Finite Group Databasebeta

← C2×#20260870 #27150913

Subgroups — C2×#20260870 #27150913

Order 12416.

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×D97C211yesmaximal, characteristic, complemented
2C22×D776C211yesmaximal, characteristic, complemented
3C22×#13095496C211yesmaximal, characteristic, complemented
4C2×#15690785C211yesmaximal, characteristic, complemented
5#20260870C218yesmaximal, direct product factor
6C2×#15690730C211yesmaximal, characteristic, complemented
7C2×#15690722C211yesmaximal, characteristic, complemented
8C2×#102×C97C211yesmaximal, characteristic, complemented
9C24×D97—21no—
10C22×D388—21no—
11C22×D388C2211yescharacteristic
12C22×D388C2211yescharacteristic, complemented
13C22×C22⋊D97—21no—
14C2×D4×D97—24no—
15C2×D4×D97—22no—
16C2×D4×D97—24no—
17C2×D4×D97C2214yescomplemented
18C22×C4×D97C2211yescharacteristic
19C2×D776—24no—
20C2×D776C2214yescomplemented
21C2×#13095496—24no—
22C2×#13095496C2214yescomplemented
23C22×#11715895C2211yescharacteristic, complemented
24#15690785—22no—
25#15690785C2214yescomplemented
26C2×#13095488—21no—
27#15690730C2214yescomplemented
28#15690722C2214yescomplemented
29C22×D4×C97C2211yescharacteristic
30C22×C8×C97C2211yescharacteristic, complemented
31C2×C4⋊C4×C97C2211yescharacteristic, complemented
32#102×C97C2214yescomplemented
33C23×D97—28no—
34C23×D97—21no—
35C23×D97—24no—
36C23×D97—21no—
37C23×D97—44no—
38C23×D97—44no—
39C23×D97D411yescharacteristic
40C2×D388—21no—
41C2×D388—22no—
42C2×D388—22no—
43C2×D388—22no—
44C2×D388—24no—
45C2×D388—21no—
46C2×D388—44no—
47C2×D388D414yescomplemented
48C2×D388C2312yes—
49C2×D388C2312yescomplemented
50C2×C22⋊D97—28no—
51C2×C22⋊D97—24no—
52D4×D97—216no—
53D4×D97—28no—
54D4×D97—28no—
55C22×Dic97D411yescharacteristic
56C2×C4×D97—22no—
57C2×C4×D97—22no—
58C2×C4×D97D412yes—
59C2×C4×D97C2312yes—
60D776—28no—
61#13095496—28no—
62C2×#11715895—22no—
63C2×#11715895C2312yescomplemented
64#13095488—24no—
65C24×C97—21no—
66C2×D4×C97—21no—
67C2×D4×C97—22no—
68C2×D4×C97D414yescomplemented
69C2×D4×C97C2312yes—
70C22×C4×C97—21no—
71C22×C4×C97C2311yescharacteristic
72C2×C8×C97—22no—
73C2×C8×C97C2312yescomplemented
74C4⋊C4×C97—21no—
75C4⋊C4×C97C2312yescomplemented
76C23×C97—24no—
77C23×C97—22no—
78C23×C97—42no—
79C23×C97—42no—
80C23×C97C2×D411yescharacteristic
81C22×D97—22no—
82C22×D97—22no—
83C22×D97—216no—
84C22×D97—24no—
85C22×D97—22no—
86C22×D97—22no—
87C22×D97—48no—
88C22×D97—432no—
89C22×D97—42no—
90C22×D97—42no—
91C22×D97—48no—
92C22×D97—42no—
93C22×D97—42no—
94C22×D97—42no—
95C22×D97D814yescomplemented
96C22×D97C2×D412yes—
97D4×C97—24no—
98D4×C97—22no—
99D4×C97C2×D414yescomplemented
100C2×C4×C97—22no—
101C2×C4×C97—42no—
102C2×C4×C97C2×D411yescharacteristic
103C2×C4×C97C2×D412yes—
