Finite Group Databasebeta

← C2×#20260867 #27150910

Subgroups — C2×#20260867 #27150910

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×#13095496C211yesmaximal, characteristic, complemented
3C2×#15690781C211yesmaximal, characteristic, complemented
4#20260867C218yesmaximal, direct product factor
5C2×#15690732C211yesmaximal, characteristic, complemented
6C2×#15690731C211yesmaximal, characteristic, complemented
7C2×#15690722C211yesmaximal, characteristic, complemented
8C22×D8×C97C211yesmaximal, characteristic, complemented
9C24×D97—21no—
10C22×D388C2211yescharacteristic
11C22×C22⋊D97—21no—
12C22×C22⋊D97—21no—
13C2×D4×D97—24no—
14C2×D4×D97—22no—
15C2×D4×D97—24no—
16C2×D4×D97C2214yescomplemented
17C22×C4×D97C2211yescharacteristic
18C2×#13095496—24no—
19C2×#13095496C2214yescomplemented
20C22×#11715895C2211yescharacteristic, complemented
21#15690781—22no—
22#15690781C2214yescomplemented
23C2×C4⋊Dic97C2211yescharacteristic, complemented
24C2×C22⋊Dic97—21no—
25#15690732C2214yescomplemented
26#15690731C2214yescomplemented
27#15690722C2214yescomplemented
28C22×D4×C97C2211yescharacteristic
29C22×D4×C97C2211yescharacteristic, complemented
30C2×D8×C97—24no—
31C2×D8×C97C2214yescomplemented
32C22×C8×C97C2211yescharacteristic, complemented
33C23×D97—28no—
34C23×D97—21no—
35C23×D97—24no—
36C23×D97—44no—
37C23×D97—44no—
38C23×D97D411yescharacteristic
39C2×D388—21no—
40C2×D388—22no—
41C2×D388D414yescomplemented
42C2×D388C2312yes—
43C2×C22⋊D97—28no—
44C2×C22⋊D97—24no—
45C2×C22⋊D97—24no—
46C2×C22⋊D97—44no—
47D4×D97—216no—
48D4×D97—28no—
49D4×D97—28no—
50C22×Dic97—21no—
51C22×Dic97D411yescharacteristic
52C2×C4×D97—22no—
53C2×C4×D97—22no—
54C2×C4×D97D412yes—
55C2×C4×D97C2312yes—
56#13095496—28no—
57C2×#11715895—22no—
58C2×#11715895C2312yescomplemented
59C4⋊Dic97—21no—
60C4⋊Dic97C2312yescomplemented
61C22⋊Dic97—24no—
62C24×C97—21no—
63C24×C97—21no—
64C2×D4×C97—21no—
65C2×D4×C97—22no—
66C2×D4×C97—22no—
67C2×D4×C97—22no—
68C2×D4×C97—21no—
69C2×D4×C97D414yescomplemented
70C2×D4×C97C2312yes—
71C2×D4×C97C2312yescomplemented
72C22×C4×C97C2311yescharacteristic
73D8×C97—28no—
74C2×C8×C97—22no—
75C2×C8×C97C2312yescomplemented
76C23×C97—24no—
77C23×C97—22no—
78C23×C97—22no—
79C23×C97—42no—
80C23×C97—42no—
81C23×C97—42no—
82C23×C97—42no—
83C23×C97—42no—
84C23×C97C2×D411yescharacteristic
85C22×D97—22no—
86C22×D97—22no—
87C22×D97—216no—
88C22×D97—24no—
89C22×D97—22no—
90C22×D97—48no—
91C22×D97—432no—
92C22×D97—42no—
93C22×D97—42no—
94C22×D97—48no—
95C22×D97D814yescomplemented
96C22×D97C2×D412yes—
97D4×C97—24no—
98D4×C97—22no—
99D4×C97—22no—
100D4×C97—22no—
101D4×C97—42no—
102D4×C97C2×D414yescomplemented
103C2×C4×C97C2×D411yescharacteristic
104C2×C4×C97C2×D412yes—
105C2×C4×C97C2×D412yes—
106C2×C4×C97C2411yescharacteristic
107C22⋊D97—216no—
108C22⋊D97—48no—
109C2×Dic97—22no—
110C2×Dic97—22no—
111C2×Dic97—42no—
112C2×Dic97C2×D412yes—
113D388—24no—
114D388—22no—
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—42no—
126C22×C97—42no—
127C22×C97—42no—
128C22×C97—84no—
129C22×C97#12012yes—
130C22×C97C2×D812yes—
131C22×C97AGL(1,ℤ/8)12yes—
132C22×C97C22×D411yescharacteristic
133C2×D97—22no—
134C2×D97—22no—
135C2×D97—28no—
136C2×D97—24no—
137C2×D97—432no—
138C2×D97—42no—
139C2×D97—42no—
140C2×D97—48no—
141C2×D97C2×D814yescomplemented
142C4×C97#12012yes—
143C4×C97C22×D411yescharacteristic
144C4×C97C22×D411yescharacteristic
145Dic97—22no—
