Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
1200 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 1200 conjugacy classes of subgroups — too many to draw. Only its 167 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 | #11724620 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | C3×#64121 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | #11723705 | C2 | 1 | 6 | yes | maximal, complemented |
| 4 | #11723651 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #11723649 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | #10594737 | — | 3 | 3 | no | maximal |
| 7 | C3×#3830 | C22 | 1 | 6 | yes | — |
| 8 | C2×#6296 | — | 2 | 2 | no | — |
| 9 | #89046 | — | 2 | 6 | no | — |
| 10 | #89042 | C22 | 1 | 1 | yes | characteristic |
| 11 | #89040 | C22 | 1 | 6 | yes | — |
| 12 | C33×D8 | C22 | 1 | 1 | yes | characteristic |
| 13 | C3×#3754 | C22 | 1 | 6 | yes | — |
| 14 | C3×#3746 | C22 | 1 | 3 | yes | — |
| 15 | C3×C8×C3⋊D3 | C22 | 1 | 3 | yes | — |
| 16 | #88812 | C22 | 1 | 6 | yes | — |
| 17 | #88809 | C22 | 1 | 3 | yes | — |
| 18 | #64309 | — | 3 | 6 | no | — |
| 19 | #64121 | D3 | 1 | 3 | yes | complemented |
| 20 | C3×#912 | — | 3 | 6 | no | — |
| 21 | C3×#912 | — | 6 | 3 | no | — |
| 22 | #63931 | — | 3 | 6 | no | — |
| 23 | #63929 | — | 3 | 12 | no | — |
| 24 | #63803 | — | 3 | 3 | no | — |
| 25 | #63802 | — | 3 | 3 | no | — |
| 26 | D3×Dic12 | — | 3 | 6 | no | — |
| 27 | C33×D4 | C23 | 1 | 2 | yes | — |
| 28 | C3×#638 | — | 2 | 6 | no | — |
| 29 | C2×C3×C3⋊Dic3 | — | 2 | 6 | no | — |
| 30 | C3×C4×C3⋊D3 | C23 | 1 | 3 | yes | — |
| 31 | C3×#634 | C23 | 1 | 6 | yes | — |
| 32 | #6296 | — | 2 | 6 | no | — |
| 33 | #6296 | D4 | 1 | 1 | yes | characteristic |
| 34 | #6295 | D4 | 1 | 3 | yes | — |
| 35 | #6294 | — | 2 | 6 | no | — |
| 36 | C33×C8 | C23 | 1 | 1 | yes | characteristic |
| 37 | C3×C3⋊(C3⋊C8) | C23 | 1 | 3 | yes | — |
| 38 | #5771 | — | 9 | 3 | no | — |
| 39 | #3830 | C2×D3 | 1 | 6 | yes | — |
| 40 | C3×#237 | — | 3 | 12 | no | — |
| 41 | C3×#237 | — | 6 | 6 | no | — |
| 42 | C2×#627 | — | 6 | 6 | no | — |
| 43 | #3805 | — | 6 | 6 | no | — |
| 44 | #3804 | — | 6 | 12 | no | — |
| 45 | #3798 | — | 3 | 3 | no | — |
| 46 | #3797 | — | 3 | 6 | no | — |
| 47 | D3×Dic6 | — | 3 | 12 | no | — |
| 48 | C32×D8 | — | 2 | 3 | no | — |
| 49 | C32×D8 | — | 4 | 1 | no | — |
| 50 | C32×D8 | C2×D3 | 1 | 3 | yes | — |
| 51 | #3754 | C2×D3 | 1 | 6 | yes | — |
| 52 | #3746 | C2×D3 | 1 | 3 | yes | — |
| 53 | C8×C3⋊D3 | C2×D3 | 1 | 3 | yes | — |
| 54 | C3×#214 | — | 3 | 12 | no | — |
| 55 | C3×#214 | — | 6 | 6 | no | — |
| 56 | C3×Dic12 | — | 3 | 6 | no | — |
| 57 | C3×Dic12 | — | 6 | 3 | no | — |
| 58 | C3×D3×C8 | — | 3 | 6 | no | — |
| 59 | C3×D3×C8 | — | 6 | 3 | no | — |
