Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
804 conjugacy classes of subgroups; only the 153 normal subgroups are shown.
This group has 804 conjugacy classes of subgroups — too many to draw. Only its 153 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×#938 | C2 | 1 | 3 | yes | maximal, complemented |
| 2 | C2×#884 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | C3×#392 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #5223 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | #4949 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | #4677 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | D3×#95 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #2656 | — | 3 | 1 | no | maximal, Sylow |
| 9 | C22×C3×D4 | C22 | 1 | 3 | yes | — |
| 10 | C22×#67 | C22 | 1 | 3 | yes | — |
| 11 | C22×#67 | C22 | 1 | 3 | yes | — |
| 12 | C22×D12 | C22 | 1 | 1 | yes | characteristic |
| 13 | C22×C4×D3 | C22 | 1 | 3 | yes | — |
| 14 | C2×C3×C22⋊C4 | C22 | 1 | 3 | yes | — |
| 15 | C2×#217 | C22 | 1 | 6 | yes | — |
| 16 | #938 | — | 2 | 12 | no | — |
| 17 | C2×(C2×D3)⋊C4 | C22 | 1 | 3 | yes | — |
| 18 | C2×C4⋊Dic3 | C22 | 1 | 3 | yes | — |
| 19 | C2×Dic3⋊C4 | C22 | 1 | 3 | yes | — |
| 20 | #884 | — | 2 | 12 | no | — |
| 21 | C3×#95 | C22 | 1 | 1 | yes | characteristic |
| 22 | C3⋊#95 | C22 | 1 | 3 | yes | — |
| 23 | C2×#121 | — | 3 | 3 | no | — |
| 24 | C2×#121 | — | 3 | 3 | no | — |
| 25 | #399 | — | 3 | 3 | no | — |
| 26 | #399 | — | 3 | 1 | no | — |
| 27 | #392 | — | 3 | 3 | no | — |
| 28 | #392 | D3 | 1 | 1 | yes | characteristic, complemented |
| 29 | C2×#95 | — | 3 | 1 | no | — |
| 30 | C24×C3 | C23 | 1 | 3 | yes | — |
| 31 | C23×D3 | C23 | 1 | 1 | yes | characteristic |
| 32 | C23×D3 | C23 | 1 | 1 | yes | characteristic |
| 33 | C2×C3×D4 | — | 2 | 6 | no | — |
| 34 | C2×C3×D4 | — | 4 | 6 | no | — |
| 35 | C22×C3×C4 | C23 | 1 | 1 | yes | characteristic |
| 36 | C22×C3×C4 | C23 | 1 | 3 | yes | — |
| 37 | C2×#67 | — | 2 | 6 | no | — |
| 38 | C2×#67 | — | 2 | 12 | no | — |
| 39 | C2×#67 | — | 2 | 6 | no | — |
| 40 | C2×#67 | — | 4 | 6 | no | — |
| 41 | C22×Dic3 | C23 | 1 | 3 | yes | — |
| 42 | C22×Dic3 | C23 | 1 | 3 | yes | — |
| 43 | C2×D12 | — | 2 | 6 | no | — |
| 44 | C2×C4×D3 | — | 4 | 6 | no | — |
| 45 | C2×C4×D3 | D4 | 1 | 12 | yes | complemented |
| 46 | C3×C22⋊C4 | — | 2 | 6 | no | — |
| 47 | #217 | — | 2 | 12 | no | — |
| 48 | (C2×D3)⋊C4 | — | 2 | 6 | no | — |
| 49 | C4⋊Dic3 | — | 2 | 6 | no | — |
| 50 | Dic3⋊C4 | — | 2 | 6 | no | — |
| 51 | C22×D4 | — | 3 | 3 | no | — |
| 52 | C22×D4 | — | 3 | 3 | no | — |
| 53 | C22×D4 | — | 3 | 1 | no | — |
| 54 | C22×D4 | C2×D3 | 1 | 3 | yes | — |
| 55 | C23×C4 | — | 3 | 3 | no | — |
| 56 | #121 | — | 6 | 12 | no | — |
| 57 | #121 | — | 6 | 12 | no | — |
