Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
336 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 336 conjugacy classes of subgroups — too many to draw. Only its 58 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 | #89040 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #89018 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #88797 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | C3×#3719 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | C3×#3712 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #63945 | — | 3 | 1 | no | maximal |
| 7 | #63801 | — | 3 | 2 | no | maximal |
| 8 | C4⋊(C3⋊(C3⋊D3)) | C22 | 1 | 1 | yes | characteristic |
| 9 | C3×C4×C3⋊D3 | C22 | 1 | 1 | yes | characteristic |
| 10 | C3×#634 | C22 | 1 | 2 | yes | — |
| 11 | C32×Dic6 | C22 | 1 | 1 | yes | characteristic |
| 12 | #6296 | — | 2 | 1 | no | — |
| 13 | #6294 | — | 2 | 1 | no | — |
| 14 | #6292 | — | 2 | 1 | no | — |
| 15 | #6289 | — | 2 | 1 | no | — |
| 16 | C32×C3⋊C8 | C22 | 1 | 2 | yes | — |
| 17 | #3832 | — | 3 | 1 | no | — |
| 18 | #3799 | — | 3 | 2 | no | — |
| 19 | #3799 | — | 6 | 1 | no | — |
| 20 | #3797 | — | 3 | 1 | no | — |
| 21 | D3×Dic6 | — | 3 | 2 | no | — |
| 22 | #3744 | — | 3 | 2 | no | — |
| 23 | C3×#216 | — | 3 | 2 | no | — |
| 24 | C3×Dic12 | — | 3 | 2 | no | — |
| 25 | C3×ΓL(1,𝔽25) | — | 3 | 2 | no | — |
| 26 | #3719 | D3 | 1 | 2 | yes | complemented |
| 27 | #3717 | — | 3 | 2 | no | — |
| 28 | #3717 | — | 3 | 2 | no | — |
| 29 | #3717 | — | 6 | 2 | no | — |
| 30 | #3712 | D3 | 1 | 1 | yes | characteristic, complemented |
| 31 | C2×C3⋊(C3⋊D3) | — | 2 | 1 | no | — |
| 32 | C2×C3×C3⋊D3 | D4 | 1 | 1 | yes | characteristic |
| 33 | C33×C4 | C23 | 1 | 1 | yes | characteristic |
| 34 | C3×C3⋊Dic3 | — | 2 | 2 | no | — |
| 35 | C3×C3⋊Dic3 | D4 | 1 | 1 | yes | characteristic |
| 36 | C32×Dic3 | — | 2 | 1 | no | — |
| 37 | #918 | — | 9 | 2 | no | — |
| 38 | #909 | — | 9 | 1 | no | — |
| 39 | C32×Q8 | — | 3 | 1 | no | — |
| 40 | C4⋊(C3⋊D3) | — | 3 | 1 | no | — |
| 41 | C4⋊(C3⋊D3) | — | 3 | 2 | no | — |
| 42 | C4⋊(C3⋊D3) | — | 6 | 1 | no | — |
| 43 | C4⋊(C3⋊D3) | — | 6 | 2 | no | — |
| 44 | C4⋊(C3⋊D3) | — | 6 | 1 | no | — |
| 45 | C4⋊(C3⋊D3) | — | 12 | 1 | no | — |
| 46 | C4×C3⋊D3 | — | 6 | 1 | no | — |
| 47 | C4×C3⋊D3 | C2×D3 | 1 | 1 | yes | characteristic |
| 48 | #634 | C2×D3 | 1 | 2 | yes | — |
| 49 | C3×C4×D3 | — | 3 | 2 | no | — |
| 50 | C3×C4×D3 | — | 6 | 1 | no | — |
| 51 | C3×Dic6 | — | 2 | 1 | no | — |
