Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
230 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 230 conjugacy classes of subgroups — too many to draw. Only its 35 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 | C2×#6296 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C3×#3805 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #88993 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #88938 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #88937 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #64239 | — | 3 | 1 | no | maximal |
| 7 | C2×C3×C3⋊Dic3 | — | 2 | 1 | no | — |
| 8 | C2×C3×C3⋊Dic3 | C22 | 1 | 1 | yes | characteristic |
| 9 | C32×#67 | — | 2 | 1 | no | — |
| 10 | #6296 | — | 2 | 1 | no | — |
| 11 | #6296 | C22 | 1 | 2 | yes | — |
| 12 | C3×#627 | C22 | 1 | 1 | yes | characteristic |
| 13 | C3×#625 | C22 | 1 | 1 | yes | characteristic |
| 14 | C3×D3×Dic3 | — | 2 | 1 | no | — |
| 15 | C3⋊#622 | C22 | 1 | 2 | yes | — |
| 16 | C3×#237 | — | 6 | 1 | no | — |
| 17 | C2×#627 | — | 3 | 1 | no | — |
| 18 | C2×#627 | — | 6 | 1 | no | — |
| 19 | #3805 | D3 | 1 | 1 | yes | characteristic |
| 20 | #3792 | — | 3 | 1 | no | — |
| 21 | #3776 | — | 3 | 2 | no | — |
| 22 | #3775 | — | 3 | 2 | no | — |
| 23 | C22×C33 | D4 | 1 | 1 | yes | characteristic |
| 24 | C2×C32×D3 | — | 2 | 1 | no | — |
| 25 | C3×C3⋊Dic3 | — | 2 | 2 | no | — |
| 26 | C3×C3⋊Dic3 | D4 | 1 | 1 | yes | characteristic |
| 27 | C3×C3⋊Dic3 | C23 | 1 | 1 | yes | characteristic |
| 28 | C32×Dic3 | — | 2 | 1 | no | — |
| 29 | #951 | — | 9 | 1 | no | maximal |
| 30 | C32×D4 | — | 6 | 1 | no | — |
| 31 | C2×C3⋊Dic3 | — | 2 | 1 | no | — |
| 32 | C2×C3⋊Dic3 | C2×D3 | 1 | 1 | yes | characteristic |
| 33 | C3×#67 | — | 2 | 1 | no | — |
| 34 | C3×#67 | — | 2 | 1 | no | — |
| 35 | C3×#67 | — | 4 | 1 | no | — |
| 36 | C3×#67 | — | 6 | 1 | no | — |
| 37 | C2×C3×Dic3 | — | 6 | 1 | no | — |
| 38 | C2×C3×Dic3 | — | 6 | 1 | no | — |
| 39 | C2×C3×Dic3 | — | 6 | 1 | no | — |
| 40 | C2×C3×Dic3 | — | 6 | 1 | no | — |
| 41 | C2×C3×Dic3 | — | 6 | 1 | no | — |
| 42 | C2×C3×Dic3 | — | 12 | 1 | no | — |
| 43 | C3×C4×D3 | — | 6 | 1 | no | — |
| 44 | C3×Dic6 | — | 6 | 1 | no | — |
| 45 | #627 | — | 3 | 2 | no | — |
| 46 | #627 | — | 6 | 2 | no | — |
| 47 | #627 | — | 6 | 2 | no | — |
| 48 | #627 | — | 6 | 1 | no | — |
| 49 | #627 | C2×D3 | 1 | 1 | yes | characteristic |
| 50 | #625 | C2×D3 | 1 | 1 | yes | characteristic |
| 51 | D3×Dic3 | — | 2 | 1 | no | — |
| 52 | #622 | — | 3 | 2 | no | — |
| 53 | C2×C33 | — | 2 | 1 | no | — |
| 54 | C2×C33 | C2×D4 | 1 | 1 | yes | characteristic |
| 55 | C32×D3 | — | 4 | 1 | no | — |
| 56 | C3×#41 | — | 9 | 1 | no | — |
| 57 | #237 | — | 6 | 1 | no | — |
| 58 | C2×Dic6 | — | 9 | 1 | no | — |
| 59 | C2×Dic6 | — | 18 | 1 | no | — |
