Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
200 conjugacy classes of subgroups; only the 31 normal subgroups are shown.
This group has 200 conjugacy classes of subgroups — too many to draw. Only its 31 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 | D4×C22⋊C27 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | (C22⋊C27)⋊Q8 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | (C22⋊C27)⋊C8 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #10588897 | — | 3 | 1 | no | maximal |
| 5 | C22×C22⋊C27 | — | 2 | 1 | no | — |
| 6 | C4×C22⋊C27 | C22 | 1 | 1 | yes | characteristic |
| 7 | (C22⋊C27)⋊C4 | — | 2 | 1 | no | — |
| 8 | #88368 | — | 4 | 1 | no | maximal |
| 9 | C22×D4×C9 | D3 | 1 | 1 | yes | characteristic |
| 10 | C2×#3672 | — | 3 | 1 | no | — |
| 11 | C2×#3672 | — | 3 | 1 | no | — |
| 12 | #63486 | — | 3 | 1 | no | — |
| 13 | #63392 | — | 3 | 1 | no | — |
| 14 | #63392 | — | 3 | 1 | no | — |
| 15 | #63389 | — | 3 | 1 | no | — |
| 16 | C2×C22⋊C27 | — | 4 | 1 | no | — |
| 17 | C2×C22⋊C27 | D4 | 1 | 1 | yes | characteristic |
| 18 | D4×C27 | — | 4 | 1 | no | — |
| 19 | Dic54 | — | 4 | 1 | no | — |
| 20 | C27⋊C8 | — | 4 | 1 | no | — |
| 21 | #5218 | — | 9 | 1 | no | — |
| 22 | C24×C9 | — | 2 | 1 | no | — |
| 23 | C2×D4×C9 | — | 3 | 1 | no | — |
| 24 | C2×D4×C9 | — | 3 | 1 | no | — |
| 25 | C2×D4×C9 | — | 6 | 1 | no | — |
| 26 | C22×C4×C9 | C2×D3 | 1 | 1 | yes | characteristic |
| 27 | C2×Dic18 | — | 3 | 1 | no | — |
| 28 | #3676 | — | 6 | 1 | no | — |
| 29 | #3672 | — | 6 | 2 | no | — |
| 30 | #3672 | — | 6 | 1 | no | — |
| 31 | #3672 | — | 6 | 1 | no | — |
| 32 | C4⋊Dic9 | — | 3 | 1 | no | — |
| 33 | Dic9⋊C4 | — | 6 | 1 | no | — |
| 34 | C2×C9⋊C8 | — | 3 | 1 | no | — |
| 35 | C2×C9⋊C8 | — | 3 | 1 | no | — |
| 36 | C22×C27 | — | 8 | 1 | no | — |
| 37 | C22⋊C27 | SD16 | 1 | 1 | yes | characteristic, complemented |
| 38 | C4×C27 | — | 4 | 1 | no | — |
| 39 | Dic27 | — | 8 | 1 | no | — |
| 40 | C22×C3×D4 | D9 | 1 | 1 | yes | characteristic |
| 41 | C2×#214 | — | 9 | 1 | no | — |
| 42 | C2×#214 | — | 9 | 1 | no | — |
| 43 | #924 | — | 9 | 1 | no | — |
| 44 | #832 | — | 9 | 1 | no | — |
| 45 | #832 | — | 9 | 1 | no | — |
| 46 | #830 | — | 9 | 1 | no | — |
| 47 | C23×C9 | — | 4 | 1 | no | — |
| 48 | C23×C9 | — | 6 | 1 | no | — |
| 49 | C23×C9 | — | 6 | 1 | no | — |
| 50 | C23×C9 | — | 12 | 1 | no | — |
| 51 | C23×C9 | #67 | 1 | 1 | yes | characteristic |
| 52 | D4×C9 | — | 3 | 2 | no | — |
| 53 | D4×C9 | — | 3 | 1 | no | — |
| 54 | D4×C9 | — | 6 | 1 | no | — |
| 55 | D4×C9 | S4 | 1 | 1 | yes | characteristic |
| 56 | C2×C4×C9 | — | 3 | 1 | no | — |
| 57 | C2×C4×C9 | — | 3 | 1 | no | — |
| 58 | C2×Dic9 | — | 6 | 1 | no | — |
| 59 | C2×Dic9 | — | 6 | 1 | no | — |
| 60 | Dic18 | — | 6 | 1 | no | — |
