Finite Group Databasebeta

← #30637870

Subgroups — #30637870

Order 16256.

206 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 65 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550255C211yesmaximal, characteristic
2#22550171C212yesmaximal
3#22549625C211yesmaximal, characteristic
4Q8⋊(C127⋊C8)C212yesmaximal
5Q82×C127C211yesmaximal, characteristic
6C2×#14523238—22no—
7#16861209C2211yescharacteristic
8#16861142—22no—
9C4⋊(C127⋊C8)C2212yes—
10#16861133—22no—
11C4×C127⋊C8C2211yescharacteristic
12C4⋊Q8×C127—22no—
13C4⋊Q8×C127—22no—
14C4⋊Q8×C127C2211yescharacteristic
15C4×Q8×C127—22no—
16C4×Q8×C127C2212yes—
17C2×Dic254—21no—
18#14523238—44no—
19C2×C127⋊C8—21no—
20C2×C127⋊C8—22no—
21C4⋊Dic127—22no—
22C2×Q8×C127—22no—
23C2×Q8×C127D412yes—
24C2×Q8×C127D412yes—
25C4⋊C4×C127—24no—
26C4⋊C4×C127—22no—
27C4⋊C4×C127—22no—
28C4⋊C4×C127D412yes—
29C42×C127—22no—
30C42×C127—22no—
31C42×C127C2311yescharacteristic
32Dic254—42no—
33C2×Dic127—41no—
34C127⋊C8—22no—
35C127⋊C8—42no—
36Q8×C127—22no—
37Q8×C127—42no—
38Q8×C127—82no—
39Q8×C127Q1614yescomplemented
40C2×C4×C127—22no—
41C2×C4×C127—22no—
42C2×C4×C127—22no—
43C2×C4×C127C2×D411yescharacteristic
44C2×C4×C127C2×D412yes—
45C22×C127#12011yescharacteristic
46C4×C127—24no—
47C4×C127—21no—
48C4×C127—42no—
49C4×C127—42no—
50C4×C127C2×Q1612yes—
51C4×C127#13712yes—
52Dic127—81no—
53C2×C127#45112yes—
54C2×C127#45311yescharacteristic
55#1633—1271nomaximal, Sylow
56C127#163311yescharacteristic, complemented, Sylow
57Q82D12711yescharacteristic
58#494—1271no—
59#477—1272no—
60#333—1271no—
61Q8⋊C8—1272no—
62C2×Q16—2542no—
63C4⋊Q8—22no—
64C4⋊Q8—22no—
65C4⋊Q8—1271no—
66C4⋊Q8C2×D12711yescharacteristic
67C4×Q8—22no—
68C4×Q8C2×D12712yes—
69#107—2542no—
70C4⋊C8—1272no—
71#103—2542no—
72C4×C8—1271no—
73C2×Q8—22no—
74C2×Q8—2541no—
75C2×Q8C22⋊D12712yes—
76C2×Q8C22⋊D12712yes—
77Q16—5084no—
78C2×C8—2542no—
79C2×C8—2541no—
80C4⋊C4—24no—
81C4⋊C4—22no—
82C4⋊C4—22no—
83C4⋊C4—2542no—
84C4⋊C4C22⋊D12712yes—
85C42—22no—
86C42—22no—
87C42C22×D12711yescharacteristic
88Q8—22no—
89Q8—42no—
90Q8—82no—
91Q8—5082no—
92Q8#1452323814yescomplemented
93C2×C4—22no—
94C2×C4—22no—
95C2×C4—22no—
96C2×C4—5081no—
97C2×C4C2×C22⋊D12711yescharacteristic
98C2×C4C2×C22⋊D12712yes—
99C8—5082no—
100C8—2542no—
101C22#1686117811yescharacteristic
102C4—24no—
103C4—21no—
104C4—42no—
105C4—42no—
106C4—10161no—
107C4C2×#1452323812yes—
108C4#1686124012yes—
109C2#2255003212yes—
110C2#2255004411yescharacteristic
subgroupG/H
centerC22#16861178
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
FittingQ82×C127C2
socle, prime radicalC22×C127#120
hypercenterQ82D127
hypocenterC127#1633
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ Q82

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
39. Q8×C127Q16semidirect
56. C127#1633semidirect
92. Q8#14523238semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#225502552
#225501712
#225496252
Q8⋊(C127⋊C8)2
Q82×C1272
#1633127