Finite Group DatabasebetaFinite Group DatabasebetaOrder 1152.
2324 conjugacy classes of subgroups; only the 337 normal subgroups are shown.
This group has 2324 conjugacy classes of subgroups — too many to draw. Only its 337 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 | #10593120 | C2 | 1 | 10 | yes | maximal, complemented |
| 2 | C2×#63700 | C2 | 1 | 5 | yes | maximal, complemented |
| 3 | C3×H(7,𝔽2) | C3 | 1 | 1 | yes | maximal, characteristic |
| 4 | C9×2−1+4 | — | 4 | 1 | no | maximal |
| 5 | 2−1+4⋊C9 | C22 | 1 | 10 | yes | complemented |
| 6 | #63700 | C22 | 1 | 10 | yes | complemented |
| 7 | C2×#41⋊C9 | C22 | 1 | 15 | yes | complemented |
| 8 | C3×#585 | — | 3 | 5 | no | — |
| 9 | C3×#585 | — | 3 | 1 | no | — |
| 10 | C3×#585 | C2×C3 | 1 | 10 | yes | — |
| 11 | C2×C3×H(5,𝔽2) | — | 3 | 10 | no | — |
| 12 | C2×C3×H(5,𝔽2) | C2×C3 | 1 | 5 | yes | — |
| 13 | C9×#41 | — | 4 | 10 | no | — |
| 14 | C2×Q8×C9 | — | 4 | 5 | no | — |
| 15 | #41⋊C9 | C23 | 1 | 10 | yes | complemented |
| 16 | C2×Q8⋊C9 | C23 | 1 | 5 | yes | complemented |
| 17 | H(7,𝔽2) | C9 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 18 | C3×2−1+4 | — | 3 | 5 | no | — |
| 19 | C3×2−1+4 | — | 3 | 10 | no | — |
| 20 | C3×2−1+4 | A4 | 1 | 1 | yes | characteristic |
| 21 | C3×2−1+4 | C22×C3 | 1 | 10 | yes | — |
| 22 | C3×H(5,𝔽2) | — | 3 | 20 | no | — |
| 23 | C3×H(5,𝔽2) | — | 3 | 30 | no | — |
| 24 | C3×H(5,𝔽2) | — | 3 | 10 | no | — |
| 25 | C3×H(5,𝔽2) | — | 3 | 30 | no | — |
| 26 | C3×H(5,𝔽2) | C22×C3 | 1 | 10 | yes | — |
| 27 | C2×C3×#41 | — | 3 | 30 | no | — |
| 28 | C2×C3×#41 | — | 3 | 20 | no | — |
| 29 | C2×C3×#41 | — | 3 | 10 | no | — |
| 30 | C2×C3×#41 | — | 3 | 5 | no | — |
| 31 | C2×C3×#41 | C22×C3 | 1 | 15 | yes | — |
| 32 | C22×C3×D4 | — | 3 | 20 | no | — |
| 33 | C22×C3×D4 | — | 3 | 15 | no | — |
| 34 | Q8×C9 | — | 4 | 10 | no | — |
| 35 | D4×C9 | — | 4 | 10 | no | — |
| 36 | C2×C4×C9 | — | 4 | 15 | no | — |
| 37 | Q8⋊C9 | C24 | 1 | 1 | yes | characteristic |
| 38 | #585 | — | 3 | 5 | no | — |
| 39 | #585 | — | 3 | 1 | no | — |
| 40 | #585 | C2×C9 | 1 | 10 | yes | complemented |
| 41 | C2×H(5,𝔽2) | — | 3 | 10 | no | — |
| 42 | C2×H(5,𝔽2) | C2×C9 | 1 | 5 | yes | complemented |
| 43 | C24×C3 | — | 3 | 10 | no | — |
| 44 | C3×#41 | — | 3 | 60 | no | — |
| 45 | C3×#41 | — | 3 | 10 | no | — |
| 46 | C3×#41 | — | 3 | 20 | no | — |
| 47 | C3×#41 | — | 3 | 10 | no | — |
| 48 | C3×#41 | — | 3 | 20 | no | — |
| 49 | C3×#41 | — | 3 | 30 | no | — |
| 50 | C3×#41 | — | 3 | 30 | no | — |
| 51 | C3×#41 | C2×A4 | 1 | 10 | yes | — |
| 52 | C3×#41 | C23×C3 | 1 | 10 | yes | — |