| 61 | Dic18 | — | 6 | 1 | no | — |
| 62 | C9⋊C8 | — | 6 | 1 | no | — |
| 63 | C9⋊C8 | — | 6 | 1 | no | — |
| 64 | #450 | — | 27 | 1 | no | Sylow |
| 65 | C2×C27 | — | 4 | 1 | no | — |
| 66 | C2×C27 | — | 16 | 1 | no | — |
| 67 | C24×C3 | — | 2 | 1 | no | — |
| 68 | C2×C3×D4 | — | 3 | 1 | no | — |
| 69 | C2×C3×D4 | — | 3 | 1 | no | — |
| 70 | C2×C3×D4 | — | 6 | 1 | no | — |
| 71 | C22×C3×C4 | C2×D9 | 1 | 1 | yes | characteristic |
| 72 | C2×Dic6 | — | 9 | 1 | no | — |
| 73 | #217 | — | 18 | 1 | no | — |
| 74 | #214 | — | 18 | 2 | no | — |
| 75 | #214 | — | 18 | 1 | no | — |
| 76 | #214 | — | 18 | 1 | no | — |
| 77 | C4⋊Dic3 | — | 9 | 1 | no | — |
| 78 | Dic3⋊C4 | — | 18 | 1 | no | — |
| 79 | C2×C3⋊C8 | — | 9 | 1 | no | — |
| 80 | C2×C3⋊C8 | — | 9 | 1 | no | — |
| 81 | C22×C9 | — | 2 | 1 | no | — |
| 82 | C22×C9 | — | 3 | 1 | no | — |
| 83 | C22×C9 | — | 3 | 1 | no | — |
| 84 | C22×C9 | — | 6 | 1 | no | — |
| 85 | C22×C9 | — | 12 | 1 | no | — |
| 86 | C22×C9 | — | 12 | 1 | no | — |
| 87 | C22×C9 | — | 12 | 1 | no | — |
| 88 | C22×C9 | — | 12 | 1 | no | — |
| 89 | C22×C9 | #214 | 1 | 1 | yes | characteristic |
| 90 | C4×C9 | — | 3 | 1 | no | — |
| 91 | C4×C9 | C2×S4 | 1 | 1 | yes | characteristic |
| 92 | Dic9 | — | 12 | 1 | no | — |
| 93 | Dic9 | — | 12 | 1 | no | — |
| 94 | C22×D4 | D27 | 1 | 1 | yes | characteristic |
| 95 | C2×SD16 | — | 27 | 1 | no | — |
| 96 | C2×SD16 | — | 27 | 1 | no | — |
| 97 | C22⋊Q8 | — | 27 | 1 | no | — |
| 98 | #102 | — | 27 | 1 | no | — |
| 99 | #102 | — | 27 | 1 | no | — |
| 100 | C22⋊C8 | — | 27 | 1 | no | — |
| 101 | C27 | — | 4 | 1 | no | Sylow |
| 102 | C23×C3 | — | 4 | 1 | no | — |
| 103 | C23×C3 | — | 6 | 1 | no | — |
| 104 | C23×C3 | — | 6 | 1 | no | — |
| 105 | C23×C3 | — | 12 | 1 | no | — |
| 106 | C23×C3 | #611 | 1 | 1 | yes | characteristic |
| 107 | C3×D4 | — | 3 | 2 | no | — |
| 108 | C3×D4 | — | 3 | 1 | no | — |
| 109 | C3×D4 | — | 6 | 1 | no | — |
| 110 | C3×D4 | (C22⋊C9)⋊C2 | 1 | 1 | yes | characteristic |
| 111 | C2×C3×C4 | — | 3 | 1 | no | — |
| 112 | C2×C3×C4 | — | 3 | 1 | no | — |
| 113 | C2×Dic3 | — | 18 | 1 | no | — |
| 114 | C2×Dic3 | — | 18 | 1 | no | — |
| 115 | Dic6 | — | 18 | 1 | no | — |
| 116 | Dic6 | — | 18 | 1 | no | — |
| 117 | C3⋊C8 | — | 18 | 1 | no | — |
| 118 | C3⋊C8 | — | 18 | 1 | no | — |
| 119 | C2×C9 | — | 3 | 1 | no | — |
| 120 | C2×C9 | — | 3 | 1 | no | — |
| 121 | C2×C9 | — | 4 | 1 | no | — |
| 122 | C2×C9 | — | 12 | 1 | no | — |
| 123 | C2×C9 | GL(2,ℤ/4) | 1 | 1 | yes | characteristic |
| 124 | C24 | — | 2 | 1 | no | — |
| 125 | C2×Q8 | — | 27 | 1 | no | — |
| 126 | C2×D4 | — | 3 | 1 | no | — |
| 127 | C2×D4 | — | 3 | 1 | no | — |
| 128 | C2×D4 | — | 6 | 1 | no | — |
| 129 | C22×C4 | C2×D27 | 1 | 1 | yes | characteristic |
| 130 | SD16 | — | 54 | 2 | no | — |