| 58 | C2×C4⋊C4 | — | 3 | 3 | no | — |
| 59 | C2×C4⋊C4 | — | 3 | 3 | no | — |
| 60 | C2×C22⋊C4 | — | 3 | 6 | no | — |
| 61 | C2×C22⋊C4 | — | 3 | 3 | no | — |
| 62 | C2×C22⋊C4 | C2×D3 | 1 | 3 | yes | — |
| 63 | #95 | — | 3 | 3 | no | — |
| 64 | #95 | C2×D3 | 1 | 1 | yes | characteristic |
| 65 | C23×C3 | — | 2 | 6 | no | — |
| 66 | C23×C3 | — | 2 | 3 | no | — |
| 67 | C23×C3 | — | 2 | 6 | no | — |
| 68 | C23×C3 | — | 2 | 3 | no | — |
| 69 | C23×C3 | — | 2 | 3 | no | — |
| 70 | C23×C3 | C24 | 1 | 1 | yes | characteristic |
| 71 | C22×D3 | — | 2 | 3 | no | — |
| 72 | C22×D3 | — | 2 | 1 | no | — |
| 73 | C22×D3 | — | 2 | 3 | no | — |
| 74 | C22×D3 | — | 2 | 3 | no | — |
| 75 | C22×D3 | — | 2 | 1 | no | — |
| 76 | C22×D3 | C2×D4 | 1 | 6 | yes | — |
| 77 | C3×D4 | — | 4 | 12 | no | — |
| 78 | C2×C3×C4 | — | 2 | 6 | no | — |
| 79 | C2×C3×C4 | — | 2 | 3 | no | — |
| 80 | C2×C3×C4 | C2×D4 | 1 | 6 | yes | — |
| 81 | #67 | — | 4 | 12 | no | — |
| 82 | #67 | — | 4 | 12 | no | — |
| 83 | C2×Dic3 | — | 2 | 6 | no | — |
| 84 | C2×Dic3 | — | 2 | 3 | no | — |
| 85 | C2×Dic3 | — | 2 | 6 | no | — |
| 86 | C2×Dic3 | C2×D4 | 1 | 6 | yes | — |
| 87 | D12 | — | 4 | 4 | no | — |
| 88 | C4×D3 | — | 4 | 12 | no | — |
| 89 | C24 | — | 3 | 1 | no | — |
| 90 | C24 | — | 3 | 1 | no | — |
| 91 | C24 | C22×D3 | 1 | 3 | yes | — |
| 92 | C2×D4 | — | 2 | 6 | no | — |
| 93 | C2×D4 | — | 4 | 6 | no | — |
| 94 | C2×D4 | — | 6 | 6 | no | — |
| 95 | C2×D4 | — | 6 | 6 | no | — |
| 96 | C2×D4 | — | 6 | 6 | no | — |
| 97 | C2×D4 | — | 6 | 12 | no | — |
| 98 | C2×D4 | — | 12 | 6 | no | — |
| 99 | C22×C4 | — | 3 | 3 | no | — |
| 100 | C22×C4 | — | 3 | 12 | no | — |
| 101 | C22×C4 | — | 3 | 3 | no | — |
| 102 | C22×C4 | — | 12 | 6 | no | — |
| 103 | C22×C4 | C22×D3 | 1 | 1 | yes | characteristic |
| 104 | C22×C4 | C22×D3 | 1 | 3 | yes | — |
| 105 | C4⋊C4 | — | 6 | 6 | no | — |
| 106 | C4⋊C4 | — | 6 | 6 | no | — |
| 107 | C22⋊C4 | — | 2 | 6 | no | — |
| 108 | C22⋊C4 | — | 6 | 12 | no | — |
| 109 | C22⋊C4 | — | 6 | 6 | no | — |
| 110 | C22×C3 | — | 4 | 3 | no | — |
| 111 | C22×C3 | — | 4 | 3 | no | — |
| 112 | C22×C3 | — | 4 | 6 | no | — |
| 113 | C22×C3 | — | 4 | 6 | no | — |
| 114 | C22×C3 | — | 4 | 3 | no | — |
| 115 | C22×C3 | C22×D4 | 1 | 3 | yes | — |
| 116 | C22×C3 | H(5,𝔽2) | 1 | 3 | yes | — |
| 117 | C22×C3 | 2−1+4 | 1 | 1 | yes | characteristic |
| 118 | C2×D3 | — | 2 | 6 | no | — |
| 119 | C2×D3 | — | 2 | 6 | no | — |
| 120 | C2×D3 | — | 4 | 1 | no | — |
| 121 | C2×D3 | — | 4 | 3 | no | — |
| 122 | C2×D3 | — | 4 | 3 | no | — |
| 123 | C2×D3 | #127 | 1 | 4 | yes | — |
| 124 | C3×C4 | — | 2 | 6 | no | — |
| 125 | C3×C4 | — | 4 | 1 | no | — |
| 126 | Dic3 | — | 2 | 6 | no | — |
| 127 | Dic3 | — | 4 | 3 | no | — |