| 53 | C2×C3×Q8 | — | 3 | 10 | no | — |
| 54 | C2×C3×Q8 | — | 3 | 10 | no | — |
| 55 | C2×C3×Q8 | C2×A4 | 1 | 5 | yes | — |
| 56 | C2×C3×Q8 | C23×C3 | 1 | 5 | yes | — |
| 57 | C2×C3×D4 | — | 3 | 15 | no | — |
| 58 | C2×C3×D4 | — | 3 | 60 | no | — |
| 59 | C2×C3×D4 | — | 3 | 60 | no | — |
| 60 | C2×C3×D4 | — | 3 | 30 | no | — |
| 61 | C2×C3×D4 | — | 3 | 30 | no | — |
| 62 | C2×C3×D4 | — | 3 | 15 | no | — |
| 63 | C22×C3×C4 | — | 3 | 20 | no | — |
| 64 | C22×C3×C4 | — | 3 | 15 | no | — |
| 65 | C22×C9 | — | 4 | 5 | no | — |
| 66 | C4×C9 | — | 4 | 10 | no | — |
| 67 | 2−1+4 | — | 3 | 5 | no | — |
| 68 | 2−1+4 | — | 3 | 10 | no | — |
| 69 | 2−1+4 | C22⋊C9 | 1 | 1 | yes | characteristic |
| 70 | 2−1+4 | C22×C9 | 1 | 10 | yes | — |
| 71 | H(5,𝔽2) | — | 3 | 20 | no | — |
| 72 | H(5,𝔽2) | — | 3 | 30 | no | — |
| 73 | H(5,𝔽2) | — | 3 | 10 | no | — |
| 74 | H(5,𝔽2) | — | 3 | 30 | no | — |
| 75 | H(5,𝔽2) | C22×C9 | 1 | 10 | yes | — |
| 76 | C2×#41 | — | 3 | 30 | no | — |
| 77 | C2×#41 | — | 3 | 20 | no | — |
| 78 | C2×#41 | — | 3 | 10 | no | — |
| 79 | C2×#41 | — | 3 | 5 | no | — |
| 80 | C2×#41 | C22×C9 | 1 | 15 | yes | — |
| 81 | C22×D4 | — | 3 | 20 | no | — |
| 82 | C22×D4 | — | 3 | 15 | no | — |
| 83 | C23×C3 | — | 3 | 20 | no | — |
| 84 | C23×C3 | — | 3 | 15 | no | — |
| 85 | C23×C3 | — | 24 | 10 | no | — |
| 86 | C3×Q8 | — | 3 | 5 | no | — |
| 87 | C3×Q8 | — | 3 | 10 | no | — |
| 88 | C3×Q8 | C22×A4 | 1 | 10 | yes | — |
| 89 | C3×Q8 | C24×C3 | 1 | 1 | yes | characteristic |
| 90 | C3×D4 | — | 3 | 10 | no | — |
| 91 | C3×D4 | — | 3 | 30 | no | — |
| 92 | C3×D4 | — | 3 | 20 | no | — |
| 93 | C3×D4 | — | 3 | 30 | no | — |
| 94 | C3×D4 | C22×A4 | 1 | 10 | yes | — |
| 95 | C2×C3×C4 | — | 3 | 20 | no | — |
| 96 | C2×C3×C4 | — | 3 | 5 | no | — |
| 97 | C2×C3×C4 | — | 3 | 30 | no | — |
| 98 | C2×C3×C4 | — | 3 | 10 | no | — |
| 99 | C2×C3×C4 | C22×A4 | 1 | 15 | yes | — |
| 100 | C2×C9 | — | 4 | 1 | no | — |
| 101 | C2×C9 | — | 8 | 5 | no | — |
| 102 | C24 | — | 3 | 10 | no | — |
| 103 | #41 | — | 3 | 60 | no | — |
| 104 | #41 | — | 3 | 10 | no | — |
| 105 | #41 | — | 3 | 20 | no | — |
| 106 | #41 | — | 3 | 10 | no | — |
| 107 | #41 | — | 3 | 20 | no | — |
| 108 | #41 | — | 3 | 30 | no | — |
| 109 | #41 | — | 3 | 30 | no | — |
| 110 | #41 | C2×C22⋊C9 | 1 | 10 | yes | — |
| 111 | #41 | C23×C9 | 1 | 10 | yes | — |
| 112 | C2×Q8 | — | 3 | 10 | no | — |
| 113 | C2×Q8 | — | 3 | 10 | no | — |
| 114 | C2×Q8 | C2×C22⋊C9 | 1 | 5 | yes | — |
| 115 | C2×Q8 | C23×C9 | 1 | 5 | yes | — |
| 116 | C2×D4 | — | 3 | 15 | no | — |
