Finite Group Databasebeta

← #30637322

Subgroups — #30637322

Order 16256.

270 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 85 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
1C4×#14523232C212yesmaximal, complemented
2#22549616C214yesmaximal, complemented
3#532×C127C211yesmaximal, characteristic
4C2×#14523232—22no—
5#16861120C2214yes—
6C4×C127⋊C8—22no—
7C2×#41×C127C412yes—
8C4⋊Q8×C127C411yescharacteristic
9#127×C127C411yescharacteristic
10#124×C127C2212yes—
11#121×C127—22no—
12C4×D4×C127—22no—
13C2×C42×C127C2211yescharacteristic
14#14523232—28no—
15C2×C127⋊C8—24no—
16#41×C127—44no—
17C2×Q8×C127C2×C412yes—
18C2×D4×C127—22no—
19C2×D4×C127—22no—
20C2×D4×C127C2×C412yes—
21C22×C4×C127—22no—
22C22×C4×C127C2×C411yescharacteristic
23C22×C4×C127D412yes—
24C4⋊C4×C127—22no—
25C22⋊C4×C127—24no—
26C42×C127C2×C411yescharacteristic
27C42×C127D412yes—
28C42×C127C2311yescharacteristic
29C127⋊C8—28no—
30C23×C127—22no—
31C23×C127C22⋊C411yescharacteristic
32Q8×C127—42no—
33D4×C127—42no—
34D4×C127—44no—
35D4×C127—44no—
36C2×C4×C127—22no—
37C2×C4×C127—24no—
38C2×C4×C127—22no—
39C2×C4×C127—44no—
40C2×C4×C127C22⋊C412yes—
41C2×C4×C127C22⋊C411yescharacteristic
42C2×C4×C127C22×C411yescharacteristic
43C2×C4×C127C2×D412yes—
44C22×C127—21no—
45C22×C127—22no—
46C22×C127—42no—
47C22×C127—44no—
48C22×C127C2×C22⋊C411yescharacteristic
49C4×C127—21no—
50C4×C127—22no—
51C4×C127—42no—
52C4×C127#10012yes—
53C4×C127#10414yes—
54C2×C127—41no—
55C2×C127—82no—
56C2×C127C2×#10011yescharacteristic
57C2×C127C2×#10412yes—
58#1525—1271nomaximal, Sylow
59C127#152511yescharacteristic, complemented, Sylow
60#532D12711yescharacteristic
61C4×M16—1272no—
62#329—1274no—
63C2×#41Dic12712yes—
64C2×M16—2542no—
65C4⋊Q8Dic12711yescharacteristic
66#127Dic12711yescharacteristic
67#124C2×D12712yes—
68#121—22no—
69C4×D4—22no—
70C2×C42C2×D12711yescharacteristic
71#97—1274no—
72C4×C8—2542no—
73#41—44no—
74C2×Q8C2×Dic12712yes—
75C2×D4—22no—
76C2×D4—22no—
77C2×D4C2×Dic12712yes—
78C22×C4—22no—
79C22×C4C2×Dic12711yescharacteristic
80C22×C4C22⋊D12712yes—
81M16—2548no—
82C2×C8—2544no—
83C4⋊C4—22no—
84C22⋊C4—24no—
85C42C2×Dic12711yescharacteristic
86C42C22⋊D12712yes—
87C42C22×D12711yescharacteristic
88C23—22no—
89C23C22⋊Dic12711yescharacteristic
90Q8—42no—
91D4—42no—
92D4—44no—
93D4—44no—
94C2×C4—22no—
95C2×C4—24no—
96C2×C4—22no—
97C2×C4—44no—
98C2×C4C22⋊Dic12712yes—
99C2×C4C22⋊Dic12711yescharacteristic
100C2×C4C22×Dic12711yescharacteristic
101C2×C4C2×C22⋊D12712yes—
102C8—2548no—
103C22—21no—
104C22—22no—
105C22—42no—
106C22—44no—
107C22C2×C22⋊Dic12711yescharacteristic
108C4—21no—
109C4—22no—
110C4—42no—
111C4#1686112612yes—
112C4#1686113614yes—
113C2—41no—
114C2—82no—
115C2C2×#1686112611yescharacteristic
116C2C2×#1686113612yes—
subgroupG/H
centerC2×C4C22⋊Dic127
commutatorC2×C4×C127C22×C4
FrattiniC42C22×D127
Fitting, prime radical#532×C127C2
socleC22×C127C2×C22⋊C4
cosocleC42×C127C23
hypercenter#532D127
hypocenterC127#1525
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #532

Complements

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

NH ≅ G/Ntype
1. C4×#14523232C2semidirect
2. #22549616C2semidirect
59. C127#1525semidirect

Maximal subgroups

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

subgroupindex
C4×#145232322
#225496162
#532×C1272
#1525127