Finite Group Databasebeta

← C2×#20265312 #27180309

Subgroups — C2×#20265312 #27180309

Order 12928.

1048 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 1048 conjugacy classes of subgroups — too many to draw. Only its 173 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×D4×D101C211yesmaximal, characteristic, complemented
2C22×D808C211yesmaximal, characteristic, complemented
3C22×#13108059C211yesmaximal, characteristic, complemented
4C2×#15692230C211yesmaximal, characteristic, complemented
5#20265312C218yesmaximal, direct product factor
6C2×#15692175C211yesmaximal, characteristic, complemented
7C2×#15692167C211yesmaximal, characteristic, complemented
8C2×#102×C101C211yesmaximal, characteristic, complemented
9C24×D101—21no—
10C22×D404—21no—
11C22×D404C2211yescharacteristic
12C22×D404C2211yescharacteristic, complemented
13C22×C22⋊D101—21no—
14C2×D4×D101—24no—
15C2×D4×D101—22no—
16C2×D4×D101—24no—
17C2×D4×D101C2214yescomplemented
18C22×C4×D101C2211yescharacteristic
19C2×D808—24no—
20C2×D808C2214yescomplemented
21C2×#13108059—24no—
22C2×#13108059C2214yescomplemented
23C22×#11717579C2211yescharacteristic, complemented
24#15692230—22no—
25#15692230C2214yescomplemented
26C2×#13108051—21no—
27#15692175C2214yescomplemented
28#15692167C2214yescomplemented
29C22×D4×C101C2211yescharacteristic
30C22×C8×C101C2211yescharacteristic, complemented
31C2×C4⋊C4×C101C2211yescharacteristic, complemented
32#102×C101C2214yescomplemented
33C23×D101—28no—
34C23×D101—21no—
35C23×D101—24no—
36C23×D101—21no—
37C23×D101—44no—
38C23×D101—44no—
39C23×D101D411yescharacteristic
40C2×D404—21no—
41C2×D404—22no—
42C2×D404—22no—
43C2×D404—22no—
44C2×D404—24no—
45C2×D404—21no—
46C2×D404—44no—
47C2×D404D414yescomplemented
48C2×D404C2312yes—
49C2×D404C2312yescomplemented
50C2×C22⋊D101—28no—
51C2×C22⋊D101—24no—
52D4×D101—216no—
53D4×D101—28no—
54D4×D101—28no—
55C22×Dic101D411yescharacteristic
56C2×C4×D101—22no—
57C2×C4×D101—22no—
58C2×C4×D101D412yes—
59C2×C4×D101C2312yes—
60D808—28no—
61#13108059—28no—
62C2×#11717579—22no—
63C2×#11717579C2312yescomplemented
64#13108051—24no—
65C24×C101—21no—
66C2×D4×C101—21no—
67C2×D4×C101—22no—
68C2×D4×C101D414yescomplemented
69C2×D4×C101C2312yes—
70C22×C4×C101—21no—
71C22×C4×C101C2311yescharacteristic
72C2×C8×C101—22no—
73C2×C8×C101C2312yescomplemented
74C4⋊C4×C101—21no—
75C4⋊C4×C101C2312yescomplemented
76C23×C101—24no—
77C23×C101—22no—
78C23×C101—42no—
79C23×C101—42no—
80C23×C101C2×D411yescharacteristic
81C22×D101—22no—
82C22×D101—22no—
83C22×D101—216no—
84C22×D101—24no—
85C22×D101—22no—
86C22×D101—22no—
87C22×D101—48no—
88C22×D101—432no—
89C22×D101—42no—
90C22×D101—42no—
91C22×D101—48no—
92C22×D101—42no—
93C22×D101—42no—
94C22×D101—42no—
95C22×D101D814yescomplemented
96C22×D101C2×D412yes—
97D4×C101—24no—
98D4×C101—22no—
99D4×C101C2×D414yescomplemented
