Finite Group Databasebeta

← C2×#20265322 #27180333

Subgroups — C2×#20265322 #27180333

Order 12928.

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

normal subgroups only

This group has 812 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×#13108061C211yesmaximal, characteristic, complemented
2C22×#13108059C211yesmaximal, characteristic, complemented
3C2×#15692229C211yesmaximal, characteristic, complemented
4#20265322C218yesmaximal, direct product factor
5C2×#15692179C211yesmaximal, characteristic, complemented
6C2×#15692176C211yesmaximal, characteristic, complemented
7C2×C22⋊#11717579C211yesmaximal, characteristic, complemented
8C22×#41×C101C211yesmaximal, characteristic, complemented
9C22×D404C2211yescharacteristic
10C22×C22⋊D101—21no—
11C2×#13108061—24no—
12C2×#13108061C2214yescomplemented
13C2×#13108059—24no—
14C2×#13108059C2214yescomplemented
15C22×#11717579C2211yescharacteristic
16C22×#11717579C2211yescharacteristic
17#15692229—22no—
18#15692229C2214yescomplemented
19C2×C4⋊Dic101C2211yescharacteristic
20C2×#13108051—21no—
21#15692179C2214yescomplemented
22#15692176C2214yescomplemented
23C22⋊#11717579C2214yescomplemented
24C2×#41×C101—24no—
25C2×#41×C101—24no—
26C2×#41×C101—22no—
27C2×#41×C101C2214yescomplemented
28C22×Q8×C101C2211yescharacteristic
29C22×D4×C101—21no—
30C22×D4×C101C2211yescharacteristic
31C23×C4×C101—21no—
32C23×C4×C101C2211yescharacteristic
33C23×D101—21no—
34C2×D404—22no—
35C2×D404—22no—
36C2×D404—21no—
37C2×D404C2312yes—
38C2×C22⋊D101—24no—
39C2×C22⋊D101—44no—
40C22×Dic101—21no—
41#13108061—28no—
42#13108059—28no—
43C2×#11717579—22no—
44C2×#11717579—22no—
45C2×#11717579C2312yes—
46C2×#11717579C2312yes—
47C4⋊Dic101—21no—
48C4⋊Dic101C2312yes—
49#13108051—24no—
50C24×C101—21no—
51C24×C101D411yescharacteristic
52#41×C101—216no—
53#41×C101—28no—
54#41×C101—28no—
55C2×Q8×C101—22no—
56C2×Q8×C101—21no—
57C2×Q8×C101D414yescomplemented
58C2×Q8×C101C2312yes—
59C2×D4×C101—28no—
60C2×D4×C101—24no—
61C2×D4×C101—21no—
62C2×D4×C101—22no—
63C2×D4×C101D414yescomplemented
64C2×D4×C101C2312yes—
65C22×C4×C101—28no—
66C22×C4×C101—21no—
67C22×C4×C101—24no—
68C22×C4×C101—22no—
69C22×C4×C101—22no—
70C22×C4×C101D411yescharacteristic
71C22×C4×C101D412yes—
72C22×C4×C101C2312yes—
73C22×C4×C101C2311yescharacteristic
74C23×C101—22no—
75C23×C101—24no—
76C23×C101—22no—
77C23×C101—22no—
78C23×C101—22no—
79C23×C101—42no—
80C23×C101—42no—
81C23×C101C2×D412yes—
82C23×C101C2×D411yescharacteristic
83C22×D101—22no—
84C22×D101—42no—
85C22×D101—42no—
86C22×D101—42no—
87Q8×C101—24no—
88Q8×C101—22no—
89Q8×C101C2×D414yescomplemented
90D4×C101—216no—
91D4×C101—24no—
92D4×C101—22no—
93D4×C101C2×D414yescomplemented
94C2×C4×C101—216no—
95C2×C4×C101—22no—
96C2×C4×C101—24no—
97C2×C4×C101—22no—
98C2×C4×C101—24no—
99C2×C4×C101—22no—
100C2×C4×C101—22no—
101C2×C4×C101C2×D412yes—
102C2×C4×C101C2×D411yescharacteristic
103C2×C4×C101C2×D412yes—
104C2×C4×C101C2×D412yes—
105C2×C4×C101C2411yescharacteristic
106C22⋊D101—48no—
107C2×Dic101—22no—
108C2×Dic101—42no—
109D404—22no—
110D404—22no—
111D404—42no—
112#11717579—22no—
113#11717579—22no—
114C22×C101—22no—
115C22×C101—24no—
116C22×C101—22no—
117C22×C101—22no—
118C22×C101—42no—
119C22×C101—42no—
120C22×C101—44no—
121C22×C101—48no—
122C22×C101#12012yes—
123C22×C101#13512yes—
124C22×C101AGL(1,ℤ/8)12yes—
125C22×C101C22×D411yescharacteristic
126C2×D101—42no—
127C2×D101—42no—
128C2×D101—42no—
129C2×D101—84no—
130C4×C101—22no—
131C4×C101—24no—
132C4×C101#12012yes—
133C4×C101C22×D411yescharacteristic
134C4×C101C22×D411yescharacteristic
135Dic101—42no—
136C2×C101—42no—
137C2×C101—44no—
138C2×C101#44914yescomplemented
139C2×C101C2×#12011yescharacteristic
140C2×C101C2×#13511yescharacteristic
141C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
142D101—82no—
143C2×#449—1011nomaximal, Sylow
144C101C2×#44911yescharacteristic, complemented, Sylow
145C22×#41D10111yescharacteristic, complemented
146C22×SD16—1011no—
