Finite Group Databasebeta

← #46822682

Subgroups — #46822682

Order 34432.

476 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 125 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#31771408C212yesmaximal, complemented
2#31771407C211yesmaximal, characteristic, complemented
3#31770980C211yesmaximal, characteristic, complemented
4#31770875C211yesmaximal, characteristic, complemented
5#31770873C212yesmaximal, complemented
6#31770872C211yesmaximal, characteristic, complemented
7#31770773C212yesmaximal, complemented
8#31770771C212yesmaximal, complemented
9#31770722C211yesmaximal, characteristic, complemented
10#31770720C211yesmaximal, characteristic, complemented
11#576×C269C211yesmaximal, characteristic, complemented
12#22575499C2212yescomplemented
13#22575480—22no—
14#22575479—22no—
15#22575479C2212yescomplemented
16#22575478—22no—
17#22575478C2212yescomplemented
18#22575478C2212yescomplemented
19#22575476—21no—
20#22575473—22no—
21#22575473C2211yescharacteristic
22C2×D2152C2211yescharacteristic, complemented
23C2×#16864387—22no—
24C2×#16864387C2212yescomplemented
25D8×D269—21no—
26#22575460C2212yescomplemented
27#22575459C2211yescharacteristic, complemented
28#22575449C2212yescomplemented
29#22575389C2211yescharacteristic, complemented
30#22575388C2211yescharacteristic, complemented
31#22575380C2211yescharacteristic, complemented
32#22575380C2212yescomplemented
33#22575379C2211yescharacteristic, complemented
34#22575370C2212yescomplemented
35#22575369C2212yescomplemented
36C8×Dic269C2211yescharacteristic, complemented
37H(5,𝔽2)×C269C2212yescomplemented
38AGL(1,ℤ/8)×C269—22no—
39AGL(1,ℤ/8)×C269C2212yescomplemented
40#135×C269—21no—
41#135×C269C2211yescharacteristic
42C2×D8×C269—21no—
43C2×D8×C269C2211yescharacteristic, complemented
44#131×C269C2211yescharacteristic, complemented
45#16864403—22no—
46#16864402—22no—
47#16864402C2311yescharacteristic
48C2×D1076—22no—
49C2×D1076C2312yes—
50C2×C22⋊D269—42no—
51D4×D269—22no—
52D4×D269—42no—
53#16864392D411yescharacteristic, complemented
54#16864390—22no—
55#16864390D411yescharacteristic, complemented
56#16864389—22no—
57#16864389—22no—
58#16864389D411yescharacteristic, complemented
59D2152—21no—
60D2152—21no—
61#16864387—22no—
62#16864387—22no—
63#16864387—22no—
64#16864387—22no—
65#16864387—22no—
66#16864387D411yescharacteristic, complemented
67#16864386—22no—
68#16864386C2312yes—
69#16864386C2311yescharacteristic
70#16864385—21no—
71C2×#14524130—22no—
72C2×#14524130C2311yescharacteristic
73C8×D269—21no—
74C4×Dic269C2311yescharacteristic
75#41×C269—22no—
76#41×C269—22no—
77#41×C269C2312yes—
78#41×C269C2311yescharacteristic
79C2×D4×C269—22no—
80C2×D4×C269—22no—
81C2×D4×C269—42no—
82C2×D4×C269C2312yes—
83Q16×C269D411yescharacteristic, complemented
84SD16×C269—22no—
85SD16×C269D412yescomplemented
86D8×C269—22no—
87D8×C269—21no—
88D8×C269—21no—
89D8×C269D411yescharacteristic, complemented
90M16×C269—21no—
91M16×C269C2311yescharacteristic
92C2×C8×C269—21no—
93C2×C8×C269C2311yescharacteristic
94C22×D269—22no—
95C22×D269—42no—
96Dic538C2×D411yescharacteristic
97D1076—22no—
98D1076—22no—
99D1076—42no—
100D1076C2×D411yescharacteristic
101C22⋊D269—21no—
102C22⋊D269—42no—
103C22⋊D269—84no—
104C2×Dic269—21no—
105C4×D269—21no—
106C4×D269—42no—
107#14524130—21no—
108#14524130—22no—
109#14524130C2×D411yescharacteristic
110#14524130C2×D411yescharacteristic
111C23×C269—22no—
112C23×C269—42no—
113Q8×C269C2×D411yescharacteristic
114Q8×C269C2×D412yes—
115D4×C269—21no—
116D4×C269—22no—
117D4×C269—22no—
118D4×C269—22no—
119D4×C269—42no—
120D4×C269—42no—
121D4×C269C2×D411yescharacteristic
122D4×C269C2×D412yes—
123C2×C4×C269—21no—
124C2×C4×C269—22no—
125C2×C4×C269—42no—
126C2×C4×C269C2411yescharacteristic
