Finite Group DatabasebetaFinite Group DatabasebetaOrder 1152.
716 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 716 conjugacy classes of subgroups — too many to draw. Only its 48 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×C3×#997 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | Dic3×H(5,𝔽2) | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C4×#997 | — | 3 | 1 | no | maximal |
| 4 | C3×#997 | C4 | 1 | 1 | yes | characteristic, complemented |
| 5 | C2×Dic3×A4 | — | 4 | 3 | no | maximal |
| 6 | Dic3×SL(2,𝔽3) | — | 4 | 2 | no | maximal |
| 7 | C2×C3×H(5,𝔽2) | C2×C3 | 1 | 1 | yes | characteristic |
| 8 | C2×#997 | — | 2 | 1 | no | — |
| 9 | C2×#997 | D3 | 1 | 1 | yes | characteristic |
| 10 | #5827 | — | 3 | 2 | no | — |
| 11 | Dic3×#41 | — | 3 | 2 | no | — |
| 12 | #5798 | — | 3 | 3 | no | — |
| 13 | C2×D4×Dic3 | — | 3 | 3 | no | — |
| 14 | C22×C3×A4 | — | 4 | 3 | no | — |
| 15 | C2×C3×SL(2,𝔽3) | — | 4 | 2 | no | — |
| 16 | Dic3×A4 | — | 4 | 6 | no | — |
| 17 | C4×H(5,𝔽2) | — | 3 | 1 | no | Sylow |
| 18 | C3×H(5,𝔽2) | — | 3 | 2 | no | — |
| 19 | C3×H(5,𝔽2) | — | 3 | 3 | no | — |
| 20 | C3×H(5,𝔽2) | C3×C4 | 1 | 1 | yes | characteristic, complemented |
| 21 | C2×C3×#41 | — | 3 | 2 | no | — |
| 22 | C22×C3×D4 | — | 3 | 3 | no | — |
| 23 | C23×Dic3 | — | 3 | 1 | no | — |
| 24 | C23×Dic3 | A4 | 1 | 3 | yes | complemented |
| 25 | #997 | — | 2 | 1 | no | — |
| 26 | #997 | Dic3 | 1 | 1 | yes | characteristic, direct product factor |
| 27 | C2×C4×A4 | — | 12 | 3 | no | — |
| 28 | C2×#217 | — | 3 | 3 | no | — |
| 29 | C2×#217 | — | 3 | 3 | no | — |
| 30 | Q8×Dic3 | — | 3 | 2 | no | — |
| 31 | Q8×Dic3 | A4 | 1 | 2 | yes | complemented |
| 32 | D4×Dic3 | — | 3 | 6 | no | — |
| 33 | D4×Dic3 | — | 3 | 12 | no | — |
| 34 | D4×Dic3 | — | 3 | 6 | no | — |
| 35 | #926 | — | 3 | 6 | no | — |
| 36 | C2×C4⋊Dic3 | — | 3 | 3 | no | — |
| 37 | C2×C4×Dic3 | — | 3 | 3 | no | — |
| 38 | C4×SL(2,𝔽3) | — | 12 | 2 | no | — |
| 39 | C2×C3×A4 | — | 4 | 3 | no | — |
| 40 | C2×C3×A4 | — | 4 | 3 | no | — |
| 41 | C2×C3×A4 | — | 4 | 3 | no | — |
| 42 | C2×C3×Dic3 | — | 16 | 1 | no | — |
| 43 | C3×SL(2,𝔽3) | — | 4 | 2 | no | — |
| 44 | C2×H(5,𝔽2) | C3×D3 | 1 | 1 | yes | characteristic |
| 45 | #520 | — | 9 | 2 | no | — |
| 46 | #518 | — | 9 | 3 | no | — |
| 47 | C4×#41 | — | 9 | 2 | no | — |
| 48 | C2×C4×D4 | — | 9 | 3 | no | — |
| 49 | C24×C3 | — | 3 | 1 | no | — |
| 50 | C24×C3 | C2×A4 | 1 | 3 | yes | — |
| 51 | C22×A4 | — | 4 | 3 | no | — |
| 52 | C22×A4 | — | 8 | 3 | no | — |
| 53 | C3×#41 | — | 3 | 6 | no | — |
| 54 | C3×#41 | — | 3 | 6 | no | — |
| 55 | C3×#41 | — | 3 | 2 | no | — |
| 56 | C3×#41 | — | 3 | 2 | no | — |