100C2×C4×C101—22no—
101C2×C4×C101—42no—
102C2×C4×C101C2×D411yescharacteristic
103C2×C4×C101C2×D412yes—
104C2×C4×C101C2×D412yes—
105C2×C4×C101C2411yescharacteristic
106C22⋊D101—216no—
107C2×Dic101—22no—
108C2×Dic101C2×D412yes—
109D404—24no—
110D404—22no—
111D404—22no—
112D404—22no—
113D404—48no—
114D404—42no—
115D404C2×D414yescomplemented
116C4×D101—24no—
117C4×D101—22no—
118C4×D101—22no—
119C8×C101—22no—
120#11717579—22no—
121C22×C101—24no—
122C22×C101—42no—
123C22×C101—42no—
124C22×C101—48no—
125C22×C101#12012yes—
126C22×C101C2×D812yes—
127C22×C101AGL(1,ℤ/8)12yes—
128C22×C101C22×D411yescharacteristic
129C2×D101—22no—
130C2×D101—22no—
131C2×D101—28no—
132C2×D101—24no—
133C2×D101—432no—
134C2×D101—42no—
135C2×D101—42no—
136C2×D101—48no—
137C2×D101—42no—
138C2×D101—42no—
139C2×D101—42no—
140C2×D101—84no—
141C2×D101C2×D814yescomplemented
142C4×C101—42no—
143C4×C101#12012yes—
144C4×C101C22×D411yescharacteristic
145C4×C101C22×D411yescharacteristic
146Dic101—22no—
147C2×C101—44no—
148C2×C101#44714yescomplemented
149C2×C101C2×#12011yescharacteristic
150C2×C101C22×D811yescharacteristic
151C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
152D101—24no—
153D101—44no—
154D101—82no—
155C2×#447—1011nomaximal, Sylow
156C101C2×#44711yescharacteristic, complemented, Sylow
157C23×D4—1011no—
158C22×D8—1011no—
159C22×D8—1011no—
160C2×#121—1011no—
161#447—1018no—
162C2×#102—1011no—
163C2×#102D10111yescharacteristic, complemented
164C2×C22⋊C8—1011no—
165C25—2021no—
166C22×D4—1014no—
167C22×D4—1011no—
168C22×D4—1011no—
169C22×D4—2021no—
170C22×D4—2022no—
171C22×D4—2021no—
172C22×D4—2024no—
173C22×D4—2024no—
174C22×D4C2×D10111yescharacteristic
175C23×C4—1011no—
176C2×D8—1014no—
177C2×D8—1014no—
178C2×D8—2024no—
179C2×D8—2024no—
180C22×C8—1011no—
181C22×C8C2×D10111yescharacteristic, complemented
182#121—1014no—
183#121—2022no—
184C2×C4⋊C4C2×D10111yescharacteristic, complemented
185C2×C22⋊C4—2021no—
186#102—1014no—
187#102C2×D10114yescomplemented
188C22⋊C8—1014no—
189C24—21no—
190C24—1011no—
191C24—4044no—
192C24—4044no—
193C24—2024no—
194C24—2028no—
195C24—2021no—
196C24—2021no—
197C2×D4—22no—
198C2×D4—21no—
199C2×D4—1014no—
200C2×D4—1012no—
201C2×D4—1012no—
202C2×D4—4044no—
203C2×D4—2024no—
204C2×D4—2024no—
205C2×D4—2028no—
206C2×D4—20216no—
207C2×D4—2028no—
208C2×D4—2028no—
209C2×D4—2021no—
210C2×D4—2022no—
211C2×D4—2021no—
212C2×D4—2022no—
213C2×D4—2022no—
214C2×D4D40414yescomplemented
215C2×D4C22×D10112yes—
216C22×C4—21no—
217C22×C4—1011no—
218C22×C4—1012no—
219C22×C4—1012no—
220C22×C4—2022no—
221C22×C4—2022no—
222C22×C4C22×D10111yescharacteristic
223D8—2028no—
224D8—2028no—
225C2×C8—22no—
226C2×C8—1012no—
227C2×C8—2022no—
228C2×C8C22×D10112yescomplemented
229C4⋊C4—21no—
230C4⋊C4C22×D10112yescomplemented
231C22⋊C4—2024no—
232C23—22no—
233C23—24no—
234C23—42no—
235C23—42no—
236C23—1012no—
