Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
251 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 251 conjugacy classes of subgroups — too many to draw. Only its 32 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⋊#226) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C3×C3⋊GL(2,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #64264 | — | 3 | 1 | no | maximal |
| 5 | #64201 | — | 3 | 3 | no | maximal |
| 6 | #63946 | — | 3 | 1 | no | maximal |
| 7 | C32×SL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic |
| 8 | #6290 | — | 4 | 1 | no | maximal |
| 9 | C3×D3×Q8 | D3 | 1 | 1 | yes | characteristic, complemented |
| 10 | C2×C3×SL(2,𝔽3) | — | 3 | 1 | no | — |
| 11 | #3795 | — | 3 | 1 | no | — |
| 12 | D3×SL(2,𝔽3) | D3 | 1 | 3 | yes | complemented |
| 13 | C3⋊GL(2,𝔽3) | D3 | 1 | 1 | yes | characteristic, complemented |
| 14 | C3⋊#226 | — | 3 | 1 | no | — |
| 15 | C3⋊#226 | — | 3 | 3 | no | — |
| 16 | C3⋊#226 | — | 6 | 1 | no | — |
| 17 | C3⋊#226 | — | 6 | 3 | no | — |
| 18 | C3×GL(2,𝔽3) | — | 3 | 3 | no | — |
| 19 | #3756 | — | 3 | 1 | no | — |
| 20 | C3×#215 | — | 3 | 1 | no | — |
| 21 | #3719 | — | 3 | 1 | no | — |
| 22 | #3715 | — | 3 | 1 | no | — |
| 23 | #3711 | — | 3 | 1 | no | — |
| 24 | C2×C3×C3⋊D3 | — | 4 | 1 | no | — |
| 25 | C2×C32×D3 | — | 4 | 1 | no | — |
| 26 | C3⋊(C3⋊Dic3) | — | 4 | 1 | no | — |
| 27 | #983 | — | 9 | 3 | no | — |
| 28 | #942 | — | 9 | 1 | no | — |
| 29 | #915 | — | 9 | 1 | no | — |
| 30 | C32×Q8 | C2×D3 | 1 | 1 | yes | characteristic |
| 31 | #638 | — | 12 | 1 | no | — |
| 32 | #634 | — | 3 | 1 | no | — |
| 33 | C3×D12 | — | 3 | 1 | no | — |
| 34 | C3×C4×D3 | — | 3 | 1 | no | — |
| 35 | C3×Dic6 | — | 3 | 1 | no | — |
| 36 | C3×SL(2,𝔽3) | — | 2 | 1 | no | — |
| 37 | C3×SL(2,𝔽3) | — | 2 | 3 | no | — |
| 38 | C3×SL(2,𝔽3) | C2×D3 | 1 | 3 | yes | — |
| 39 | C3×SL(2,𝔽3) | C2×D3 | 1 | 1 | yes | characteristic |
| 40 | #625 | — | 6 | 1 | no | — |
| 41 | #625 | — | 12 | 3 | no | — |
| 42 | D3×Dic3 | — | 6 | 1 | no | — |
| 43 | C3⋊(C3⋊C8) | — | 3 | 1 | no | — |
| 44 | C3×C3⋊C8 | — | 3 | 1 | no | — |
| 45 | C2×C33 | — | 4 | 1 | no | — |
| 46 | C3×C3⋊D3 | — | 8 | 1 | no | — |
| 47 | C32×D3 | — | 8 | 1 | no | — |
| 48 | C2×C3×Q8 | — | 3 | 1 | no | — |
| 49 | D3×Q8 | C3⋊D3 | 1 | 1 | yes | characteristic, complemented |
| 50 | #237 | — | 9 | 1 | no | — |
| 51 | #235 | — | 9 | 1 | no | — |
| 52 | C2×SL(2,𝔽3) | — | 3 | 3 | no | — |
| 53 | GL(2,𝔽3) | — | 3 | 3 | no | — |
| 54 | #226 | — | 9 | 3 | no | — |
| 55 | #226 | — | 18 | 3 | no | — |
| 56 | C3×SD16 | — | 9 | 1 | no | — |
| 57 | #216 | — | 9 | 1 | no | — |