| 128 | C23 | — | 2 | 3 | no | — |
| 129 | C23 | — | 2 | 6 | no | — |
| 130 | C23 | — | 2 | 3 | no | — |
| 131 | C23 | — | 2 | 6 | no | — |
| 132 | C23 | — | 2 | 3 | no | — |
| 133 | C23 | — | 3 | 6 | no | — |
| 134 | C23 | — | 6 | 3 | no | — |
| 135 | C23 | — | 6 | 1 | no | — |
| 136 | C23 | — | 6 | 3 | no | — |
| 137 | C23 | — | 6 | 3 | no | — |
| 138 | C23 | — | 6 | 1 | no | — |
| 139 | C23 | C23×D3 | 1 | 1 | yes | characteristic |
| 140 | D4 | — | 4 | 12 | no | — |
| 141 | D4 | — | 12 | 4 | no | — |
| 142 | D4 | — | 12 | 12 | no | — |
| 143 | D4 | — | 12 | 12 | no | — |
| 144 | C2×C4 | — | 2 | 6 | no | — |
| 145 | C2×C4 | — | 2 | 3 | no | — |
| 146 | C2×C4 | — | 3 | 6 | no | — |
| 147 | C2×C4 | — | 6 | 6 | no | — |
| 148 | C2×C4 | — | 6 | 6 | no | — |
| 149 | C2×C4 | — | 6 | 3 | no | — |
| 150 | C2×C4 | — | 12 | 12 | no | — |
| 151 | C2×C4 | D3×D4 | 1 | 6 | yes | — |
| 152 | C2×C3 | — | 8 | 3 | no | — |
| 153 | C2×C3 | C2×#127 | 1 | 1 | yes | characteristic |
| 154 | C2×C3 | #534 | 1 | 3 | yes | — |
| 155 | C2×C3 | #537 | 1 | 3 | yes | — |
| 156 | D3 | — | 2 | 4 | no | — |
| 157 | D3 | — | 8 | 1 | no | — |
| 158 | C22 | — | 3 | 4 | no | — |
| 159 | C22 | — | 4 | 3 | no | — |
| 160 | C22 | — | 4 | 6 | no | — |
| 161 | C22 | — | 4 | 3 | no | — |
| 162 | C22 | — | 4 | 3 | no | — |
| 163 | C22 | — | 4 | 6 | no | — |
| 164 | C22 | — | 6 | 6 | no | — |
| 165 | C22 | — | 6 | 6 | no | — |
| 166 | C22 | — | 12 | 1 | no | — |
| 167 | C22 | — | 12 | 3 | no | — |
| 168 | C22 | — | 12 | 3 | no | — |
| 169 | C22 | C2×D3×D4 | 1 | 3 | yes | — |
| 170 | C22 | #1004 | 1 | 3 | yes | — |
| 171 | C22 | #1010 | 1 | 1 | yes | characteristic |
| 172 | C4 | — | 2 | 6 | no | — |
| 173 | C4 | — | 4 | 1 | no | — |
| 174 | C4 | — | 6 | 6 | no | — |
| 175 | C4 | — | 12 | 3 | no | — |
| 176 | C3 | #2656 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 177 | C2 | — | 6 | 4 | no | — |
| 178 | C2 | — | 8 | 3 | no | — |
| 179 | C2 | — | 24 | 1 | no | — |
| 180 | C2 | #5589 | 1 | 3 | yes | — |
| 181 | C2 | #5608 | 1 | 3 | yes | — |
| 182 | C2 | D3×#127 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×#392 | C2 |
| hypercenter | #392 | D3 |
| hypocenter | C3 | #2656 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #392 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#938 | C2 | semidirect |
| 2. C2×#884 | C2 | semidirect |
| 3. C3×#392 | C2 | semidirect |
| 4. #5223 | C2 | semidirect |
| 5. #4949 | C2 | semidirect |
| 6. #4677 | C2 | semidirect |
| 7. D3×#95 | C2 | semidirect |
| 28. #392 | D3 | semidirect |
| 45. C2×C4×D3 | D4 | semidirect |
| 176. C3 | #2656 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#938 | 2 |
| C2×#884 | 2 |
| C3×#392 | 2 |
| #5223 | 2 |
| #4949 | 2 |
| #4677 | 2 |
| D3×#95 | 2 |
| #2656 | 3 |