Finite Group Databasebeta

← C2×#20260879 #27150936

Subgroups — C2×#20260879 #27150936

Order 12416.

812 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 812 conjugacy classes of subgroups — too many to draw. Only its 165 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×#13095512C211yesmaximal, characteristic, complemented
2C22×#13095496C211yesmaximal, characteristic, complemented
3C2×#15690781C211yesmaximal, characteristic, complemented
4#20260879C218yesmaximal, direct product factor
5C2×#15690734C211yesmaximal, characteristic, complemented
6C2×#15690732C211yesmaximal, characteristic, complemented
7C2×#15690722C211yesmaximal, characteristic, complemented
8C22×SD16×C97C211yesmaximal, characteristic, complemented
9C2×#13095512—24no—
10C2×#13095512—22no—
11C2×#13095512—24no—
12C2×#13095512C2214yescomplemented
13C22×D388—21no—
14C22×D388C2211yescharacteristic
15C22×C22⋊D97—21no—
16C22×C4×D97—21no—
17C22×C4×D97C2211yescharacteristic
18C2×#13095496—24no—
19C2×#13095496C2214yescomplemented
20C22×#11715895C2211yescharacteristic
21#15690781—22no—
22#15690781C2214yescomplemented
23C2×C4⋊Dic97C2211yescharacteristic
24C2×C22⋊Dic97—21no—
25#15690734C2214yescomplemented
26#15690732C2214yescomplemented
27#15690722C2214yescomplemented
28C22×Q8×C97C2211yescharacteristic
29C22×D4×C97C2211yescharacteristic
30C2×SD16×C97—24no—
31C2×SD16×C97C2214yescomplemented
32C22×C8×C97C2211yescharacteristic
33C23×D97—21no—
34C23×D97D411yescharacteristic
35#13095512—216no—
36#13095512—28no—
37#13095512—28no—
38C2×D388—28no—
39C2×D388—24no—
40C2×D388—21no—
41C2×D388—22no—
42C2×D388D414yescomplemented
43C2×D388C2312yes—
44C2×C22⋊D97—24no—
45C2×C22⋊D97—44no—
46C22×Dic97—21no—
47C22×Dic97D411yescharacteristic
48C2×C4×D97—28no—
49C2×C4×D97—22no—
50C2×C4×D97—22no—
51C2×C4×D97—24no—
52C2×C4×D97D412yes—
53C2×C4×D97C2312yes—
54#13095496—28no—
55C2×#11715895—22no—
56C2×#11715895C2312yes—
57C4⋊Dic97—21no—
58C4⋊Dic97C2312yes—
59C22⋊Dic97—24no—
60C24×C97—21no—
61C2×Q8×C97—22no—
62C2×Q8×C97—21no—
63C2×Q8×C97D414yescomplemented
64C2×Q8×C97C2312yes—
65C2×D4×C97—22no—
66C2×D4×C97—22no—
67C2×D4×C97—21no—
68C2×D4×C97C2312yes—
69C22×C4×C97—21no—
70C22×C4×C97C2311yescharacteristic
71SD16×C97—28no—
72C2×C8×C97—22no—
73C2×C8×C97C2312yes—
74C23×C97—22no—
75C23×C97—42no—
76C23×C97—42no—
77C23×C97—42no—
78C23×C97C2×D411yescharacteristic
79C22×D97—22no—
80C22×D97—24no—
81C22×D97—22no—
82C22×D97—22no—
83C22×D97—22no—
84C22×D97—42no—
85C22×D97—42no—
86C22×D97C2×D412yes—
87Q8×C97—24no—
88Q8×C97—22no—
89Q8×C97C2×D414yescomplemented
90D4×C97—22no—
91D4×C97—22no—
92D4×C97—42no—
93C2×C4×C97—24no—
94C2×C4×C97—22no—
95C2×C4×C97C2×D411yescharacteristic
96C2×C4×C97C2×D412yes—
97C2×C4×C97C2×D412yes—
98C2×C4×C97C2411yescharacteristic
99C22⋊D97—48no—
100C2×Dic97—22no—
101C2×Dic97—22no—
102C2×Dic97—42no—
103C2×Dic97C2×D412yes—
104D388—216no—
105D388—24no—
106D388—22no—
107D388C2×D414yescomplemented
108C4×D97—216no—
109C4×D97—24no—
110C4×D97—22no—
111C4×D97—22no—
112C8×C97—22no—
113#11715895—22no—
114C22×C97—42no—
115C22×C97—42no—
116C22×C97—42no—
117C22×C97—84no—
118C22×C97#12012yes—
119C22×C97#13512yes—
120C22×C97AGL(1,ℤ/8)12yes—
121C22×C97C22×D411yescharacteristic
122C2×D97—22no—
123C2×D97—24no—
124C2×D97—22no—
125C2×D97—22no—
126C2×D97—42no—
127C2×D97—42no—
128C2×D97—44no—
129C2×D97—48no—
130C4×C97—24no—
131C4×C97#12012yes—
132C4×C97C22×D411yescharacteristic
133C4×C97C22×D411yescharacteristic
134Dic97—22no—
135Dic97—42no—
136C2×C97—82no—
137C2×C97#44914yescomplemented
138C2×C97C2×#12011yescharacteristic
139C2×C97C2×#13511yescharacteristic
140C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
141D97—42no—
142D97—44no—
143C2×#449—971nomaximal, Sylow
144C97C2×#44911yescharacteristic, complemented, Sylow