| 58 | #216 | — | 9 | 1 | no | — |
| 59 | #216 | — | 9 | 1 | no | — |
| 60 | #216 | — | 18 | 1 | no | — |
| 61 | #215 | — | 3 | 1 | no | — |
| 62 | #215 | — | 9 | 1 | no | — |
| 63 | #214 | — | 9 | 1 | no | — |
| 64 | #208 | — | 9 | 1 | no | — |
| 65 | Dic12 | — | 9 | 1 | no | — |
| 66 | ΓL(1,𝔽25) | — | 9 | 1 | no | — |
| 67 | C22×C32 | — | 12 | 1 | no | — |
| 68 | C2×C3⋊D3 | — | 4 | 1 | no | — |
| 69 | C2×C3×D3 | — | 4 | 3 | no | — |
| 70 | C2×C3×D3 | — | 6 | 1 | no | — |
| 71 | C2×C3×D3 | — | 12 | 3 | no | — |
| 72 | C2×C3×D3 | S4 | 1 | 1 | yes | characteristic |
| 73 | C32×C4 | — | 3 | 1 | no | — |
| 74 | C3⋊Dic3 | — | 6 | 1 | no | — |
| 75 | C3⋊Dic3 | — | 12 | 1 | no | — |
| 76 | C3⋊Dic3 | — | 12 | 3 | no | — |
| 77 | C3⋊Dic3 | — | 24 | 1 | no | — |
| 78 | C3⋊Dic3 | — | 24 | 3 | no | — |
| 79 | C3×Dic3 | — | 3 | 1 | no | — |
| 80 | #137 | — | 27 | 1 | no | Sylow |
| 81 | C33 | — | 4 | 1 | no | Sylow |
| 82 | C3×Q8 | — | 2 | 1 | no | — |
| 83 | C3×Q8 | — | 9 | 1 | no | — |
| 84 | C3×Q8 | D32 | 1 | 1 | yes | characteristic |
| 85 | C3×Q8 | C2×C3⋊D3 | 1 | 1 | yes | characteristic |
| 86 | C3×D4 | — | 9 | 1 | no | — |
| 87 | C2×C3×C4 | — | 9 | 1 | no | — |
| 88 | #67 | — | 18 | 1 | no | — |
| 89 | #67 | — | 18 | 1 | no | — |
| 90 | #67 | — | 36 | 3 | no | — |
| 91 | C2×Dic3 | — | 18 | 1 | no | — |
| 92 | D12 | — | 3 | 1 | no | — |
| 93 | C4×D3 | — | 3 | 1 | no | — |
| 94 | C4×D3 | — | 18 | 1 | no | — |
| 95 | Dic6 | — | 3 | 1 | no | — |
| 96 | Dic6 | — | 9 | 1 | no | — |
| 97 | Dic6 | — | 9 | 1 | no | — |
| 98 | Dic6 | — | 18 | 1 | no | — |
| 99 | SL(2,𝔽3) | — | 2 | 3 | no | — |
| 100 | SL(2,𝔽3) | D32 | 1 | 3 | yes | — |
| 101 | C3×C8 | — | 9 | 1 | no | — |
| 102 | C3⋊C8 | — | 3 | 1 | no | — |
| 103 | C3⋊C8 | — | 9 | 1 | no | — |
| 104 | C3⋊C8 | — | 9 | 1 | no | — |
| 105 | C3⋊C8 | — | 18 | 1 | no | — |
| 106 | C2×C32 | — | 4 | 1 | no | — |
| 107 | C2×C32 | — | 4 | 3 | no | — |
| 108 | C2×C32 | — | 8 | 3 | no | — |
| 109 | C2×C32 | — | 8 | 1 | no | — |
| 110 | C2×C32 | — | 24 | 1 | no | — |
| 111 | C2×C32 | C2×S4 | 1 | 1 | yes | characteristic |
| 112 | C3⋊D3 | — | 8 | 1 | no | — |
| 113 | C3×D3 | — | 2 | 1 | no | — |
| 114 | C3×D3 | — | 8 | 3 | no | — |
| 115 | C3×D3 | — | 12 | 1 | no | — |
| 116 | C3×D3 | — | 24 | 3 | no | — |
| 117 | #41 | — | 27 | 1 | no | — |
| 118 | C2×Q8 | — | 3 | 1 | no | — |
| 119 | Q16 | — | 27 | 1 | no | — |
| 120 | Q16 | — | 27 | 1 | no | — |
| 121 | SD16 | — | 9 | 1 | no | — |
| 122 | SD16 | — | 27 | 1 | no | — |
| 123 | M16 | — | 27 | 1 | no | — |
| 124 | C22×C3 | — | 3 | 1 | no | — |
| 125 | C22×C3 | — | 12 | 3 | no | — |