237C23—1014no—
238C23—4048no—
239C23—40432no—
240C23—4048no—
241C23—4042no—
242C23—4042no—
243C23—4042no—
244C23—4042no—
245C23—4042no—
246C23—20216no—
247C23—2022no—
248C23—2022no—
249C23—2024no—
250C23—2022no—
251C23—2022no—
252C23D4×D10111yescharacteristic
253D4—22no—
254D4—24no—
255D4—1014no—
256D4—4048no—
257D4—4042no—
258D4—20216no—
259D4—2022no—
260D4—2022no—
261D4—2022no—
262D4—2024no—
263D4C2×D40414yescomplemented
264C2×C4—22no—
265C2×C4—42no—
266C2×C4—1012no—
267C2×C4—2022no—
268C2×C4—2022no—
269C2×C4—2022no—
270C2×C4—2024no—
271C2×C4D4×D10112yes—
272C2×C4D4×D10111yescharacteristic
273C2×C4C2×D40412yes—
274C2×C4C23×D10111yescharacteristic
275C8—22no—
276C8—2022no—
277C22—24no—
278C22—48no—
279C22—42no—
280C22—42no—
281C22—8084no—
282C22—1014no—
283C22—40432no—
284C22—4042no—
285C22—4042no—
286C22—4042no—
287C22—4048no—
288C22—4042no—
289C22—4042no—
290C22—2024no—
291C22—2028no—
292C22—2022no—
293C22—2022no—
294C22#1569222512yes—
295C22D8×D10112yes—
296C22#1569228212yes—
297C22C2×D4×D10111yescharacteristic
298C4—42no—
299C4—2022no—
300C4#1569222512yes—
301C4C2×D4×D10111yescharacteristic
302C4C22×D40411yescharacteristic
303C2—44no—
304C2—8082no—
305C2—4044no—
306C2—2024no—
307C2#2026531214yesdirect product factor
308C2C2×#1569222511yescharacteristic
309C2C2×D8×D10111yescharacteristic
310C2C2×#1569228211yescharacteristic
subgroupG/H
centerC23D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC2×#102×C101C2
socleC23×C101C2×D4
hypercenterC2×#102D101
hypocenterC101C2×#447
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. C22×D4×D101C2semidirect
2. C22×D808C2semidirect
3. C22×#13108059C2semidirect
4. C2×#15692230C2semidirect
5. #20265312C2direct (N × H)
6. C2×#15692175C2semidirect
7. C2×#15692167C2semidirect
8. C2×#102×C101C2semidirect
12. C22×D404C22semidirect
17. C2×D4×D101C22semidirect
20. C2×D808C22semidirect
22. C2×#13108059C22semidirect
23. C22×#11717579C22semidirect
25. #15692230C22semidirect
27. #15692175C22semidirect
28. #15692167C22semidirect
30. C22×C8×C101C22semidirect
31. C2×C4⋊C4×C101C22semidirect
32. #102×C101C22semidirect
47. C2×D404D4semidirect
49. C2×D404C23semidirect
63. C2×#11717579C23semidirect
68. C2×D4×C101D4semidirect
73. C2×C8×C101C23semidirect
75. C4⋊C4×C101C23semidirect
95. C22×D101D8semidirect
99. D4×C101C2×D4semidirect
115. D404C2×D4semidirect
141. C2×D101C2×D8semidirect
148. C2×C101#447semidirect
156. C101C2×#447semidirect
163. C2×#102D101semidirect
181. C22×C8C2×D101semidirect
184. C2×C4⋊C4C2×D101semidirect
187. #102C2×D101semidirect
214. C2×D4D404semidirect
228. C2×C8C22×D101semidirect
230. C4⋊C4C22×D101semidirect
263. D4C2×D404semidirect
307. C2#20265312direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D4×D1012
C22×D8082
C22×#131080592
C2×#156922302
#202653122
C2×#156921752
C2×#156921672
C2×#102×C1012
C2×#447101