Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
310 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 310 conjugacy classes of subgroups — too many to draw. Only its 53 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 | #5510 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #4991 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #4732 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | C3×#324 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #4470 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #1598 | — | 3 | 1 | no | maximal, Sylow |
| 7 | C2×#235 | — | 2 | 1 | no | — |
| 8 | C22×D12 | C22 | 1 | 1 | yes | characteristic |
| 9 | C2×C3×C4⋊C4 | C22 | 1 | 1 | yes | characteristic |
| 10 | #930 | — | 2 | 1 | no | — |
| 11 | #930 | C22 | 1 | 2 | yes | — |
| 12 | C2×(C2×D3)⋊C4 | — | 2 | 1 | no | — |
| 13 | C2×(C2×D3)⋊C4 | — | 2 | 1 | no | — |
| 14 | C2×D24 | — | 2 | 2 | no | — |
| 15 | #897 | — | 2 | 2 | no | — |
| 16 | #895 | — | 2 | 2 | no | — |
| 17 | C3×C22⋊C8 | C22 | 1 | 2 | yes | — |
| 18 | C3⋊#95 | C22 | 1 | 1 | yes | characteristic |
| 19 | #821 | — | 2 | 2 | no | — |
| 20 | #537 | — | 3 | 1 | no | — |
| 21 | #447 | — | 3 | 2 | no | — |
| 22 | #394 | — | 3 | 1 | no | — |
| 23 | #327 | — | 3 | 2 | no | — |
| 24 | #324 | D3 | 1 | 1 | yes | characteristic, complemented |
| 25 | C23×D3 | — | 2 | 1 | no | — |
| 26 | C22×C3×C4 | D4 | 1 | 1 | yes | characteristic |
| 27 | C22×C3×C4 | D4 | 1 | 1 | yes | characteristic |
| 28 | C22×C3×C4 | C23 | 1 | 1 | yes | characteristic |
| 29 | C2×#67 | — | 2 | 2 | no | — |
| 30 | C22×Dic3 | — | 2 | 1 | no | — |
| 31 | #235 | — | 4 | 2 | no | — |
| 32 | #235 | — | 4 | 2 | no | — |
| 33 | C2×D12 | — | 2 | 2 | no | — |
| 34 | C2×D12 | — | 2 | 1 | no | — |
| 35 | C2×D12 | — | 2 | 1 | no | — |
| 36 | C2×D12 | — | 4 | 2 | no | — |
| 37 | C2×D12 | D4 | 1 | 2 | yes | — |
| 38 | C2×C4×D3 | — | 2 | 2 | no | — |
| 39 | C2×Dic6 | — | 2 | 1 | no | — |
| 40 | C2×C3×C8 | — | 2 | 2 | no | — |
| 41 | C3×C4⋊C4 | — | 2 | 2 | no | — |
| 42 | (C2×D3)⋊C4 | — | 2 | 2 | no | — |
| 43 | (C2×D3)⋊C4 | — | 2 | 2 | no | — |
| 44 | (C2×D3)⋊C4 | — | 4 | 2 | no | — |
| 45 | (C2×D3)⋊C4 | — | 4 | 2 | no | — |
| 46 | C4⋊Dic3 | — | 2 | 1 | no | — |
| 47 | C4⋊Dic3 | D4 | 1 | 2 | yes | — |
| 48 | D24 | — | 4 | 4 | no | — |
| 49 | C2×#41 | — | 6 | 1 | no | — |
| 50 | C22×D4 | — | 3 | 1 | no | — |
| 51 | C2×D8 | — | 6 | 2 | no | — |
| 52 | C22⋊Q8 | — | 6 | 2 | no | — |
| 53 | #121 | — | 3 | 2 | no | — |
| 54 | #121 | — | 6 | 2 | no | — |
| 55 | #121 | — | 6 | 1 | no | — |
| 56 | C2×C4⋊C4 | C2×D3 | 1 | 1 | yes | characteristic |
| 57 | C2×C22⋊C4 | — | 6 | 1 | no | — |
| 58 | C2×C22⋊C4 | — | 6 | 1 | no | — |
| 59 | #102 | — | 6 | 2 | no | — |
| 60 | C22⋊C8 | C2×D3 | 1 | 2 | yes | — |
| 61 | #95 | — | 3 | 1 | no | — |
| 62 | C23×C3 | C2×D4 | 1 | 1 | yes | characteristic |
| 63 | C22×D3 | — | 2 | 2 | no | — |
| 64 | C22×D3 | — | 2 | 2 | no | — |
| 65 | C22×D3 | — | 4 | 1 | no | — |
| 66 | C22×D3 | — | 4 | 1 | no | — |