| 117 | C2×D4 | — | 3 | 60 | no | — |
| 118 | C2×D4 | — | 3 | 60 | no | — |
| 119 | C2×D4 | — | 3 | 30 | no | — |
| 120 | C2×D4 | — | 3 | 30 | no | — |
| 121 | C2×D4 | — | 3 | 15 | no | — |
| 122 | C22×C4 | — | 3 | 20 | no | — |
| 123 | C22×C4 | — | 3 | 15 | no | — |
| 124 | C22×C3 | — | 3 | 10 | no | — |
| 125 | C22×C3 | — | 12 | 15 | no | — |
| 126 | C22×C3 | — | 12 | 20 | no | — |
| 127 | C22×C3 | C23×A4 | 1 | 5 | yes | — |
| 128 | C3×C4 | — | 3 | 5 | no | — |
| 129 | C3×C4 | — | 3 | 1 | no | — |
| 130 | C3×C4 | C23×A4 | 1 | 10 | yes | — |
| 131 | C9 | — | 4 | 1 | no | Sylow |
| 132 | C23 | — | 3 | 20 | no | — |
| 133 | C23 | — | 3 | 15 | no | — |
| 134 | C23 | — | 24 | 10 | no | — |
| 135 | Q8 | — | 3 | 5 | no | — |
| 136 | Q8 | — | 3 | 10 | no | — |
| 137 | Q8 | C22×C22⋊C9 | 1 | 10 | yes | — |
| 138 | Q8 | C24×C9 | 1 | 1 | yes | characteristic |
| 139 | D4 | — | 3 | 10 | no | — |
| 140 | D4 | — | 3 | 30 | no | — |
| 141 | D4 | — | 3 | 20 | no | — |
| 142 | D4 | — | 3 | 30 | no | — |
| 143 | D4 | C22×C22⋊C9 | 1 | 10 | yes | — |
| 144 | C2×C4 | — | 3 | 20 | no | — |
| 145 | C2×C4 | — | 3 | 5 | no | — |
| 146 | C2×C4 | — | 3 | 10 | no | — |
| 147 | C2×C4 | — | 3 | 30 | no | — |
| 148 | C2×C4 | C22×C22⋊C9 | 1 | 15 | yes | — |
| 149 | C2×C3 | — | 2 | 5 | no | — |
| 150 | C2×C3 | — | 6 | 10 | no | — |
| 151 | C2×C3 | C24×A4 | 1 | 1 | yes | characteristic |
| 152 | C22 | — | 3 | 10 | no | — |
| 153 | C22 | — | 12 | 15 | no | — |
| 154 | C22 | — | 12 | 20 | no | — |
| 155 | C22 | C23×C22⋊C9 | 1 | 5 | yes | — |
| 156 | C4 | — | 3 | 1 | no | — |
| 157 | C4 | — | 3 | 5 | no | — |
| 158 | C4 | C23×C22⋊C9 | 1 | 10 | yes | — |
| 159 | C3 | #87536 | 1 | 1 | yes | characteristic |
| 160 | C2 | — | 2 | 5 | no | — |
| 161 | C2 | — | 6 | 10 | no | — |
| 162 | C2 | C24×C22⋊C9 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C2×C3 | C24×A4 |
| commutator, hypocenter | Q8 | C24×C9 |
| Fitting, prime radical | C3×H(7,𝔽2) | C3 |
| cosocle | C3×Q8 | C24×C3 |
| hypercenter | C3×2−1+4 | A4 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C2×C3 ◁ C3×2−1+4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #10593120 | C2 | semidirect |
| 2. C2×#63700 | C2 | semidirect |
| 5. 2−1+4⋊C9 | C22 | semidirect |
| 6. #63700 | C22 | semidirect |
| 7. C2×#41⋊C9 | C22 | semidirect |
| 15. #41⋊C9 | C23 | semidirect |
| 16. C2×Q8⋊C9 | C23 | semidirect |
| 17. H(7,𝔽2) | C9 | semidirect |
| 40. #585 | C2×C9 | semidirect |
| 42. C2×H(5,𝔽2) | C2×C9 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10593120 | 2 |
| C2×#63700 | 2 |
| C3×H(7,𝔽2) | 3 |
| C9×2−1+4 | 4 |