Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
1077 conjugacy classes of subgroups; only the 41 normal subgroups are shown.
This group has 1077 conjugacy classes of subgroups — too many to draw. Only its 41 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 | #5950 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×#997 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | H(7,𝔽2) | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Sylow |
| 4 | #997 | C22 | 1 | 1 | yes | characteristic, complemented |
| 5 | 2−1+4⋊C3 | — | 4 | 2 | no | maximal |
| 6 | D4×A4 | — | 4 | 3 | no | maximal |
| 7 | #585 | — | 3 | 4 | no | — |
| 8 | #585 | — | 3 | 2 | no | — |
| 9 | #585 | — | 3 | 3 | no | — |
| 10 | #585 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 11 | C2×H(5,𝔽2) | — | 3 | 2 | no | — |
| 12 | C2×H(5,𝔽2) | — | 3 | 3 | no | — |
| 13 | C2×H(5,𝔽2) | — | 3 | 6 | no | — |
| 14 | C2×H(5,𝔽2) | C2×C3 | 1 | 2 | yes | complemented |
| 15 | C22×A4 | — | 4 | 6 | no | — |
| 16 | #41⋊C3 | — | 4 | 2 | no | — |
| 17 | C2×SL(2,𝔽3) | — | 4 | 4 | no | — |
| 18 | C4×A4 | — | 4 | 3 | no | — |
| 19 | 2−1+4 | — | 3 | 6 | no | — |
| 20 | 2−1+4 | — | 3 | 6 | no | — |
| 21 | 2−1+4 | — | 3 | 4 | no | — |
| 22 | 2−1+4 | — | 3 | 2 | no | — |
| 23 | 2−1+4 | A4 | 1 | 2 | yes | complemented |
| 24 | H(5,𝔽2) | — | 3 | 4 | no | — |
| 25 | H(5,𝔽2) | — | 3 | 12 | no | — |
| 26 | H(5,𝔽2) | — | 3 | 6 | no | — |
| 27 | H(5,𝔽2) | — | 3 | 12 | no | — |
| 28 | H(5,𝔽2) | — | 3 | 12 | no | — |
| 29 | H(5,𝔽2) | — | 3 | 24 | no | — |
| 30 | H(5,𝔽2) | — | 3 | 6 | no | — |
| 31 | H(5,𝔽2) | — | 3 | 6 | no | — |
| 32 | H(5,𝔽2) | — | 3 | 2 | no | — |
| 33 | H(5,𝔽2) | — | 3 | 3 | no | — |
| 34 | H(5,𝔽2) | — | 3 | 6 | no | — |
| 35 | H(5,𝔽2) | C22×C3 | 1 | 1 | yes | characteristic |
| 36 | C2×#41 | — | 3 | 6 | no | — |
| 37 | C2×#41 | — | 3 | 24 | no | — |
| 38 | C2×#41 | — | 3 | 12 | no | — |
| 39 | C2×#41 | — | 3 | 4 | no | — |
| 40 | C2×#41 | — | 3 | 3 | no | — |
| 41 | C2×#41 | — | 3 | 6 | no | — |
| 42 | C2×#41 | — | 3 | 3 | no | — |
| 43 | C2×#41 | — | 3 | 2 | no | — |
| 44 | C2×#41 | — | 3 | 4 | no | — |
| 45 | C2×#41 | — | 3 | 3 | no | — |
| 46 | C2×#41 | — | 3 | 3 | no | — |
| 47 | C22×D4 | — | 3 | 12 | no | — |
| 48 | C22×D4 | — | 3 | 3 | no | — |
| 49 | C22×D4 | — | 3 | 6 | no | — |
| 50 | C22×D4 | — | 3 | 1 | no | — |
| 51 | C22×D4 | — | 3 | 6 | no | — |
| 52 | C22×D4 | — | 3 | 6 | no | — |
| 53 | C22×D4 | A4 | 1 | 3 | yes | complemented |
| 54 | C2×A4 | — | 4 | 3 | no | — |
| 55 | C2×A4 | — | 8 | 6 | no | — |
| 56 | C3×D4 | — | 16 | 1 | no | — |
| 57 | SL(2,𝔽3) | — | 4 | 2 | no | — |