| 57 | C2×C3×Q8 | C2×A4 | 1 | 2 | yes | — |
| 58 | C2×C3×D4 | — | 3 | 6 | no | — |
| 59 | C2×C3×D4 | — | 3 | 3 | no | — |
| 60 | C2×C3×D4 | — | 3 | 12 | no | — |
| 61 | C2×C3×D4 | — | 3 | 3 | no | — |
| 62 | C2×C3×D4 | — | 3 | 3 | no | — |
| 63 | C2×C3×D4 | — | 3 | 3 | no | — |
| 64 | C22×C3×C4 | — | 3 | 3 | no | — |
| 65 | C22×Dic3 | — | 3 | 3 | no | — |
| 66 | C22×Dic3 | — | 3 | 3 | no | — |
| 67 | C22×Dic3 | — | 3 | 3 | no | — |
| 68 | C22×Dic3 | — | 3 | 3 | no | — |
| 69 | C22×Dic3 | — | 4 | 6 | no | — |
| 70 | C22×Dic3 | — | 12 | 2 | no | — |
| 71 | C2×SL(2,𝔽3) | — | 4 | 2 | no | — |
| 72 | C2×SL(2,𝔽3) | — | 8 | 2 | no | — |
| 73 | C4×A4 | — | 12 | 6 | no | — |
| 74 | #217 | — | 3 | 12 | no | — |
| 75 | C4⋊Dic3 | — | 3 | 6 | no | — |
| 76 | C4⋊Dic3 | — | 3 | 2 | no | — |
| 77 | C4×Dic3 | — | 3 | 6 | no | — |
| 78 | C4×Dic3 | — | 3 | 2 | no | — |
| 79 | C22×C32 | — | 16 | 1 | no | — |
| 80 | C3×A4 | — | 4 | 3 | no | — |
| 81 | C3×Dic3 | — | 16 | 2 | no | — |
| 82 | H(5,𝔽2) | — | 3 | 2 | no | — |
| 83 | H(5,𝔽2) | — | 3 | 3 | no | — |
| 84 | H(5,𝔽2) | C3×Dic3 | 1 | 1 | yes | characteristic, complemented |
| 85 | C2×#41 | — | 3 | 2 | no | — |
| 86 | C22×D4 | — | 3 | 3 | no | — |
| 87 | C23×C4 | — | 3 | 3 | no | — |
| 88 | C23×C4 | — | 9 | 1 | no | — |
| 89 | C4×Q8 | — | 3 | 2 | no | — |
| 90 | C4×Q8 | — | 9 | 2 | no | — |
| 91 | C4×D4 | — | 9 | 6 | no | — |
| 92 | C4×D4 | — | 9 | 12 | no | — |
| 93 | C4×D4 | — | 9 | 6 | no | — |
| 94 | #117 | — | 9 | 6 | no | — |
| 95 | C2×C4⋊C4 | — | 9 | 3 | no | — |
| 96 | C2×C22⋊C4 | — | 9 | 3 | no | — |
| 97 | C2×C22⋊C4 | — | 9 | 3 | no | — |
| 98 | C2×C42 | — | 9 | 3 | no | — |
| 99 | C23×C3 | — | 3 | 3 | no | — |
| 100 | C23×C3 | — | 3 | 1 | no | — |
| 101 | C23×C3 | — | 3 | 3 | no | — |
| 102 | C23×C3 | — | 3 | 3 | no | — |
| 103 | C23×C3 | — | 4 | 3 | no | — |
| 104 | C23×C3 | — | 4 | 3 | no | — |
| 105 | C23×C3 | — | 12 | 1 | no | — |
| 106 | C23×C3 | — | 12 | 1 | no | — |
| 107 | C23×C3 | C4×A4 | 1 | 3 | yes | complemented |
| 108 | C2×A4 | — | 4 | 3 | no | — |
| 109 | C2×A4 | — | 4 | 3 | no | — |
| 110 | C2×A4 | — | 4 | 3 | no | — |
| 111 | C2×A4 | — | 8 | 3 | no | — |
| 112 | C2×A4 | — | 8 | 3 | no | — |
| 113 | C2×A4 | — | 8 | 3 | no | — |
| 114 | C3×Q8 | — | 3 | 2 | no | — |
| 115 | C3×Q8 | C4×A4 | 1 | 2 | yes | complemented |
| 116 | C3×D4 | — | 3 | 12 | no | — |
| 117 | C3×D4 | — | 3 | 6 | no | — |
| 118 | C3×D4 | — | 3 | 6 | no | — |
| 119 | C2×C3×C4 | — | 3 | 3 | no | — |
| 120 | C2×C3×C4 | — | 3 | 6 | no | — |
| 121 | C2×C3×C4 | — | 3 | 3 | no | — |
| 122 | C2×C3×C4 | — | 3 | 2 | no | — |
| 123 | C2×C3×C4 | — | 48 | 1 | no | — |
| 124 | C2×Dic3 | — | 3 | 2 | no | — |
