Finite Group Databasebeta

← #27150585

Subgroups — #27150585

Order 12416.

528 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 175 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
2#20261207C212yesmaximal, complemented
3#20260848C211yesmaximal, characteristic, complemented
4#20260847C211yesmaximal, characteristic, complemented
5#20260846C211yesmaximal, characteristic, complemented
6C4×#13095496C212yesmaximal, complemented
7#20260815C212yesmaximal, complemented
8#20260770C211yesmaximal, characteristic, complemented
9#20260738C212yesmaximal, complemented
10#20260692C211yesmaximal, characteristic, complemented
11#442×C97C211yesmaximal, characteristic, complemented
12C2×D4×D97C2212yes—
13C2×D776C2211yescharacteristic
14C2×#13095496C2212yes—
15D8×D97C418yescomplemented
16C2×#13095494C2211yescharacteristic
17C2×C8×D97C2211yescharacteristic
18C4×D388C2212yescomplemented
19D4×Dic97C2212yescomplemented
20#15690772C2212yescomplemented
21C4×C22⋊D97—22no—
22#15690770—22no—
23#15690768C2212yes—
24#15690767C2211yescharacteristic
25C2×#13095488—22no—
26C22⋊C4×D97—22no—
27#15690743C2212yescomplemented
28#15690742C2211yescharacteristic, complemented
29#15690732C2212yescomplemented
30#15690731C2212yescomplemented
31#15690730C2212yescomplemented
32#15690720C2211yescharacteristic, complemented
33C4×#11715895C2211yescharacteristic, complemented
34C8×Dic97C2211yescharacteristic, 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×D97C2311yescharacteristic
49D776C2×C414yescomplemented
50#13095496C2×C418yescomplemented
51#13095494—22no—
52C2×#11715895C2311yescharacteristic
53C2×#11715895C2311yescharacteristic
54C8×D97C2×C414yescomplemented
55C4⋊Dic97C2312yes—
56#13095490D412yes—
57C22⋊Dic97—22no—
58#13095488—22no—
59#13095488—22no—
60#13095488—22no—
61#13095488D412yes—
62C4×Dic97—22no—
63C4×Dic97C2311yescharacteristic
64C2×D4×C97C2312yes—
65C22×C4×C97—22no—
66D8×C97C2×C414yescomplemented
67C2×C8×C97C2311yescharacteristic
68C2×C8×C97C2311yescharacteristic
69C4⋊C4×C97C2312yes—
70C22⋊C4×C97—22no—
71C42×C97C2311yescharacteristic
72C23×C97—22no—
73C22×D97—22no—
74C22×D97—28no—
75C22×D97—82no—
76C22×D97—82no—
77C22×D97C2×D411yescharacteristic
78D4×C97—22no—
79D4×C97C22×C414yes—
80C2×C4×C97—22no—
81C2×C4×C97—82no—
82C2×C4×C97C2×D412yes—
83C2×C4×C97C2411yescharacteristic
84C22⋊D97—28no—
85C2×Dic97—22no—
86C2×Dic97—82no—
87C2×Dic97C2×D412yes—
88C2×Dic97C2×D411yescharacteristic
89D388—22no—
90D388C22×C414yes—
91C4×D97—82no—
92C4×D97—82no—
93C4×D97C22×C412yes—
94C4×D97#4112yes—
95C8×C97—21no—
96C8×C97C22×C412yes—
97#11715895—21no—
98#11715895C22×C412yes—
99C22×C97—24no—
100C22×C97—42no—
101C22×C97—42no—
102C22×C97C22×D411yescharacteristic
103C2×D97—22no—
104C2×D97—24no—
105C2×D97—42no—
106C2×D97—42no—
107C2×D97—88no—
108C2×D97C4×D412yes—
109C4×C97—22no—
110C4×C97—42no—
111C4×C97C23×C411yescharacteristic
112C4×C97C2×#4111yescharacteristic
113Dic97—22no—
114Dic97—42no—
115Dic97C4×D412yes—
116C2×C97—44no—
117C2×C97C2×C4×D411yescharacteristic
118C2×C97#57612yes—
119D97—22no—
120D97—44no—
121#2948—971nomaximal, Sylow
122C97#294811yescharacteristic, complemented, Sylow
123C22×D8—971no—
124#518—972no—
125#442—971no—
126#442—971no—
127#442—971no—
128#442D9711yescharacteristic, complemented
129C4×D8—972no—
130C4×D8—972no—
131#427—971no—
132#418—972no—
133#405—971no—
134C22×D4—972no—