| 58 | C24 | — | 3 | 6 | no | — |
| 59 | C24 | — | 3 | 2 | no | — |
| 60 | C24 | C2×A4 | 1 | 6 | yes | complemented |
| 61 | #41 | — | 3 | 6 | no | — |
| 62 | #41 | — | 3 | 12 | no | — |
| 63 | #41 | — | 3 | 12 | no | — |
| 64 | #41 | — | 3 | 12 | no | — |
| 65 | #41 | — | 3 | 24 | no | — |
| 66 | #41 | — | 3 | 24 | no | — |
| 67 | #41 | — | 3 | 12 | no | — |
| 68 | #41 | — | 3 | 12 | no | — |
| 69 | #41 | — | 3 | 12 | no | — |
| 70 | #41 | — | 3 | 6 | no | — |
| 71 | #41 | — | 3 | 12 | no | — |
| 72 | #41 | — | 3 | 12 | no | — |
| 73 | #41 | — | 3 | 12 | no | — |
| 74 | #41 | — | 3 | 6 | no | — |
| 75 | #41 | — | 3 | 2 | no | — |
| 76 | #41 | — | 3 | 4 | no | — |
| 77 | #41 | — | 3 | 2 | no | — |
| 78 | #41 | — | 3 | 4 | no | — |
| 79 | #41 | C2×A4 | 1 | 2 | yes | complemented |
| 80 | C2×Q8 | — | 3 | 2 | no | — |
| 81 | C2×Q8 | — | 3 | 12 | no | — |
| 82 | C2×Q8 | — | 3 | 3 | no | — |
| 83 | C2×Q8 | — | 3 | 3 | no | — |
| 84 | C2×Q8 | — | 3 | 2 | no | — |
| 85 | C2×Q8 | C2×A4 | 1 | 4 | yes | complemented |
| 86 | C2×D4 | — | 3 | 12 | no | — |
| 87 | C2×D4 | — | 3 | 6 | no | — |
| 88 | C2×D4 | — | 3 | 12 | no | — |
| 89 | C2×D4 | — | 3 | 24 | no | — |
| 90 | C2×D4 | — | 3 | 12 | no | — |
| 91 | C2×D4 | — | 3 | 24 | no | — |
| 92 | C2×D4 | — | 3 | 6 | no | — |
| 93 | C2×D4 | — | 3 | 24 | no | — |
| 94 | C2×D4 | — | 3 | 6 | no | — |
| 95 | C2×D4 | — | 3 | 6 | no | — |
| 96 | C2×D4 | — | 3 | 24 | no | — |
| 97 | C2×D4 | — | 3 | 6 | no | — |
| 98 | C2×D4 | — | 3 | 12 | no | — |
| 99 | C2×D4 | — | 3 | 6 | no | — |
| 100 | C2×D4 | — | 3 | 3 | no | — |
| 101 | C2×D4 | — | 3 | 3 | no | — |
| 102 | C2×D4 | — | 3 | 3 | no | — |
| 103 | C2×D4 | — | 3 | 6 | no | — |
| 104 | C2×D4 | — | 3 | 3 | no | — |
| 105 | C2×D4 | — | 3 | 3 | no | — |
| 106 | C2×D4 | — | 3 | 3 | no | — |
| 107 | C2×D4 | — | 3 | 6 | no | — |
| 108 | C22×C4 | — | 3 | 12 | no | — |
| 109 | C22×C4 | — | 3 | 6 | no | — |
| 110 | C22×C4 | — | 3 | 6 | no | — |
| 111 | C22×C4 | — | 3 | 6 | no | — |
| 112 | C22×C4 | — | 3 | 3 | no | — |
| 113 | C22×C4 | — | 3 | 1 | no | — |
| 114 | C22×C4 | C2×A4 | 1 | 3 | yes | complemented |
| 115 | C22×C3 | — | 16 | 2 | no | — |
| 116 | A4 | — | 4 | 3 | no | — |
| 117 | C3×C4 | — | 16 | 1 | no | — |
| 118 | C23 | — | 3 | 6 | no | — |
| 119 | C23 | — | 3 | 12 | no | — |
| 120 | C23 | — | 3 | 1 | no | — |
| 121 | C23 | — | 3 | 3 | no | — |
| 122 | C23 | — | 3 | 6 | no | — |
| 123 | C23 | — | 3 | 6 | no | — |
| 124 | C23 | — | 8 | 6 | no | — |
| 125 | C23 | — | 24 | 2 | no | — |
| 126 | C23 | — | 24 | 6 | no | — |
| 127 | C23 | C22×A4 | 1 | 3 | yes | — |
| 128 | Q8 | — | 3 | 4 | no | — |