146Dic97—42no—
147C2×C97—44no—
148C2×C97—82no—
149C2×C97#44714yescomplemented
150C2×C97C2×#12011yescharacteristic
151C2×C97C22×D811yescharacteristic
152C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
153D97—24no—
154D97—44no—
155C2×#447—971nomaximal, Sylow
156C97C2×#44711yescharacteristic, complemented, Sylow
157C23×D4—971no—
158C22×D8—971no—
159C22×D8D9711yescharacteristic, complemented
160C2×#121—971no—
161#447—978no—
162C2×#102—971no—
163C2×#102—971no—
164C2×C22⋊C8—971no—
165C25—1941no—
166C22×D4—974no—
167C22×D4—971no—
168C22×D4—1942no—
169C22×D4—1941no—
170C22×D4—1944no—
171C22×D4—1941no—
172C22×D4—1944no—
173C22×D4C2×D9711yescharacteristic
174C22×D4C2×D9711yescharacteristic, complemented
175C23×C4—971no—
176C2×D8—24no—
177C2×D8—974no—
178C2×D8—1944no—
179C2×D8C2×D9714yescomplemented
180C22×C8—971no—
181C22×C8C2×D9711yescharacteristic, complemented
182#121—974no—
183#121—1942no—
184C2×C4⋊C4—971no—
185C2×C22⋊C4—1941no—
186#102—974no—
187#102—974no—
188C22⋊C8—974no—
189C24—21no—
190C24—21no—
191C24—971no—
192C24—3884no—
193C24—3884no—
194C24—1944no—
195C24—1948no—
196C24—1941no—
197C2×D4—21no—
198C2×D4—22no—
199C2×D4—22no—
200C2×D4—21no—
201C2×D4—22no—
202C2×D4—974no—
203C2×D4—972no—
204C2×D4—3884no—
205C2×D4—1944no—
206C2×D4—1948no—
207C2×D4—19416no—
208C2×D4—1944no—
209C2×D4—1948no—
210C2×D4—1948no—
211C2×D4—1941no—
212C2×D4—1942no—
213C2×D4C22⋊D9714yescomplemented
214C2×D4C22×D9712yes—
215C2×D4C22×D9712yescomplemented
216C22×C4—971no—
217C22×C4—972no—
218C22×C4—972no—
219C22×C4—1941no—
220C22×C4—1942no—
221C22×C4—1942no—
222C22×C4C22×D9711yescharacteristic
223D8—28no—
224D8—1948no—
225C2×C8—22no—
226C2×C8—972no—
227C2×C8—1942no—
228C2×C8C22×D9712yescomplemented
229C4⋊C4—972no—
230C4⋊C4—1941no—
231C22⋊C4—1944no—
232C23—24no—
233C23—22no—
234C23—22no—
235C23—42no—
236C23—42no—
237C23—42no—
238C23—42no—
239C23—42no—
240C23—974no—
241C23—972no—
242C23—3888no—
243C23—38832no—
244C23—3888no—
245C23—3882no—
246C23—3882no—
247C23—19416no—
248C23—1944no—
249C23—1942no—
250C23—1942no—
251C23—1942no—
252C23D4×D9711yescharacteristic
253D4—24no—
254D4—22no—
255D4—22no—
256D4—22no—
257D4—42no—
258D4—974no—
259D4—3888no—
260D4—19416no—
261D4—1944no—
262D4—1942no—
263D4C2×C22⋊D9714yescomplemented
264C2×C4—972no—
265C2×C4—3882no—
266C2×C4—1942no—
267C2×C4—1942no—
268C2×C4—1944no—
269C2×C4—1942no—
270C2×C4—1942no—
271C2×C4D4×D9711yescharacteristic
272C2×C4D4×D9712yes—
273C2×C4C2×C22⋊D9712yes—
274C2×C4C23×D9711yescharacteristic
275C8—22no—
276C8—1942no—
277C22—24no—
278C22—48no—
279C22—42no—
280C22—42no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22—84no—
285C22—974no—
286C22—38832no—
287C22—3888no—
288C22—3882no—
289C22—3882no—
290C22—1944no—
291C22—1948no—
292C22—1942no—
293C22—1942no—
294C22#1569077812yes—
295C22D8×D9712yes—
296C22#1569083412yes—
297C22C2×D4×D9711yescharacteristic
298C4—3882no—
299C4—1942no—
300C4#1569077812yes—
301C4C2×D4×D9711yescharacteristic
302C4C22×C22⋊D9711yescharacteristic
303C2—44no—
304C2—82no—
305C2—3884no—
306C2—1944no—
307C2#2026086714yesdirect product factor
308C2C2×#1569077811yescharacteristic
309C2C2×D8×D9711yescharacteristic
310C2C2×#1569083411yescharacteristic
subgroupG/H
centerC23D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC22×D8×C97C2
socleC23×C97C2×D4
hypercenterC22×D8D97
hypocenterC97C2×#447
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×D8

