Finite Group Databasebeta

← #27140304

Subgroups — #27140304

Order 12416.

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

normal subgroups only

This group has 340 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
1C2×C22⋊#11715895C211yesmaximal, characteristic, complemented
2#20260595C211yesmaximal, characteristic, complemented
3#20260469C214yesmaximal, complemented
4C2×#121×C97C211yesmaximal, characteristic
5C22×#11715895—21no—
6C2×C4⋊Dic97—21no—
7C2×C22⋊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—
21C2×#11715895—22no—
22C2×#11715895—42no—
23C4⋊Dic97—42no—
24C22⋊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—42no—
32C2×D4×C97—42no—
33C2×D4×C97C2×C412yes—
34C22×C4×C97—21no—
35C22×C4×C97—42no—
36C22×C4×C97C2×C411yescharacteristic
37C22×C4×C97C2×C411yescharacteristic
38C22×C4×C97D411yescharacteristic
39C22×C4×C97D411yescharacteristic
40C22×C4×C97D411yescharacteristic
41C22×C4×C97C2311yescharacteristic
42C4⋊C4×C97—21no—
43C4⋊C4×C97C2×C412yes—
44C22⋊C4×C97—22no—
45C22⋊C4×C97—22no—
46C23×C97—22no—
47C23×C97—41no—
48C23×C97—41no—
49C23×C97—42no—
50C23×C97—42no—
51C23×C97—42no—
52C23×C97C22⋊C411yescharacteristic
53C23×C97C22⋊C411yescharacteristic
54C23×C97D812yes—
55C23×C97SD1612yes—
56C23×C97C2×D411yescharacteristic
57D4×C97—42no—
58D4×C97—44no—
59D4×C97—42no—
60D4×C97—42no—
61D4×C97—42no—
62C2×C4×C97—22no—
63C2×C4×C97—22no—
64C2×C4×C97—22no—
65C2×C4×C97—22no—
66C2×C4×C97—41no—
67C2×C4×C97—41no—
68C2×C4×C97—41no—
69C2×C4×C97—42no—
70C2×C4×C97C22⋊C411yescharacteristic
71C2×C4×C97C22⋊C411yescharacteristic
72C2×C4×C97C22×C411yescharacteristic
73C2×C4×C97C2×D411yescharacteristic
74C2×Dic97—42no—
75C2×Dic97—42no—
76C2×Dic97—42no—
77#11715895—42no—
78C22×C97—21no—
79C22×C97—21no—
80C22×C97—21no—
81C22×C97—21no—
82C22×C97—21no—
83C22×C97—21no—
84C22×C97—42no—
85C22×C97—42no—
86C22×C97—42no—
87C22×C97—42no—
88C22×C97—42no—
89C22×C97—84no—
90C22×C97#10212yes—
91C22×C97#10212yes—
92C22×C97C2×C22⋊C411yescharacteristic
93C22×C97C2×D811yescharacteristic
94C22×C97C2×SD1611yescharacteristic
95C4×C97—21no—
96C4×C97—21no—
97C4×C97—21no—
98C4×C97—42no—
99C4×C97#9912yes—
100Dic97—82no—
101C2×C97—21no—
102C2×C97—22no—
103C2×C97—22no—
104C2×C97—21no—
105C2×C97—82no—
106C2×C97C2×#9911yescharacteristic
107C2×C97C2×#10211yescharacteristic
108C2×C97#42111yescharacteristic
109#1519—971nomaximal, Sylow
110C97#151911yescharacteristic, complemented, Sylow
111C2×#121D9711yescharacteristic
112C2×C22⋊C8—971no—
113#380—971no—
114#327—974no—
115C22×D4—21no—
116C22×D4Dic9711yescharacteristic, complemented
117C23×C4C2×D9711yescharacteristic
118C22×C8—1941no—
119#121—21no—
120#121—21no—
121#121Dic9712yescomplemented
122#121C2×D9712yes—
123C2×C4⋊C4—1941no—
124C2×C4⋊C4Dic9711yescharacteristic, complemented
125C2×C22⋊C4—21no—
126C2×C22⋊C4—1941no—
127C22⋊C8—972no—
128C22⋊C8—1941no—
129#95—972no—
130C24—21no—
131C24C22⋊D9711yescharacteristic
132C2×D4—22no—
133C2×D4—22no—
134C2×D4—21no—
135C2×D4—42no—
136C2×D4—42no—
137C2×D4—42no—
138C2×D4C2×Dic9712yes—
139C22×C4—21no—
140C22×C4—42no—
141C22×C4—1942no—
142C22×C4C2×Dic9711yescharacteristic
143C22×C4C2×Dic9711yescharacteristic
144C22×C4C22⋊D9711yescharacteristic
145C22×C4C22⋊D9711yescharacteristic
146C22×C4C22⋊D9711yescharacteristic
147C22×C4C22×D9711yescharacteristic
148C2×C8—3882no—
149C2×C8—1942no—
150C4⋊C4—21no—
151C4⋊C4—3882no—
152C4⋊C4C2×Dic9712yes—
153C22⋊C4—22no—
154C22⋊C4—22no—
155C22⋊C4—3882no—
156C23—22no—
157C23—41no—
158C23—41no—
159C23—42no—
160C23—42no—
161C23—42no—
162C23C22⋊Dic9711yescharacteristic
163C23C22⋊Dic9711yescharacteristic
164C23#1309549612yes—
165C23#1309549912yes—
166C23C2×C22⋊D9711yescharacteristic
167D4—42no—
168D4—42no—
169D4—42no—
170D4—44no—
171D4—42no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—42no—
180C2×C4—3882no—
181C2×C4—3882no—
182C2×C4—3882no—
183C2×C4C22⋊Dic9711yescharacteristic
184C2×C4C22⋊Dic9711yescharacteristic
185C2×C4C22×Dic9711yescharacteristic
186C2×C4C2×C22⋊D9711yescharacteristic
187C8—3882no—
188C22—21no—
189C22—21no—
190C22—21no—
191C22—21no—
192C22—21no—
193C22—21no—
194C22—42no—
195C22—42no—
196C22—42no—
197C22—42no—
198C22—42no—
199C22—84no—
200C22#1569073112yes—
201C22#1569073112yes—
202C22C2×C22⋊Dic9711yescharacteristic
203C22C2×#1309549611yescharacteristic
204C22C2×#1309549911yescharacteristic
205C4—21no—
206C4—21no—
207C4—21no—
208C4—42no—
209C4—7762no—
210C4#1569072512yes—
211C2—21no—
212C2—22no—
213C2—21no—
214C2—22no—
215C2—82no—
216C2C2×#1569072511yescharacteristic
217C2C2×#1569073111yescharacteristic
218C2#2026075211yescharacteristic
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#1519
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. C2×C22⋊#11715895C2semidirect
2. #20260595C2semidirect
3. #20260469C2semidirect
9. C22⋊#11715895C22semidirect
10. #15690717C22semidirect
12. C22×D4×C97C4semidirect
16. #121×C97C4semidirect
18. C2×C4⋊C4×C97C4semidirect
110. C97#1519semidirect
116. C22×D4Dic97semidirect
121. #121Dic97semidirect
124. C2×C4⋊C4Dic97semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊#117158952
#202605952
#202604692
C2×#121×C972
#151997