | no | — |
| 14 | #63579 | — | 3 | 6 | no | — |
| 15 | #63579 | — | 3 | 3 | no | — |
| 16 | C23×C33 | — | 2 | 3 | no | — |
| 17 | C23×C33 | — | 2 | 3 | no | — |
| 18 | C23×C33 | Q8 | 1 | 2 | yes | complemented |
| 19 | C23×C33 | C23 | 1 | 1 | yes | characteristic |
| 20 | C2×C3×C3⋊Dic3 | — | 2 | 6 | no | — |
| 21 | C2×C3×C3⋊Dic3 | — | 2 | 3 | no | — |
| 22 | C2×C3×C3⋊Dic3 | — | 2 | 6 | no | — |
| 23 | C2×C3×C3⋊Dic3 | D4 | 1 | 6 | yes | — |
| 24 | #6296 | — | 2 | 8 | no | — |
| 25 | #4947 | — | 9 | 3 | no | — |
| 26 | C24×C32 | — | 2 | 3 | no | — |
| 27 | C24×C32 | — | 4 | 1 | no | — |
| 28 | C24×C32 | C2×D3 | 1 | 3 | yes | — |
| 29 | C22×C3⋊Dic3 | C2×D3 | 1 | 3 | yes | — |
| 30 | C22×C3⋊Dic3 | C2×D3 | 1 | 3 | yes | — |
| 31 | C22×C3×Dic3 | — | 3 | 6 | no | — |
| 32 | C22×C3×Dic3 | — | 3 | 6 | no | — |
| 33 | C22×C3×Dic3 | — | 6 | 3 | no | — |
| 34 | C22×C3×Dic3 | — | 6 | 3 | no | — |
| 35 | C2×#627 | — | 6 | 12 | no | — |
| 36 | C2×#627 | — | 6 | 6 | no | — |
| 37 | #3757 | — | 2 | 6 | no | — |
| 38 | C3×#217 | — | 6 | 12 | no | — |
| 39 | C3×#217 | — | 6 | 12 | no | — |
| 40 | C22×C33 | — | 2 | 6 | no | — |
| 41 | C22×C33 | — | 4 | 1 | no | — |
| 42 | C22×C33 | — | 4 | 3 | no | — |
| 43 | C22×C33 | C2×D4 | 1 | 3 | yes | — |
| 44 | C22×C33 | C2×Q8 | 1 | 1 | yes | characteristic |
| 45 | C22×C33 | #41 | 1 | 3 | yes | — |
| 46 | C3×C3⋊Dic3 | — | 2 | 6 | no | — |
| 47 | C3×C3⋊Dic3 | — | 4 | 3 | no | — |
| 48 | C22×Dic6 | — | 9 | 3 | no | — |
| 49 | C2×C3×C22⋊C4 | — | 9 | 3 | no | — |
| 50 | C2×#217 | — | 3 | 6 | no | — |
| 51 | C2×#217 | — | 6 | 3 | no | — |
| 52 | C3⋊#95 | — | 9 | 6 | no | — |
| 53 | C3⋊#95 | — | 9 | 3 | no | — |
| 54 | C23×C32 | — | 2 | 3 | no | — |
| 55 | C23×C32 | — | 2 | 3 | no | — |
| 56 | C23×C32 | — | 2 | 6 | no | — |
| 57 | C23×C32 | — | 2 | 6 | no | — |
| 58 | C23×C32 | — | 2 | 6 | no | — |
| 59 | C23×C32 | — | 2 | 6 | no | — |
| 60 | C23×C32 | — | 2 | 6 | no | — |
| 61 | C23×C32 | — | 2 | 3 | no | — |
| 62 | C23×C32 | — | 4 | 2 | no | — |
| 63 | C23×C32 | — | 4 | 1 | no | — |
| 64 | C23×C32 | — | 4 | 6 | no | — |
| 65 | C23×C32 | — | 4 | 6 | no | — |
| 66 | C23×C32 | — | 4 | 6 | no | — |
| 67 | C23×C32 | — | 4 | 6 | no | — |
| 68 | C23×C32 | Dic6 | 1 | 6 | yes | complemented |
| 69 | C23×C32 | C22×D3 | 1 | 3 | yes | — |
| 70 | C2×C3⋊Dic3 | — | 2 | 3 | no | — |
| 71 | C2×C3⋊Dic3 | — | 2 | 6 | no | — |
| 72 | C2×C3⋊Dic3 | — | 2 | 6 | no | — |
| 73 | C2×C3⋊Dic3 | #67 | 1 | 6 | yes | — |
| 74 | C2×C3×Dic3 | — | 3 | 12 | no | — |
| 75 | C2×C3×Dic3 | — | 6 | 6 | no | — |
| 76 | C2×C3×Dic3 | — | 6 | 6 | no | — |
| 77 | C2×C3×Dic3 | — | 6 | 6 | no | — |
| 78 | C2×C3×Dic3 | — | 6 | 6 | no | — |
| 79 | C2×C3×Dic3 | — | 6 | 6 | no | — |