127C8×C269—21no—
128C8×C269C2×D411yescharacteristic
129C8×C269C2×D411yescharacteristic
130C22×C269—21no—
131C22×C269—22no—
132C22×C269—42no—
133C22×C269—82no—
134C22×C269—162no—
135C22×C269H(5,𝔽2)11yescharacteristic
136C2×D269—21no—
137C2×D269—42no—
138C2×D269—82no—
139C2×D269—162no—
140C4×C269—21no—
141C4×C269—22no—
142C4×C269C22×D411yescharacteristic
143C4×C269C22×D411yescharacteristic
144Dic269—22no—
145Dic269—21no—
146C2×C269—21no—
147C2×C269—42no—
148C2×C269—41no—
149C2×C269—82no—
150C2×C269D4211yescharacteristic
151D269—41no—
152D269—82no—
153C269#329211yescharacteristic, complemented, Sylow
154#3292—2691nomaximal, Sylow
155#576—2692no—
156#576—2691no—
157#576D26911yescharacteristic, complemented
158#493—2691no—
159#472—2692no—
160#472—2691no—
161#472—2691no—
162#453—2692no—
163#453—2692no—
164#443—2691no—
165#443—2691no—
166H(5,𝔽2)—2692no—
167H(5,𝔽2)C2×D26912yescomplemented
168AGL(1,ℤ/8)—22no—
169AGL(1,ℤ/8)—2692no—
170AGL(1,ℤ/8)—2692no—
171AGL(1,ℤ/8)—2692no—
172AGL(1,ℤ/8)—5382no—
173AGL(1,ℤ/8)—5382no—
174AGL(1,ℤ/8)—5382no—
175AGL(1,ℤ/8)C2×D26912yescomplemented
176#135—21no—
177#135—2691no—
178#135—5382no—
179#135—5381no—
180#135C2×D26911yescharacteristic
181C2×D8—21no—
182C2×D8—2692no—
183C2×D8—2691no—
184C2×D8—5382no—
185C2×D8—5381no—
186C2×D8C2×D26911yescharacteristic, complemented
187#131—2692no—
188#131—2691no—
189#131C2×D26911yescharacteristic, complemented
190#127—2692no—
191#108—2691no—
192#108—2691no—
193#104—2691no—
194#104—2692no—
195#104—2691no—
196#100—2692no—
197#100—2692no—
198C4×C8—2691no—
199#41—22no—
200#41—22no—
201#41—2691no—
202#41—5382no—
203#41—5382no—
204#41C22×D26911yescharacteristic
205#41C22×D26912yes—
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—42no—
209C2×D4—2692no—
210C2×D4—5382no—
211C2×D4—5382no—
212C2×D4—10762no—
213C2×D4—10762no—
214C2×D4C22×D26912yes—
215Q16—2691no—
216Q16C22⋊D26911yescharacteristic, complemented
217SD16—22no—
218SD16—2691no—
219SD16—2691no—
220SD16—5382no—
221SD16—5382no—
222SD16—5382no—
223SD16C22⋊D26912yescomplemented
224D8—22no—
225D8—21no—
226D8—21no—
227D8—2691no—
228D8—5382no—
229D8—5382no—
230D8—5382no—
231D8—5382no—
232D8—5382no—
233D8—5381no—
234D8—5381no—
235D8C22⋊D26911yescharacteristic, complemented
236M16—21no—
237M16—2692no—
238M16—2691no—
239M16—5382no—
240M16—5381no—
241M16C22×D26911yescharacteristic
242C2×C8—21no—
243C2×C8—2691no—
244C2×C8—5382no—
245C2×C8—5381no—
246C2×C8C22×D26911yescharacteristic
247C42—2691no—
248C23—22no—
249C23—42no—
250C23—5382no—
251C23—10762no—
252Q8—2691no—
253Q8D4×D26911yescharacteristic
254Q8C2×C22⋊D26912yes—
255D4—21no—
256D4—22no—
257D4—22no—
258D4—22no—
259D4—42no—
260D4—42no—
261D4—2691no—
262D4—5381no—
263D4—5382no—
264D4—5382no—
265D4—10762no—
266D4—10762no—
267D4—21524no—
268D4D4×D26911yescharacteristic
269D4C2×C22⋊D26912yes—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—42no—
273C2×C4—5381no—
274C2×C4—5381no—
275C2×C4—10762no—
276C2×C4C23×D26911yescharacteristic
277C8—21no—
278C8—2691no—
279C8—2691no—
280C8—5382no—
281C8—5381no—
282C8C2×C22⋊D26911yescharacteristic
283C8C2×C22⋊D26911yescharacteristic
284C22—21no—
285C22—22no—
286C22—42no—
287C22—82no—
288C22—162no—
289C22—5381no—
290C22—10762no—
291C22—21522no—
292C22—43042no—
293C22#2257549811yescharacteristic
294C4—21no—
295C4—22no—
296C4—5382no—
297C4—5381no—
298C4C2×D4×D26911yescharacteristic
299C4C22×C22⋊D26911yescharacteristic
300C2—21no—
301C2—41no—
302C2—42no—
303C2—82no—
304C2—10761no—
305C2—21522no—
306C2D4×C22⋊D26911yescharacteristic
subgroupG/H
centerC2D4×C22⋊D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#576×C269C2
socleC2×C269D42
hypercenter#576D269
hypocenterC269#3292
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C2 ◁ C2×C4 ◁ #576