| 125 | C2×Dic3 | — | 3 | 3 | no | — |
| 126 | C2×Dic3 | — | 6 | 6 | no | — |
| 127 | C2×Dic3 | — | 12 | 3 | no | — |
| 128 | C2×Dic3 | — | 12 | 3 | no | — |
| 129 | C2×Dic3 | — | 12 | 3 | no | — |
| 130 | C2×Dic3 | C22⋊A4 | 1 | 1 | yes | characteristic |
| 131 | SL(2,𝔽3) | — | 4 | 2 | no | — |
| 132 | SL(2,𝔽3) | — | 8 | 2 | no | — |
| 133 | C2×C32 | — | 16 | 1 | no | — |
| 134 | C2×C32 | — | 16 | 1 | no | — |
| 135 | C2×C32 | — | 16 | 1 | no | — |
| 136 | C24 | — | 3 | 1 | no | — |
| 137 | C24 | D3×A4 | 1 | 3 | yes | — |
| 138 | #41 | — | 3 | 6 | no | — |
| 139 | #41 | — | 3 | 6 | no | — |
| 140 | #41 | — | 3 | 2 | no | — |
| 141 | #41 | — | 3 | 2 | no | — |
| 142 | C2×Q8 | D3×A4 | 1 | 2 | yes | — |
| 143 | C2×D4 | — | 3 | 6 | no | — |
| 144 | C2×D4 | — | 3 | 3 | no | — |
| 145 | C2×D4 | — | 3 | 12 | no | — |
| 146 | C2×D4 | — | 3 | 3 | no | — |
| 147 | C2×D4 | — | 3 | 3 | no | — |
| 148 | C2×D4 | — | 3 | 3 | no | — |
| 149 | C22×C4 | — | 3 | 3 | no | — |
| 150 | C22×C4 | — | 9 | 3 | no | — |
| 151 | C22×C4 | — | 9 | 3 | no | — |
| 152 | C22×C4 | — | 9 | 3 | no | — |
| 153 | C22×C4 | — | 9 | 3 | no | — |
| 154 | C22×C4 | — | 12 | 6 | no | — |
| 155 | C22×C4 | — | 36 | 2 | no | — |
| 156 | C4⋊C4 | — | 9 | 2 | no | — |
| 157 | C4⋊C4 | — | 9 | 6 | no | — |
| 158 | C22⋊C4 | — | 9 | 12 | no | — |
| 159 | C42 | — | 9 | 2 | no | — |
| 160 | C42 | — | 9 | 6 | no | — |
| 161 | C22×C3 | — | 3 | 3 | no | — |
| 162 | C22×C3 | — | 3 | 3 | no | — |
| 163 | C22×C3 | — | 4 | 3 | no | — |
| 164 | C22×C3 | — | 6 | 3 | no | — |
| 165 | C22×C3 | — | 6 | 3 | no | — |
| 166 | C22×C3 | — | 12 | 3 | no | — |
| 167 | C22×C3 | — | 12 | 1 | no | — |
| 168 | C22×C3 | — | 12 | 3 | no | — |
| 169 | C22×C3 | — | 16 | 1 | no | — |
| 170 | C22×C3 | — | 32 | 1 | no | — |
| 171 | C22×C3 | C2×C22⋊A4 | 1 | 1 | yes | characteristic |
| 172 | A4 | — | 4 | 3 | no | — |
| 173 | A4 | — | 8 | 3 | no | — |
| 174 | C3×C4 | — | 3 | 2 | no | — |
| 175 | C3×C4 | — | 3 | 2 | no | — |
| 176 | C3×C4 | — | 48 | 2 | no | — |
| 177 | Dic3 | — | 6 | 3 | no | — |
| 178 | Dic3 | — | 6 | 2 | no | — |
| 179 | Dic3 | #997 | 1 | 2 | yes | direct product factor |
| 180 | C32 | — | 16 | 1 | no | Sylow |
| 181 | C23 | — | 3 | 1 | no | — |
| 182 | C23 | — | 3 | 3 | no | — |
| 183 | C23 | — | 3 | 3 | no | — |
| 184 | C23 | — | 3 | 3 | no | — |
| 185 | C23 | — | 4 | 3 | no | — |
| 186 | C23 | — | 4 | 3 | no | — |
| 187 | C23 | — | 12 | 1 | no | — |
| 188 | C23 | — | 12 | 1 | no | — |
| 189 | C23 | Dic3×A4 | 1 | 3 | yes | complemented |
| 190 | Q8 | — | 3 | 2 | no | — |
| 191 | Q8 | Dic3×A4 | 1 | 2 | yes | complemented |
| 192 | D4 | — | 3 | 6 | no | — |
| 193 | D4 | — | 3 | 6 | no | — |