| 80 | C2×C3×Dic3 | — | 6 | 6 | no | — |
| 81 | C2×C3×Dic3 | — | 6 | 6 | no | — |
| 82 | C2×C3×Dic3 | — | 12 | 6 | no | — |
| 83 | C2×C3×Dic3 | — | 12 | 6 | no | — |
| 84 | #627 | — | 6 | 24 | no | — |
| 85 | #393 | — | 27 | 1 | no | Sylow |
| 86 | C2×C33 | — | 4 | 2 | no | — |
| 87 | C2×C33 | #120 | 1 | 1 | yes | characteristic |
| 88 | C2×C33 | C22⋊Q8 | 1 | 3 | yes | — |
| 89 | C2×C33 | #124 | 1 | 3 | yes | — |
| 90 | C24×C3 | — | 2 | 3 | no | — |
| 91 | C24×C3 | — | 4 | 1 | no | — |
| 92 | C24×C3 | D32 | 1 | 3 | yes | — |
| 93 | C22×C3×C4 | — | 9 | 3 | no | — |
| 94 | C22×C3×C4 | — | 9 | 3 | no | — |
| 95 | C22×Dic3 | — | 3 | 6 | no | — |
| 96 | C22×Dic3 | — | 3 | 6 | no | — |
| 97 | C22×Dic3 | — | 6 | 3 | no | — |
| 98 | C22×Dic3 | — | 6 | 3 | no | — |
| 99 | C2×Dic6 | — | 18 | 12 | no | — |
| 100 | C2×Dic6 | — | 18 | 6 | no | — |
| 101 | C3×C22⋊C4 | — | 18 | 6 | no | — |
| 102 | #217 | — | 6 | 12 | no | — |
| 103 | #217 | — | 6 | 12 | no | — |
| 104 | C22×C32 | — | 2 | 6 | no | — |
| 105 | C22×C32 | — | 2 | 12 | no | — |
| 106 | C22×C32 | — | 2 | 12 | no | — |
| 107 | C22×C32 | — | 2 | 6 | no | — |
| 108 | C22×C32 | — | 2 | 3 | no | — |
| 109 | C22×C32 | — | 2 | 3 | no | — |
| 110 | C22×C32 | — | 2 | 6 | no | — |
| 111 | C22×C32 | — | 2 | 3 | no | — |
| 112 | C22×C32 | — | 4 | 3 | no | — |
| 113 | C22×C32 | — | 4 | 1 | no | — |
| 114 | C22×C32 | — | 4 | 3 | no | — |
| 115 | C22×C32 | — | 4 | 12 | no | — |
| 116 | C22×C32 | — | 4 | 12 | no | — |
| 117 | C22×C32 | — | 4 | 6 | no | — |
| 118 | C22×C32 | — | 4 | 12 | no | — |
| 119 | C22×C32 | — | 4 | 6 | no | — |
| 120 | C22×C32 | — | 4 | 4 | no | — |
| 121 | C22×C32 | — | 4 | 12 | no | — |
| 122 | C22×C32 | — | 4 | 6 | no | — |
| 123 | C22×C32 | — | 4 | 3 | no | — |
| 124 | C22×C32 | — | 4 | 3 | no | — |
| 125 | C22×C32 | C2×Dic6 | 1 | 3 | yes | — |
| 126 | C22×C32 | #235 | 1 | 3 | yes | — |
| 127 | C22×C32 | D3×D4 | 1 | 6 | yes | — |
| 128 | C22×C32 | #237 | 1 | 6 | yes | — |
| 129 | C22×C32 | C2×#67 | 1 | 3 | yes | — |
| 130 | C3⋊Dic3 | — | 2 | 6 | no | — |
| 131 | C3⋊Dic3 | — | 4 | 3 | no | — |
| 132 | C3×Dic3 | — | 6 | 12 | no | — |
| 133 | C3×Dic3 | — | 12 | 6 | no | — |
| 134 | C3×Dic3 | — | 12 | 6 | no | — |
| 135 | C3×Dic3 | — | 12 | 6 | no | — |
| 136 | C22×Q8 | — | 27 | 1 | no | — |
| 137 | C2×C22⋊C4 | — | 9 | 3 | no | — |
| 138 | #95 | — | 27 | 3 | no | — |
| 139 | C33 | #393 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 140 | C23×C3 | — | 2 | 6 | no | — |
| 141 | C23×C3 | — | 2 | 6 | no | — |
| 142 | C23×C3 | — | 2 | 6 | no | — |
| 143 | C23×C3 | — | 2 | 3 | no | — |
| 144 | C23×C3 | — | 2 | 6 | no | — |
| 145 | C23×C3 | — | 2 | 3 | no | — |
| 146 | C23×C3 | — | 2 | 6 | no | — |
| 147 | C23×C3 | — | 2 | 3 | no | — |