135C2×D8—972no—
136C2×D8—978no—
137C2×D8—971no—
138C2×D8C2×D9711yescharacteristic
139C2×M16—971no—
140C22×C8—971no—
141C4×D4—972no—
142C4×D4—972no—
143C4×D4—972no—
144C4×D4—1942no—
145C4×D4—1942no—
146C4×D4C2×D9712yescomplemented
147#117—971no—
148#117—972no—
149C2×C22⋊C4—1942no—
150C2×C22⋊C4—1942no—
151#107—972no—
152#106—971no—
153#106C2×D9711yescharacteristic, complemented
154#102—972no—
155#102—972no—
156#102—972no—
157#102C2×D9712yescomplemented
158#97—971no—
159#97C2×D9711yescharacteristic, complemented
160C4×C8—971no—
161C4×C8—971no—
162C24—1942no—
163C2×D4—978no—
164C2×D4—972no—
165C2×D4—1944no—
166C2×D4—1942no—
167C2×D4C22×D9712yes—
168C22×C4—22no—
169C22×C4—971no—
170C22×C4—1942no—
171C22×C4—1942no—
172C22×C4—1942no—
173D8—978no—
174D8—974no—
175D8C4×D9714yescomplemented
176M16—1942no—
177C2×C8—971no—
178C2×C8—974no—
179C2×C8—971no—
180C2×C8C22×D9711yescharacteristic
181C2×C8C22×D9711yescharacteristic
182C4⋊C4—972no—
183C4⋊C4—972no—
184C4⋊C4C22×D9712yes—
185C22⋊C4—22no—
186C22⋊C4—972no—
187C22⋊C4—1942no—
188C22⋊C4—1942no—
189C22⋊C4—1942no—
190C22⋊C4—1942no—
191C42—971no—
192C42—1942no—
193C42C22×D9711yescharacteristic
194C23—22no—
195C23—7762no—
196C23—7762no—
197C23—971no—
198C23—1948no—
199C23—1942no—
200D4—22no—
201D4—974no—
202D4—1948no—
203D4—1942no—
204D4C2×C4×D9714yes—
205C2×C4—22no—
206C2×C4—82no—
207C2×C4—7762no—
208C2×C4—7762no—
209C2×C4—7762no—
210C2×C4—971no—
211C2×C4—972no—
212C2×C4—972no—
213C2×C4—972no—
214C2×C4—1942no—
215C2×C4D4×D9712yes—
216C2×C4C23×D9711yescharacteristic
217C8—21no—
218C8—972no—
219C8—1941no—
220C8C2×C4×D9712yes—
221C22—24no—
222C22—42no—
223C22—42no—
224C22—7768no—
225C22—972no—
226C22—3882no—
227C22—3882no—
228C22—1944no—
229C22—1942no—
230C22C2×D4×D9711yescharacteristic
231C4—22no—
232C4—42no—
233C4—972no—
234C4—3882no—
235C4—1942no—
236C4C22×C4×D9711yescharacteristic
237C4#41×D9711yescharacteristic
238C2—44no—
239C2—3884no—
240C2—1942no—
241C2C4×D4×D9711yescharacteristic
242C2#2026153912yes—
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
hypocenterC97#2948
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. #20261207C2semidirect
3. #20260848C2semidirect
4. #20260847C2semidirect
5. #20260846C2semidirect
6. C4×#13095496C2semidirect
7. #20260815C2semidirect
8. #20260770C2semidirect
9. #20260738C2semidirect
10. #20260692C2semidirect
11. #442×C97C2semidirect
15. D8×D97C4semidirect
18. C4×D388C22semidirect
19. D4×Dic97C22semidirect
20. #15690772C22semidirect
27. #15690743C22semidirect
28. #15690742C22semidirect
29. #15690732C22semidirect
30. #15690731C22semidirect
31. #15690730C22semidirect
32. #15690720C22semidirect
33. C4×#11715895C22semidirect
34. C8×Dic97C22semidirect
36. C4×D4×C97C22semidirect
37. #106×C97C22semidirect
38. #102×C97C22semidirect
39. #97×C97C22semidirect
44. D4×D97C2×C4semidirect
49. D776C2×C4semidirect
50. #13095496C2×C4semidirect
54. C8×D97C2×C4semidirect
66. D8×C97C2×C4semidirect
122. C97#2948semidirect
128. #442D97semidirect
146. C4×D4C2×D97semidirect
153. #106C2×D97semidirect
157. #102C2×D97semidirect
159. #97C2×D97semidirect
175. D8C4×D97semidirect

Maximal subgroups

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

subgroupindex
C2×D8×D972
#202612072
#202608482
#202608472
#202608462
C4×#130954962
#202608152
#202607702
#202607382
#202606922
#442×C972
#294897