Finite Group Databasebeta

← #442×D97 #27150525

Subgroups — #442×D97 #27150525

Order 12416.

608 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 608 conjugacy classes of subgroups — too many to draw. Only its 183 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
1C2×D8×D97C211yesmaximal, characteristic
2C4×D4×D97C212yesmaximal, complemented
3#20260848C211yesmaximal, characteristic, complemented
4#20260847C211yesmaximal, characteristic, complemented
5#20260846C211yesmaximal, characteristic, complemented
6#20260845C212yesmaximal, complemented
7#20260844C212yesmaximal, complemented
8#106×D97C211yesmaximal, characteristic, complemented
9#102×D97C212yesmaximal, complemented
10#97×D97C211yesmaximal, characteristic, complemented
11#442×C97C211yesmaximal, characteristic, complemented
12C2×D4×D97C2212yes—
13C22×C4×D97—22no—
14C2×D776C2211yescharacteristic
15C2×#13095496C2212yes—
16D8×D97C418yescomplemented
17C2×C8×D97C2211yescharacteristic
18C2×C8×D97C2211yescharacteristic
19C4×D388C2212yescomplemented
20D4×Dic97C2212yescomplemented
21#15690772C2212yescomplemented
22C4×C22⋊D97—22no—
23#15690770—22no—
24C4⋊C4×D97C2212yes—
25C22⋊C4×D97—22no—
26C42×D97C2211yescharacteristic
27#15690742C2211yescharacteristic, complemented
28#15690741C2212yescomplemented
29#15690732C2212yescomplemented
30#15690731C2212yescomplemented
31#15690730C2212yescomplemented
32#15690721C2211yescharacteristic, complemented
33#15690720C2211yescharacteristic, complemented
34#15690720C2211yescharacteristic, complemented
35C2×D8×C97C2211yescharacteristic
36C4×D4×C97C2212yescomplemented
37#106×C97C2211yescharacteristic, complemented
38#102×C97C2212yescomplemented
39#97×C97C2211yescharacteristic, complemented
40C23×D97—22no—
41C2×D388C2312yes—
42C2×C22⋊D97—22no—
43D4×D97—24no—
44D4×D97C2×C418yescomplemented
45C22×Dic97—22no—
46C2×C4×D97—22no—
47C2×C4×D97—22no—
48C2×C4×D97—22no—
49C2×C4×D97—48no—
50C2×C4×D97D412yes—
51C2×C4×D97C2311yescharacteristic
52D776C2×C414yescomplemented
53#13095496C2×C418yescomplemented
54C2×#11715895C2311yescharacteristic
55C2×#11715895C2311yescharacteristic
56C8×D97—22no—
57C8×D97C2×C414yescomplemented
58C4⋊Dic97C2312yes—
59#13095490—22no—
60C22⋊Dic97—22no—
61#13095488—22no—
62#13095488—22no—
63C4×Dic97D412yes—
64C4×Dic97C2311yescharacteristic
65C2×D4×C97C2312yes—
66C22×C4×C97—22no—
67D8×C97C2×C414yescomplemented
68C2×C8×C97C2311yescharacteristic
69C2×C8×C97C2311yescharacteristic
70C4⋊C4×C97C2312yes—
71C22⋊C4×C97—22no—
72C42×C97C2311yescharacteristic
73C23×C97—22no—
74C22×D97—22no—
75C22×D97—28no—
76C22×D97—44no—
77C22×D97—44no—
78C22×D97C2×D411yescharacteristic
79D4×C97—22no—
80D4×C97C22×C414yescomplemented
81C2×C4×C97—22no—
82C2×C4×C97—44no—
83C2×C4×C97C2×D412yes—
84C2×C4×C97C2411yescharacteristic
85C22⋊D97—28no—
86C2×Dic97—22no—
87C2×Dic97—44no—
88C2×Dic97C2×D412yes—
89C2×Dic97C2×D411yescharacteristic
90D388—22no—
91D388C22×C414yescomplemented
92C4×D97—24no—
93C4×D97—44no—
94C4×D97—44no—
95C4×D97—44no—
96C4×D97C22×C412yes—
97C4×D97#4112yes—
98C8×C97—21no—
99C8×C97C22×C412yescomplemented
100#11715895—21no—
101#11715895C22×C412yescomplemented
102C22×C97—24no—
103C22×C97—42no—
104C22×C97—42no—
105C22×C97C22×D411yescharacteristic
106C2×D97—24no—
107C2×D97—42no—
108C2×D97—42no—
109C2×D97—416no—
110C2×D97C4×D412yes—
111C2×D97AGL(1,ℤ/8)14yes—
112C4×C97—22no—
113C4×C97—42no—
114C4×C97C23×C411yescharacteristic
115C4×C97C2×#4111yescharacteristic
116Dic97—22no—
117Dic97—42no—
118Dic97C4×D412yes—
119C2×C97—44no—
120C2×C97C2×C4×D411yescharacteristic
121C2×C97C2×AGL(1,ℤ/8)12yes—
122D97—44no—
123D97#44214yesdirect product factor
124C2×#442—971nomaximal, Sylow
125C97C2×#44211yescharacteristic, complemented, Sylow
126C22×D8—971no—
127C2×C4×D4—972no—
128#442—971no—
129#442—972no—
130#442—971no—
131#442—972no—
132#442—971no—
