Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
204 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 204 conjugacy classes of subgroups — too many to draw. Only its 24 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 | C3⋊D3×SL(2,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C3×C3⋊GL(2,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C3×C3⋊#226 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #64202 | — | 3 | 1 | no | maximal |
| 5 | #64201 | — | 3 | 1 | no | maximal |
| 6 | #63951 | — | 3 | 1 | no | maximal |
| 7 | C32×SL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic |
| 8 | #6295 | — | 4 | 1 | no | maximal |
| 9 | Q8×C3⋊D3 | D3 | 1 | 1 | yes | characteristic, complemented |
| 10 | #3796 | — | 3 | 1 | no | — |
| 11 | D3×SL(2,𝔽3) | — | 3 | 1 | no | — |
| 12 | D3×SL(2,𝔽3) | — | 3 | 1 | no | — |
| 13 | D3×SL(2,𝔽3) | — | 6 | 1 | no | — |
| 14 | C3⋊GL(2,𝔽3) | D3 | 1 | 1 | yes | characteristic, complemented |
| 15 | C3⋊#226 | D3 | 1 | 1 | yes | characteristic, complemented |
| 16 | C3×GL(2,𝔽3) | — | 3 | 1 | no | — |
| 17 | C3×GL(2,𝔽3) | — | 6 | 1 | no | — |
| 18 | C3×#226 | — | 3 | 1 | no | — |
| 19 | C3×#226 | — | 6 | 1 | no | — |
| 20 | C3×#216 | — | 3 | 1 | no | — |
| 21 | C3×#215 | — | 3 | 1 | no | — |
| 22 | #3719 | — | 3 | 1 | no | — |
| 23 | #3716 | — | 3 | 1 | no | — |
| 24 | #3712 | — | 3 | 1 | no | — |
| 25 | C2×C3×C3⋊D3 | — | 4 | 1 | no | — |
| 26 | C2×C3×C3⋊D3 | — | 4 | 1 | no | — |
| 27 | C3×C3⋊Dic3 | — | 4 | 1 | no | — |
| 28 | #983 | — | 9 | 1 | no | — |
| 29 | #918 | — | 9 | 1 | no | — |
| 30 | #915 | — | 9 | 1 | no | — |
| 31 | C32×Q8 | C2×D3 | 1 | 1 | yes | characteristic |
| 32 | C4×C3⋊D3 | — | 3 | 1 | no | — |
| 33 | #634 | — | 3 | 1 | no | — |
| 34 | C3×D12 | — | 3 | 1 | no | — |
| 35 | C3×Dic6 | — | 3 | 1 | no | — |
| 36 | C3×SL(2,𝔽3) | — | 2 | 1 | no | — |
| 37 | C3×SL(2,𝔽3) | — | 2 | 1 | no | — |
| 38 | C3×SL(2,𝔽3) | — | 2 | 1 | no | — |
| 39 | C3×SL(2,𝔽3) | — | 4 | 1 | no | — |
| 40 | C3×SL(2,𝔽3) | C2×D3 | 1 | 1 | yes | characteristic |
| 41 | C3×SL(2,𝔽3) | C2×D3 | 1 | 1 | yes | characteristic |
| 42 | #626 | — | 6 | 1 | no | — |
| 43 | #626 | — | 12 | 1 | no | — |
| 44 | #625 | — | 12 | 1 | no | — |
| 45 | D3×Dic3 | — | 6 | 1 | no | — |
| 46 | C3×C3⋊C8 | — | 3 | 1 | no | — |
| 47 | C3×C3⋊C8 | — | 3 | 1 | no | — |
| 48 | C2×C33 | — | 4 | 1 | no | — |
| 49 | C3×C3⋊D3 | — | 8 | 1 | no | — |
| 50 | C3×C3⋊D3 | — | 8 | 1 | no | — |
| 51 | #239 | — | 9 | 1 | no | — |
| 52 | D3×Q8 | — | 3 | 1 | no | — |
| 53 | D3×Q8 | — | 3 | 1 | no | — |
| 54 | D3×Q8 | — | 6 | 1 | no | — |
| 55 | #237 | — | 9 | 1 | no | — |
| 56 | C2×SL(2,𝔽3) | — | 9 | 1 | no | — |
| 57 | GL(2,𝔽3) | — | 3 | 1 | no | — |
| 58 | GL(2,𝔽3) | — | 6 | 1 | no | — |
| 59 | #226 | — | 3 | 1 | no | — |
| 60 | #226 | — | 6 | 1 | no | — |
| 61 | C3×Q16 | — | 9 | 1 | no | — |
| 62 | C3×SD16 | — | 9 | 1 | no | — |
| 63 | #216 | — | 3 | 1 | no | — |
| 64 | #216 | — | 9 | 1 | no | — |