104C2×C4×C97C2×D412yes—
105C2×C4×C97C2411yescharacteristic
106C22⋊D97—216no—
107C2×Dic97—22no—
108C2×Dic97C2×D412yes—
109D388—24no—
110D388—22no—
111D388—22no—
112D388—22no—
113D388—48no—
114D388—42no—
115D388C2×D414yescomplemented
116C4×D97—24no—
117C4×D97—22no—
118C4×D97—22no—
119C8×C97—22no—
120#11715895—22no—
121C22×C97—24no—
122C22×C97—42no—
123C22×C97—42no—
124C22×C97—48no—
125C22×C97#12012yes—
126C22×C97C2×D812yes—
127C22×C97AGL(1,ℤ/8)12yes—
128C22×C97C22×D411yescharacteristic
129C2×D97—22no—
130C2×D97—22no—
131C2×D97—28no—
132C2×D97—24no—
133C2×D97—432no—
134C2×D97—42no—
135C2×D97—42no—
136C2×D97—48no—
137C2×D97—42no—
138C2×D97—42no—
139C2×D97—42no—
140C2×D97—84no—
141C2×D97C2×D814yescomplemented
142C4×C97—42no—
143C4×C97#12012yes—
144C4×C97C22×D411yescharacteristic
145C4×C97C22×D411yescharacteristic
146Dic97—22no—
147C2×C97—44no—
148C2×C97#44714yescomplemented
149C2×C97C2×#12011yescharacteristic
150C2×C97C22×D811yescharacteristic
151C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
152D97—24no—
153D97—44no—
154D97—82no—
155C2×#447—971nomaximal, Sylow
156C97C2×#44711yescharacteristic, complemented, Sylow
157C23×D4—971no—
158C22×D8—971no—
159C22×D8—971no—
160C2×#121—971no—
161#447—978no—
162C2×#102—971no—
163C2×#102D9711yescharacteristic, complemented
164C2×C22⋊C8—971no—
165C25—1941no—
166C22×D4—974no—
167C22×D4—971no—
168C22×D4—971no—
169C22×D4—1941no—
170C22×D4—1942no—
171C22×D4—1941no—
172C22×D4—1944no—
173C22×D4—1944no—
174C22×D4C2×D9711yescharacteristic
175C23×C4—971no—
176C2×D8—974no—
177C2×D8—974no—
178C2×D8—1944no—
179C2×D8—1944no—
180C22×C8—971no—
181C22×C8C2×D9711yescharacteristic, complemented
182#121—974no—
183#121—1942no—
184C2×C4⋊C4C2×D9711yescharacteristic, complemented
185C2×C22⋊C4—1941no—
186#102—974no—
187#102C2×D9714yescomplemented
188C22⋊C8—974no—
189C24—21no—
190C24—971no—
191C24—3884no—
192C24—3884no—
193C24—1944no—
194C24—1948no—
195C24—1941no—
196C24—1941no—
197C2×D4—21no—
198C2×D4—22no—
199C2×D4—974no—
200C2×D4—972no—
201C2×D4—972no—
202C2×D4—3884no—
203C2×D4—1944no—
204C2×D4—1944no—
205C2×D4—1948no—
206C2×D4—19416no—
207C2×D4—1948no—
208C2×D4—1948no—
209C2×D4—1941no—
210C2×D4—1942no—
211C2×D4—1941no—
212C2×D4—1942no—
213C2×D4—1942no—
214C2×D4D38814yescomplemented
215C2×D4C22×D9712yes—
216C22×C4—21no—
217C22×C4—971no—
218C22×C4—972no—
219C22×C4—972no—
220C22×C4—1942no—
221C22×C4—1942no—
222C22×C4C22×D9711yescharacteristic
223D8—1948no—
224D8—1948no—
225C2×C8—22no—
226C2×C8—972no—
227C2×C8—1942no—
228C2×C8C22×D9712yescomplemented
229C4⋊C4—21no—
230C4⋊C4C22×D9712yescomplemented
231C22⋊C4—1944no—
232C23—24no—
233C23—22no—
234C23—42no—
235C23—42no—
236C23—974no—
237C23—972no—
238C23—3888no—
239C23—38832no—
240C23—3888no—