| 129 | Q8 | — | 3 | 6 | no | — |
| 130 | Q8 | — | 3 | 6 | no | — |
| 131 | Q8 | — | 3 | 2 | no | — |
| 132 | Q8 | C22×A4 | 1 | 2 | yes | — |
| 133 | D4 | — | 3 | 2 | no | — |
| 134 | D4 | — | 3 | 24 | no | — |
| 135 | D4 | — | 3 | 12 | no | — |
| 136 | D4 | — | 3 | 12 | no | — |
| 137 | D4 | — | 3 | 6 | no | — |
| 138 | D4 | — | 3 | 6 | no | — |
| 139 | D4 | — | 3 | 6 | no | — |
| 140 | D4 | — | 3 | 12 | no | — |
| 141 | D4 | — | 3 | 4 | no | — |
| 142 | D4 | — | 3 | 6 | no | — |
| 143 | D4 | — | 3 | 3 | no | — |
| 144 | D4 | C22⋊A4 | 1 | 1 | yes | characteristic |
| 145 | C2×C4 | — | 3 | 6 | no | — |
| 146 | C2×C4 | — | 3 | 4 | no | — |
| 147 | C2×C4 | — | 3 | 12 | no | — |
| 148 | C2×C4 | — | 3 | 24 | no | — |
| 149 | C2×C4 | — | 3 | 3 | no | — |
| 150 | C2×C4 | — | 3 | 6 | no | — |
| 151 | C2×C4 | — | 3 | 3 | no | — |
| 152 | C2×C4 | — | 3 | 4 | no | — |
| 153 | C2×C4 | — | 3 | 2 | no | — |
| 154 | C2×C4 | — | 3 | 3 | no | — |
| 155 | C2×C4 | — | 3 | 3 | no | — |
| 156 | C2×C3 | — | 16 | 1 | no | — |
| 157 | C2×C3 | — | 32 | 2 | no | — |
| 158 | C22 | — | 3 | 6 | no | — |
| 159 | C22 | — | 3 | 3 | no | — |
| 160 | C22 | — | 3 | 2 | no | — |
| 161 | C22 | — | 4 | 3 | no | — |
| 162 | C22 | — | 12 | 6 | no | — |
| 163 | C22 | — | 12 | 6 | no | — |
| 164 | C22 | — | 12 | 1 | no | — |
| 165 | C22 | — | 12 | 6 | no | — |
| 166 | C22 | — | 12 | 3 | no | — |
| 167 | C22 | — | 12 | 12 | no | — |
| 168 | C22 | C2×C22⋊A4 | 1 | 2 | yes | — |
| 169 | C4 | — | 3 | 4 | no | — |
| 170 | C4 | — | 3 | 2 | no | — |
| 171 | C4 | — | 3 | 3 | no | — |
| 172 | C4 | C2×C22⋊A4 | 1 | 1 | yes | characteristic |
| 173 | C3 | — | 16 | 1 | no | Sylow |
| 174 | C2 | — | 2 | 2 | no | — |
| 175 | C2 | — | 6 | 3 | no | — |
| 176 | C2 | — | 6 | 6 | no | — |
| 177 | C2 | — | 6 | 2 | no | — |
| 178 | C2 | C22×C22⋊A4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C2 | C22×C22⋊A4 |
| commutator, cosocle, hypocenter | H(5,𝔽2) | C22×C3 |
| Fitting | H(7,𝔽2) | C3 |
| hypercenter | D4 | C22⋊A4 |
| LCS | G ▷ H(5,𝔽2) | |
| UCS | C1 ◁ C2 ◁ D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #5950 | C2 | semidirect |
| 2. C2×#997 | C2 | semidirect |
| 3. H(7,𝔽2) | C3 | semidirect |
| 4. #997 | C22 | semidirect |
| 10. #585 | C2×C3 | semidirect |
| 14. C2×H(5,𝔽2) | C2×C3 | semidirect |
| 23. 2−1+4 | A4 | semidirect |
| 53. C22×D4 | A4 | semidirect |
| 60. C24 | C2×A4 | semidirect |
| 79. #41 | C2×A4 | semidirect |
| 85. C2×Q8 | C2×A4 | semidirect |
| 114. C22×C4 | C2×A4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #5950 | 2 |
| C2×#997 | 2 |
| H(7,𝔽2) | 3 |
| 2−1+4⋊C3 | 4 |
| D4×A4 | 4 |