147C22×D8—1011no—
148C2×#121—1011no—
149#449—1018no—
150C2×#103—1011no—
151C2×#102—1011no—
152C2×C22⋊C8—1011no—
153C2×#41—24no—
154C2×#41—24no—
155C2×#41—22no—
156C2×#41C2×D10114yescomplemented
157C22×Q8C2×D10111yescharacteristic
158C22×D4—21no—
159C22×D4—1011no—
160C22×D4—2021no—
161C22×D4C2×D10111yescharacteristic
162C23×C4—21no—
163C23×C4C2×D10111yescharacteristic
164C2×SD16—1014no—
165C2×SD16—2024no—
166C2×D8—1014no—
167C2×D8—2024no—
168C22×C8—1011no—
169C22×C8—1011no—
170#121—1014no—
171#121—2022no—
172C2×C4⋊C4—1011no—
173C2×C22⋊C4—2021no—
174#103—1014no—
175#102—1014no—
176C22⋊C8—1014no—
177C24—21no—
178C24—2021no—
179C24C22⋊D10111yescharacteristic
180#41—216no—
181#41—28no—
182#41—28no—
183C2×Q8—21no—
184C2×Q8—22no—
185C2×Q8C22⋊D10114yescomplemented
186C2×Q8C22×D10112yes—
187C2×D4—28no—
188C2×D4—24no—
189C2×D4—22no—
190C2×D4—21no—
191C2×D4—1012no—
192C2×D4—4044no—
193C2×D4—2024no—
194C2×D4—2021no—
195C2×D4—2022no—
196C2×D4—2022no—
197C2×D4C22⋊D10114yescomplemented
198C2×D4C22×D10112yes—
199C22×C4—28no—
200C22×C4—21no—
201C22×C4—24no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—2021no—
205C22×C4C22⋊D10112yes—
206C22×C4C22⋊D10111yescharacteristic
207C22×C4C22×D10112yes—
208C22×C4C22×D10111yescharacteristic
209SD16—2028no—
210D8—2028no—
211C2×C8—1012no—
212C2×C8—1012no—
213C2×C8—2022no—
214C2×C8—2022no—
215C4⋊C4—1012no—
216C4⋊C4—2021no—
217C22⋊C4—2024no—
218C23—22no—
219C23—22no—
220C23—24no—
221C23—22no—
222C23—22no—
223C23—42no—
224C23—42no—
225C23—4042no—
226C23—4042no—
227C23—4042no—
228C23—2022no—
229C23C2×C22⋊D10112yes—
230C23C2×C22⋊D10111yescharacteristic
231Q8—22no—
232Q8—24no—
233Q8C2×C22⋊D10114yescomplemented
234D4—216no—
235D4—22no—
236D4—24no—
237D4—4048no—
238D4—4042no—
239D4—2022no—
240D4—2022no—
241D4C2×C22⋊D10114yescomplemented
242C2×C4—216no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—24no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—24no—
249C2×C4—4042no—
250C2×C4—2022no—
251C2×C4C2×C22⋊D10112yes—
252C2×C4C2×C22⋊D10112yes—
253C2×C4C2×C22⋊D10112yes—
254C2×C4C2×C22⋊D10111yescharacteristic
255C2×C4C23×D10111yescharacteristic
256C8—2022no—
257C8—2022no—
258C22—22no—
259C22—24no—
260C22—22no—
261C22—22no—
262C22—44no—
263C22—48no—
264C22—42no—
265C22—42no—
266C22—8084no—
267C22—4042no—
268C22—4042no—
269C22—4042no—
270C22#1569222412yes—
271C22#1569227512yes—
272C22#1569228012yes—
273C22C22×C22⋊D10111yescharacteristic
274C4—22no—
275C4—24no—
276C4—4042no—
277C4#1569222412yes—
278C4C22×C22⋊D10111yescharacteristic
279C4C22×C22⋊D10111yescharacteristic
280C2—42no—
281C2—44no—
282C2—8082no—
283C2#2026532214yesdirect product factor
284C2C2×#1569222411yescharacteristic
285C2C2×#1569227511yescharacteristic
286C2C2×#1569228011yescharacteristic
subgroupG/H
centerC23C2×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC22×#41×C101C2
socleC23×C101C2×D4
hypercenterC22×#41D101
hypocenterC101C2×#449
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C22×#13108061C2semidirect
2. C22×#13108059C2semidirect
3. C2×#15692229C2semidirect
4. #20265322C2direct (N × H)
5. C2×#15692179C2semidirect
6. C2×#15692176C2semidirect
7. C2×C22⋊#11717579C2semidirect
8. C22×#41×C101C2semidirect
12. C2×#13108061C22semidirect
14. C2×#13108059C22semidirect
18. #15692229C22semidirect
21. #15692179C22semidirect
22. #15692176C22semidirect
23. C22⋊#11717579C22semidirect
27. C2×#41×C101C22semidirect
57. C2×Q8×C101D4semidirect
63. C2×D4×C101D4semidirect
89. Q8×C101C2×D4semidirect
93. D4×C101C2×D4semidirect
138. C2×C101#449semidirect
144. C101C2×#449semidirect
145. C22×#41D101semidirect
156. C2×#41C2×D101semidirect
185. C2×Q8C22⋊D101semidirect
197. C2×D4C22⋊D101semidirect
233. Q8C2×C22⋊D101semidirect
241. D4C2×C22⋊D101semidirect
283. C2#20265322direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131080612
C22×#131080592
C2×#156922292
#202653222
C2×#156921792
C2×#156921762
C2×C22⋊#117175792
C22×#41×C1012
C2×#449101