Finite Group Databasebeta

← C2×#20265406 #27180692

Subgroups — C2×#20265406 #27180692

Order 12928.

564 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 181 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
1C22×#13108063C211yesmaximal, characteristic, complemented
2C2×#15692234C211yesmaximal, characteristic, complemented
3C2×#15692229C211yesmaximal, characteristic
4#20265406C218yesmaximal, direct product factor
5C2×#15692187C211yesmaximal, characteristic, complemented
6C2×#15692180C211yesmaximal, characteristic, complemented
7C2×#15692177C211yesmaximal, characteristic, complemented
8C23×C8×C101C211yesmaximal, characteristic, complemented
9C22×Dic202C2211yescharacteristic
10C22×D404C2211yescharacteristic
11C22×C22⋊D101—21no—
12C2×#13108063—24no—
13C2×#13108063C2214yescomplemented
14#15692234—22no—
15#15692234C2214yescomplemented
16#15692229—22no—
17#15692229C2214yes—
18C2×C4⋊Dic101C2211yescharacteristic
19C2×C4⋊Dic101C2211yescharacteristic
20C2×#13108053—21no—
21C2×C22⋊Dic101—21no—
22C2×#13108051—21no—
23#15692187C2214yescomplemented
24#15692180C2214yescomplemented
25#15692177C2214yescomplemented
26C23×C4×C101C2211yescharacteristic
27C22×C8×C101—24no—
28C22×C8×C101C2211yescharacteristic
29C22×C8×C101C2211yescharacteristic
30C22×C8×C101C2214yescomplemented
31C23×D101—21no—
32C2×Dic202—22no—
33C2×Dic202—22no—
34C2×Dic202—21no—
35C2×Dic202C2312yes—
36C2×D404—22no—
37C2×D404—22no—
38C2×D404—21no—
39C2×D404C2312yes—
40C2×C22⋊D101—24no—
41C2×C22⋊D101—44no—
42C22×Dic101—21no—
43C22×Dic101—21no—
44C22×Dic101—21no—
45#13108063—28no—
46C4⋊Dic101—21no—
47C4⋊Dic101—21no—
48C4⋊Dic101C2312yes—
49C4⋊Dic101C2312yes—
50#13108053—24no—
51C22⋊Dic101—24no—
52#13108051—24no—
53C24×C101D411yescharacteristic
54C22×C4×C101—22no—
55C22×C4×C101—22no—
56C22×C4×C101D411yescharacteristic
57C22×C4×C101D412yes—
58C22×C4×C101C2312yes—
59C22×C4×C101C2311yescharacteristic
60C2×C8×C101—22no—
61C2×C8×C101—28no—
62C2×C8×C101D414yescomplemented
63C2×C8×C101C2312yes—
64C2×C8×C101C2312yes—
65C23×C101—22no—
66C23×C101—22no—
67C23×C101SD1614yescomplemented
68C23×C101C2×D412yes—
69C23×C101C2×D411yescharacteristic
70C22×D101—22no—
71C22×D101—42no—
72C22×D101—42no—
73C22×D101—42no—
74C2×C4×C101—22no—
75C2×C4×C101—24no—
76C2×C4×C101—22no—
77C2×C4×C101—22no—
78C2×C4×C101C2×D412yes—
79C2×C4×C101C2×D412yes—
80C2×C4×C101C2×D411yescharacteristic
81C2×C4×C101#4112yes—
82C2×C4×C101C2411yescharacteristic
83C22⋊D101—48no—
84C2×Dic101—22no—
85C2×Dic101—22no—
86C2×Dic101—22no—
87C2×Dic101—42no—
88C2×Dic101—42no—
89C2×Dic101—42no—
90D404—22no—
91D404—22no—
92D404—42no—
93Dic202—22no—
94Dic202—22no—
95Dic202—42no—
96C8×C101—22no—
97C8×C101C2×D414yescomplemented
98C22×C101—22no—
99C22×C101—22no—
100C22×C101—28no—
101C22×C101#12112yes—
102C22×C101C2×SD1612yes—
103C22×C101C2×SD1614yescomplemented
104C22×C101#13512yes—
105C22×C101C22×D411yescharacteristic
106C2×D101—42no—
107C2×D101—42no—
108C2×D101—42no—
109C2×D101—84no—
110C4×C101—22no—
111C4×C101#12112yes—
112C4×C101C22×D411yescharacteristic
113C4×C101C2×#4111yescharacteristic
114Dic101—42no—
115Dic101—42no—
116Dic101—42no—
117C2×C101—24no—
118C2×C101#46514yescomplemented
119C2×C101C2×#12111yescharacteristic
120C2×C101C22×SD1611yescharacteristic
121C2×C101C2×#13511yescharacteristic
122D101—82no—
123C2×#465—1011nomaximal, Sylow
124C101C2×#46511yescharacteristic, complemented, Sylow
125C22×SD16—1011no—
126C23×C8D10111yescharacteristic, complemented
127C2×C22⋊Q8—1011no—
128C2×#121—1011no—
129#465—1018no—
130C2×#106—1011no—
131C2×#103—1011no—
132C2×#102—1011no—