| 60 | #3719 | — | 3 | 12 | no | — |
| 61 | #3718 | — | 3 | 6 | no | — |
| 62 | #3712 | — | 3 | 3 | no | — |
| 63 | #3711 | — | 3 | 6 | no | — |
| 64 | C22×C33 | — | 2 | 2 | no | — |
| 65 | C2×C3×C3⋊D3 | C2×D4 | 1 | 3 | yes | — |
| 66 | C33×C4 | C24 | 1 | 1 | yes | characteristic |
| 67 | C3×C3⋊Dic3 | — | 2 | 6 | no | — |
| 68 | C3×C3⋊Dic3 | C2×D4 | 1 | 3 | yes | — |
| 69 | #1010 | — | 9 | 6 | no | — |
| 70 | C3×#135 | — | 9 | 3 | no | — |
| 71 | #951 | — | 9 | 6 | no | — |
| 72 | D3×Q16 | — | 9 | 6 | no | — |
| 73 | #915 | — | 9 | 12 | no | — |
| 74 | #912 | — | 3 | 6 | no | — |
| 75 | #912 | — | 6 | 3 | no | — |
| 76 | C2×Dic12 | — | 9 | 3 | no | — |
| 77 | #901 | — | 9 | 3 | no | — |
| 78 | C32×D4 | — | 2 | 6 | no | — |
| 79 | C32×D4 | — | 4 | 2 | no | — |
| 80 | C32×D4 | C22×D3 | 1 | 6 | yes | — |
| 81 | #638 | — | 2 | 6 | no | — |
| 82 | C2×C3⋊Dic3 | — | 2 | 6 | no | — |
| 83 | C4×C3⋊D3 | C22×D3 | 1 | 3 | yes | — |
| 84 | #634 | C22×D3 | 1 | 6 | yes | — |
| 85 | C3×#67 | — | 6 | 12 | no | — |
| 86 | C3×#67 | — | 12 | 6 | no | — |
| 87 | C2×C3×Dic3 | — | 6 | 12 | no | — |
| 88 | C2×C3×Dic3 | — | 12 | 6 | no | — |
| 89 | C3×C4×D3 | — | 3 | 6 | no | — |
| 90 | C3×C4×D3 | — | 6 | 3 | no | — |
| 91 | C3×Dic6 | — | 3 | 12 | no | — |
| 92 | C3×Dic6 | — | 6 | 6 | no | — |
| 93 | #627 | — | 3 | 3 | no | — |
| 94 | #627 | — | 6 | 12 | no | — |
| 95 | #627 | — | 6 | 6 | no | — |
| 96 | #626 | — | 3 | 6 | no | — |
| 97 | #625 | — | 3 | 3 | no | — |
| 98 | #624 | — | 6 | 6 | no | — |
| 99 | D3×Dic3 | — | 6 | 12 | no | — |
| 100 | C32×C8 | — | 2 | 3 | no | — |
| 101 | C32×C8 | — | 4 | 1 | no | — |
| 102 | C32×C8 | C22×D3 | 1 | 3 | yes | — |
| 103 | C3⋊(C3⋊C8) | C22×D3 | 1 | 3 | yes | — |
| 104 | C3×C3⋊C8 | — | 3 | 6 | no | — |
| 105 | C3×C3⋊C8 | — | 6 | 3 | no | — |
| 106 | #578 | — | 27 | 1 | no | Sylow |
| 107 | C2×C33 | — | 4 | 2 | no | — |
| 108 | C2×C33 | C22×D4 | 1 | 1 | yes | characteristic |
| 109 | C3×C3⋊D3 | — | 2 | 3 | no | — |
| 110 | C3×#41 | — | 9 | 6 | no | — |
| 111 | D3×Q8 | — | 9 | 12 | no | — |
| 112 | #237 | — | 3 | 12 | no | — |
| 113 | #237 | — | 6 | 6 | no | — |
| 114 | #237 | — | 18 | 12 | no | — |
| 115 | #235 | — | 9 | 3 | no | — |
| 116 | #235 | — | 18 | 6 | no | — |
| 117 | C2×Dic6 | — | 9 | 6 | no | — |
| 118 | C2×Dic6 | — | 18 | 6 | no | — |
| 119 | C3×Q16 | — | 9 | 3 | no | — |
| 120 | C3×SD16 | — | 9 | 6 | no | — |
| 121 | C3×D8 | — | 2 | 3 | no | — |
| 122 | C3×D8 | — | 4 | 1 | no | — |
| 123 | C3×D8 | D32 | 1 | 3 | yes | — |
| 124 | C2×C3×C8 | — | 9 | 3 | no | — |
| 125 | #216 | — | 9 | 12 | no | — |
| 126 | #214 | — | 3 | 12 | no | — |
| 127 | #214 | — | 6 | 6 | no | — |
| 128 | #208 | — | 9 | 3 | no | — |
| 129 | Dic12 | — | 3 | 6 | no | — |
| 130 | Dic12 | — | 6 | 3 | no | — |
