Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
248 conjugacy classes of subgroups; only the 27 normal subgroups are shown.
This group has 248 conjugacy classes of subgroups — too many to draw. Only its 27 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×A42 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C22×C3⋊S4 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | A4×S4 | — | 3 | 3 | no | maximal |
| 4 | A4×#67 | — | 3 | 1 | no | maximal |
| 5 | A4×C3⋊D3 | — | 4 | 1 | no | maximal |
| 6 | C3×C3⋊S4 | — | 4 | 1 | no | maximal |
| 7 | C3×C22⋊A4 | — | 2 | 1 | no | — |
| 8 | C22×C3×A4 | D3 | 1 | 1 | yes | characteristic, complemented |
| 9 | C22×C3×A4 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 10 | C2×D3×A4 | — | 3 | 1 | no | — |
| 11 | C2×C3⋊S4 | — | 3 | 1 | no | — |
| 12 | C2×C3⋊S4 | — | 3 | 1 | no | — |
| 13 | A42 | — | 2 | 3 | no | — |
| 14 | A42 | D3 | 1 | 3 | yes | complemented |
| 15 | Dic3×A4 | — | 3 | 1 | no | — |
| 16 | C32×A4 | — | 4 | 1 | no | — |
| 17 | C32×A4 | — | 4 | 1 | no | — |
| 18 | C22×S4 | — | 3 | 3 | no | — |
| 19 | C22×#67 | — | 3 | 1 | no | — |
| 20 | D4×A4 | — | 9 | 1 | no | — |
| 21 | C22×C3⋊D3 | — | 4 | 1 | no | — |
| 22 | C2×C3×A4 | — | 3 | 1 | no | — |
| 23 | C2×C3×A4 | — | 3 | 1 | no | — |
| 24 | D3×A4 | — | 6 | 1 | no | — |
| 25 | D3×A4 | — | 12 | 3 | no | — |
| 26 | C3⋊S4 | — | 3 | 1 | no | — |
| 27 | C3⋊S4 | A4 | 1 | 1 | yes | characteristic, direct product factor |
| 28 | C3×S4 | — | 12 | 3 | no | — |
| 29 | C3×#67 | — | 12 | 1 | no | — |
| 30 | C3×C3⋊D3 | — | 16 | 1 | no | — |
| 31 | C24×C3 | C3×D3 | 1 | 1 | yes | characteristic, complemented |
| 32 | C23×D3 | — | 3 | 1 | no | — |
| 33 | C22⋊A4 | — | 2 | 3 | no | — |
| 34 | C22×A4 | — | 2 | 1 | no | — |
| 35 | C22×A4 | — | 9 | 1 | no | — |
| 36 | C22×A4 | C3×D3 | 1 | 3 | yes | complemented |
| 37 | C22×A4 | C3⋊D3 | 1 | 1 | yes | characteristic, complemented |
| 38 | C2×S4 | — | 9 | 3 | no | — |
| 39 | C2×S4 | — | 9 | 3 | no | — |
| 40 | C2×#67 | — | 9 | 1 | no | — |
| 41 | C2×#67 | — | 9 | 1 | no | — |
| 42 | C2×#67 | — | 9 | 1 | no | — |
| 43 | C2×#67 | — | 9 | 1 | no | — |
| 44 | C22×Dic3 | — | 3 | 1 | no | — |
| 45 | C4×A4 | — | 9 | 1 | no | — |
| 46 | C22×C32 | — | 4 | 1 | no | — |
| 47 | C22×C32 | — | 4 | 1 | no | — |
| 48 | C2×C3⋊D3 | — | 12 | 1 | no | — |
| 49 | C2×C3⋊D3 | — | 12 | 1 | no | — |
| 50 | C2×C3×D3 | — | 12 | 1 | no | — |
| 51 | C3×A4 | — | 4 | 3 | no | — |
| 52 | C3×A4 | — | 4 | 3 | no | — |
| 53 | C3×A4 | — | 8 | 3 | no | — |
| 54 | C3×A4 | — | 8 | 1 | no | — |
| 55 | C3×A4 | — | 8 | 1 | no | — |
| 56 | C3×A4 | — | 8 | 3 | no | — |
| 57 | C3×A4 | — | 24 | 1 | no | — |
| 58 | C3×A4 | S4 | 1 | 1 | yes | characteristic, complemented |
| 59 | C3×A4 | C2×A4 | 1 | 1 | yes | characteristic, complemented |
