Finite Group Databasebeta

← #27140252

Subgroups — #27140252

Order 12416.

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

normal subgroups only

This group has 346 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
1#20260699C211yesmaximal, characteristic, complemented
2C4×C22⋊Dic97C211yesmaximal, characteristic, complemented
3#20260469C214yesmaximal, complemented
4C2×#121×C97C211yesmaximal, characteristic
5C2×#13095495—21no—
6C2×C22⋊Dic97—21no—
7C2×C4×Dic97—21no—
8C22⋊#11715895—21no—
9C22⋊#11715895C2212yescomplemented
10#15690717C2212yescomplemented
11C22×D4×C97—21no—
12C22×D4×C97C411yescharacteristic, complemented
13C23×C4×C97C2211yescharacteristic
14#121×C97—21no—
15#121×C97—21no—
16#121×C97C412yescomplemented
17#121×C97C2212yes—
18C2×C4⋊C4×C97C411yescharacteristic, complemented
19C2×C22⋊C4×C97—21no—
20C22×Dic97—22no—
21#13095495—42no—
22C2×#11715895—22no—
23C22⋊Dic97—24no—
24C4×Dic97—42no—
25C24×C97—21no—
26C24×C97D411yescharacteristic
27C2×D4×C97—22no—
28C2×D4×C97—22no—
29C2×D4×C97—21no—
30C2×D4×C97—42no—
31C2×D4×C97—44no—
32C2×D4×C97C2×C412yes—
33C22×C4×C97—21no—
34C22×C4×C97—42no—
35C22×C4×C97C2×C411yescharacteristic
36C22×C4×C97C2×C411yescharacteristic
37C22×C4×C97D411yescharacteristic
38C22×C4×C97D411yescharacteristic
39C22×C4×C97D411yescharacteristic
40C22×C4×C97C2311yescharacteristic
41C4⋊C4×C97—21no—
42C4⋊C4×C97C2×C412yes—
43C22⋊C4×C97—22no—
44C22⋊C4×C97—22no—
45C23×C97—22no—
46C23×C97—22no—
47C23×C97—22no—
48C23×C97—41no—
49C23×C97—42no—
50C23×C97—42no—
51C23×C97—42no—
52C23×C97C22⋊C411yescharacteristic
53C23×C97C22⋊C411yescharacteristic
54C23×C97C2×D411yescharacteristic
55D4×C97—42no—
56D4×C97—44no—
57D4×C97—44no—
58D4×C97—42no—
59C2×C4×C97—22no—
60C2×C4×C97—22no—
61C2×C4×C97—22no—
62C2×C4×C97—22no—
63C2×C4×C97—41no—
64C2×C4×C97—41no—
65C2×C4×C97—41no—
66C2×C4×C97—42no—
67C2×C4×C97C22⋊C411yescharacteristic
68C2×C4×C97C22⋊C411yescharacteristic
69C2×C4×C97C22×C411yescharacteristic
70C2×C4×C97C2×D411yescharacteristic
71C2×Dic97—24no—
72C2×Dic97—42no—
73C2×Dic97—42no—
74#11715895—42no—
75C22×C97—21no—
76C22×C97—21no—
77C22×C97—21no—
78C22×C97—21no—
79C22×C97—21no—
80C22×C97—21no—
81C22×C97—21no—
82C22×C97—44no—
83C22×C97—42no—
84C22×C97—42no—
85C22×C97—42no—
86C22×C97—84no—
87C22×C97#10412yes—
88C22×C97C2×C22⋊C411yescharacteristic
89C22×C97AGL(1,ℤ/8)12yes—
90C4×C97—21no—
91C4×C97—21no—
92C4×C97—21no—
93C4×C97—42no—
94C4×C97#9912yes—
95Dic97—44no—
96C2×C97—22no—
97C2×C97—22no—
98C2×C97—41no—
99C2×C97—82no—
100C2×C97C2×#9911yescharacteristic
101C2×C97#41811yescharacteristic
102C2×C97C2×#10411yescharacteristic
103#1507—971nomaximal, Sylow
104C97#150711yescharacteristic, complemented, Sylow
105C2×#121D9711yescharacteristic
