Finite Group Databasebeta

← #27140283

Subgroups — #27140283

Order 12416.

252 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 77 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#20260792C211yesmaximal, characteristic, complemented
2#20260603C211yesmaximal, characteristic, complemented
3#20260469C214yesmaximal, complemented
4#532×C97C211yesmaximal, characteristic
5C4⋊#11715895—21no—
6C22⋊#11715895C2212yes—
7#15690721—21no—
8#15690717—21no—
9#15690717—21no—
10#15690717C2212yes—
11C2×#41×C97—21no—
12C4⋊Q8×C97C411yescharacteristic, complemented
13#127×C97C411yescharacteristic, complemented
14#124×C97—21no—
15#121×C97—21no—
16#121×C97C2212yes—
17C4×D4×C97—21no—
18C4×D4×C97C412yescomplemented
19C2×C42×C97C2211yescharacteristic
20C22×Dic97—22no—
21C2×#11715895—22no—
22#41×C97—44no—
23C2×Q8×C97—21no—
24C2×D4×C97—22no—
25C2×D4×C97—21no—
26C2×D4×C97—21no—
27C2×D4×C97C2×C412yes—
28C22×C4×C97—22no—
29C22×C4×C97D411yescharacteristic
30C22×C4×C97D411yescharacteristic
31C22×C4×C97C2311yescharacteristic
32C4⋊C4×C97—21no—
33C4⋊C4×C97C2×C412yes—
34C22⋊C4×C97—22no—
35C22⋊C4×C97—22no—
36C42×C97C2×C411yescharacteristic
37C42×C97C2×C411yescharacteristic
38C42×C97D411yescharacteristic
39C42×C97D411yescharacteristic
40C23×C97—22no—
41C23×C97C2×D411yescharacteristic
42Q8×C97—42no—
43D4×C97—42no—
44D4×C97—42no—
45D4×C97—42no—
46D4×C97—44no—
47C2×C4×C97—22no—
48C2×C4×C97—22no—
49C2×C4×C97—41no—
50C2×C4×C97—41no—
51C2×C4×C97—44no—
52C2×C4×C97C22⋊C411yescharacteristic
53C2×C4×C97C22⋊C411yescharacteristic
54C2×C4×C97C22⋊C411yescharacteristic
55C2×C4×C97C22⋊C411yescharacteristic
56C2×C4×C97C22×C411yescharacteristic
57C2×C4×C97C2×D411yescharacteristic
58C2×Dic97—42no—
59C2×Dic97—42no—
60C2×Dic97—42no—
61#11715895—42no—
62C22×C97—21no—
63C22×C97—21no—
64C22×C97—42no—
65C22×C97—42no—
66C22×C97—42no—
67C22×C97C2×C22⋊C411yescharacteristic
68C22×C97AGL(1,ℤ/8)12yes—
69C4×C97—21no—
70C4×C97—21no—
71C4×C97—21no—
72C4×C97—21no—
73C4×C97—21no—
74C4×C97—21no—
75C4×C97—42no—
76Dic97—82no—
77C2×C97—22no—
78C2×C97—82no—
79C2×C97#41011yescharacteristic
80C2×C97#41811yescharacteristic
81C2×C97#42111yescharacteristic
82#1514—971nomaximal, Sylow
83C97#151411yescharacteristic, complemented, Sylow
84#532D9711yescharacteristic
85#432—971no—
86#383—971no—
87#327—974no—
88C2×#41—21no—
89C4⋊Q8Dic9711yescharacteristic, complemented
90#127Dic9711yescharacteristic, complemented
91#124—21no—
92#121—21no—
93#121C2×D9712yes—
94C4×D4—21no—
95C4×D4Dic9712yescomplemented
96C2×C42C2×D9711yescharacteristic
97C4⋊C8—1941no—
98C22⋊C8—972no—
99#97—1941no—
100#95—972no—
101#95—1941no—
102#95—1941no—
103#41—44no—
104C2×Q8—21no—
105C2×D4—22no—
106C2×D4—21no—
107C2×D4—21no—
108C2×D4C2×Dic9712yes—
109C22×C4—22no—
110C22×C4—1942no—
111C22×C4C22⋊D9711yescharacteristic
112C22×C4C22⋊D9711yescharacteristic
113C22×C4C22×D9711yescharacteristic
114C2×C8—1942no—
115C4⋊C4—21no—
116C4⋊C4C2×Dic9712yes—
117C22⋊C4—22no—
118C22⋊C4—22no—
119C42C2×Dic9711yescharacteristic
120C42C2×Dic9711yescharacteristic
121C42C22⋊D9711yescharacteristic
122C42C22⋊D9711yescharacteristic
123C23—22no—
124C23C2×C22⋊D9711yescharacteristic
125Q8—42no—
126D4—42no—
127D4—42no—
128D4—42no—
129D4—44no—
130C2×C4—22no—
131C2×C4—22no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—44no—
135C2×C4—3882no—
136C2×C4—3882no—
137C2×C4—3882no—
138C2×C4C22⋊Dic9711yescharacteristic
139C2×C4C22⋊Dic9711yescharacteristic
140C2×C4C22⋊Dic9711yescharacteristic
141C2×C4C22⋊Dic9711yescharacteristic
142C2×C4C22×Dic9711yescharacteristic
143C2×C4C2×C22⋊D9711yescharacteristic
144C8—3882no—
145C22—21no—
146C22—21no—
147C22—42no—
148C22—42no—
149C22—42no—
150C22C2×C22⋊Dic9711yescharacteristic
151C22#1569083612yes—
152C4—21no—
153C4—21no—
154C4—21no—
155C4—21no—
156C4—21no—
157C4—21no—
158C4—42no—
159C4—7762no—
160C2—22no—
161C2—82no—
162C2#2026070911yescharacteristic
163C2#2026074011yescharacteristic
164C2#2026075211yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic97
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting, prime radical#532×C97C2
socleC22×C97C2×C22⋊C4
cosocleC22×C4×C97C23
hypercenter#532D97
hypocenterC97#1514
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C2×C42 ◁ #532

Complements

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

NH ≅ G/Ntype
1. #20260792C2semidirect
2. #20260603C2semidirect
3. #20260469C2semidirect
12. C4⋊Q8×C97C4semidirect
13. #127×C97C4semidirect
18. C4×D4×C97C4semidirect
83. C97#1514semidirect
89. C4⋊Q8Dic97semidirect
90. #127Dic97semidirect
95. C4×D4Dic97semidirect

Maximal subgroups

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

subgroupindex
#202607922
#202606032
#202604692
#532×C972
#151497