| 52 | C3×Dic6 | — | 3 | 2 | no | — |
| 53 | C3×Dic6 | — | 3 | 2 | no | — |
| 54 | C3×Dic6 | — | 6 | 2 | no | — |
| 55 | C3×Dic6 | C2×D3 | 1 | 2 | yes | — |
| 56 | #627 | — | 6 | 2 | no | — |
| 57 | #627 | — | 6 | 1 | no | — |
| 58 | #626 | — | 6 | 2 | no | — |
| 59 | #626 | — | 6 | 2 | no | — |
| 60 | #624 | — | 6 | 2 | no | — |
| 61 | #624 | — | 6 | 2 | no | — |
| 62 | #624 | — | 6 | 1 | no | — |
| 63 | D3×Dic3 | — | 6 | 2 | no | — |
| 64 | C32×C8 | — | 3 | 2 | no | — |
| 65 | C3×C3⋊C8 | — | 2 | 2 | no | — |
| 66 | C3×C3⋊C8 | C2×D3 | 1 | 2 | yes | — |
| 67 | C3×C3⋊C8 | C2×D3 | 1 | 2 | yes | — |
| 68 | C2×C33 | C2×D4 | 1 | 1 | yes | characteristic |
| 69 | C3⋊(C3⋊D3) | — | 4 | 1 | no | — |
| 70 | C3×C3⋊D3 | — | 2 | 1 | no | — |
| 71 | #239 | — | 9 | 2 | no | — |
| 72 | #239 | — | 18 | 1 | no | — |
| 73 | D3×Q8 | — | 9 | 2 | no | — |
| 74 | #235 | — | 9 | 1 | no | — |
| 75 | C2×Dic6 | — | 9 | 1 | no | — |
| 76 | C3×Q16 | — | 9 | 2 | no | — |
| 77 | C3×M16 | — | 9 | 1 | no | — |
| 78 | #216 | — | 3 | 2 | no | — |
| 79 | #215 | — | 9 | 2 | no | — |
| 80 | Dic12 | — | 3 | 2 | no | — |
| 81 | #204 | — | 9 | 2 | no | — |
| 82 | #204 | — | 9 | 2 | no | — |
| 83 | #204 | — | 18 | 2 | no | — |
| 84 | ΓL(1,𝔽25) | — | 3 | 2 | no | — |
| 85 | C2×C3⋊D3 | — | 6 | 1 | no | — |
| 86 | C2×C3⋊D3 | — | 6 | 2 | no | — |
| 87 | C2×C3⋊D3 | — | 6 | 2 | no | — |
| 88 | C2×C3⋊D3 | — | 12 | 2 | no | — |
| 89 | C2×C3⋊D3 | — | 12 | 2 | no | — |
| 90 | C2×C3⋊D3 | D12 | 1 | 1 | yes | characteristic |
| 91 | C2×C3×D3 | — | 3 | 2 | no | — |
| 92 | C2×C3×D3 | — | 6 | 1 | no | — |
| 93 | C32×C4 | — | 2 | 2 | no | — |
| 94 | C32×C4 | — | 2 | 1 | no | — |
| 95 | C32×C4 | — | 4 | 1 | no | — |
| 96 | C32×C4 | — | 6 | 1 | no | — |
| 97 | C32×C4 | C22×D3 | 1 | 1 | yes | characteristic |
| 98 | C32×C4 | C22×D3 | 1 | 2 | yes | — |
| 99 | C3⋊Dic3 | — | 2 | 2 | no | — |
| 100 | C3⋊Dic3 | D12 | 1 | 1 | yes | characteristic |
| 101 | C3×Dic3 | — | 2 | 2 | no | — |
| 102 | C3×Dic3 | — | 2 | 2 | no | — |
| 103 | C3×Dic3 | — | 3 | 2 | no | — |
| 104 | C3×Dic3 | — | 6 | 1 | no | — |
| 105 | C3×Dic3 | — | 6 | 2 | no | — |
| 106 | C3×Dic3 | — | 6 | 2 | no | — |
| 107 | C3×Dic3 | — | 12 | 2 | no | — |
| 108 | #137 | — | 27 | 1 | no | Sylow |
| 109 | C33 | #137 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 110 | C3×Q8 | — | 3 | 2 | no | — |
| 111 | C3×Q8 | — | 6 | 1 | no | — |
| 112 | C3×Q8 | — | 9 | 2 | no | — |
| 113 | C2×C3×C4 | — | 9 | 1 | no | — |