106#407—971no—
107C4×C22⋊C4—971no—
108#327—974no—
109C22×D4—21no—
110C22×D4Dic9711yescharacteristic, complemented
111C23×C4C2×D9711yescharacteristic
112C2×M16—1941no—
113#121—21no—
114#121—21no—
115#121Dic9712yescomplemented
116#121C2×D9712yes—
117C2×C4⋊C4Dic9711yescharacteristic, complemented
118C2×C22⋊C4—21no—
119C2×C22⋊C4—1941no—
120C2×C42—1941no—
121C22⋊C8—972no—
122C22⋊C8—1941no—
123#95—972no—
124C24—21no—
125C24C22⋊D9711yescharacteristic
126C2×D4—22no—
127C2×D4—22no—
128C2×D4—21no—
129C2×D4—42no—
130C2×D4—44no—
131C2×D4C2×Dic9712yes—
132C22×C4—21no—
133C22×C4—42no—
134C22×C4—1942no—
135C22×C4C2×Dic9711yescharacteristic
136C22×C4C2×Dic9711yescharacteristic
137C22×C4C22⋊D9711yescharacteristic
138C22×C4C22⋊D9711yescharacteristic
139C22×C4C22⋊D9711yescharacteristic
140C22×C4C22×D9711yescharacteristic
141M16—3882no—
142C2×C8—1942no—
143C4⋊C4—21no—
144C4⋊C4C2×Dic9712yes—
145C22⋊C4—22no—
146C22⋊C4—22no—
147C22⋊C4—1944no—
148C42—3882no—
149C23—22no—
150C23—22no—
151C23—22no—
152C23—41no—
153C23—42no—
154C23—42no—
155C23—42no—
156C23C22⋊Dic9711yescharacteristic
157C23C22⋊Dic9711yescharacteristic
158C23C2×C22⋊D9711yescharacteristic
159D4—42no—
160D4—44no—
161D4—44no—
162D4—42no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—42no—
171C2×C4—3882no—
172C2×C4—3882no—
173C2×C4—1944no—
174C2×C4C22⋊Dic9711yescharacteristic
175C2×C4C22⋊Dic9711yescharacteristic
176C2×C4C22×Dic9711yescharacteristic
177C2×C4C2×C22⋊D9711yescharacteristic
178C8—3882no—
179C22—21no—
180C22—21no—
181C22—21no—
182C22—21no—
183C22—21no—
184C22—21no—
185C22—21no—
186C22—44no—
187C22—42no—
188C22—42no—
189C22—42no—
190C22—84no—
191C22#1569073712yes—
192C22C2×C22⋊Dic9711yescharacteristic
193C22#1569083612yes—
194C4—21no—
195C4—21no—
196C4—21no—
197C4—42no—
198C4—3884no—
199C4#1569072512yes—
200C2—22no—
201C2—22no—
202C2—41no—
203C2—82no—
204C2C2×#1569072511yescharacteristic
205C2#2026074011yescharacteristic
206C2C2×#1569073711yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic97
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting, prime radicalC2×#121×C97C2
socleC22×C97C2×C22⋊C4
cosocleC22×C4×C97C23
hypercenterC2×#121D97
hypocenterC97#1507
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. #20260699C2semidirect
2. C4×C22⋊Dic97C2semidirect
3. #20260469C2semidirect
9. C22⋊#11715895C22semidirect
10. #15690717C22semidirect
12. C22×D4×C97C4semidirect
16. #121×C97C4semidirect
18. C2×C4⋊C4×C97C4semidirect
104. C97#1507semidirect
110. C22×D4Dic97semidirect
115. #121Dic97semidirect
117. C2×C4⋊C4Dic97semidirect

Maximal subgroups

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

subgroupindex
#202606992
C4×C22⋊Dic972
#202604692
C2×#121×C972
#150797