| 67 | C22×D3 | — | 4 | 1 | no | — |
| 68 | C22×D3 | — | 4 | 1 | no | — |
| 69 | C22×D3 | — | 4 | 1 | no | — |
| 70 | C22×D3 | — | 4 | 1 | no | — |
| 71 | C2×C3×C4 | — | 2 | 1 | no | — |
| 72 | C2×C3×C4 | — | 2 | 1 | no | — |
| 73 | C2×C3×C4 | — | 2 | 1 | no | — |
| 74 | C2×C3×C4 | — | 4 | 1 | no | — |
| 75 | C2×C3×C4 | — | 4 | 1 | no | — |
| 76 | C2×C3×C4 | D8 | 1 | 2 | yes | complemented |
| 77 | C2×C3×C4 | C2×D4 | 1 | 2 | yes | — |
| 78 | #67 | — | 4 | 2 | no | — |
| 79 | #67 | — | 4 | 2 | no | — |
| 80 | C2×Dic3 | — | 2 | 2 | no | — |
| 81 | C2×Dic3 | — | 4 | 1 | no | — |
| 82 | C2×Dic3 | — | 4 | 1 | no | — |
| 83 | C2×Dic3 | — | 4 | 1 | no | — |
| 84 | D12 | — | 2 | 4 | no | — |
| 85 | D12 | — | 4 | 2 | no | — |
| 86 | D12 | — | 4 | 2 | no | — |
| 87 | D12 | — | 4 | 2 | no | — |
| 88 | D12 | — | 8 | 1 | no | — |
| 89 | C4×D3 | — | 4 | 4 | no | — |
| 90 | Dic6 | — | 4 | 2 | no | — |
| 91 | C3×C8 | — | 4 | 2 | no | — |
| 92 | C24 | — | 6 | 1 | no | — |
| 93 | #41 | — | 12 | 2 | no | — |
| 94 | #41 | — | 12 | 2 | no | — |
| 95 | C2×Q8 | — | 6 | 1 | no | — |
| 96 | C2×D4 | — | 3 | 2 | no | — |
| 97 | C2×D4 | — | 6 | 1 | no | — |
| 98 | C2×D4 | — | 6 | 1 | no | — |
| 99 | C2×D4 | — | 6 | 2 | no | — |
| 100 | C2×D4 | — | 6 | 2 | no | — |
| 101 | C2×D4 | — | 12 | 2 | no | — |
| 102 | C22×C4 | — | 6 | 2 | no | — |
| 103 | C22×C4 | — | 6 | 1 | no | — |
| 104 | C22×C4 | D12 | 1 | 1 | yes | characteristic |
| 105 | C22×C4 | D12 | 1 | 1 | yes | characteristic |
| 106 | C22×C4 | C22×D3 | 1 | 1 | yes | characteristic |
| 107 | D8 | — | 12 | 4 | no | — |
| 108 | C2×C8 | — | 2 | 2 | no | — |
| 109 | C4⋊C4 | — | 2 | 2 | no | — |
| 110 | C4⋊C4 | — | 3 | 2 | no | — |
| 111 | C4⋊C4 | — | 6 | 1 | no | — |
| 112 | C22⋊C4 | — | 6 | 2 | no | — |
| 113 | C22⋊C4 | — | 6 | 2 | no | — |
| 114 | C22⋊C4 | — | 12 | 2 | no | — |
| 115 | C22⋊C4 | — | 12 | 2 | no | — |
| 116 | C22×C3 | — | 2 | 1 | no | — |
| 117 | C22×C3 | — | 2 | 1 | no | — |
| 118 | C22×C3 | #120 | 1 | 1 | yes | characteristic |
| 119 | C22×C3 | C2×D8 | 1 | 1 | yes | characteristic |
| 120 | C22×C3 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 121 | C2×D3 | — | 4 | 2 | no | — |
| 122 | C2×D3 | — | 4 | 2 | no | — |
| 123 | C2×D3 | — | 4 | 2 | no | — |
| 124 | C2×D3 | — | 4 | 2 | no | — |
| 125 | C2×D3 | — | 4 | 2 | no | — |
| 126 | C2×D3 | — | 4 | 2 | no | — |
| 127 | C2×D3 | — | 8 | 1 | no | — |
| 128 | C2×D3 | — | 8 | 1 | no | — |
| 129 | C2×D3 | — | 8 | 1 | no | — |
| 130 | C2×D3 | — | 8 | 1 | no | — |
| 131 | C3×C4 | — | 2 | 2 | no | — |
| 132 | C3×C4 | — | 2 | 2 | no | — |
| 133 | C3×C4 | — | 4 | 1 | no | — |
| 134 | Dic3 | — | 4 | 2 | no | — |
| 135 | Dic3 | — | 8 | 1 | no | — |
| 136 | C23 | — | 6 | 2 | no | — |
| 137 | C23 | — | 6 | 2 | no | — |
| 138 | C23 | — | 12 | 1 | no | — |
| 139 | C23 | — | 12 | 1 | no | — |
| 140 | C23 | — | 12 | 1 | no | — |
| 141 | C23 | — | 12 | 1 | no | — |
| 142 | C23 | — | 12 | 1 | no | — |
| 143 | C23 | — | 12 | 1 | no | — |
| 144 | C23 | C2×D12 | 1 | 1 | yes | characteristic |