| 114 | #67 | — | 18 | 1 | no | — |
| 115 | C2×Dic3 | — | 18 | 1 | no | — |
| 116 | D12 | — | 9 | 1 | no | — |
| 117 | D12 | — | 9 | 2 | no | — |
| 118 | D12 | — | 18 | 1 | no | — |
| 119 | D12 | — | 18 | 2 | no | — |
| 120 | D12 | — | 18 | 2 | no | — |
| 121 | D12 | — | 18 | 2 | no | — |
| 122 | D12 | — | 36 | 1 | no | — |
| 123 | C4×D3 | — | 3 | 2 | no | — |
| 124 | C4×D3 | — | 6 | 1 | no | — |
| 125 | C4×D3 | — | 18 | 1 | no | — |
| 126 | C4×D3 | — | 18 | 2 | no | — |
| 127 | C4×D3 | — | 18 | 2 | no | — |
| 128 | C4×D3 | — | 18 | 2 | no | — |
| 129 | Dic6 | — | 3 | 2 | no | — |
| 130 | Dic6 | — | 3 | 2 | no | — |
| 131 | Dic6 | — | 6 | 2 | no | — |
| 132 | Dic6 | — | 18 | 2 | no | — |
| 133 | Dic6 | — | 18 | 1 | no | — |
| 134 | Dic6 | D32 | 1 | 1 | yes | characteristic |
| 135 | C3×C8 | — | 3 | 2 | no | — |
| 136 | C3×C8 | — | 3 | 2 | no | — |
| 137 | C3×C8 | — | 6 | 2 | no | — |
| 138 | C3⋊C8 | D32 | 1 | 2 | yes | — |
| 139 | C2×C32 | — | 2 | 2 | no | — |
| 140 | C2×C32 | — | 2 | 1 | no | — |
| 141 | C2×C32 | — | 4 | 1 | no | — |
| 142 | C2×C32 | C2×D12 | 1 | 1 | yes | characteristic |
| 143 | C2×C32 | D3×D4 | 1 | 2 | yes | — |
| 144 | C3⋊D3 | — | 2 | 1 | no | — |
| 145 | C3⋊D3 | — | 12 | 1 | no | — |
| 146 | C3⋊D3 | — | 12 | 2 | no | — |
| 147 | C3⋊D3 | — | 12 | 2 | no | — |
| 148 | C3⋊D3 | — | 24 | 2 | no | — |
| 149 | C3⋊D3 | — | 24 | 2 | no | — |
| 150 | C3×D3 | — | 6 | 2 | no | — |
| 151 | C3×D3 | — | 12 | 1 | no | — |
| 152 | #41 | — | 27 | 1 | no | — |
| 153 | C2×Q8 | — | 27 | 1 | no | — |
| 154 | Q16 | — | 9 | 2 | no | — |
| 155 | SD16 | — | 27 | 2 | no | — |
| 156 | M16 | — | 9 | 1 | no | — |
| 157 | C22×C3 | — | 9 | 1 | no | — |
| 158 | C2×D3 | — | 3 | 2 | no | — |
| 159 | C2×D3 | — | 6 | 1 | no | — |
| 160 | C2×D3 | — | 18 | 2 | no | — |
| 161 | C2×D3 | — | 18 | 2 | no | — |
| 162 | C2×D3 | — | 18 | 1 | no | — |
| 163 | C2×D3 | — | 36 | 2 | no | — |
| 164 | C2×D3 | — | 36 | 2 | no | — |
| 165 | C3×C4 | — | 2 | 2 | no | — |
| 166 | C3×C4 | — | 2 | 1 | no | — |
| 167 | C3×C4 | — | 4 | 1 | no | — |
| 168 | C3×C4 | — | 6 | 2 | no | — |
| 169 | C3×C4 | — | 6 | 2 | no | — |
| 170 | C3×C4 | — | 9 | 1 | no | — |
| 171 | C3×C4 | — | 18 | 2 | no | — |
| 172 | C3×C4 | C2×D32 | 1 | 2 | yes | — |
| 173 | C3×C4 | C2×D32 | 1 | 1 | yes | characteristic |
| 174 | Dic3 | — | 2 | 1 | no | — |
| 175 | Dic3 | — | 3 | 2 | no | — |
| 176 | Dic3 | — | 6 | 1 | no | — |
| 177 | Dic3 | — | 6 | 2 | no | — |
| 178 | Dic3 | — | 6 | 2 | no | — |