| 148 | C23×C3 | — | 4 | 6 | no | — |
| 149 | C23×C3 | — | 4 | 6 | no | — |
| 150 | C23×C3 | — | 4 | 6 | no | — |
| 151 | C23×C3 | — | 4 | 6 | no | — |
| 152 | C23×C3 | — | 4 | 2 | no | — |
| 153 | C23×C3 | — | 4 | 1 | no | — |
| 154 | C23×C3 | #627 | 1 | 6 | yes | complemented |
| 155 | C23×C3 | C2×D32 | 1 | 3 | yes | — |
| 156 | C2×C3×C4 | — | 9 | 6 | no | — |
| 157 | C2×C3×C4 | — | 18 | 6 | no | — |
| 158 | C2×C3×C4 | — | 18 | 6 | no | — |
| 159 | C2×C3×C4 | — | 18 | 3 | no | — |
| 160 | C2×Dic3 | — | 3 | 12 | no | — |
| 161 | C2×Dic3 | — | 6 | 6 | no | — |
| 162 | C2×Dic3 | — | 6 | 6 | no | — |
| 163 | C2×Dic3 | — | 6 | 6 | no | — |
| 164 | C2×Dic3 | — | 6 | 6 | no | — |
| 165 | C2×Dic3 | — | 6 | 6 | no | — |
| 166 | C2×Dic3 | — | 6 | 6 | no | — |
| 167 | C2×Dic3 | — | 6 | 6 | no | — |
| 168 | C2×Dic3 | — | 12 | 6 | no | — |
| 169 | C2×Dic3 | — | 12 | 6 | no | — |
| 170 | Dic6 | — | 18 | 24 | no | — |
| 171 | C2×C32 | — | 2 | 3 | no | — |
| 172 | C2×C32 | — | 2 | 6 | no | — |
| 173 | C2×C32 | — | 2 | 6 | no | — |
| 174 | C2×C32 | — | 2 | 3 | no | — |
| 175 | C2×C32 | — | 2 | 3 | no | — |
| 176 | C2×C32 | — | 4 | 1 | no | — |
| 177 | C2×C32 | — | 4 | 3 | no | — |
| 178 | C2×C32 | — | 4 | 3 | no | — |
| 179 | C2×C32 | — | 4 | 12 | no | — |
| 180 | C2×C32 | — | 4 | 8 | no | — |
| 181 | C2×C32 | — | 4 | 6 | no | — |
| 182 | C2×C32 | #878 | 1 | 6 | yes | — |
| 183 | C2×C32 | #885 | 1 | 6 | yes | — |
| 184 | C2×C32 | #924 | 1 | 3 | yes | — |
| 185 | C2×C32 | #936 | 1 | 3 | yes | — |
| 186 | C2×C32 | #937 | 1 | 3 | yes | — |
| 187 | C24 | #1113 | 1 | 1 | yes | characteristic |
| 188 | C2×Q8 | — | 54 | 6 | no | — |
| 189 | C22×C4 | — | 9 | 3 | no | — |
| 190 | C22×C4 | — | 9 | 3 | no | — |
| 191 | C22⋊C4 | — | 18 | 6 | no | — |
| 192 | C22×C3 | — | 2 | 12 | no | — |
| 193 | C22×C3 | — | 2 | 12 | no | — |
| 194 | C22×C3 | — | 2 | 6 | no | — |
| 195 | C22×C3 | — | 2 | 6 | no | — |
| 196 | C22×C3 | — | 2 | 3 | no | — |
| 197 | C22×C3 | — | 2 | 3 | no | — |
| 198 | C22×C3 | — | 2 | 3 | no | — |
| 199 | C22×C3 | — | 2 | 6 | no | — |
| 200 | C22×C3 | — | 4 | 12 | no | — |
| 201 | C22×C3 | — | 4 | 12 | no | — |
| 202 | C22×C3 | — | 4 | 6 | no | — |
| 203 | C22×C3 | — | 4 | 12 | no | — |
| 204 | C22×C3 | — | 4 | 6 | no | — |
| 205 | C22×C3 | — | 4 | 12 | no | — |
| 206 | C22×C3 | — | 4 | 4 | no | — |
| 207 | C22×C3 | — | 4 | 3 | no | — |
| 208 | C22×C3 | — | 4 | 6 | no | — |
| 209 | C22×C3 | — | 4 | 3 | no | — |
| 210 | C22×C3 | — | 4 | 3 | no | — |
| 211 | C22×C3 | — | 4 | 1 | no | — |
| 212 | C22×C3 | — | 4 | 3 | no | — |
| 213 | C22×C3 | #3804 | 1 | 6 | yes | — |
| 214 | C22×C3 | #3805 | 1 | 3 | yes | — |
| 215 | C22×C3 | C2×#627 | 1 | 3 | yes | — |
