Finite Group Databasebeta

← #27153563

Subgroups — #27153563

Order 12416.

462 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 123 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
1#20261445C211yesmaximal, characteristic, complemented
2#20261370C211yesmaximal, characteristic
3#20261081C211yesmaximal, characteristic, complemented
4#20261052C212yesmaximal, complemented
5#20261026C211yesmaximal, characteristic, complemented
6#20260961C212yesmaximal, complemented
7#20260884C212yesmaximal, complemented
8C4×#13095498C211yesmaximal, characteristic, complemented
9#20260800C211yesmaximal, characteristic, complemented
10#102×D97C212yesmaximal, complemented
11#460×C97C211yesmaximal, characteristic, complemented
12C2×D4×D97—21no—
13C2×D4×D97C2212yes—
14C2×#13095498C2211yescharacteristic
15C2×C8×D97C2212yescomplemented
16#15690810C2211yescharacteristic, complemented
17#15690802—21no—
18#15690788—22no—
19#15690786C2212yescomplemented
20#15690781—21no—
21#15690781C2212yes—
22#15690778—22no—
23C4×D388—21no—
24C4×D388C2211yescharacteristic
25C4×D388C2211yescharacteristic
26C4⋊C4×D97C2212yescomplemented
27#15690742C2212yescomplemented
28#15690741C2211yescharacteristic, complemented
29#15690734C2211yescharacteristic, complemented
30#15690732C2212yes—
31#15690731C2212yescomplemented
32#15690731C2212yescomplemented
33#15690730C2211yescharacteristic
34#15690722C2212yescomplemented
35C4×#11715895C2211yescharacteristic, complemented
36C2×SD16×C97C2212yescomplemented
37#127×C97C2211yescharacteristic
38C4×Q8×C97C2211yescharacteristic, complemented
39C4⋊C8×C97C2211yescharacteristic, complemented
40#102×C97C2212yescomplemented
41C23×D97—22no—
42C2×Dic194—21no—
43C2×D388D411yescharacteristic
44C2×D388C2311yescharacteristic
45C2×C22⋊D97—22no—
46C2×C22⋊D97—22no—
47D4×D97—24no—
48D4×D97—42no—
49D4×D97—44no—
50C2×C4×D97—21no—
51C2×C4×D97—22no—
52C2×C4×D97C2312yes—
53#13095498—22no—
54C2×#11715895C2311yescharacteristic
55C2×#11715895C2311yescharacteristic
56C8×D97—42no—
57C4⋊Dic97—21no—
58C4⋊Dic97D411yescharacteristic
59C4⋊Dic97C2312yes—
60C4⋊Dic97C2311yescharacteristic
61#13095490—22no—
62#13095490—22no—
63C22⋊Dic97—22no—
64#13095488—22no—
65#13095488—21no—
66#13095488D412yes—
67C2×Q8×C97C2311yescharacteristic
68C2×D4×C97—21no—
69C2×D4×C97—21no—
70C2×D4×C97C2312yes—
71SD16×C97—42no—
72C2×C8×C97C2312yes—
73C4⋊C4×C97—21no—
74C4⋊C4×C97C2311yescharacteristic
75C42×C97—21no—
76C42×C97C2311yescharacteristic
77C23×C97—22no—
78C22×D97—21no—
79C22×D97—44no—
80C22×D97—44no—
81C22×D97—82no—
82C22×D97C2×D412yes—
83Q8×C97—21no—
84Q8×C97—21no—
85D4×C97—22no—
86D4×C97—22no—
87D4×C97—42no—
88D4×C97—81no—
89C2×C4×C97—21no—
90C2×C4×C97—21no—
91C2×C4×C97C2×D411yescharacteristic
92C2×C4×C97C2×D411yescharacteristic
93C2×C4×C97C2411yescharacteristic
94C22⋊D97—44no—
95C22⋊D97—44no—
96C2×Dic97—21no—
97C2×Dic97—22no—
98C2×Dic97C2×D412yes—
99D388—21no—
100D388SD1614yescomplemented
101D388#4112yes—
102C4×D97—22no—
103C4×D97—22no—
104C4×D97—42no—
105C4×D97—82no—
106Dic194—81no—
107C8×C97—22no—
108#11715895—21no—
109#11715895—21no—
110C22×C97—42no—
111C22×C97—42no—
112C22×C97—42no—
113C22×C97C22×D411yescharacteristic
114C2×D97—22no—
115C2×D97—22no—
116C2×D97—22no—
117C2×D97—41no—
118C2×D97—41no—
119C2×D97—88no—
120C2×D97C2×SD1614yes—
121C4×C97—21no—
122C4×C97—41no—
123C4×C97—41no—
124C4×C97C2×SD1612yes—
125C4×C97C2×#4111yescharacteristic
126C4×C97H(5,𝔽2)11yescharacteristic
127Dic97—22no—
128Dic97—41no—
129Dic97—42no—
130C2×C97—82no—
131C2×C97#54611yescharacteristic
132C2×C97C22×SD1611yescharacteristic
133C2×C97#57611yescharacteristic
134D97—24no—
135D97—42no—
136#3295—971nomaximal, Sylow
137C97#329511yescharacteristic, complemented, Sylow
