Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
158 conjugacy classes of subgroups; only the 34 normal subgroups are shown.
This group has 158 conjugacy classes of subgroups — too many to draw. Only its 34 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 | #543 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #450 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #392 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #328 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #323 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C22×D4 | — | 2 | 1 | no | — |
| 7 | C22×D4 | C22 | 1 | 1 | yes | characteristic |
| 8 | C2×SD16 | — | 2 | 2 | no | — |
| 9 | C22⋊Q8 | — | 2 | 2 | no | — |
| 10 | C22⋊Q8 | — | 2 | 1 | no | — |
| 11 | C22⋊Q8 | — | 2 | 2 | no | — |
| 12 | C22⋊Q8 | C22 | 1 | 2 | yes | — |
| 13 | C2×C22⋊C4 | — | 2 | 1 | no | — |
| 14 | C2×C22⋊C4 | — | 2 | 1 | no | — |
| 15 | C2×C22⋊C4 | C22 | 1 | 1 | yes | characteristic |
| 16 | #102 | — | 2 | 2 | no | — |
| 17 | C22⋊C8 | C22 | 1 | 2 | yes | — |
| 18 | #95 | C22 | 1 | 1 | yes | characteristic |
| 19 | C24 | — | 2 | 1 | no | — |
| 20 | C24 | D4 | 1 | 1 | yes | characteristic |
| 21 | C2×Q8 | — | 2 | 1 | no | — |
| 22 | C2×Q8 | D4 | 1 | 2 | yes | complemented |
| 23 | C2×D4 | — | 2 | 2 | no | — |
| 24 | C2×D4 | — | 4 | 2 | no | — |
| 25 | C2×D4 | — | 4 | 2 | no | — |
| 26 | C2×D4 | — | 4 | 2 | no | — |
| 27 | C2×D4 | — | 4 | 2 | no | — |
| 28 | C22×C4 | — | 2 | 1 | no | — |
| 29 | C22×C4 | — | 2 | 1 | no | — |
| 30 | C22×C4 | — | 2 | 1 | no | — |
| 31 | C22×C4 | D4 | 1 | 1 | yes | characteristic |
| 32 | C22×C4 | C23 | 1 | 1 | yes | characteristic |
| 33 | SD16 | — | 4 | 4 | no | — |
| 34 | C2×C8 | — | 2 | 2 | no | — |
| 35 | C4⋊C4 | — | 2 | 1 | no | — |
| 36 | C4⋊C4 | — | 2 | 2 | no | — |
| 37 | C4⋊C4 | — | 2 | 2 | no | — |
| 38 | C4⋊C4 | D4 | 1 | 2 | yes | complemented |
| 39 | C22⋊C4 | — | 2 | 2 | no | — |
| 40 | C22⋊C4 | — | 2 | 2 | no | — |
| 41 | C22⋊C4 | — | 2 | 2 | no | — |
| 42 | C22⋊C4 | — | 4 | 2 | no | — |
| 43 | C23 | — | 2 | 1 | no | — |
| 44 | C23 | — | 2 | 1 | no | — |
| 45 | C23 | — | 2 | 2 | no | — |
| 46 | C23 | — | 4 | 1 | no | — |
| 47 | C23 | — | 4 | 1 | no | — |
| 48 | C23 | — | 4 | 1 | no | — |
| 49 | C23 | — | 4 | 1 | no | — |
| 50 | C23 | — | 4 | 1 | no | — |
| 51 | C23 | — | 4 | 1 | no | — |
| 52 | C23 | — | 4 | 1 | no | — |
| 53 | C23 | — | 4 | 1 | no | — |
| 54 | C23 | SD16 | 1 | 2 | yes | complemented |
| 55 | C23 | C2×D4 | 1 | 1 | yes | characteristic |
| 56 | Q8 | — | 4 | 2 | no | — |
| 57 | Q8 | — | 8 | 1 | no | — |
| 58 | D4 | — | 2 | 4 | no | — |
| 59 | D4 | — | 4 | 2 | no | — |
| 60 | D4 | — | 4 | 8 | no | — |
| 61 | C2×C4 | — | 2 | 2 | no | — |
| 62 | C2×C4 | — | 2 | 2 | no | — |
| 63 | C2×C4 | — | 2 | 1 | no | — |
| 64 | C2×C4 | — | 2 | 2 | no | — |
| 65 | C2×C4 | — | 4 | 1 | no | — |
| 66 | C2×C4 | — | 4 | 1 | no | — |
| 67 | C2×C4 | — | 4 | 1 | no | — |
| 68 | C2×C4 | — | 4 | 1 | no | — |
| 69 | C2×C4 | — | 8 | 1 | no | — |
| 70 | C2×C4 | — | 8 | 1 | no | — |
| 71 | C2×C4 | C2×D4 | 1 | 2 | yes | — |
| 72 | C8 | — | 4 | 2 | no | — |
| 73 | C22 | — | 2 | 1 | no | — |
| 74 | C22 | — | 2 | 1 | no | — |
| 75 | C22 | — | 2 | 2 | no | — |
| 76 | C22 | — | 4 | 2 | no | — |
| 77 | C22 | — | 4 | 2 | no | — |
| 78 | C22 | — | 4 | 2 | no | — |
| 79 | C22 | — | 4 | 1 | no | — |
| 80 | C22 | — | 4 | 1 | no | — |
| 81 | C22 | — | 4 | 2 | no | — |
| 82 | C22 | — | 8 | 1 | no | — |
| 83 | C22 | — | 8 | 1 | no | — |
| 84 | C22 | — | 8 | 1 | no | — |
| 85 | C22 | — | 8 | 1 | no | — |
| 86 | C22 | — | 8 | 1 | no | — |
| 87 | C22 | #120 | 1 | 1 | yes | characteristic |
| 88 | C22 | C2×SD16 | 1 | 1 | yes | characteristic |
| 89 | C22 | #137 | 1 | 1 | yes | characteristic |
| 90 | C4 | — | 2 | 2 | no | — |
| 91 | C4 | — | 4 | 2 | no | — |
| 92 | C4 | — | 4 | 1 | no | — |
| 93 | C4 | — | 4 | 2 | no | — |
| 94 | C4 | — | 4 | 2 | no | — |
| 95 | C2 | — | 2 | 1 | no | — |
| 96 | C2 | — | 2 | 1 | no | — |
| 97 | C2 | — | 4 | 2 | no | — |
| 98 | C2 | — | 8 | 2 | no | — |
| 99 | C2 | #448 | 1 | 1 | yes | characteristic |
| 100 | C2 | #453 | 1 | 1 | yes | characteristic |
| 101 | C2 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #120 |
| commutator, Frattini, cosocle | C22×C4 | C23 |
| prime radical | #392 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #543 | C2 | semidirect |
| 2. #450 | C2 | semidirect |
| 4. #328 | C2 | semidirect |
| 5. #323 | C2 | semidirect |
| 22. C2×Q8 | D4 | semidirect |
| 38. C4⋊C4 | D4 | semidirect |
| 54. C23 | SD16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #543 | 2 |
| #450 | 2 |
| #392 | 2 |
| #328 | 2 |
| #323 | 2 |