Finite Group Databasebeta

← #30640395

Subgroups — #30640395

Order 16256.

258 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 73 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#22549850C212yesmaximal, complemented
2#22549671C212yesmaximal, complemented
3#22549669C212yesmaximal, complemented
4#532×C127C211yesmaximal, characteristic
5#16861168—22no—
6#16861124—22no—
7#16861124C2214yes—
8C2×#41×C127C411yescharacteristic, complemented
9C2×#41×C127C2211yescharacteristic
10C4⋊Q8×C127C411yescharacteristic, complemented
11#127×C127C2211yescharacteristic
12#124×C127C411yescharacteristic, complemented
13#124×C127C2211yescharacteristic
14#121×C127—22no—
15C4×D4×C127—22no—
16C2×C42×C127C411yescharacteristic, complemented
17C4⋊Dic127—42no—
18C22⋊Dic127—24no—
19C4×Dic127—42no—
20#41×C127—41no—
21#41×C127—41no—
22#41×C127—41no—
23#41×C127—41no—
24C2×Q8×C127C2×C411yescharacteristic
25C2×Q8×C127C2×C411yescharacteristic
26C2×D4×C127—21no—
27C2×D4×C127—22no—
28C2×D4×C127C2×C411yescharacteristic
29C2×D4×C127D412yes—
30C2×D4×C127C2311yescharacteristic
31C22×C4×C127—21no—
32C22×C4×C127—21no—
33C22×C4×C127C2×C411yescharacteristic
34C22×C4×C127D412yes—
35C4⋊C4×C127—22no—
36C22⋊C4×C127—22no—
37C22⋊C4×C127—22no—
38C42×C127—21no—
39C42×C127C2×C411yescharacteristic
40C42×C127C2×C411yescharacteristic
41C2×Dic127—44no—
42C23×C127—21no—
43C23×C127C22⋊C411yescharacteristic
44C23×C127C2×D412yes—
45Q8×C127—41no—
46Q8×C127—41no—
47D4×C127—42no—
48D4×C127—41no—
49D4×C127—41no—
50D4×C127—44no—
51D4×C127—42no—
52C2×C4×C127—21no—
53C2×C4×C127—21no—
54C2×C4×C127—21no—
55C2×C4×C127—22no—
56C2×C4×C127—22no—
57C2×C4×C127—21no—
58C2×C4×C127—21no—
59C2×C4×C127—42no—
60C2×C4×C127—41no—
61C2×C4×C127—41no—
62C2×C4×C127C22⋊C412yes—
63C2×C4×C127C22⋊C411yescharacteristic
64C2×C4×C127C22×C411yescharacteristic
65C22×C127—21no—
66C22×C127—22no—
67C22×C127—41no—
68C22×C127—42no—
69C22×C127—42no—
70C22×C127—41no—
71C22×C127C2×C22⋊C411yescharacteristic
72C4×C127—22no—
73C4×C127—22no—
74C4×C127—21no—
75C4×C127—41no—
76C4×C127—41no—
77C4×C127#9911yescharacteristic
78C4×C127#9911yescharacteristic
79Dic127—84no—
80C2×C127—21no—
81C2×C127—42no—
82C2×C127—41no—
83C2×C127—81no—
84C2×C127C2×#9911yescharacteristic
85#2126—1271nomaximal, Sylow
86C127#212611yescharacteristic, complemented, Sylow
87#532D12711yescharacteristic
88#410—1272no—
89#354—1272no—
90#353—1272no—
91C2×#41Dic12711yescharacteristic, complemented
92C2×#41C2×D12711yescharacteristic
93C4⋊Q8Dic12711yescharacteristic, complemented
94#127C2×D12711yescharacteristic
95#124Dic12711yescharacteristic, complemented
96#124C2×D12711yescharacteristic
97#121—22no—
98C4×D4—22no—
99#117—2542no—
100C2×C42Dic12711yescharacteristic, complemented
101#99—1274no—
102#99—2542no—
103#41—41no—
104#41—41no—
105#41—41no—
106#41—41no—
107C2×Q8C2×Dic12711yescharacteristic
108C2×Q8C2×Dic12711yescharacteristic
109C2×D4—21no—
110C2×D4—22no—
111C2×D4C2×Dic12711yescharacteristic
112C2×D4C22⋊D12712yes—
113C2×D4C22×D12711yescharacteristic
114C22×C4—21no—
115C22×C4—21no—
116C22×C4C2×Dic12711yescharacteristic
117C22×C4C22⋊D12712yes—
118C4⋊C4—22no—
119C4⋊C4—5082no—
120C22⋊C4—22no—
121C22⋊C4—22no—
122C22⋊C4—2544no—
123C42—21no—
124C42—5082no—
125C42C2×Dic12711yescharacteristic
126C42C2×Dic12711yescharacteristic
127C23—21no—
128C23C22⋊Dic12711yescharacteristic
129C23C2×C22⋊D12712yes—
130Q8—41no—
131Q8—41no—
132D4—42no—
133D4—41no—
134D4—41no—
135D4—44no—
136D4—42no—
137C2×C4—21no—
138C2×C4—22no—
139C2×C4—21no—
140C2×C4—22no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—42no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—5084no—
148C2×C4C22⋊Dic12712yes—
149C2×C4C22⋊Dic12711yescharacteristic
150C2×C4C22×Dic12711yescharacteristic
151C22—22no—
152C22—21no—
153C22—41no—
154C22—42no—
155C22—42no—
156C22—41no—
157C22C2×C22⋊Dic12711yescharacteristic
158C4—22no—
159C4—22no—
160C4—21no—
161C4—41no—
162C4—41no—
163C4—10164no—
164C4#1686112411yescharacteristic
165C4#1686112411yescharacteristic
166C2—21no—
167C2—42no—
168C2—41no—
169C2—81no—
170C2C2×#1686112411yescharacteristic
subgroupG/H
centerC4#16861124
commutatorC2×C4×C127C22×C4
FrattiniC2×D4C22×D127
Fitting, prime radical#532×C127C2
socleC2×C127C2×#99
cosocleC2×D4×C127C23
hypercenter#532D127
hypocenterC127#2126
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #532

Complements

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

NH ≅ G/Ntype
1. #22549850C2semidirect
2. #22549671C2semidirect
3. #22549669C2semidirect
8. C2×#41×C127C4semidirect
10. C4⋊Q8×C127C4semidirect
12. #124×C127C4semidirect
16. C2×C42×C127C4semidirect
86. C127#2126semidirect
91. C2×#41Dic127semidirect
93. C4⋊Q8Dic127semidirect
95. #124Dic127semidirect
100. C2×C42Dic127semidirect

Maximal subgroups

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

subgroupindex
#225498502
#225496712
#225496692
#532×C1272
#2126127