Finite Group DatabasebetaFinite Group DatabasebetaOrder 2176.
852 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 852 conjugacy classes of subgroups — too many to draw. Only its 119 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×C22⋊#3633 | C2 | 1 | 6 | yes | maximal, complemented |
| 2 | C24×#595 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C24×D17 | C4 | 1 | 1 | yes | characteristic |
| 4 | C23×#595 | — | 2 | 12 | no | — |
| 5 | C23×#595 | C4 | 1 | 3 | yes | — |
| 6 | C23×#595 | C22 | 1 | 3 | yes | — |
| 7 | C22×#3633 | C22 | 1 | 4 | yes | complemented |
| 8 | C22⋊#3633 | — | 2 | 12 | no | — |
| 9 | C24×C17 | C8 | 1 | 1 | yes | characteristic, complemented |
| 10 | C23×D17 | — | 2 | 6 | no | — |
| 11 | C23×D17 | — | 2 | 6 | no | — |
| 12 | C23×D17 | C8 | 1 | 3 | yes | complemented |
| 13 | C23×D17 | C2×C4 | 1 | 3 | yes | — |
| 14 | C22×#595 | — | 2 | 32 | no | — |
| 15 | C22×#595 | — | 2 | 6 | no | — |
| 16 | C22×#595 | — | 2 | 12 | no | — |
| 17 | C22×#595 | — | 2 | 12 | no | — |
| 18 | C22×#595 | C2×C4 | 1 | 3 | yes | — |
| 19 | C22×#595 | D4 | 1 | 12 | yes | — |
| 20 | C22×#595 | C23 | 1 | 1 | yes | characteristic |
| 21 | C2×#3633 | — | 2 | 12 | no | — |
| 22 | C23×C17 | — | 2 | 6 | no | — |
| 23 | C23×C17 | C2×C8 | 1 | 3 | yes | — |
| 24 | C22×D17 | — | 2 | 8 | no | — |
| 25 | C22×D17 | — | 2 | 24 | no | — |
| 26 | C22×D17 | — | 2 | 6 | no | — |
| 27 | C22×D17 | — | 2 | 12 | no | — |
| 28 | C22×D17 | — | 2 | 6 | no | — |
| 29 | C22×D17 | — | 2 | 6 | no | — |
| 30 | C22×D17 | C22⋊C4 | 1 | 6 | yes | — |
| 31 | C22×D17 | C2×C8 | 1 | 3 | yes | — |
| 32 | C22×D17 | M16 | 1 | 6 | yes | — |
| 33 | C22×D17 | C22×C4 | 1 | 1 | yes | characteristic |
| 34 | C2×#595 | — | 2 | 24 | no | — |
| 35 | C2×#595 | — | 2 | 6 | no | — |
| 36 | C2×#595 | — | 2 | 24 | no | — |
| 37 | C2×#595 | C22⋊C4 | 1 | 6 | yes | — |
| 38 | C2×#595 | C2×D4 | 1 | 6 | yes | — |
| 39 | #3633 | — | 4 | 4 | no | — |
| 40 | #1779 | — | 17 | 1 | no | maximal, Sylow |
| 41 | C22×C17 | — | 2 | 8 | no | — |
| 42 | C22×C17 | — | 2 | 6 | no | — |
| 43 | C22×C17 | C22⋊C8 | 1 | 6 | yes | complemented |
| 44 | C22×C17 | C22×C8 | 1 | 1 | yes | characteristic |
| 45 | C2×D17 | — | 2 | 24 | no | — |
| 46 | C2×D17 | — | 2 | 6 | no | — |
| 47 | C2×D17 | — | 2 | 6 | no | — |
| 48 | C2×D17 | — | 2 | 6 | no | — |
| 49 | C2×D17 | — | 2 | 12 | no | — |
| 50 | C2×D17 | C22⋊C8 | 1 | 6 | yes | complemented |
| 51 | C2×D17 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 52 | C2×D17 | C2×M16 | 1 | 3 | yes | — |
| 53 | #595 | — | 2 | 6 | no | — |
| 54 | #595 | #120 | 1 | 4 | yes | — |
| 55 | C24×C4 | — | 17 | 1 | no | — |
| 56 | C2×C22⋊C8 | — | 17 | 6 | no | — |
| 57 | C2×C17 | — | 2 | 6 | no | — |
| 58 | C2×C17 | C2×C22⋊C8 | 1 | 3 | yes | — |
| 59 | D17 | — | 2 | 6 | no | — |
| 60 | D17 | #379 | 1 | 1 | yes | characteristic |
| 61 | D17 | #407 | 1 | 3 | yes | — |
| 62 | C25 | — | 17 | 1 | no | — |