Complements

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

NH ≅ G/Ntype
1. #31771408C2semidirect
2. #31771407C2semidirect
3. #31770980C2semidirect
4. #31770875C2semidirect
5. #31770873C2semidirect
6. #31770872C2semidirect
7. #31770773C2semidirect
8. #31770771C2semidirect
9. #31770722C2semidirect
10. #31770720C2semidirect
11. #576×C269C2semidirect
12. #22575499C22semidirect
15. #22575479C22semidirect
17. #22575478C22semidirect
18. #22575478C22semidirect
22. C2×D2152C22semidirect
24. C2×#16864387C22semidirect
26. #22575460C22semidirect
27. #22575459C22semidirect
28. #22575449C22semidirect
29. #22575389C22semidirect
30. #22575388C22semidirect
31. #22575380C22semidirect
32. #22575380C22semidirect
33. #22575379C22semidirect
34. #22575370C22semidirect
35. #22575369C22semidirect
36. C8×Dic269C22semidirect
37. H(5,𝔽2)×C269C22semidirect
39. AGL(1,ℤ/8)×C269C22semidirect
43. C2×D8×C269C22semidirect
44. #131×C269C22semidirect
53. #16864392D4semidirect
55. #16864390D4semidirect
58. #16864389D4semidirect
66. #16864387D4semidirect
83. Q16×C269D4semidirect
85. SD16×C269D4semidirect
89. D8×C269D4semidirect
153. C269#3292semidirect
157. #576D269semidirect
167. H(5,𝔽2)C2×D269semidirect
175. AGL(1,ℤ/8)C2×D269semidirect
186. C2×D8C2×D269semidirect
189. #131C2×D269semidirect
216. Q16C22⋊D269semidirect
223. SD16C22⋊D269semidirect
235. D8C22⋊D269semidirect

Maximal subgroups

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

subgroupindex
#317714082
#317714072
#317709802
#317708752
#317708732
#317708722
#317707732
#317707712
#317707222
#317707202
#576×C2692
#3292269