241C23—3882no—
242C23—3882no—
243C23—3882no—
244C23—3882no—
245C23—3882no—
246C23—19416no—
247C23—1942no—
248C23—1944no—
249C23—1942no—
250C23—1942no—
251C23—1942no—
252C23D4×D9711yescharacteristic
253D4—24no—
254D4—22no—
255D4—974no—
256D4—3888no—
257D4—3882no—
258D4—19416no—
259D4—1944no—
260D4—1942no—
261D4—1942no—
262D4—1942no—
263D4C2×D38814yescomplemented
264C2×C4—22no—
265C2×C4—42no—
266C2×C4—972no—
267C2×C4—1942no—
268C2×C4—1944no—
269C2×C4—1942no—
270C2×C4—1942no—
271C2×C4D4×D9711yescharacteristic
272C2×C4D4×D9712yes—
273C2×C4C2×D38812yes—
274C2×C4C23×D9711yescharacteristic
275C8—22no—
276C8—1942no—
277C22—24no—
278C22—48no—
279C22—42no—
280C22—42no—
281C22—7764no—
282C22—974no—
283C22—38832no—
284C22—3882no—
285C22—3882no—
286C22—3882no—
287C22—3888no—
288C22—3882no—
289C22—3882no—
290C22—1944no—
291C22—1948no—
292C22—1942no—
293C22—1942no—
294C22#1569078012yes—
295C22D8×D9712yes—
296C22#1569083712yes—
297C22C2×D4×D9711yescharacteristic
298C4—42no—
299C4—1942no—
300C4#1569078012yes—
301C4C2×D4×D9711yescharacteristic
302C4C22×D38811yescharacteristic
303C2—44no—
304C2—7762no—
305C2—3884no—
306C2—1944no—
307C2#2026087014yesdirect product factor
308C2C2×#1569078011yescharacteristic
309C2C2×D8×D9711yescharacteristic
310C2C2×#1569083711yescharacteristic
subgroupG/H
centerC23D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC2×#102×C97C2
socleC23×C97C2×D4
hypercenterC2×#102D97
hypocenterC97C2×#447
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. C22×D4×D97C2semidirect
2. C22×D776C2semidirect
3. C22×#13095496C2semidirect
4. C2×#15690785C2semidirect
5. #20260870C2direct (N × H)
6. C2×#15690730C2semidirect
7. C2×#15690722C2semidirect
8. C2×#102×C97C2semidirect
12. C22×D388C22semidirect
17. C2×D4×D97C22semidirect
20. C2×D776C22semidirect
22. C2×#13095496C22semidirect
23. C22×#11715895C22semidirect
25. #15690785C22semidirect
27. #15690730C22semidirect
28. #15690722C22semidirect
30. C22×C8×C97C22semidirect
31. C2×C4⋊C4×C97C22semidirect
32. #102×C97C22semidirect
47. C2×D388D4semidirect
49. C2×D388C23semidirect
63. C2×#11715895C23semidirect
68. C2×D4×C97D4semidirect
73. C2×C8×C97C23semidirect
75. C4⋊C4×C97C23semidirect
95. C22×D97D8semidirect
99. D4×C97C2×D4semidirect
115. D388C2×D4semidirect
141. C2×D97C2×D8semidirect
148. C2×C97#447semidirect
156. C97C2×#447semidirect
163. C2×#102D97semidirect
181. C22×C8C2×D97semidirect
184. C2×C4⋊C4C2×D97semidirect
187. #102C2×D97semidirect
214. C2×D4D388semidirect
228. C2×C8C22×D97semidirect
230. C4⋊C4C22×D97semidirect
263. D4C2×D388semidirect
307. C2#20260870direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D4×D972
C22×D7762
C22×#130954962
C2×#156907852
#202608702
C2×#156907302
C2×#156907222
C2×#102×C972
C2×#44797