| 63 | C23×C4 | — | 17 | 3 | no | — |
| 64 | C23×C4 | — | 17 | 3 | no | — |
| 65 | C23×C4 | — | 34 | 12 | no | — |
| 66 | C22×C8 | — | 17 | 4 | no | — |
| 67 | C22⋊C8 | — | 34 | 12 | no | — |
| 68 | C17 | #1779 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 69 | C24 | — | 17 | 3 | no | — |
| 70 | C24 | — | 17 | 3 | no | — |
| 71 | C24 | — | 34 | 6 | no | — |
| 72 | C24 | — | 34 | 6 | no | — |
| 73 | C24 | #3633 | 1 | 1 | yes | characteristic, complemented |
| 74 | C22×C4 | — | 17 | 1 | no | — |
| 75 | C22×C4 | — | 17 | 3 | no | — |
| 76 | C22×C4 | — | 17 | 12 | no | — |
| 77 | C22×C4 | — | 34 | 12 | no | — |
| 78 | C22×C4 | — | 34 | 12 | no | — |
| 79 | C22×C4 | — | 34 | 6 | no | — |
| 80 | C22×C4 | — | 34 | 32 | no | — |
| 81 | C2×C8 | — | 34 | 12 | no | — |
| 82 | C23 | — | 2 | 6 | no | — |
| 83 | C23 | — | 17 | 1 | no | — |
| 84 | C23 | — | 17 | 6 | no | — |
| 85 | C23 | — | 17 | 6 | no | — |
| 86 | C23 | — | 17 | 3 | no | — |
| 87 | C23 | — | 34 | 6 | no | — |
| 88 | C23 | — | 34 | 6 | no | — |
| 89 | C23 | — | 34 | 12 | no | — |
| 90 | C23 | — | 34 | 8 | no | — |
| 91 | C23 | — | 34 | 24 | no | — |
| 92 | C23 | — | 34 | 6 | no | — |
| 93 | C23 | C2×#3633 | 1 | 3 | yes | — |
| 94 | C2×C4 | — | 17 | 6 | no | — |
| 95 | C2×C4 | — | 17 | 6 | no | — |
| 96 | C2×C4 | — | 34 | 24 | no | — |
| 97 | C2×C4 | — | 34 | 6 | no | — |
| 98 | C2×C4 | — | 34 | 24 | no | — |
| 99 | C8 | — | 68 | 4 | no | — |
| 100 | C22 | — | 2 | 8 | no | — |
| 101 | C22 | — | 2 | 6 | no | — |
| 102 | C22 | — | 17 | 3 | no | — |
| 103 | C22 | — | 17 | 3 | no | — |
| 104 | C22 | — | 17 | 6 | no | — |
| 105 | C22 | — | 34 | 6 | no | — |
| 106 | C22 | — | 34 | 6 | no | — |
| 107 | C22 | — | 34 | 12 | no | — |
| 108 | C22 | — | 34 | 24 | no | — |
| 109 | C22 | — | 34 | 6 | no | — |
| 110 | C22 | C22⋊#3633 | 1 | 6 | yes | complemented |
| 111 | C22 | C22×#3633 | 1 | 1 | yes | characteristic |
| 112 | C4 | — | 17 | 4 | no | — |
| 113 | C4 | — | 34 | 6 | no | — |
| 114 | C2 | — | 2 | 6 | no | — |
| 115 | C2 | — | 17 | 1 | no | — |
| 116 | C2 | — | 17 | 3 | no | — |
| 117 | C2 | — | 34 | 6 | no | — |
| 118 | C2 | C2×C22⋊#3633 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×#3633 |
| commutator, socle | C22×C17 | C22×C8 |
| Fitting | C24×C17 | C8 |
| prime radical | C24×D17 | C4 |
| cosocle | C22×#595 | C23 |
| hypercenter | C24 | #3633 |
| hypocenter | C17 | #1779 |
| LCS | G ▷ C22×C17 ▷ C17 | |
| UCS | C1 ◁ C22 ◁ C24 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C22⋊#3633 | C2 | semidirect |
| 7. C22×#3633 | C22 | semidirect |
| 9. C24×C17 | C8 | semidirect |
| 12. C23×D17 | C8 | semidirect |
| 43. C22×C17 | C22⋊C8 | semidirect |
| 50. C2×D17 | C22⋊C8 | semidirect |
| 68. C17 | #1779 | semidirect |
| 73. C24 | #3633 | semidirect |
| 110. C22 | C22⋊#3633 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C22⋊#3633 | 2 |
| C24×#595 | 2 |
| #1779 | 17 |