138#554—971no—
139D42—971no—
140#486—971no—
141#481—972no—
142#476—971no—
143#465—972no—
144#460D9711yescharacteristic, complemented
145#450—972no—
146C4×SD16—971no—
147C8×D4—971no—
148C2×#102—972no—
149C22×D4—972no—
150C22×D4—1941no—
151C2×SD16—971no—
152C2×SD16C2×D9712yescomplemented
153C22×C8—972no—
154C4⋊Q8—971no—
155#127C2×D9711yescharacteristic
156#125—1941no—
157C22⋊Q8—972no—
158C22⋊Q8—1942no—
159#121—972no—
160#121—1941no—
161#120—1942no—
162C4×Q8C2×D9711yescharacteristic, complemented
163C4×D4—971no—
164C4×D4—971no—
165C4×D4—1941no—
166C2×C4⋊C4—972no—
167#106—972no—
168#106—971no—
169C4⋊C8C2×D9711yescharacteristic, complemented
170#103—971no—
171#102—972no—
172#102—972no—
173#102—972no—
174#102—971no—
175#102C2×D9712yescomplemented
176C22⋊C8—972no—
177C4×C8—971no—
178C24—1942no—
179C2×Q8—1941no—
180C2×Q8C22×D9711yescharacteristic
181C2×D4—21no—
182C2×D4—21no—
183C2×D4—971no—
184C2×D4—971no—
185C2×D4—3884no—
186C2×D4—3882no—
187C2×D4—1942no—
188C2×D4—1944no—
189C2×D4—1942no—
190C2×D4C22×D9712yes—
191C22×C4—972no—
192C22×C4—1942no—
193C22×C4—1941no—
194SD16—42no—
195SD16—1942no—
196C2×C8—971no—
197C2×C8—971no—
198C2×C8—3882no—
199C2×C8C22×D9712yes—
200C4⋊C4—21no—
201C4⋊C4—971no—
202C4⋊C4—972no—
203C4⋊C4—971no—
204C4⋊C4—1942no—
205C4⋊C4—1942no—
206C4⋊C4—1941no—
207C4⋊C4C22×D9711yescharacteristic
208C22⋊C4—972no—
209C22⋊C4—1942no—
210C22⋊C4—1942no—
211C22⋊C4—1941no—
212C42—21no—
213C42C22×D9711yescharacteristic
214C23—22no—
215C23—7762no—
216C23—972no—
217C23—3884no—
218C23—3884no—
219C23—1941no—
220Q8—21no—
221Q8—21no—
222Q8—7761no—
223D4—22no—
224D4—22no—
225D4—42no—
226D4—81no—
227D4—974no—
228D4—972no—
229D4—3884no—
230D4—3884no—
231D4—1941no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—7762no—
235C2×C4—972no—
236C2×C4—3882no—
237C2×C4—1941no—
238C2×C4—1942no—
239C2×C4—1942no—
240C2×C4—1942no—
241C2×C4D4×D9711yescharacteristic
242C2×C4D4×D9711yescharacteristic
243C2×C4C23×D9711yescharacteristic
244C8—22no—
245C8—1941no—
246C8—1941no—
247C22—42no—
248C22—42no—
249C22—42no—
250C22—7768no—
251C22—974no—
252C22—3881no—
253C22—3881no—
254C22—1942no—
255C22—1942no—
256C22—1942no—
257C22C2×D4×D9711yescharacteristic
258C4—21no—
259C4—41no—
260C4—41no—
261C4—3881no—
262C4—3882no—
263C4—1942no—
264C4SD16×D9712yes—
265C4#41×D9711yescharacteristic
266C4#1569085511yescharacteristic
267C2—82no—
268C2—3882no—
269C2—1944no—
270C2#2026138211yescharacteristic
271C2C2×SD16×D9711yescharacteristic
272C2#2026153911yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#460×C97C2
socleC22×C97C22×D4
prime radical#20261370C2
hypercenter#460D97
hypocenterC97#3295
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. #20261445C2semidirect
3. #20261081C2semidirect
4. #20261052C2semidirect
5. #20261026C2semidirect
6. #20260961C2semidirect
7. #20260884C2semidirect
8. C4×#13095498C2semidirect
9. #20260800C2semidirect
10. #102×D97C2semidirect
11. #460×C97C2semidirect
15. C2×C8×D97C22semidirect
16. #15690810C22semidirect
19. #15690786C22semidirect
26. C4⋊C4×D97C22semidirect
27. #15690742C22semidirect
28. #15690741C22semidirect
29. #15690734C22semidirect
31. #15690731C22semidirect
32. #15690731C22semidirect
34. #15690722C22semidirect
35. C4×#11715895C22semidirect
36. C2×SD16×C97C22semidirect
38. C4×Q8×C97C22semidirect
39. C4⋊C8×C97C22semidirect
40. #102×C97C22semidirect
100. D388SD16semidirect
137. C97#3295semidirect
144. #460D97semidirect
152. C2×SD16C2×D97semidirect
162. C4×Q8C2×D97semidirect
169. C4⋊C8C2×D97semidirect
175. #102C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202614452
#202613702
#202610812
#202610522
#202610262
#202609612
#202608842
C4×#130954982
#202608002
#102×D972
#460×C972
#329597