| 131 | Dic12 | — | 9 | 6 | no | — |
| 132 | ΓL(1,𝔽25) | — | 9 | 6 | no | — |
| 133 | D3×C8 | — | 3 | 6 | no | — |
| 134 | D3×C8 | — | 6 | 3 | no | — |
| 135 | C22×C32 | — | 2 | 6 | no | — |
| 136 | C22×C32 | — | 4 | 6 | no | — |
| 137 | C22×C32 | — | 8 | 2 | no | — |
| 138 | C2×C3⋊D3 | D3×D4 | 1 | 3 | yes | — |
| 139 | C2×C3×D3 | — | 3 | 6 | no | — |
| 140 | C2×C3×D3 | — | 6 | 3 | no | — |
| 141 | C32×C4 | — | 2 | 3 | no | — |
| 142 | C32×C4 | — | 4 | 1 | no | — |
| 143 | C32×C4 | C23×D3 | 1 | 3 | yes | — |
| 144 | C3⋊Dic3 | — | 2 | 6 | no | — |
| 145 | C3⋊Dic3 | D3×D4 | 1 | 3 | yes | — |
| 146 | C3×Dic3 | — | 3 | 6 | no | — |
| 147 | C3×Dic3 | — | 6 | 3 | no | — |
| 148 | C3×Dic3 | — | 6 | 12 | no | — |
| 149 | C3×Dic3 | — | 12 | 6 | no | — |
| 150 | 2−1+4 | — | 27 | 2 | no | — |
| 151 | #137 | — | 27 | 6 | no | — |
| 152 | #135 | — | 9 | 3 | no | — |
| 153 | C2×Q16 | — | 27 | 3 | no | — |
| 154 | #131 | — | 27 | 1 | no | — |
| 155 | C33 | #578 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 156 | C3×Q8 | — | 9 | 6 | no | — |
| 157 | C3×D4 | — | 2 | 6 | no | — |
| 158 | C3×D4 | — | 4 | 2 | no | — |
| 159 | C3×D4 | — | 18 | 6 | no | — |
| 160 | C3×D4 | C2×D32 | 1 | 6 | yes | — |
| 161 | C2×C3×C4 | — | 9 | 3 | no | — |
| 162 | C2×C3×C4 | — | 18 | 6 | no | — |
| 163 | #67 | — | 6 | 12 | no | — |
| 164 | #67 | — | 9 | 6 | no | — |
| 165 | #67 | — | 12 | 6 | no | — |
| 166 | C2×Dic3 | — | 6 | 12 | no | — |
| 167 | C2×Dic3 | — | 12 | 6 | no | — |
| 168 | C2×Dic3 | — | 18 | 6 | no | — |
| 169 | D12 | — | 9 | 3 | no | — |
| 170 | C4×D3 | — | 3 | 6 | no | — |
| 171 | C4×D3 | — | 6 | 3 | no | — |
| 172 | C4×D3 | — | 18 | 12 | no | — |
| 173 | Dic6 | — | 3 | 12 | no | — |
| 174 | Dic6 | — | 6 | 6 | no | — |
| 175 | Dic6 | — | 9 | 3 | no | — |
| 176 | Dic6 | — | 18 | 12 | no | — |
| 177 | Dic6 | — | 18 | 6 | no | — |
| 178 | C3×C8 | — | 2 | 3 | no | — |
| 179 | C3×C8 | — | 4 | 1 | no | — |
| 180 | C3×C8 | — | 9 | 3 | no | — |
| 181 | C3×C8 | C2×D32 | 1 | 3 | yes | — |
| 182 | C3⋊C8 | — | 3 | 6 | no | — |
| 183 | C3⋊C8 | — | 6 | 3 | no | — |
| 184 | C2×C32 | — | 2 | 3 | no | — |
| 185 | C2×C32 | — | 4 | 1 | no | — |
| 186 | C2×C32 | — | 4 | 6 | no | — |
| 187 | C2×C32 | — | 8 | 6 | no | — |
| 188 | C2×C32 | — | 16 | 2 | no | — |
| 189 | C2×C32 | C2×D3×D4 | 1 | 3 | yes | — |
| 190 | C3⋊D3 | — | 2 | 3 | no | — |
| 191 | C3×D3 | — | 6 | 6 | no | — |
| 192 | C3×D3 | — | 12 | 3 | no | — |
| 193 | #41 | — | 9 | 6 | no | — |
| 194 | #41 | — | 27 | 1 | no | — |
| 195 | #41 | — | 54 | 6 | no | — |
| 196 | C2×Q8 | — | 27 | 6 | no | — |
| 197 | C2×Q8 | — | 54 | 2 | no | — |
| 198 | Q16 | — | 9 | 3 | no | — |
| 199 | Q16 | — | 27 | 6 | no | — |
| 200 | SD16 | — | 9 | 6 | no | — |
| 201 | D8 | #1113 | 1 | 1 | yes | characteristic |