| 65 | #215 | — | 3 | 1 | no | — |
| 66 | #214 | — | 9 | 1 | no | — |
| 67 | Dic12 | — | 9 | 1 | no | — |
| 68 | #204 | — | 9 | 1 | no | — |
| 69 | ΓL(1,𝔽25) | — | 9 | 1 | no | — |
| 70 | ΓL(1,𝔽25) | — | 9 | 1 | no | — |
| 71 | C2×C3⋊D3 | — | 4 | 1 | no | — |
| 72 | C2×C3⋊D3 | S4 | 1 | 1 | yes | characteristic |
| 73 | C2×C3×D3 | — | 6 | 1 | no | — |
| 74 | C2×C3×D3 | — | 12 | 1 | no | — |
| 75 | C2×C3×D3 | — | 12 | 1 | no | — |
| 76 | C2×C3×D3 | — | 12 | 1 | no | — |
| 77 | C2×C3×D3 | — | 24 | 1 | no | — |
| 78 | C2×C3×D3 | — | 24 | 1 | no | — |
| 79 | C32×C4 | — | 3 | 1 | no | — |
| 80 | C3⋊Dic3 | — | 3 | 1 | no | — |
| 81 | C3⋊Dic3 | — | 4 | 1 | no | — |
| 82 | C3×Dic3 | — | 6 | 1 | no | — |
| 83 | C3×Dic3 | — | 12 | 1 | no | — |
| 84 | C3×Dic3 | — | 24 | 1 | no | — |
| 85 | #137 | — | 27 | 1 | no | Sylow |
| 86 | C33 | — | 4 | 1 | no | Sylow |
| 87 | C3×Q8 | — | 2 | 1 | no | — |
| 88 | C3×Q8 | — | 9 | 1 | no | — |
| 89 | C3×Q8 | D32 | 1 | 1 | yes | characteristic |
| 90 | C3×Q8 | D32 | 1 | 1 | yes | characteristic |
| 91 | C3×D4 | — | 9 | 1 | no | — |
| 92 | #67 | — | 18 | 1 | no | — |
| 93 | #67 | — | 36 | 1 | no | — |
| 94 | C2×Dic3 | — | 18 | 1 | no | — |
| 95 | D12 | — | 3 | 1 | no | — |
| 96 | D12 | — | 18 | 1 | no | — |
| 97 | C4×D3 | — | 9 | 1 | no | — |
| 98 | C4×D3 | — | 9 | 1 | no | — |
| 99 | C4×D3 | — | 18 | 1 | no | — |
| 100 | C4×D3 | — | 18 | 1 | no | — |
| 101 | Dic6 | — | 3 | 1 | no | — |
| 102 | Dic6 | — | 9 | 1 | no | — |
| 103 | Dic6 | — | 9 | 1 | no | — |
| 104 | Dic6 | — | 18 | 1 | no | — |
| 105 | SL(2,𝔽3) | — | 2 | 1 | no | — |
| 106 | SL(2,𝔽3) | — | 2 | 1 | no | — |
| 107 | SL(2,𝔽3) | — | 4 | 1 | no | — |
| 108 | SL(2,𝔽3) | D32 | 1 | 1 | yes | characteristic |
| 109 | C3×C8 | — | 9 | 1 | no | — |
| 110 | C3×C8 | — | 9 | 1 | no | — |
| 111 | C3⋊C8 | — | 3 | 1 | no | — |
| 112 | C3⋊C8 | — | 3 | 1 | no | — |
| 113 | C2×C32 | — | 4 | 1 | no | — |
| 114 | C2×C32 | — | 4 | 1 | no | — |
| 115 | C2×C32 | — | 8 | 1 | no | — |
| 116 | C2×C32 | — | 8 | 1 | no | — |
| 117 | C2×C32 | — | 8 | 1 | no | — |
| 118 | C2×C32 | — | 16 | 1 | no | — |
| 119 | C2×C32 | C2×S4 | 1 | 1 | yes | characteristic |
| 120 | C3⋊D3 | — | 2 | 1 | no | — |
| 121 | C3⋊D3 | — | 8 | 1 | no | — |
| 122 | C3×D3 | — | 12 | 1 | no | — |
| 123 | C3×D3 | — | 24 | 1 | no | — |
| 124 | C3×D3 | — | 24 | 1 | no | — |
| 125 | C3×D3 | — | 24 | 2 | no | — |
| 126 | C3×D3 | — | 24 | 1 | no | — |
| 127 | C3×D3 | — | 24 | 2 | no | — |
| 128 | #41 | — | 27 | 1 | no | — |
| 129 | C2×Q8 | — | 9 | 1 | no | — |
| 130 | Q16 | — | 9 | 1 | no | — |
| 131 | Q16 | — | 27 | 1 | no | — |
| 132 | SD16 | — | 9 | 1 | no | — |
| 133 | SD16 | — | 27 | 1 | no | — |
| 134 | M16 | — | 27 | 1 | no | — |
| 135 | C22×C3 | — | 18 | 1 | no | — |
| 136 | C22×C3 | — | 36 | 1 | no | — |
| 137 | C2×D3 | — | 3 | 1 | no | — |
| 138 | C2×D3 | — | 3 | 1 | no | — |
| 139 | C2×D3 | — | 6 | 1 | no | — |
| 140 | C2×D3 | — | 6 | 1 | no | — |