| 179 | Dic3 | — | 12 | 2 | no | — |
| 180 | C32 | — | 2 | 2 | no | — |
| 181 | C32 | — | 2 | 1 | no | — |
| 182 | C32 | — | 4 | 1 | no | — |
| 183 | C32 | #909 | 1 | 1 | yes | characteristic, complemented |
| 184 | C32 | #918 | 1 | 2 | yes | complemented |
| 185 | Q8 | — | 3 | 1 | no | — |
| 186 | Q8 | — | 9 | 2 | no | — |
| 187 | Q8 | — | 54 | 1 | no | — |
| 188 | D4 | — | 27 | 1 | no | — |
| 189 | D4 | — | 54 | 1 | no | — |
| 190 | C2×C4 | — | 9 | 1 | no | — |
| 191 | C2×C4 | — | 54 | 1 | no | — |
| 192 | C2×C4 | — | 54 | 1 | no | — |
| 193 | C8 | — | 3 | 2 | no | — |
| 194 | C2×C3 | — | 2 | 1 | no | — |
| 195 | C2×C3 | — | 2 | 2 | no | — |
| 196 | C2×C3 | — | 4 | 1 | no | — |
| 197 | C2×C3 | — | 18 | 1 | no | — |
| 198 | C2×C3 | D3×D12 | 1 | 2 | yes | — |
| 199 | C2×C3 | #3811 | 1 | 1 | yes | characteristic |
| 200 | D3 | — | 6 | 2 | no | — |
| 201 | D3 | — | 12 | 1 | no | — |
| 202 | D3 | — | 36 | 2 | no | — |
| 203 | D3 | — | 36 | 2 | no | — |
| 204 | D3 | — | 36 | 1 | no | — |
| 205 | D3 | — | 72 | 2 | no | — |
| 206 | D3 | — | 72 | 2 | no | — |
| 207 | C22 | — | 9 | 1 | no | — |
| 208 | C22 | — | 54 | 1 | no | — |
| 209 | C4 | — | 6 | 1 | no | — |
| 210 | C4 | — | 9 | 1 | no | — |
| 211 | C4 | — | 18 | 2 | no | — |
| 212 | C4 | D33 | 1 | 1 | yes | characteristic |
| 213 | C3 | — | 2 | 1 | no | — |
| 214 | C3 | — | 2 | 2 | no | — |
| 215 | C3 | — | 4 | 1 | no | — |
| 216 | C3 | #63801 | 1 | 2 | yes | complemented |
| 217 | C3 | #63945 | 1 | 1 | yes | characteristic, complemented |
| 218 | C2 | — | 18 | 1 | no | — |
| 219 | C2 | — | 108 | 1 | no | — |
| 220 | C2 | #88955 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #88955 |
| commutator, Fitting, cosocle | C33×C4 | C23 |
| Frattini, hypercenter | C4 | D33 |
| socle | C2×C33 | C2×D4 |
| prime radical | #89018 | C2 |
| hypocenter | C33 | #137 |
| LCS | G ▷ C33×C4 ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2 ◁ C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #89040 | C2 | semidirect |
| 3. #88797 | C2 | semidirect |
| 4. C3×#3719 | C2 | semidirect |
| 5. C3×#3712 | C2 | semidirect |
| 26. #3719 | D3 | semidirect |
| 30. #3712 | D3 | semidirect |
| 109. C33 | #137 | semidirect |
| 183. C32 | #909 | semidirect |
| 184. C32 | #918 | semidirect |
| 216. C3 | #63801 | semidirect |
| 217. C3 | #63945 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #89040 | 2 |
| #89018 | 2 |
| #88797 | 2 |
| C3×#3719 | 2 |
| C3×#3712 | 2 |
| #63945 | 3 |
| #63801 | 3 |