| 60 | C3×Dic3 | — | 12 | 1 | no | — |
| 61 | C22×D4 | — | 9 | 1 | no | Sylow |
| 62 | C33 | — | 16 | 1 | no | Sylow |
| 63 | C23×C3 | — | 3 | 1 | no | — |
| 64 | C23×C3 | — | 3 | 1 | no | — |
| 65 | C23×C3 | — | 9 | 1 | no | — |
| 66 | C22×D3 | — | 6 | 1 | no | — |
| 67 | C22×D3 | — | 9 | 1 | no | — |
| 68 | C22×D3 | — | 9 | 1 | no | — |
| 69 | C22×D3 | — | 12 | 3 | no | — |
| 70 | C22×D3 | — | 18 | 1 | no | — |
| 71 | C2×A4 | — | 3 | 1 | no | — |
| 72 | C2×A4 | — | 3 | 3 | no | — |
| 73 | C2×A4 | — | 6 | 1 | no | — |
| 74 | C2×A4 | — | 18 | 1 | no | — |
| 75 | S4 | — | 3 | 3 | no | — |
| 76 | S4 | — | 9 | 3 | no | — |
| 77 | C3×D4 | — | 36 | 1 | no | — |
| 78 | #67 | — | 3 | 1 | no | — |
| 79 | #67 | — | 9 | 1 | no | — |
| 80 | #67 | — | 9 | 1 | no | — |
| 81 | #67 | — | 9 | 2 | no | — |
| 82 | #67 | — | 9 | 1 | no | — |
| 83 | C2×Dic3 | — | 9 | 1 | no | — |
| 84 | C2×Dic3 | — | 9 | 1 | no | — |
| 85 | C2×C32 | — | 12 | 1 | no | — |
| 86 | C2×C32 | — | 12 | 1 | no | — |
| 87 | C3⋊D3 | — | 4 | 1 | no | — |
| 88 | C3⋊D3 | — | 12 | 1 | no | — |
| 89 | C3×D3 | — | 24 | 1 | no | — |
| 90 | C3×D3 | — | 48 | 3 | no | — |
| 91 | C24 | — | 9 | 1 | no | — |
| 92 | C24 | C3×C3⋊D3 | 1 | 1 | yes | characteristic, complemented |
| 93 | C2×D4 | — | 27 | 1 | no | — |
| 94 | C2×D4 | — | 27 | 1 | no | — |
| 95 | C2×D4 | — | 27 | 1 | no | — |
| 96 | C2×D4 | — | 27 | 1 | no | — |
| 97 | C22×C4 | — | 9 | 1 | no | — |
| 98 | C22×C3 | — | 4 | 3 | no | — |
| 99 | C22×C3 | — | 4 | 1 | no | — |
| 100 | C22×C3 | — | 6 | 1 | no | — |
| 101 | C22×C3 | — | 8 | 1 | no | — |
| 102 | C22×C3 | — | 9 | 1 | no | — |
| 103 | C22×C3 | — | 9 | 1 | no | — |
| 104 | C22×C3 | — | 9 | 1 | no | — |
| 105 | C22×C3 | — | 36 | 1 | no | — |
| 106 | C22×C3 | C3×S4 | 1 | 1 | yes | characteristic, complemented |
| 107 | C22×C3 | D3×A4 | 1 | 1 | yes | characteristic, complemented |
| 108 | C2×D3 | — | 3 | 1 | no | — |
| 109 | C2×D3 | — | 9 | 1 | no | — |
| 110 | C2×D3 | — | 18 | 1 | no | — |
| 111 | C2×D3 | — | 18 | 1 | no | — |
| 112 | C2×D3 | — | 18 | 1 | no | — |
| 113 | C2×D3 | — | 18 | 1 | no | — |
| 114 | C2×D3 | — | 36 | 3 | no | — |
| 115 | C2×D3 | — | 36 | 3 | no | — |
| 116 | A4 | — | 2 | 1 | no | — |
| 117 | A4 | — | 8 | 3 | no | — |
| 118 | A4 | — | 8 | 3 | no | — |
| 119 | A4 | — | 24 | 3 | no | — |
| 120 | A4 | C3⋊S4 | 1 | 1 | yes | characteristic, direct product factor |
| 121 | A4 | D3×A4 | 1 | 3 | yes | complemented |
| 122 | C3×C4 | — | 36 | 1 | no | — |
| 123 | Dic3 | — | 3 | 1 | no | — |
| 124 | Dic3 | — | 9 | 1 | no | — |
| 125 | C32 | — | 4 | 1 | no | — |
| 126 | C32 | — | 4 | 1 | no | — |
| 127 | C32 | — | 16 | 3 | no | — |
| 128 | C32 | — | 32 | 3 | no | — |
| 129 | C32 | — | 32 | 1 | no | — |
| 130 | C23 | — | 3 | 1 | no | — |