| 60 | #216 | — | 9 | 2 | no | — |
| 61 | #214 | — | 9 | 2 | no | — |
| 62 | #208 | — | 9 | 1 | no | — |
| 63 | C22×C32 | — | 2 | 1 | no | — |
| 64 | C22×C32 | — | 2 | 1 | no | — |
| 65 | C22×C32 | — | 4 | 1 | no | — |
| 66 | C22×C32 | — | 4 | 1 | no | — |
| 67 | C22×C32 | — | 6 | 1 | no | — |
| 68 | C22×C32 | #67 | 1 | 1 | yes | characteristic |
| 69 | C2×C3×D3 | — | 2 | 1 | no | — |
| 70 | C2×C3×D3 | — | 2 | 1 | no | — |
| 71 | C2×C3×D3 | — | 4 | 1 | no | — |
| 72 | C32×C4 | — | 6 | 1 | no | — |
| 73 | C3⋊Dic3 | — | 2 | 2 | no | — |
| 74 | C3⋊Dic3 | #67 | 1 | 1 | yes | characteristic |
| 75 | C3⋊Dic3 | C22×D3 | 1 | 1 | yes | characteristic |
| 76 | C3×Dic3 | — | 2 | 1 | no | — |
| 77 | C3×Dic3 | — | 2 | 1 | no | — |
| 78 | C3×Dic3 | — | 4 | 1 | no | — |
| 79 | C3×Dic3 | — | 6 | 1 | no | — |
| 80 | C3×Dic3 | — | 6 | 1 | no | — |
| 81 | C3×Dic3 | — | 6 | 1 | no | — |
| 82 | C3×Dic3 | — | 6 | 1 | no | — |
| 83 | C3×Dic3 | — | 6 | 2 | no | — |
| 84 | C3×Dic3 | — | 6 | 2 | no | — |
| 85 | C3×Dic3 | — | 12 | 2 | no | — |
| 86 | #137 | — | 27 | 1 | no | Sylow |
| 87 | C33 | #137 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 88 | C3×Q8 | — | 9 | 1 | no | — |
| 89 | C3×D4 | — | 6 | 1 | no | — |
| 90 | C3×D4 | — | 6 | 1 | no | — |
| 91 | C3×D4 | — | 9 | 1 | no | — |
| 92 | C3×D4 | — | 12 | 1 | no | — |
| 93 | C2×C3×C4 | — | 9 | 1 | no | — |
| 94 | C2×C3×C4 | — | 18 | 1 | no | — |
| 95 | C2×C3×C4 | — | 18 | 1 | no | — |
| 96 | #67 | — | 2 | 1 | no | — |
| 97 | #67 | — | 6 | 1 | no | — |
| 98 | C2×Dic3 | — | 6 | 1 | no | — |
| 99 | C2×Dic3 | — | 6 | 1 | no | — |
| 100 | C2×Dic3 | — | 6 | 1 | no | — |
| 101 | C2×Dic3 | — | 6 | 1 | no | — |
| 102 | C2×Dic3 | — | 6 | 1 | no | — |
| 103 | C2×Dic3 | — | 12 | 1 | no | — |
| 104 | C4×D3 | — | 6 | 1 | no | — |
| 105 | Dic6 | — | 6 | 1 | no | — |
| 106 | Dic6 | — | 9 | 2 | no | — |
| 107 | Dic6 | — | 18 | 2 | no | — |
| 108 | Dic6 | — | 18 | 1 | no | — |
| 109 | Dic6 | — | 18 | 2 | no | — |
| 110 | C3⋊C8 | — | 9 | 2 | no | — |
| 111 | C2×C32 | — | 2 | 2 | no | — |
| 112 | C2×C32 | — | 2 | 1 | no | — |
| 113 | C2×C32 | — | 2 | 1 | no | — |
| 114 | C2×C32 | — | 2 | 1 | no | — |
| 115 | C2×C32 | — | 4 | 1 | no | — |
| 116 | C2×C32 | — | 4 | 1 | no | — |
| 117 | C2×C32 | — | 4 | 1 | no | — |
| 118 | C2×C32 | — | 4 | 2 | no | — |
| 119 | C2×C32 | — | 8 | 1 | no | — |
| 120 | C2×C32 | — | 12 | 1 | no | — |
| 121 | C2×C32 | C2×#67 | 1 | 1 | yes | characteristic |
| 122 | C3×D3 | — | 4 | 1 | no | — |
| 123 | C3×D3 | — | 4 | 1 | no | — |
| 124 | C3×D3 | — | 8 | 1 | no | — |
| 125 | #41 | — | 9 | 1 | no | — |
| 126 | C2×Q8 | — | 27 | 1 | no | — |
| 127 | Q16 | — | 27 | 2 | no | — |
| 128 | SD16 | — | 27 | 2 | no | — |
| 129 | M16 | — | 27 | 1 | no | — |
| 130 | C22×C3 | — | 2 | 1 | no | — |
| 131 | C22×C3 | — | 2 | 1 | no | — |
