Finite Group Databasebeta

← C2×#20265311 #27180308

Subgroups — C2×#20265311 #27180308

Order 12928.

1048 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 1048 conjugacy classes of subgroups — too many to draw. Only its 197 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×#13108059C212yesmaximal, complemented
2C2×#15692229C211yesmaximal, characteristic, complemented
3#20265311C218yesmaximal, direct product factor
4C2×#15692176C212yesmaximal, complemented
5C2×C22⋊#11717579C211yesmaximal, characteristic, complemented
6C23×D4×C101C211yesmaximal, characteristic, complemented
7C22×D404C2211yescharacteristic, complemented
8C22×C22⋊D101—21no—
9C2×#13108059—28no—
10C2×#13108059C2218yescomplemented
11C22×#11717579C2212yescomplemented
12#15692229—22no—
13#15692229C2214yescomplemented
14C2×C4⋊Dic101C2211yescharacteristic, complemented
15C2×#13108051—21no—
16#15692176C2218yescomplemented
17C22⋊#11717579C2214yescomplemented
18C25×C101—21no—
19C22×D4×C101—28no—
20C22×D4×C101—21no—
21C22×D4×C101—22no—
22C22×D4×C101C2212yes—
23C22×D4×C101C2214yescomplemented
24C23×C4×C101C2211yescharacteristic
25C23×D101—21no—
26C2×D404—24no—
27C2×D404—21no—
28C2×D404C2312yescomplemented
29C2×C22⋊D101—24no—
30C2×C22⋊D101—44no—
31C22×Dic101—21no—
32#13108059—216no—
33C2×#11717579—24no—
34C2×#11717579C2314yescomplemented
35C4⋊Dic101—21no—
36C4⋊Dic101C2312yescomplemented
37#13108051—24no—
38C24×C101—28no—
39C24×C101—22no—
40C24×C101—24no—
41C24×C101—44no—
42C24×C101—44no—
43C24×C101D411yescharacteristic
44C2×D4×C101—28no—
45C2×D4×C101—216no—
46C2×D4×C101—216no—
47C2×D4×C101—22no—
48C2×D4×C101—24no—
49C2×D4×C101—24no—
50C2×D4×C101D418yescomplemented
51C2×D4×C101C2314yes—
52C22×C4×C101—24no—
53C22×C4×C101D411yescharacteristic
54C22×C4×C101D412yes—
55C22×C4×C101C2312yes—
56C22×C4×C101C2311yescharacteristic
57C23×C101—22no—
58C23×C101—22no—
59C23×C101—216no—
60C23×C101—28no—
61C23×C101—24no—
62C23×C101—48no—
63C23×C101—432no—
64C23×C101—44no—
65C23×C101—48no—
66C23×C101—44no—
67C23×C101D814yescomplemented
68C23×C101C2×D412yes—
69C23×C101C2×D411yescharacteristic
70C22×D101—22no—
71C22×D101—42no—
72C22×D101—42no—
73C22×D101—42no—
74D4×C101—216no—
75D4×C101—28no—
76D4×C101—24no—
77D4×C101C2×D418yescomplemented
78C2×C4×C101—22no—
79C2×C4×C101—24no—
80C2×C4×C101—24no—
81C2×C4×C101C2×D412yes—
82C2×C4×C101C2×D411yescharacteristic
83C2×C4×C101C2×D414yes—
84C2×C4×C101C2411yescharacteristic
85C22⋊D101—48no—
86C2×Dic101—22no—
87C2×Dic101—42no—
88D404—24no—
89D404—42no—
90#11717579—24no—
91C22×C101—22no—
92C22×C101—22no—
93C22×C101—28no—
94C22×C101—28no—
95C22×C101—432no—
96C22×C101—44no—
97C22×C101—44no—
98C22×C101—416no—
99C22×C101#12012yes—
100C22×C101C2×D812yes—
101C22×C101C2×D814yescomplemented
102C22×C101AGL(1,ℤ/8)12yes—
103C22×C101C22×D411yescharacteristic
104C2×D101—42no—
105C2×D101—42no—
106C2×D101—42no—
107C2×D101—84no—
108C4×C101—22no—
109C4×C101#12012yes—
110C4×C101C22×D412yes—
111Dic101—42no—
112C2×C101—24no—
113C2×C101—48no—
114C2×C101#44714yescomplemented
115C2×C101C2×#12011yescharacteristic
116C2×C101C22×D811yescharacteristic
117C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
118D101—82no—
119C2×#447—1011nomaximal, Sylow
120C101C2×#44711yescharacteristic, complemented, Sylow
121C23×D4D10111yescharacteristic, complemented
122C22×D8—1012no—
123C2×#121—1011no—
124#447—1018no—
125C2×#102—1012no—