| 131 | C23 | — | 3 | 1 | no | — |
| 132 | C23 | — | 9 | 1 | no | — |
| 133 | C23 | — | 18 | 1 | no | — |
| 134 | C23 | — | 27 | 1 | no | — |
| 135 | C23 | — | 27 | 1 | no | — |
| 136 | C23 | — | 54 | 1 | no | — |
| 137 | D4 | — | 9 | 1 | no | — |
| 138 | D4 | — | 27 | 1 | no | — |
| 139 | D4 | — | 27 | 1 | no | — |
| 140 | D4 | — | 27 | 2 | no | — |
| 141 | D4 | — | 27 | 1 | no | — |
| 142 | C2×C4 | — | 27 | 1 | no | — |
| 143 | C2×C4 | — | 27 | 1 | no | — |
| 144 | C2×C3 | — | 3 | 1 | no | — |
| 145 | C2×C3 | — | 3 | 1 | no | — |
| 146 | C2×C3 | — | 9 | 1 | no | — |
| 147 | C2×C3 | — | 12 | 1 | no | — |
| 148 | C2×C3 | — | 12 | 3 | no | — |
| 149 | C2×C3 | — | 24 | 1 | no | — |
| 150 | C2×C3 | — | 72 | 1 | no | — |
| 151 | D3 | — | 6 | 1 | no | — |
| 152 | D3 | — | 12 | 3 | no | — |
| 153 | D3 | — | 18 | 1 | no | — |
| 154 | D3 | — | 36 | 3 | no | — |
| 155 | C22 | — | 6 | 1 | no | — |
| 156 | C22 | — | 9 | 1 | no | — |
| 157 | C22 | — | 9 | 1 | no | — |
| 158 | C22 | — | 9 | 1 | no | — |
| 159 | C22 | — | 9 | 1 | no | — |
| 160 | C22 | — | 27 | 1 | no | — |
| 161 | C22 | — | 54 | 1 | no | — |
| 162 | C22 | — | 54 | 1 | no | — |
| 163 | C22 | — | 54 | 1 | no | — |
| 164 | C22 | — | 54 | 1 | no | — |
| 165 | C22 | C3×C3⋊S4 | 1 | 1 | yes | characteristic, complemented |
| 166 | C22 | A4×C3⋊D3 | 1 | 1 | yes | characteristic, complemented |
| 167 | C4 | — | 9 | 1 | no | — |
| 168 | C4 | — | 27 | 1 | no | — |
| 169 | C3 | — | 4 | 3 | no | — |
| 170 | C3 | — | 4 | 1 | no | — |
| 171 | C3 | — | 8 | 1 | no | — |
| 172 | C3 | — | 32 | 3 | no | — |
| 173 | C3 | A4×S4 | 1 | 1 | yes | characteristic, complemented |
| 174 | C2 | — | 3 | 1 | no | — |
| 175 | C2 | — | 3 | 1 | no | — |
| 176 | C2 | — | 9 | 1 | no | — |
| 177 | C2 | — | 18 | 1 | no | — |
| 178 | C2 | — | 54 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C22×C3×A4 | C2×C3 |
| Fitting, socle | C24×C3 | C3×D3 |
| LCS | G ▷ C22×C3×A4 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×A42 | C2 | semidirect |
| 2. C22×C3⋊S4 | C3 | semidirect |
| 8. C22×C3×A4 | D3 | semidirect |
| 9. C22×C3×A4 | C2×C3 | semidirect |
| 14. A42 | D3 | semidirect |
| 27. C3⋊S4 | A4 | direct (N × H) |
| 31. C24×C3 | C3×D3 | semidirect |
| 36. C22×A4 | C3×D3 | semidirect |
| 37. C22×A4 | C3⋊D3 | semidirect |
| 58. C3×A4 | S4 | semidirect |
| 59. C3×A4 | C2×A4 | semidirect |
| 92. C24 | C3×C3⋊D3 | semidirect |
| 106. C22×C3 | C3×S4 | semidirect |
| 107. C22×C3 | D3×A4 | semidirect |
| 120. A4 | C3⋊S4 | direct (N × H) |
| 121. A4 | D3×A4 | semidirect |
| 165. C22 | C3×C3⋊S4 | semidirect |
| 166. C22 | A4×C3⋊D3 | semidirect |
| 173. C3 | A4×S4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×A42 | 2 |
| C22×C3⋊S4 | 3 |
| A4×S4 | 3 |
| A4×#67 | 3 |
| A4×C3⋊D3 | 4 |
| C3×C3⋊S4 | 4 |