| 194 | D4 | — | 3 | 12 | no | — |
| 195 | C2×C4 | — | 3 | 3 | no | — |
| 196 | C2×C4 | — | 3 | 6 | no | — |
| 197 | C2×C4 | — | 3 | 3 | no | — |
| 198 | C2×C4 | — | 3 | 2 | no | — |
| 199 | C2×C4 | — | 3 | 1 | no | — |
| 200 | C2×C4 | — | 9 | 2 | no | — |
| 201 | C2×C4 | — | 9 | 3 | no | — |
| 202 | C2×C4 | — | 18 | 6 | no | — |
| 203 | C2×C4 | — | 36 | 3 | no | — |
| 204 | C2×C4 | — | 36 | 3 | no | — |
| 205 | C2×C4 | — | 36 | 3 | no | — |
| 206 | C2×C3 | — | 6 | 3 | no | — |
| 207 | C2×C3 | — | 6 | 3 | no | — |
| 208 | C2×C3 | — | 16 | 1 | no | — |
| 209 | C2×C3 | — | 16 | 1 | no | — |
| 210 | C2×C3 | — | 16 | 1 | no | — |
| 211 | C2×C3 | — | 32 | 1 | no | — |
| 212 | C2×C3 | — | 32 | 1 | no | — |
| 213 | C2×C3 | — | 32 | 1 | no | — |
| 214 | C2×C3 | C4×C22⋊A4 | 1 | 1 | yes | characteristic |
| 215 | C2×C3 | C2×#997 | 1 | 1 | yes | characteristic |
| 216 | C2×C3 | #5950 | 1 | 1 | yes | characteristic |
| 217 | C22 | — | 3 | 3 | no | — |
| 218 | C22 | — | 3 | 3 | no | — |
| 219 | C22 | — | 4 | 3 | no | — |
| 220 | C22 | — | 6 | 3 | no | — |
| 221 | C22 | — | 6 | 3 | no | — |
| 222 | C22 | — | 12 | 3 | no | — |
| 223 | C22 | — | 12 | 1 | no | — |
| 224 | C22 | — | 12 | 3 | no | — |
| 225 | C22 | D3×C22⋊A4 | 1 | 1 | yes | characteristic |
| 226 | C4 | — | 3 | 2 | no | — |
| 227 | C4 | — | 3 | 2 | no | — |
| 228 | C4 | — | 3 | 2 | no | — |
| 229 | C4 | — | 18 | 2 | no | — |
| 230 | C4 | — | 18 | 3 | no | — |
| 231 | C3 | — | 16 | 1 | no | — |
| 232 | C3 | — | 32 | 1 | no | — |
| 233 | C3 | C4×#997 | 1 | 1 | yes | characteristic, complemented |
| 234 | C2 | — | 6 | 3 | no | — |
| 235 | C2 | — | 6 | 3 | no | — |
| 236 | C2 | C3⋊#5950 | 1 | 1 | yes | characteristic |
| 237 | C2 | D3×#997 | 1 | 1 | yes | characteristic |
| 238 | C2 | Dic3×C22⋊A4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C22 | D3×C22⋊A4 |
| commutator, hypocenter | C3×H(5,𝔽2) | C3×C4 |
| Fitting, cosocle | C2×C3×H(5,𝔽2) | C2×C3 |
| socle | C22×C3 | C2×C22⋊A4 |
| prime radical | C2×C3×#997 | C2 |
| LCS | G ▷ C3×H(5,𝔽2) | |
| UCS | C1 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. Dic3×H(5,𝔽2) | C3 | semidirect |
| 4. C3×#997 | C4 | semidirect |
| 20. C3×H(5,𝔽2) | C3×C4 | semidirect |
| 24. C23×Dic3 | A4 | semidirect |
| 26. #997 | Dic3 | direct (N × H) |
| 31. Q8×Dic3 | A4 | semidirect |
| 84. H(5,𝔽2) | C3×Dic3 | semidirect |
| 107. C23×C3 | C4×A4 | semidirect |
| 115. C3×Q8 | C4×A4 | semidirect |
| 179. Dic3 | #997 | direct (N × H) |
| 189. C23 | Dic3×A4 | semidirect |
| 191. Q8 | Dic3×A4 | semidirect |
| 233. C3 | C4×#997 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×#997 | 2 |
| Dic3×H(5,𝔽2) | 3 |
| C4×#997 | 3 |
| C2×Dic3×A4 | 4 |
| Dic3×SL(2,𝔽3) | 4 |