126C2×C22⋊C8—1011no—
127C25—21no—
128C22×D4—28no—
129C22×D4—22no—
130C22×D4—21no—
131C22×D4—1011no—
132C22×D4—2021no—
133C22×D4C2×D10112yes—
134C22×D4C2×D10114yescomplemented
135C23×C4C2×D10111yescharacteristic
136C2×D8—1018no—
137C2×D8—2028no—
138C22×C8—1012no—
139#121—1014no—
140#121—2022no—
141C2×C4⋊C4—1011no—
142C2×C22⋊C4—2021no—
143#102—1018no—
144C22⋊C8—1014no—
145C24—28no—
146C24—24no—
147C24—22no—
148C24—44no—
149C24—44no—
150C24—2021no—
151C24C22⋊D10111yescharacteristic
152C2×D4—28no—
153C2×D4—216no—
154C2×D4—216no—
155C2×D4—24no—
156C2×D4—22no—
157C2×D4—24no—
158C2×D4—1012no—
159C2×D4—4044no—
160C2×D4—2024no—
161C2×D4—2021no—
162C2×D4—2024no—
163C2×D4C22⋊D10118yescomplemented
164C2×D4C22×D10114yes—
165C22×C4—24no—
166C22×C4—2021no—
167C22×C4C22⋊D10112yes—
168C22×C4C22⋊D10111yescharacteristic
169C22×C4C22×D10112yes—
170C22×C4C22×D10111yescharacteristic
171D8—20216no—
172C2×C8—1014no—
173C2×C8—2024no—
174C4⋊C4—1012no—
175C4⋊C4—2021no—
176C22⋊C4—2024no—
177C23—22no—
178C23—22no—
179C23—216no—
180C23—24no—
181C23—28no—
182C23—432no—
183C23—48no—
184C23—44no—
185C23—48no—
186C23—44no—
187C23—4042no—
188C23—4042no—
189C23—4042no—
190C23—2022no—
191C23#1310805914yescomplemented
192C23C2×C22⋊D10112yes—
193C23C2×C22⋊D10111yescharacteristic
194D4—216no—
195D4—24no—
196D4—28no—
197D4—4048no—
198D4—4042no—
199D4—2024no—
200D4C2×C22⋊D10118yescomplemented
201C2×C4—22no—
202C2×C4—24no—
203C2×C4—24no—
204C2×C4—4042no—
205C2×C4—2022no—
206C2×C4C2×C22⋊D10112yes—
207C2×C4C2×C22⋊D10114yes—
208C2×C4C2×C22⋊D10111yescharacteristic
209C2×C4C23×D10111yescharacteristic
210C8—2024no—
211C22—28no—
212C22—22no—
213C22—22no—
214C22—28no—
215C22—432no—
216C22—416no—
217C22—44no—
218C22—44no—
219C22—8084no—
220C22—4042no—
221C22—4042no—
222C22—4042no—
223C22#1569222412yes—
224C22C2×#1310805912yes—
225C22C2×#1310805914yescomplemented
226C22#1569228112yes—
227C22C22×C22⋊D10111yescharacteristic
228C4—22no—
229C4—4042no—
230C4#1569222412yes—
231C4C22×C22⋊D10112yes—
232C2—24no—
233C2—48no—
234C2—8082no—
235C2#2026531114yesdirect product factor
236C2C2×#1569222411yescharacteristic
237C2C22×#1310805911yescharacteristic
238C2C2×#1569228111yescharacteristic
subgroupG/H
centerC23C2×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC23×D4×C101C2
socleC23×C101C2×D4
hypercenterC23×D4D101
hypocenterC101C2×#447
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C22×#13108059C2semidirect
2. C2×#15692229C2semidirect
3. #20265311C2direct (N × H)
4. C2×#15692176C2semidirect
5. C2×C22⋊#11717579C2semidirect
6. C23×D4×C101C2semidirect
7. C22×D404C22semidirect
10. C2×#13108059C22semidirect
11. C22×#11717579C22semidirect
13. #15692229C22semidirect
14. C2×C4⋊Dic101C22semidirect
16. #15692176C22semidirect
17. C22⋊#11717579C22semidirect
23. C22×D4×C101C22semidirect
28. C2×D404C23semidirect
34. C2×#11717579C23semidirect
36. C4⋊Dic101C23semidirect
50. C2×D4×C101D4semidirect
67. C23×C101D8semidirect
77. D4×C101C2×D4semidirect
101. C22×C101C2×D8semidirect
114. C2×C101#447semidirect
120. C101C2×#447semidirect
121. C23×D4D101semidirect
134. C22×D4C2×D101semidirect
163. C2×D4C22⋊D101semidirect
191. C23#13108059semidirect
200. D4C2×C22⋊D101semidirect
225. C22C2×#13108059semidirect
235. C2#20265311direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131080592
C2×#156922292
#202653112
C2×#156921762
C2×C22⋊#117175792
C23×D4×C1012
C2×#447101