Finite Group Databasebeta

← C2×#20260872 #27150922

Subgroups — C2×#20260872 #27150922

Order 12416.

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

normal subgroups only

This group has 792 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×Q8×D97C211yesmaximal, characteristic, complemented
2C22×#13095499C211yesmaximal, characteristic, complemented
3C2×#15690781C211yesmaximal, characteristic
4#20260872C218yesmaximal, direct product factor
5C2×#15690735C211yesmaximal, characteristic, complemented
6C2×#15690734C211yesmaximal, characteristic, complemented
7C2×#15690722C211yesmaximal, characteristic, complemented
8C22×SD16×C97C211yesmaximal, characteristic, complemented
9C22×Dic194—21no—
10C22×Dic194C2211yescharacteristic
11C2×Q8×D97—24no—
12C2×Q8×D97—22no—
13C2×Q8×D97—24no—
14C2×Q8×D97C2214yescomplemented
15C22×C22⋊D97—21no—
16C22×C4×D97—21no—
17C22×C4×D97C2211yescharacteristic
18C2×#13095499—24no—
19C2×#13095499C2214yescomplemented
20C22×#11715895C2211yescharacteristic
21#15690781—22no—
22#15690781C2214yes—
23C2×C4⋊Dic97C2211yescharacteristic
24C2×C22⋊Dic97—21no—
25#15690735C2214yescomplemented
26#15690734C2214yescomplemented
27#15690722C2214yescomplemented
28C22×Q8×C97C2211yescharacteristic
29C22×D4×C97C2211yescharacteristic
30C2×SD16×C97—24no—
31C2×SD16×C97C2214yescomplemented
32C22×C8×C97C2211yescharacteristic
33C23×D97D411yescharacteristic
34C2×Dic194—28no—
35C2×Dic194—21no—
36C2×Dic194—22no—
37C2×Dic194—24no—
38C2×Dic194D414yescomplemented
39C2×Dic194C2312yes—
40Q8×D97—216no—
41Q8×D97—28no—
42Q8×D97—28no—
43C2×C22⋊D97—24no—
44C2×C22⋊D97—44no—
45C22×Dic97—21no—
46C22×Dic97—21no—
47C22×Dic97D411yescharacteristic
48C2×C4×D97—28no—
49C2×C4×D97—24no—
50C2×C4×D97—22no—
51C2×C4×D97—22no—
52C2×C4×D97D412yes—
53C2×C4×D97C2312yes—
54#13095499—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—22no—
81C22×D97SD1614yescomplemented
82C22×D97C2×D412yes—
83Q8×C97—24no—
84Q8×C97—22no—
85Q8×C97C2×D414yescomplemented
86D4×C97—22no—
87D4×C97—22no—
88D4×C97—42no—
89C2×C4×C97—24no—
90C2×C4×C97—22no—
91C2×C4×C97C2×D411yescharacteristic
92C2×C4×C97C2×D412yes—
93C2×C4×C97C2×D412yes—
94C2×C4×C97C2411yescharacteristic
95C22⋊D97—48no—
96C2×Dic97—24no—
97C2×Dic97—22no—
98C2×Dic97—22no—
99C2×Dic97—22no—
100C2×Dic97—42no—
101C2×Dic97C2×D412yes—
102C4×D97—216no—
103C4×D97—24no—
104C4×D97—22no—
105C4×D97—22no—
106Dic194—216no—
107Dic194—24no—
108Dic194—22no—
109Dic194C2×D414yescomplemented
110C8×C97—22no—
111#11715895—22no—
112C22×C97—42no—
113C22×C97—42no—
114C22×C97—42no—
115C22×C97—84no—
116C22×C97#12012yes—
117C22×C97C2×SD1612yes—
118C22×C97#13712yes—
119C22×C97C22×D411yescharacteristic
120C2×D97—22no—
121C2×D97—22no—
122C2×D97—28no—
123C2×D97C2×SD1614yescomplemented
124C4×C97—24no—
125C4×C97#12012yes—
126C4×C97C22×D411yescharacteristic
127C4×C97C22×D411yescharacteristic
128Dic97—22no—
129Dic97—24no—
130Dic97—42no—
131C2×C97—82no—
132C2×C97#44814yescomplemented
133C2×C97C2×#12011yescharacteristic
134C2×C97C22×SD1611yescharacteristic
135C2×C97C2×#13711yescharacteristic
136D97—24no—
137C2×#448—971nomaximal, Sylow
138C97C2×#44811yescharacteristic, complemented, Sylow
139C23×Q8—971no—
140C22×SD16—971no—
141C22×SD16D9711yescharacteristic, complemented
142C2×#121—971no—
143#448—978no—
144C2×#103—971no—
145C2×#103—971no—
146C2×C22⋊C8—971no—
147C22×Q8—974no—
148C22×Q8—971no—
149C22×Q8—1941no—
150C22×Q8—1942no—
151C22×Q8—1944no—
152C22×Q8—1944no—
153C22×Q8C2×D9711yescharacteristic