| 126 | C22×C3 | — | 18 | 1 | no | — |
| 127 | C2×D3 | — | 6 | 1 | no | — |
| 128 | C2×D3 | — | 12 | 3 | no | — |
| 129 | C2×D3 | C3⋊S4 | 1 | 1 | yes | characteristic |
| 130 | C3×C4 | — | 3 | 1 | no | — |
| 131 | C3×C4 | — | 3 | 1 | no | — |
| 132 | C3×C4 | — | 6 | 1 | no | — |
| 133 | C3×C4 | — | 9 | 1 | no | — |
| 134 | Dic3 | — | 3 | 1 | no | — |
| 135 | Dic3 | — | 18 | 1 | no | — |
| 136 | Dic3 | — | 18 | 1 | no | — |
| 137 | Dic3 | — | 36 | 1 | no | — |
| 138 | Dic3 | — | 36 | 3 | no | — |
| 139 | Dic3 | — | 72 | 3 | no | — |
| 140 | C32 | — | 4 | 1 | no | — |
| 141 | C32 | — | 4 | 3 | no | — |
| 142 | C32 | — | 8 | 3 | no | — |
| 143 | C32 | — | 8 | 1 | no | — |
| 144 | C32 | #983 | 1 | 1 | yes | characteristic, complemented |
| 145 | Q8 | — | 9 | 1 | no | — |
| 146 | Q8 | — | 27 | 1 | no | — |
| 147 | Q8 | D3×C3⋊D3 | 1 | 1 | yes | characteristic |
| 148 | D4 | — | 9 | 1 | no | — |
| 149 | D4 | — | 54 | 1 | no | — |
| 150 | C2×C4 | — | 9 | 1 | no | — |
| 151 | C2×C4 | — | 54 | 1 | no | — |
| 152 | C8 | — | 9 | 1 | no | — |
| 153 | C8 | — | 27 | 1 | no | — |
| 154 | C2×C3 | — | 2 | 1 | no | — |
| 155 | C2×C3 | — | 4 | 3 | no | — |
| 156 | C2×C3 | — | 6 | 1 | no | — |
| 157 | C2×C3 | — | 8 | 3 | no | — |
| 158 | C2×C3 | — | 24 | 3 | no | — |
| 159 | C2×C3 | — | 36 | 1 | no | — |
| 160 | C2×C3 | D3×S4 | 1 | 1 | yes | characteristic |
| 161 | C2×C3 | C2×C3⋊S4 | 1 | 1 | yes | characteristic |
| 162 | D3 | — | 2 | 1 | no | — |
| 163 | D3 | — | 12 | 1 | no | — |
| 164 | D3 | — | 24 | 3 | no | — |
| 165 | C22 | — | 3 | 1 | no | — |
| 166 | C22 | — | 18 | 1 | no | — |
| 167 | C4 | — | 3 | 1 | no | — |
| 168 | C4 | — | 9 | 1 | no | — |
| 169 | C4 | — | 54 | 1 | no | — |
| 170 | C3 | — | 2 | 1 | no | — |
| 171 | C3 | — | 4 | 3 | no | — |
| 172 | C3 | — | 8 | 3 | no | — |
| 173 | C3 | #64201 | 1 | 1 | yes | characteristic, complemented |
| 174 | C3 | #64264 | 1 | 1 | yes | characteristic, complemented |
| 175 | C2 | — | 6 | 1 | no | — |
| 176 | C2 | — | 36 | 1 | no | — |
| 177 | C2 | D3×C3⋊S4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D3×C3⋊S4 |
| 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⋊#226) | C2 | semidirect |
| 3. C3×C3⋊GL(2,𝔽3) | C2 | semidirect |
| 9. C3×D3×Q8 | D3 | semidirect |
| 12. D3×SL(2,𝔽3) | D3 | semidirect |
| 13. C3⋊GL(2,𝔽3) | D3 | semidirect |
| 49. D3×Q8 | C3⋊D3 | semidirect |
| 144. C32 | #983 | semidirect |
| 173. C3 | #64201 | semidirect |
| 174. C3 | #64264 | 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⋊#226) | 2 |
| C3×C3⋊GL(2,𝔽3) | 2 |
| #64264 | 3 |
| #64201 | 3 |
| #63946 | 3 |
| #6290 | 4 |