| 145 | Q8 | — | 12 | 2 | no | — |
| 146 | D4 | — | 6 | 4 | no | — |
| 147 | D4 | — | 12 | 2 | no | — |
| 148 | D4 | — | 12 | 2 | no | — |
| 149 | D4 | — | 12 | 2 | no | — |
| 150 | D4 | — | 12 | 2 | no | — |
| 151 | D4 | — | 12 | 2 | no | — |
| 152 | D4 | — | 24 | 1 | no | — |
| 153 | C2×C4 | — | 2 | 1 | no | — |
| 154 | C2×C4 | — | 2 | 1 | no | — |
| 155 | C2×C4 | — | 2 | 1 | no | — |
| 156 | C2×C4 | — | 4 | 1 | no | — |
| 157 | C2×C4 | — | 4 | 1 | no | — |
| 158 | C2×C4 | — | 6 | 2 | no | — |
| 159 | C2×C4 | — | 12 | 4 | no | — |
| 160 | C2×C4 | — | 12 | 1 | no | — |
| 161 | C2×C4 | — | 12 | 1 | no | — |
| 162 | C2×C4 | — | 12 | 1 | no | — |
| 163 | C2×C4 | D24 | 1 | 2 | yes | complemented |
| 164 | C2×C4 | D3×D4 | 1 | 2 | yes | — |
| 165 | C8 | — | 4 | 2 | no | — |
| 166 | C2×C3 | — | 2 | 1 | no | — |
| 167 | C2×C3 | — | 2 | 1 | no | — |
| 168 | C2×C3 | #447 | 1 | 1 | yes | characteristic |
| 169 | C2×C3 | #454 | 1 | 1 | yes | characteristic |
| 170 | C2×C3 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 171 | D3 | — | 8 | 2 | no | — |
| 172 | D3 | — | 8 | 2 | no | — |
| 173 | C22 | — | 2 | 1 | no | — |
| 174 | C22 | — | 2 | 1 | no | — |
| 175 | C22 | — | 12 | 2 | no | — |
| 176 | C22 | — | 12 | 2 | no | — |
| 177 | C22 | — | 12 | 2 | no | — |
| 178 | C22 | — | 12 | 2 | no | — |
| 179 | C22 | — | 12 | 2 | no | — |
| 180 | C22 | — | 12 | 2 | no | — |
| 181 | C22 | — | 24 | 1 | no | — |
| 182 | C22 | — | 24 | 1 | no | — |
| 183 | C22 | — | 24 | 1 | no | — |
| 184 | C22 | — | 24 | 1 | no | — |
| 185 | C22 | #882 | 1 | 1 | yes | characteristic |
| 186 | C22 | C2×D24 | 1 | 1 | yes | characteristic |
| 187 | C22 | #908 | 1 | 1 | yes | characteristic |
| 188 | C4 | — | 2 | 2 | no | — |
| 189 | C4 | — | 2 | 2 | no | — |
| 190 | C4 | — | 4 | 1 | no | — |
| 191 | C4 | — | 12 | 2 | no | — |
| 192 | C4 | — | 24 | 1 | no | — |
| 193 | C3 | #1598 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 194 | C2 | — | 2 | 1 | no | — |
| 195 | C2 | — | 2 | 1 | no | — |
| 196 | C2 | — | 24 | 2 | no | — |
| 197 | C2 | — | 24 | 2 | no | — |
| 198 | C2 | #4732 | 1 | 1 | yes | characteristic |
| 199 | C2 | #4741 | 1 | 1 | yes | characteristic |
| 200 | C2 | #4753 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C22 | #882 |
| commutator, cosocle | C22×C3×C4 | C23 |
| Frattini | C22×C4 | C22×D3 |
| Fitting | C3×#324 | C2 |
| socle | C22×C3 | #120 |
| hypercenter | #324 | D3 |
| hypocenter | C3 | #1598 |
| LCS | G ▷ C22×C3×C4 ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #324 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #5510 | C2 | semidirect |
| 2. #4991 | C2 | semidirect |
| 3. #4732 | C2 | semidirect |
| 4. C3×#324 | C2 | semidirect |
| 5. #4470 | C2 | semidirect |
| 24. #324 | D3 | semidirect |
| 76. C2×C3×C4 | D8 | semidirect |
| 163. C2×C4 | D24 | semidirect |
| 193. C3 | #1598 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #5510 | 2 |
| #4991 | 2 |
| #4732 | 2 |
| C3×#324 | 2 |
| #4470 | 2 |
| #1598 | 3 |