Finite Group Databasebeta

← #30637229

Subgroups — #30637229

Order 16256.

240 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 97 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#22549929C211yesmaximal, characteristic
2C4×C4⋊Dic127C211yesmaximal, characteristic
3#22549613C214yesmaximal
4#533×C127C211yesmaximal, characteristic
5C2×C4⋊Dic127—21no—
6C2×C4×Dic127—21no—
7C4⋊(C127⋊C8)—21no—
8C22⋊(C127⋊C8)C2212yes—
9C4×C127⋊C8—21no—
10C127⋊#95C2212yes—
11C4⋊Q8×C127C411yescharacteristic, complemented
12C4⋊Q8×C127C411yescharacteristic, complemented
13#125×C127—21no—
14C22⋊Q8×C127—21no—
15C22⋊Q8×C127C2212yes—
16C4×Q8×C127—21no—
17C4×Q8×C127C412yescomplemented
18#117×C127—21no—
19C2×C42×C127C2211yescharacteristic
20C22×Dic127—22no—
21C2×C127⋊C8—22no—
22C4⋊Dic127—24no—
23C4×Dic127—42no—
24C2×Q8×C127—21no—
25C2×Q8×C127C2×C412yes—
26C22×C4×C127D411yescharacteristic
27C22×C4×C127D411yescharacteristic
28C22×C4×C127C2311yescharacteristic
29C4⋊C4×C127—21no—
30C4⋊C4×C127—21no—
31C4⋊C4×C127—21no—
32C4⋊C4×C127—22no—
33C4⋊C4×C127—22no—
34C4⋊C4×C127C2×C412yes—
35C22⋊C4×C127—22no—
36C42×C127—22no—
37C42×C127C2×C411yescharacteristic
38C42×C127C2×C411yescharacteristic
39C42×C127D411yescharacteristic
40C42×C127D411yescharacteristic
41C2×Dic127—24no—
42C2×Dic127—42no—
43C2×Dic127—42no—
44C127⋊C8—42no—
45C23×C127C2×D411yescharacteristic
46Q8×C127—42no—
47Q8×C127—42no—
48C2×C4×C127—22no—
49C2×C4×C127—22no—
50C2×C4×C127—22no—
51C2×C4×C127—41no—
52C2×C4×C127C22⋊C411yescharacteristic
53C2×C4×C127C22⋊C411yescharacteristic
54C2×C4×C127C22⋊C411yescharacteristic
55C2×C4×C127C22⋊C411yescharacteristic
56C2×C4×C127SD1612yes—
57C2×C4×C127Q1612yes—
58C2×C4×C127C22×C411yescharacteristic
59C2×C4×C127C2×D411yescharacteristic
60C22×C127—21no—
61C22×C127—21no—
62C22×C127C2×C22⋊C411yescharacteristic
63C22×C127C2×SD1611yescharacteristic
64C22×C127C2×Q1611yescharacteristic
65C4×C127—21no—
66C4×C127—21no—
67C4×C127—21no—
68C4×C127—42no—
69C4×C127—42no—
70C4×C127#10312yes—
71C4×C127#10312yes—
72C4×C127#10412yes—
73Dic127—44no—
74C2×C127—21no—
75C2×C127—21no—
76C2×C127#41011yescharacteristic
77C2×C127C2×#10311yescharacteristic
78C2×C127C2×#10411yescharacteristic
79#1503—1271nomaximal, Sylow
80C127#150311yescharacteristic, complemented, Sylow
81#533D12711yescharacteristic
82#431—1271no—
83C4×C4⋊C4—1271no—
84#328—1274no—
85C4⋊Q8Dic12711yescharacteristic, complemented
86C4⋊Q8Dic12711yescharacteristic, complemented
87#125—21no—
88C22⋊Q8—21no—
89C22⋊Q8C2×D12712yes—
90C4×Q8—21no—
91C4×Q8Dic12712yescomplemented
92#117—21no—
93C2×C4⋊C4—2541no—
94C2×C42—2541no—
95C2×C42C2×D12711yescharacteristic
96C4⋊C8—2541no—
97C22⋊C8—1272no—
98C4×C8—2541no—
99#95—1272no—
100C2×Q8—21no—
101C2×Q8C2×Dic12712yes—
102C22×C4—2542no—
103C22×C4C22⋊D12711yescharacteristic
104C22×C4C22⋊D12711yescharacteristic
105C22×C4C22×D12711yescharacteristic
106C2×C8—2542no—
107C4⋊C4—21no—
108C4⋊C4—21no—
109C4⋊C4—21no—
110C4⋊C4—22no—
111C4⋊C4—22no—
112C4⋊C4—2544no—
113C4⋊C4C2×Dic12712yes—
114C22⋊C4—22no—
115C42—22no—
116C42—5082no—
117C42C2×Dic12711yescharacteristic
118C42C2×Dic12711yescharacteristic
119C42C22⋊D12711yescharacteristic
120C42C22⋊D12711yescharacteristic
121C23C2×C22⋊D12711yescharacteristic
122Q8—42no—
123Q8—42no—
124C2×C4—22no—
125C2×C4—22no—
126C2×C4—22no—
127C2×C4—41no—
128C2×C4—5082no—
129C2×C4—5082no—
130C2×C4—2544no—
131C2×C4C22⋊Dic12711yescharacteristic
132C2×C4C22⋊Dic12711yescharacteristic
133C2×C4C22⋊Dic12711yescharacteristic
134C2×C4C22⋊Dic12711yescharacteristic
135C2×C4#1452323512yes—
136C2×C4#1452323812yes—
137C2×C4C22×Dic12711yescharacteristic
138C2×C4C2×C22⋊D12711yescharacteristic
139C8—5082no—
140C22—21no—
141C22—21no—
142C22C2×C22⋊Dic12711yescharacteristic
143C22C2×#1452323511yescharacteristic
144C22C2×#1452323811yescharacteristic
145C4—21no—
146C4—21no—
147C4—21no—
148C4—42no—
149C4—42no—
150C4—5084no—
151C4#1686113312yes—
152C4#1686113312yes—
153C4#1686113612yes—
154C2—21no—
155C2—21no—
156C2#2254985011yescharacteristic
157C2C2×#1686113311yescharacteristic
158C2C2×#1686113611yescharacteristic
subgroupG/H
centerC2×C4C22⋊Dic127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#533×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC23×C127C2×D4
cosocleC22×C4×C127C23
hypercenter#533D127
hypocenterC127#1503
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #533

Complements

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

NH ≅ G/Ntype
11. C4⋊Q8×C127C4semidirect
12. C4⋊Q8×C127C4semidirect
17. C4×Q8×C127C4semidirect
80. C127#1503semidirect
85. C4⋊Q8Dic127semidirect
86. C4⋊Q8Dic127semidirect
91. C4×Q8Dic127semidirect

Maximal subgroups

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

subgroupindex
#225499292
C4×C4⋊Dic1272
#225496132
#533×C1272
#1503127