| 132 | C22×C3 | — | 4 | 1 | no | — |
| 133 | C22×C3 | — | 4 | 1 | no | — |
| 134 | C22×C3 | — | 6 | 1 | no | — |
| 135 | C22×C3 | — | 6 | 1 | no | — |
| 136 | C22×C3 | — | 12 | 1 | no | — |
| 137 | C22×C3 | GO+(4,𝔽2) | 1 | 1 | yes | characteristic |
| 138 | C2×D3 | — | 2 | 1 | no | — |
| 139 | C3×C4 | — | 6 | 1 | no | — |
| 140 | C3×C4 | — | 6 | 1 | no | — |
| 141 | C3×C4 | — | 9 | 1 | no | — |
| 142 | C3×C4 | — | 9 | 1 | no | — |
| 143 | C3×C4 | — | 12 | 1 | no | — |
| 144 | C3×C4 | — | 18 | 2 | no | — |
| 145 | Dic3 | — | 2 | 1 | no | — |
| 146 | Dic3 | — | 6 | 1 | no | — |
| 147 | Dic3 | — | 6 | 1 | no | — |
| 148 | Dic3 | — | 6 | 1 | no | — |
| 149 | Dic3 | — | 6 | 1 | no | — |
| 150 | Dic3 | — | 6 | 2 | no | — |
| 151 | Dic3 | — | 6 | 2 | no | — |
| 152 | Dic3 | — | 12 | 2 | no | — |
| 153 | C32 | — | 2 | 1 | no | — |
| 154 | C32 | — | 2 | 1 | no | — |
| 155 | C32 | — | 4 | 1 | no | — |
| 156 | C32 | — | 4 | 1 | no | — |
| 157 | C32 | #951 | 1 | 1 | yes | characteristic, complemented |
| 158 | Q8 | — | 9 | 1 | no | — |
| 159 | Q8 | — | 27 | 2 | no | — |
| 160 | Q8 | — | 54 | 1 | no | — |
| 161 | D4 | — | 6 | 1 | no | — |
| 162 | D4 | — | 9 | 1 | no | — |
| 163 | C2×C4 | — | 9 | 1 | no | — |
| 164 | C2×C4 | — | 18 | 1 | no | — |
| 165 | C2×C4 | — | 18 | 1 | no | — |
| 166 | C8 | — | 27 | 2 | no | — |
| 167 | C2×C3 | — | 2 | 2 | no | — |
| 168 | C2×C3 | — | 2 | 1 | no | — |
| 169 | C2×C3 | — | 2 | 1 | no | — |
| 170 | C2×C3 | — | 2 | 1 | no | — |
| 171 | C2×C3 | — | 4 | 1 | no | — |
| 172 | C2×C3 | — | 4 | 2 | no | — |
| 173 | C2×C3 | — | 4 | 1 | no | — |
| 174 | C2×C3 | — | 4 | 1 | no | — |
| 175 | C2×C3 | — | 8 | 1 | no | — |
| 176 | C2×C3 | — | 12 | 1 | no | — |
| 177 | C2×C3 | — | 12 | 1 | no | — |
| 178 | C2×C3 | — | 24 | 1 | no | — |
| 179 | C2×C3 | C2×GO+(4,𝔽2) | 1 | 1 | yes | characteristic |
| 180 | D3 | — | 4 | 1 | no | — |
| 181 | C22 | — | 6 | 1 | no | — |
| 182 | C22 | D32⋊D3 | 1 | 1 | yes | characteristic |
| 183 | C4 | — | 6 | 1 | no | — |
| 184 | C4 | — | 9 | 1 | no | — |
| 185 | C4 | — | 9 | 1 | no | — |
| 186 | C4 | — | 18 | 2 | no | — |
| 187 | C3 | — | 2 | 1 | no | — |
| 188 | C3 | — | 2 | 1 | no | — |
| 189 | C3 | — | 4 | 1 | no | — |
| 190 | C3 | — | 4 | 1 | no | — |
| 191 | C3 | #64239 | 1 | 1 | yes | characteristic, complemented |
| 192 | C2 | — | 2 | 1 | no | — |
| 193 | C2 | — | 12 | 1 | no | — |
| 194 | C2 | C2×D32⋊D3 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#6296 | C2 | semidirect |
| 3. #88993 | C2 | semidirect |
| 4. #88938 | C2 | semidirect |
| 5. #88937 | C2 | semidirect |
| 87. C33 | #137 | semidirect |
| 157. C32 | #951 | semidirect |
| 191. C3 | #64239 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#6296 | 2 |
| C3×#3805 | 2 |
| #88993 | 2 |
| #88938 | 2 |
| #88937 | 2 |
| #64239 | 3 |
| #951 | 9 |