Complements

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

NH ≅ G/Ntype
1. C22×D4×D97C2semidirect
2. C22×#13095496C2semidirect
3. C2×#15690781C2semidirect
4. #20260867C2direct (N × H)
5. C2×#15690732C2semidirect
6. C2×#15690731C2semidirect
7. C2×#15690722C2semidirect
8. C22×D8×C97C2semidirect
16. C2×D4×D97C22semidirect
19. C2×#13095496C22semidirect
20. C22×#11715895C22semidirect
22. #15690781C22semidirect
23. C2×C4⋊Dic97C22semidirect
25. #15690732C22semidirect
26. #15690731C22semidirect
27. #15690722C22semidirect
29. C22×D4×C97C22semidirect
31. C2×D8×C97C22semidirect
32. C22×C8×C97C22semidirect
41. C2×D388D4semidirect
58. C2×#11715895C23semidirect
60. C4⋊Dic97C23semidirect
69. C2×D4×C97D4semidirect
71. C2×D4×C97C23semidirect
75. C2×C8×C97C23semidirect
95. C22×D97D8semidirect
102. D4×C97C2×D4semidirect
115. D388C2×D4semidirect
141. C2×D97C2×D8semidirect
149. C2×C97#447semidirect
156. C97C2×#447semidirect
159. C22×D8D97semidirect
174. C22×D4C2×D97semidirect
179. C2×D8C2×D97semidirect
181. C22×C8C2×D97semidirect
213. C2×D4C22⋊D97semidirect
215. C2×D4C22×D97semidirect
228. C2×C8C22×D97semidirect
263. D4C2×C22⋊D97semidirect
307. C2#20260867direct (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×#130954962
C2×#156907812
#202608672
C2×#156907322
C2×#156907312
C2×#156907222
C22×D8×C972
C2×#44797