133#442D9711yescharacteristic, direct product factor
134C2×#106—971no—
135C2×#102—972no—
136C2×#97—971no—
137C22×D4—972no—
138C23×C4—1942no—
139C2×D8—972no—
140C2×D8—978no—
141C2×D8—971no—
142C2×D8C2×D9711yescharacteristic
143C22×C8—971no—
144C22×C8—971no—
145C4×D4—972no—
146C4×D4—972no—
147C4×D4—972no—
148C4×D4—1942no—
149C4×D4—1942no—
150C4×D4C2×D9712yescomplemented
151C2×C4⋊C4—972no—
152C2×C22⋊C4—1942no—
153C2×C42—971no—
154#106—971no—
155#106—972no—
156#106C2×D9711yescharacteristic, complemented
157#102—972no—
158#102—972no—
159#102—972no—
160#102C2×D9712yescomplemented
161#97—971no—
162#97—971no—
163#97—971no—
164#97C2×D9711yescharacteristic, complemented
165C24—1942no—
166C2×D4—978no—
167C2×D4—972no—
168C2×D4—1944no—
169C2×D4—1942no—
170C2×D4C22×D9712yes—
171C22×C4—22no—
172C22×C4—971no—
173C22×C4—972no—
174C22×C4—3888no—
175C22×C4—1942no—
176C22×C4—1942no—
177C22×C4—1942no—
178C22×C4—1942no—
179D8—978no—
180D8—974no—
181D8C4×D9714yescomplemented
182C2×C8—971no—
183C2×C8—974no—
184C2×C8—971no—
185C2×C8—1942no—
186C2×C8C22×D9711yescharacteristic
187C2×C8C22×D9711yescharacteristic
188C4⋊C4—972no—
189C4⋊C4—1942no—
190C4⋊C4C22×D9712yes—
191C22⋊C4—22no—
192C22⋊C4—1942no—
193C22⋊C4—1942no—
194C22⋊C4—1942no—
195C42—972no—
196C42—971no—
197C42C22×D9711yescharacteristic
198C23—22no—
199C23—971no—
200C23—3884no—
201C23—3884no—
202C23—1948no—
203C23—1942no—
204D4—22no—
205D4—974no—
206D4—1948no—
207D4—1942no—
208D4C2×C4×D9714yescomplemented
209C2×C4—22no—
210C2×C4—44no—
211C2×C4—971no—
212C2×C4—972no—
213C2×C4—972no—
214C2×C4—972no—
215C2×C4—3884no—
216C2×C4—3884no—
217C2×C4—3884no—
218C2×C4—3884no—
219C2×C4—1944no—
220C2×C4—1942no—
221C2×C4D4×D9712yes—
222C2×C4C23×D9711yescharacteristic
223C8—21no—
224C8—972no—
225C8—1941no—
226C8C2×C4×D9712yescomplemented
227C22—24no—
228C22—42no—
229C22—42no—
230C22—972no—
231C22—974no—
232C22—38816no—
233C22—3882no—
234C22—3882no—
235C22—1944no—
236C22C2×D4×D9711yescharacteristic
237C4—22no—
238C4—42no—
239C4—972no—
240C4—3882no—
241C4—1942no—
242C4C22×C4×D9711yescharacteristic
243C4#41×D9711yescharacteristic
244C2—44no—
245C2—974no—
246C2—3884no—
247C2C4×D4×D9711yescharacteristic
248C2AGL(1,ℤ/8)×D9712yes—
subgroupG/H
centerC22C2×D4×D97
commutatorC4×C97C23×C4
FrattiniC2×C4C23×D97
Fitting#442×C97C2
socleC22×C97C22×D4
prime radicalC2×D8×D97C2
cosocleC2×C4×C97C24
hypercenter#442D97
hypocenterC97C2×#442
LCSG ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
2. C4×D4×D97C2semidirect
3. #20260848C2semidirect
4. #20260847C2semidirect
5. #20260846C2semidirect
6. #20260845C2semidirect
7. #20260844C2semidirect
8. #106×D97C2semidirect
9. #102×D97C2semidirect
10. #97×D97C2semidirect
11. #442×C97C2semidirect
16. D8×D97C4semidirect
19. C4×D388C22semidirect
20. D4×Dic97C22semidirect
21. #15690772C22semidirect
27. #15690742C22semidirect
28. #15690741C22semidirect
29. #15690732C22semidirect
30. #15690731C22semidirect
31. #15690730C22semidirect
32. #15690721C22semidirect
33. #15690720C22semidirect
34. #15690720C22semidirect
36. C4×D4×C97C22semidirect
37. #106×C97C22semidirect
38. #102×C97C22semidirect
39. #97×C97C22semidirect
44. D4×D97C2×C4semidirect
52. D776C2×C4semidirect
53. #13095496C2×C4semidirect
57. C8×D97C2×C4semidirect
67. D8×C97C2×C4semidirect
80. D4×C97C22×C4semidirect
91. D388C22×C4semidirect
99. C8×C97C22×C4semidirect
101. #11715895C22×C4semidirect
123. D97#442direct (N × H)
125. C97C2×#442semidirect
133. #442D97direct (N × H)
150. C4×D4C2×D97semidirect
156. #106C2×D97semidirect
160. #102C2×D97semidirect
164. #97C2×D97semidirect
181. D8C4×D97semidirect
208. D4C2×C4×D97semidirect
226. C8C2×C4×D97semidirect

Maximal subgroups

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

subgroupindex
C2×D8×D972
C4×D4×D972
#202608482
#202608472
#202608462
#202608452
#202608442
#106×D972
#102×D972
#97×D972
#442×C972
C2×#44297