| 216 | C22×C3 | D3×#67 | 1 | 6 | yes | — |
| 217 | C22×C3 | #3811 | 1 | 3 | yes | — |
| 218 | C3×C4 | — | 18 | 6 | no | — |
| 219 | C3×C4 | — | 36 | 3 | no | — |
| 220 | Dic3 | — | 6 | 12 | no | — |
| 221 | Dic3 | — | 12 | 6 | no | — |
| 222 | Dic3 | — | 12 | 6 | no | — |
| 223 | Dic3 | — | 12 | 6 | no | — |
| 224 | C32 | — | 2 | 3 | no | — |
| 225 | C32 | — | 4 | 1 | no | — |
| 226 | C32 | #4947 | 1 | 3 | yes | complemented |
| 227 | C23 | — | 2 | 3 | no | — |
| 228 | C23 | — | 2 | 3 | no | — |
| 229 | C23 | #6296 | 1 | 2 | yes | complemented |
| 230 | C23 | C2×#1113 | 1 | 1 | yes | characteristic |
| 231 | Q8 | — | 54 | 8 | no | — |
| 232 | C2×C4 | — | 9 | 6 | no | — |
| 233 | C2×C4 | — | 18 | 6 | no | — |
| 234 | C2×C4 | — | 18 | 6 | no | — |
| 235 | C2×C4 | — | 18 | 3 | no | — |
| 236 | C2×C3 | — | 2 | 3 | no | — |
| 237 | C2×C3 | — | 2 | 6 | no | — |
| 238 | C2×C3 | — | 2 | 3 | no | — |
| 239 | C2×C3 | — | 2 | 6 | no | — |
| 240 | C2×C3 | — | 2 | 3 | no | — |
| 241 | C2×C3 | — | 4 | 12 | no | — |
| 242 | C2×C3 | — | 4 | 8 | no | — |
| 243 | C2×C3 | — | 4 | 6 | no | — |
| 244 | C2×C3 | — | 4 | 3 | no | — |
| 245 | C2×C3 | — | 4 | 3 | no | — |
| 246 | C2×C3 | — | 4 | 1 | no | — |
| 247 | C2×C3 | #63953 | 1 | 3 | yes | — |
| 248 | C2×C3 | #63959 | 1 | 6 | yes | — |
| 249 | C2×C3 | #63964 | 1 | 6 | yes | — |
| 250 | C2×C3 | #63982 | 1 | 3 | yes | — |
| 251 | C2×C3 | #63983 | 1 | 3 | yes | — |
| 252 | C22 | — | 2 | 6 | no | — |
| 253 | C22 | — | 4 | 3 | no | — |
| 254 | C22 | — | 4 | 1 | no | — |
| 255 | C22 | #89046 | 1 | 3 | yes | — |
| 256 | C22 | C2×#6296 | 1 | 1 | yes | characteristic |
| 257 | C22 | #89049 | 1 | 3 | yes | — |
| 258 | C4 | — | 18 | 6 | no | — |
| 259 | C4 | — | 36 | 3 | no | — |
| 260 | C3 | — | 2 | 3 | no | — |
| 261 | C3 | — | 4 | 1 | no | — |
| 262 | C3 | #10591692 | 1 | 3 | yes | complemented |
| 263 | C2 | — | 4 | 2 | no | — |
| 264 | C2 | #11723718 | 1 | 3 | yes | — |
| 265 | C2 | #11723721 | 1 | 3 | yes | — |
| 266 | C2 | #11723729 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C2×#1113 |
| commutator, socle, cosocle | C23×C33 | C23 |
| Fitting, prime radical | C24×C33 | C22 |
| hypercenter | C24 | #1113 |
| hypocenter | C33 | #393 |
| LCS | G ▷ C23×C33 ▷ C33 | |
| UCS | C1 ◁ C23 ◁ C24 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#6296 | C2 | semidirect |
| 3. #11722082 | C2 | semidirect |
| 18. C23×C33 | Q8 | semidirect |
| 68. C23×C32 | Dic6 | semidirect |
| 139. C33 | #393 | semidirect |
| 154. C23×C3 | #627 | semidirect |
| 226. C32 | #4947 | semidirect |
| 229. C23 | #6296 | semidirect |
| 262. C3 | #10591692 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#6296 | 2 |
| C2×C3×#3757 | 2 |
| #11722082 | 2 |
| #10591692 | 3 |