| 131 | SD16 | — | 54 | 1 | no | — |
| 132 | SD16 | — | 54 | 1 | no | — |
| 133 | C2×C8 | — | 27 | 1 | no | — |
| 134 | C2×C8 | — | 27 | 1 | no | — |
| 135 | C4⋊C4 | — | 27 | 1 | no | — |
| 136 | C4⋊C4 | — | 54 | 1 | no | — |
| 137 | C22⋊C4 | — | 54 | 1 | no | — |
| 138 | C22×C3 | — | 2 | 1 | no | — |
| 139 | C22×C3 | — | 3 | 1 | no | — |
| 140 | C22×C3 | — | 3 | 1 | no | — |
| 141 | C22×C3 | — | 6 | 1 | no | — |
| 142 | C22×C3 | — | 12 | 1 | no | — |
| 143 | C22×C3 | — | 12 | 1 | no | — |
| 144 | C22×C3 | — | 12 | 1 | no | — |
| 145 | C22×C3 | — | 12 | 1 | no | — |
| 146 | C22×C3 | #3672 | 1 | 1 | yes | characteristic |
| 147 | C3×C4 | — | 3 | 1 | no | — |
| 148 | C3×C4 | C2×(C22⋊C9)⋊C2 | 1 | 1 | yes | characteristic |
| 149 | Dic3 | — | 36 | 1 | no | — |
| 150 | Dic3 | — | 36 | 1 | no | — |
| 151 | C9 | #5414 | 1 | 1 | yes | characteristic |
| 152 | C23 | — | 4 | 1 | no | — |
| 153 | C23 | — | 6 | 1 | no | — |
| 154 | C23 | — | 6 | 1 | no | — |
| 155 | C23 | — | 12 | 1 | no | — |
| 156 | C23 | #6171 | 1 | 1 | yes | characteristic |
| 157 | Q8 | — | 54 | 1 | no | — |
| 158 | Q8 | — | 54 | 1 | no | — |
| 159 | D4 | — | 3 | 2 | no | — |
| 160 | D4 | — | 3 | 1 | no | — |
| 161 | D4 | — | 6 | 1 | no | — |
| 162 | D4 | (C22⋊C27)⋊C2 | 1 | 1 | yes | characteristic |
| 163 | C2×C4 | — | 3 | 1 | no | — |
| 164 | C2×C4 | — | 3 | 1 | no | — |
| 165 | C2×C4 | — | 54 | 1 | no | — |
| 166 | C2×C4 | — | 54 | 1 | no | — |
| 167 | C8 | — | 54 | 1 | no | — |
| 168 | C8 | — | 54 | 1 | no | — |
| 169 | C2×C3 | — | 3 | 1 | no | — |
| 170 | C2×C3 | — | 3 | 1 | no | — |
| 171 | C2×C3 | — | 4 | 1 | no | — |
| 172 | C2×C3 | — | 12 | 1 | no | — |
| 173 | C2×C3 | #63694 | 1 | 1 | yes | characteristic |
| 174 | C22 | — | 2 | 1 | no | — |
| 175 | C22 | — | 3 | 1 | no | — |
| 176 | C22 | — | 3 | 1 | no | — |
| 177 | C22 | — | 6 | 1 | no | — |
| 178 | C22 | — | 12 | 1 | no | — |
| 179 | C22 | — | 12 | 1 | no | — |
| 180 | C22 | — | 12 | 1 | no | — |
| 181 | C22 | — | 12 | 1 | no | — |
| 182 | C22 | #88368 | 1 | 1 | yes | characteristic, complemented |
| 183 | C4 | — | 3 | 1 | no | — |
| 184 | C4 | — | 108 | 1 | no | — |
| 185 | C4 | — | 108 | 1 | no | — |
| 186 | C4 | C2×(C22⋊C27)⋊C2 | 1 | 1 | yes | characteristic |
| 187 | C3 | #10589516 | 1 | 1 | yes | characteristic |
| 188 | C2 | — | 3 | 1 | no | — |
| 189 | C2 | — | 3 | 1 | no | — |
| 190 | C2 | — | 4 | 1 | no | — |
| 191 | C2 | — | 12 | 1 | no | — |
| 192 | C2 | #11720654 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. (C22⋊C27)⋊Q8 | C2 | semidirect |
| 3. (C22⋊C27)⋊C8 | C2 | semidirect |
| 37. C22⋊C27 | SD16 | semidirect |
| 182. C22 | #88368 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| D4×C22⋊C27 | 2 |
| (C22⋊C27)⋊Q8 | 2 |
| (C22⋊C27)⋊C8 | 2 |
| #10588897 | 3 |
| #88368 | 4 |