145C22×#41—971no—
146C22×SD16D9711yescharacteristic, complemented
147C22×D8—971no—
148C2×#121—971no—
149#449—978no—
150C2×#103—971no—
151C2×#102—971no—
152C2×C22⋊C8—971no—
153C2×#41—974no—
154C2×#41—1942no—
155C2×#41—1944no—
156C2×#41—1944no—
157C22×Q8C2×D9711yescharacteristic
158C22×D4—971no—
159C22×D4—1941no—
160C22×D4—1941no—
161C22×D4C2×D9711yescharacteristic
162C23×C4—971no—
163C23×C4—1941no—
164C2×SD16—24no—
165C2×SD16C2×D9714yescomplemented
166C2×D8—974no—
167C2×D8—1944no—
168C22×C8—971no—
169C22×C8C2×D9711yescharacteristic
170#121—974no—
171#121—1942no—
172C2×C4⋊C4—971no—
173C2×C22⋊C4—1941no—
174#103—974no—
175#102—974no—
176C22⋊C8—974no—
177C24—21no—
178C24—971no—
179C24—1941no—
180#41—1948no—
181#41—19416no—
182#41—1948no—
183C2×Q8—22no—
184C2×Q8—21no—
185C2×Q8C22⋊D9714yescomplemented
186C2×Q8C22×D9712yes—
187C2×D4—22no—
188C2×D4—21no—
189C2×D4—22no—
190C2×D4—974no—
191C2×D4—972no—
192C2×D4—3884no—
193C2×D4—1944no—
194C2×D4—1948no—
195C2×D4—1944no—
196C2×D4—1941no—
197C2×D4—1942no—
198C2×D4C22×D9712yes—
199C22×C4—21no—
200C22×C4—971no—
201C22×C4—972no—
202C22×C4—972no—
203C22×C4—1944no—
204C22×C4—1948no—
205C22×C4—1941no—
206C22×C4—1942no—
207C22×C4—1942no—
208C22×C4C22×D9711yescharacteristic
209SD16—28no—
210D8—1948no—
211C2×C8—22no—
212C2×C8—972no—
213C2×C8—1942no—
214C2×C8C22×D9712yes—
215C4⋊C4—972no—
216C4⋊C4—1941no—
217C22⋊C4—1944no—
218C23—22no—
219C23—42no—
220C23—42no—
221C23—42no—
222C23—972no—
223C23—3882no—
224C23—3882no—
225C23—1944no—
226C23—1942no—
227C23—1942no—
228C23—1942no—
229C23—1942no—
230C23D4×D9711yescharacteristic
231Q8—24no—
232Q8—22no—
233Q8C2×C22⋊D9714yescomplemented
234D4—22no—
235D4—22no—
236D4—42no—
237D4—974no—
238D4—3888no—
239D4—19416no—
240D4—1944no—
241D4—1942no—
242C2×C4—24no—
243C2×C4—22no—
244C2×C4—972no—
245C2×C4—3882no—
246C2×C4—19416no—
247C2×C4—1942no—
248C2×C4—1942no—
249C2×C4—1944no—
250C2×C4—1942no—
251C2×C4—1942no—
252C2×C4D4×D9711yescharacteristic
253C2×C4D4×D9712yes—
254C2×C4C2×C22⋊D9712yes—
255C2×C4C23×D9711yescharacteristic
256C8—22no—
257C8—1942no—
258C22—42no—
259C22—42no—
260C22—42no—
261C22—84no—
262C22—3888no—
263C22—3884no—
264C22—3882no—
265C22—3882no—
266C22—1944no—
267C22—1942no—
268C22—1942no—
269C22—1942no—
270C22#1569077812yes—
271C22#1569083112yes—
272C22#1569083712yes—
273C22C2×D4×D9711yescharacteristic
274C4—24no—
275C4—3882no—
276C4—1942no—
277C4#1569077812yes—
278C4C2×D4×D9711yescharacteristic
279C4C22×C22⋊D9711yescharacteristic
280C2—82no—
281C2—3884no—
282C2—3882no—
283C2#2026087914yesdirect product factor
284C2C2×#1569077811yescharacteristic
285C2C2×#1569083111yescharacteristic
286C2C2×#1569083711yescharacteristic
subgroupG/H
centerC23D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC22×SD16×C97C2
socleC23×C97C2×D4
hypercenterC22×SD16D97
hypocenterC97C2×#449
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C22×#13095512C2semidirect
2. C22×#13095496C2semidirect
3. C2×#15690781C2semidirect
4. #20260879C2direct (N × H)
5. C2×#15690734C2semidirect
6. C2×#15690732C2semidirect
7. C2×#15690722C2semidirect
8. C22×SD16×C97C2semidirect
12. C2×#13095512C22semidirect
19. C2×#13095496C22semidirect
22. #15690781C22semidirect
25. #15690734C22semidirect
26. #15690732C22semidirect
27. #15690722C22semidirect
31. C2×SD16×C97C22semidirect
42. C2×D388D4semidirect
63. C2×Q8×C97D4semidirect
89. Q8×C97C2×D4semidirect
107. D388C2×D4semidirect
137. C2×C97#449semidirect
144. C97C2×#449semidirect
146. C22×SD16D97semidirect
165. C2×SD16C2×D97semidirect
185. C2×Q8C22⋊D97semidirect
233. Q8C2×C22⋊D97semidirect
283. C2#20260879direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130955122
C22×#130954962
C2×#156907812
#202608792
C2×#156907342
C2×#156907322
C2×#156907222
C22×SD16×C972
C2×#44997