154C22×D4—1941no—
155C22×D4C2×D9711yescharacteristic
156C23×C4—971no—
157C23×C4—1941no—
158C2×SD16—24no—
159C2×SD16—974no—
160C2×SD16—1944no—
161C2×SD16C2×D9714yescomplemented
162C22×C8—971no—
163C22×C8C2×D9711yescharacteristic
164#121—974no—
165#121—1942no—
166C2×C4⋊C4—971no—
167C2×C22⋊C4—1941no—
168#103—974no—
169#103—974no—
170C22⋊C8—974no—
171C24—21no—
172C24—971no—
173C2×Q8—22no—
174C2×Q8—21no—
175C2×Q8—974no—
176C2×Q8—972no—
177C2×Q8—1944no—
178C2×Q8—1948no—
179C2×Q8—1948no—
180C2×Q8—19416no—
181C2×Q8—1948no—
182C2×Q8—1941no—
183C2×Q8—1942no—
184C2×Q8C22⋊D9714yescomplemented
185C2×Q8C22×D9712yes—
186C2×D4—22no—
187C2×D4—21no—
188C2×D4—22no—
189C2×D4—3884no—
190C2×D4—1944no—
191C2×D4C22×D9712yes—
192C22×C4—21no—
193C22×C4—971no—
194C22×C4—972no—
195C22×C4—972no—
196C22×C4—1944no—
197C22×C4—1948no—
198C22×C4—1941no—
199C22×C4—1941no—
200C22×C4—1942no—
201C22×C4—1942no—
202C22×C4C22×D9711yescharacteristic
203SD16—28no—
204SD16—1948no—
205C2×C8—22no—
206C2×C8—972no—
207C2×C8—1942no—
208C2×C8C22×D9712yes—
209C4⋊C4—972no—
210C4⋊C4—1941no—
211C22⋊C4—1944no—
212C23—22no—
213C23—42no—
214C23—42no—
215C23—42no—
216C23—974no—
217C23—972no—
218C23—1942no—
219C23—1942no—
220C23D4×D9711yescharacteristic
221Q8—24no—
222Q8—22no—
223Q8—974no—
224Q8—19416no—
225Q8—1944no—
226Q8—1942no—
227Q8C2×C22⋊D9714yescomplemented
228D4—22no—
229D4—22no—
230D4—42no—
231D4—3888no—
232C2×C4—24no—
233C2×C4—22no—
234C2×C4—972no—
235C2×C4—3882no—
236C2×C4—19416no—
237C2×C4—1942no—
238C2×C4—1942no—
239C2×C4—1944no—
240C2×C4—1942no—
241C2×C4—1944no—
242C2×C4—1942no—
243C2×C4—1942no—
244C2×C4D4×D9711yescharacteristic
245C2×C4D4×D9712yes—
246C2×C4C2×C22⋊D9712yes—
247C2×C4C23×D9711yescharacteristic
248C8—22no—
249C8—1942no—
250C22—42no—
251C22—42no—
252C22—42no—
253C22—84no—
254C22—974no—
255C22—1948no—
256C22—1942no—
257C22—1942no—
258C22#1569077812yes—
259C22SD16×D9712yes—
260C22#1569083912yes—
261C22C2×D4×D9711yescharacteristic
262C4—24no—
263C4—3882no—
264C4—1942no—
265C4—1944no—
266C4#1569077812yes—
267C4C2×D4×D9711yescharacteristic
268C4C22×C22⋊D9711yescharacteristic
269C2—82no—
270C2—1944no—
271C2#2026087214yesdirect product factor
272C2C2×#1569077811yescharacteristic
273C2C2×SD16×D9711yescharacteristic
274C2C2×#1569083911yescharacteristic
subgroupG/H
centerC23D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC22×SD16×C97C2
socleC23×C97C2×D4
prime radicalC2×#15690781C2
hypercenterC22×SD16D97
hypocenterC97C2×#448
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×Q8×D97C2semidirect
2. C22×#13095499C2semidirect
4. #20260872C2direct (N × H)
5. C2×#15690735C2semidirect
6. C2×#15690734C2semidirect
7. C2×#15690722C2semidirect
8. C22×SD16×C97C2semidirect
14. C2×Q8×D97C22semidirect
19. C2×#13095499C22semidirect
25. #15690735C22semidirect
26. #15690734C22semidirect
27. #15690722C22semidirect
31. C2×SD16×C97C22semidirect
38. C2×Dic194D4semidirect
63. C2×Q8×C97D4semidirect
81. C22×D97SD16semidirect
85. Q8×C97C2×D4semidirect
109. Dic194C2×D4semidirect
123. C2×D97C2×SD16semidirect
132. C2×C97#448semidirect
138. C97C2×#448semidirect
141. C22×SD16D97semidirect
161. C2×SD16C2×D97semidirect
184. C2×Q8C22⋊D97semidirect
227. Q8C2×C22⋊D97semidirect
271. C2#20260872direct (N × H)

Maximal subgroups

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

subgroupindex
C22×Q8×D972
C22×#130954992
C2×#156907812
#202608722
C2×#156907352
C2×#156907342
C2×#156907222
C22×SD16×C972
C2×#44897