| 202 | M16 | — | 27 | 3 | no | — |
| 203 | C2×C8 | — | 9 | 3 | no | — |
| 204 | C22×C3 | — | 2 | 6 | no | — |
| 205 | C22×C3 | — | 4 | 6 | no | — |
| 206 | C22×C3 | — | 8 | 2 | no | — |
| 207 | C22×C3 | — | 9 | 3 | no | — |
| 208 | C2×D3 | — | 3 | 6 | no | — |
| 209 | C2×D3 | — | 6 | 3 | no | — |
| 210 | C3×C4 | — | 2 | 3 | no | — |
| 211 | C3×C4 | — | 4 | 1 | no | — |
| 212 | C3×C4 | — | 9 | 3 | no | — |
| 213 | C3×C4 | — | 18 | 6 | no | — |
| 214 | C3×C4 | C22×D32 | 1 | 3 | yes | — |
| 215 | Dic3 | — | 3 | 6 | no | — |
| 216 | Dic3 | — | 6 | 3 | no | — |
| 217 | Dic3 | — | 6 | 12 | no | — |
| 218 | Dic3 | — | 12 | 6 | no | — |
| 219 | C32 | — | 2 | 3 | no | — |
| 220 | C32 | — | 4 | 1 | no | — |
| 221 | C32 | #5771 | 1 | 3 | yes | complemented |
| 222 | Q8 | — | 9 | 6 | no | — |
| 223 | Q8 | — | 27 | 1 | no | — |
| 224 | Q8 | — | 54 | 6 | no | — |
| 225 | D4 | — | 18 | 6 | no | — |
| 226 | D4 | — | 27 | 3 | no | — |
| 227 | D4 | C2×#1113 | 1 | 2 | yes | — |
| 228 | C2×C4 | — | 9 | 3 | no | — |
| 229 | C2×C4 | — | 18 | 6 | no | — |
| 230 | C2×C4 | — | 54 | 6 | no | — |
| 231 | C8 | — | 9 | 3 | no | — |
| 232 | C8 | C2×#1113 | 1 | 1 | yes | characteristic |
| 233 | C2×C3 | — | 2 | 3 | no | — |
| 234 | C2×C3 | — | 4 | 6 | no | — |
| 235 | C2×C3 | — | 4 | 1 | no | — |
| 236 | C2×C3 | — | 8 | 6 | no | — |
| 237 | C2×C3 | — | 16 | 2 | no | — |
| 238 | C2×C3 | — | 18 | 3 | no | — |
| 239 | C2×C3 | D32×D4 | 1 | 3 | yes | — |
| 240 | D3 | — | 6 | 6 | no | — |
| 241 | D3 | — | 12 | 3 | no | — |
| 242 | C22 | — | 2 | 2 | no | — |
| 243 | C22 | — | 9 | 3 | no | — |
| 244 | C4 | — | 9 | 3 | no | — |
| 245 | C4 | — | 18 | 6 | no | — |
| 246 | C4 | C22×#1113 | 1 | 1 | yes | characteristic |
| 247 | C3 | — | 2 | 3 | no | — |
| 248 | C3 | — | 4 | 1 | no | — |
| 249 | C3 | #10594737 | 1 | 3 | yes | complemented |
| 250 | C2 | — | 4 | 2 | no | — |
| 251 | C2 | — | 18 | 3 | no | — |
| 252 | C2 | D4×#1113 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D4×#1113 |
| commutator, cosocle | C33×C4 | C24 |
| Frattini | C4 | C22×#1113 |
| Fitting | C33×D8 | C22 |
| socle | C2×C33 | C22×D4 |
| hypercenter | D8 | #1113 |
| hypocenter | C33 | #578 |
| LCS | G ▷ C33×C4 ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ D8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #11724620 | C2 | semidirect |
| 2. C3×#64121 | C2 | semidirect |
| 3. #11723705 | C2 | semidirect |
| 4. #11723651 | C2 | semidirect |
| 5. #11723649 | C2 | semidirect |
| 19. #64121 | D3 | semidirect |
| 155. C33 | #578 | semidirect |
| 221. C32 | #5771 | semidirect |
| 249. C3 | #10594737 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11724620 | 2 |
| C3×#64121 | 2 |
| #11723705 | 2 |
| #11723651 | 2 |
| #11723649 | 2 |
| #10594737 | 3 |