133C22×Q8—1011no—
134C22×D4—1011no—
135C22×D4—2021no—
136C23×C4C2×D10111yescharacteristic
137C2×SD16—1014no—
138C2×SD16—2024no—
139C22×C8—24no—
140C22×C8C2×D10111yescharacteristic
141C22×C8C2×D10111yescharacteristic
142C22×C8C2×D10114yescomplemented
143C22⋊Q8—1014no—
144C22⋊Q8—2022no—
145#121—1014no—
146#121—2022no—
147C2×C4⋊C4—1011no—
148C2×C4⋊C4—1011no—
149C2×C4⋊C4—2021no—
150C2×C22⋊C4—2021no—
151C2×C22⋊C4—2021no—
152#106—1014no—
153#103—1014no—
154#102—1014no—
155C24—2021no—
156C24D40411yescharacteristic
157C2×Q8—1012no—
158C2×Q8—2022no—
159C2×Q8—2022no—
160C2×Q8—2021no—
161C2×D4—1012no—
162C2×D4—4044no—
163C2×D4—2024no—
164C2×D4—2022no—
165C2×D4—2021no—
166C2×D4—2022no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—2021no—
170C22×C4—2021no—
171C22×C4—2021no—
172C22×C4D40412yes—
173C22×C4D40411yescharacteristic
174C22×C4C22×D10112yes—
175C22×C4C22×D10111yescharacteristic
176SD16—2028no—
177C2×C8—22no—
178C2×C8—28no—
179C2×C8C22⋊D10114yescomplemented
180C2×C8C22×D10112yes—
181C2×C8C22×D10112yes—
182C4⋊C4—1012no—
183C4⋊C4—1012no—
184C4⋊C4—2024no—
185C4⋊C4—2021no—
186C4⋊C4—2021no—
187C22⋊C4—2024no—
188C22⋊C4—2024no—
189C23—22no—
190C23—22no—
191C23—4042no—
192C23—4042no—
193C23—4042no—
194C23—2022no—
195C23#1310806314yescomplemented
196C23C2×D40412yes—
197C23C2×D40411yescharacteristic
198Q8—4042no—
199Q8—2022no—
200Q8—2022no—
201D4—4048no—
202D4—4042no—
203D4—2022no—
204D4—2022no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—24no—
209C2×C4—4042no—
210C2×C4—4042no—
211C2×C4—4042no—
212C2×C4—2022no—
213C2×C4—2022no—
214C2×C4—2022no—
215C2×C4C2×C22⋊D10112yes—
216C2×C4C2×D40412yes—
217C2×C4C2×D40411yescharacteristic
218C2×C4#1310807412yes—
219C2×C4C23×D10111yescharacteristic
220C8—22no—
221C8C2×C22⋊D10114yescomplemented
222C22—28no—
223C22—22no—
224C22—22no—
225C22—8084no—
226C22—4042no—
227C22—4042no—
228C22—4042no—
229C22#1569222912yes—
230C22C2×#1310806312yes—
231C22C2×#1310806314yescomplemented
232C22#1569227712yes—
233C22C22×D40411yescharacteristic
234C4—22no—
235C4—4042no—
236C4—4042no—
237C4—4042no—
238C4#1569222912yes—
239C4C22×C22⋊D10111yescharacteristic
240C4C2×#1310807411yescharacteristic
241C2—24no—
242C2—8082no—
243C2#2026540614yesdirect product factor
244C2C2×#1569222911yescharacteristic
245C2C22×#1310806311yescharacteristic
246C2C2×#1569227711yescharacteristic
subgroupG/H
centerC23C2×D404
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC23×C8×C101C2
socleC23×C101C2×D4
prime radicalC2×#15692229C2
hypercenterC23×C8D101
hypocenterC101C2×#465
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×C8

Complements

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

NH ≅ G/Ntype
1. C22×#13108063C2semidirect
2. C2×#15692234C2semidirect
4. #20265406C2direct (N × H)
5. C2×#15692187C2semidirect
6. C2×#15692180C2semidirect
7. C2×#15692177C2semidirect
8. C23×C8×C101C2semidirect
13. C2×#13108063C22semidirect
15. #15692234C22semidirect
23. #15692187C22semidirect
24. #15692180C22semidirect
25. #15692177C22semidirect
30. C22×C8×C101C22semidirect
62. C2×C8×C101D4semidirect
67. C23×C101SD16semidirect
97. C8×C101C2×D4semidirect
103. C22×C101C2×SD16semidirect
118. C2×C101#465semidirect
124. C101C2×#465semidirect
126. C23×C8D101semidirect
142. C22×C8C2×D101semidirect
179. C2×C8C22⋊D101semidirect
195. C23#13108063semidirect
221. C8C2×C22⋊D101semidirect
231. C22C2×#13108063semidirect
243. C2#20265406direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131080632
C2×#156922342
C2×#156922292
#202654062
C2×#156921872
C2×#156921802
C2×#156921772
C23×C8×C1012
C2×#465101