| 141 | C2×D3 | — | 12 | 1 | no | — |
| 142 | C2×D3 | — | 24 | 1 | no | — |
| 143 | C3×C4 | — | 3 | 1 | no | — |
| 144 | C3×C4 | — | 3 | 1 | no | — |
| 145 | C3×C4 | — | 6 | 1 | no | — |
| 146 | C3×C4 | — | 18 | 1 | no | — |
| 147 | Dic3 | — | 6 | 1 | no | — |
| 148 | Dic3 | — | 9 | 1 | no | — |
| 149 | Dic3 | — | 9 | 1 | no | — |
| 150 | Dic3 | — | 12 | 1 | no | — |
| 151 | Dic3 | — | 18 | 1 | no | — |
| 152 | Dic3 | — | 24 | 1 | no | — |
| 153 | C32 | — | 4 | 1 | no | — |
| 154 | C32 | — | 4 | 1 | no | — |
| 155 | C32 | — | 8 | 1 | no | — |
| 156 | C32 | — | 8 | 1 | no | — |
| 157 | C32 | — | 8 | 1 | no | — |
| 158 | C32 | — | 16 | 1 | no | — |
| 159 | C32 | #983 | 1 | 1 | yes | characteristic, complemented |
| 160 | Q8 | — | 9 | 1 | no | — |
| 161 | Q8 | — | 27 | 1 | no | — |
| 162 | Q8 | #1113 | 1 | 1 | yes | characteristic |
| 163 | D4 | — | 9 | 1 | no | — |
| 164 | D4 | — | 54 | 1 | no | — |
| 165 | C2×C4 | — | 27 | 1 | no | — |
| 166 | C2×C4 | — | 54 | 1 | no | — |
| 167 | C8 | — | 9 | 1 | no | — |
| 168 | C8 | — | 9 | 1 | no | — |
| 169 | C2×C3 | — | 2 | 1 | no | — |
| 170 | C2×C3 | — | 4 | 1 | no | — |
| 171 | C2×C3 | — | 8 | 1 | no | — |
| 172 | C2×C3 | — | 8 | 1 | no | — |
| 173 | C2×C3 | — | 16 | 1 | no | — |
| 174 | C2×C3 | — | 36 | 1 | no | — |
| 175 | C2×C3 | — | 72 | 1 | no | — |
| 176 | C2×C3 | D3×S4 | 1 | 1 | yes | characteristic |
| 177 | C2×C3 | D3×S4 | 1 | 1 | yes | characteristic |
| 178 | D3 | — | 6 | 2 | no | — |
| 179 | D3 | — | 6 | 1 | no | — |
| 180 | D3 | — | 6 | 1 | no | — |
| 181 | D3 | — | 12 | 1 | no | — |
| 182 | D3 | — | 24 | 1 | no | — |
| 183 | D3 | — | 24 | 2 | no | — |
| 184 | C22 | — | 9 | 1 | no | — |
| 185 | C22 | — | 18 | 1 | no | — |
| 186 | C4 | — | 3 | 1 | no | — |
| 187 | C4 | — | 18 | 1 | no | — |
| 188 | C4 | — | 27 | 1 | no | — |
| 189 | C3 | — | 2 | 1 | no | — |
| 190 | C3 | — | 4 | 1 | no | — |
| 191 | C3 | — | 8 | 1 | no | — |
| 192 | C3 | — | 8 | 1 | no | — |
| 193 | C3 | — | 16 | 1 | no | — |
| 194 | C3 | #64201 | 1 | 1 | yes | characteristic, complemented |
| 195 | C3 | #64202 | 1 | 1 | yes | characteristic, complemented |
| 196 | C2 | — | 18 | 1 | no | — |
| 197 | C2 | — | 36 | 1 | no | — |
| 198 | C2 | #89101 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | #89101 |
| commutator, cosocle, hypocenter | C32×SL(2,𝔽3) | C22 |
| Fitting | C32×Q8 | C2×D3 |
| socle | C2×C32 | C2×S4 |
| LCS | G ▷ C32×SL(2,𝔽3) | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3⋊D3×SL(2,𝔽3) | C2 | semidirect |
| 2. C3×C3⋊GL(2,𝔽3) | C2 | semidirect |
| 3. C3×C3⋊#226 | C2 | semidirect |
| 9. Q8×C3⋊D3 | D3 | semidirect |
| 14. C3⋊GL(2,𝔽3) | D3 | semidirect |
| 15. C3⋊#226 | D3 | semidirect |
| 159. C32 | #983 | semidirect |
| 194. C3 | #64201 | semidirect |
| 195. C3 | #64202 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3⋊D3×SL(2,𝔽3) | 2 |
| C3×C3⋊GL(2,𝔽3) | 2 |
| C3×C3⋊#226 | 2 |
| #64202 | 3 |
| #64201 | 3 |
| #63951 | 3 |
| #6295 | 4 |