Finite Group Databasebeta

← #46822683

Subgroups — #46822683

Order 34432.

476 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 117 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#31771410C211yesmaximal, characteristic, complemented
2#31771409C212yesmaximal, complemented
3#31771406C211yesmaximal, characteristic, complemented
4#31770980C211yesmaximal, characteristic, complemented
5#31770876C211yesmaximal, characteristic, complemented
6#31770874C212yesmaximal, complemented
7#31770775C212yesmaximal, complemented
8#31770771C212yesmaximal, complemented
9#31770721C211yesmaximal, characteristic, complemented
10#31770720C211yesmaximal, characteristic, complemented
11#472×C269C211yesmaximal, characteristic, complemented
12#22575499C2212yescomplemented
13#22575498C2212yescomplemented
14#22575481—22no—
15#22575481C2212yescomplemented
16#22575481C2212yescomplemented
17#22575480—22no—
18#22575479C2212yescomplemented
19#22575477—22no—
20#22575477—22no—
21#22575475—21no—
22#22575475C2211yescharacteristic
23#22575474—22no—
24#22575473C2211yescharacteristic
25#22575472—21no—
26C2×D2152—21no—
27C2×D2152C2211yescharacteristic, complemented
28C2×#16864387C2212yescomplemented
29D8×D269—21no—
30D8×D269—22no—
31#22575461C2212yescomplemented
32#22575459C2211yescharacteristic, complemented
33#22575449C2212yescomplemented
34#22575388C2211yescharacteristic, complemented
35#22575380C2211yescharacteristic, complemented
36#22575379C2211yescharacteristic, complemented
37#22575379C2212yescomplemented
38#22575369C2212yescomplemented
39C8×Dic269C2211yescharacteristic, complemented
40AGL(1,ℤ/8)×C269C2212yescomplemented
41C2×D8×C269C2211yescharacteristic, complemented
42#131×C269C2211yescharacteristic, complemented
43#108×C269C2211yescharacteristic, complemented
44#100×C269C2212yescomplemented
45#16864403—22no—
46#16864402—22no—
47#16864402C2311yescharacteristic
48#16864402C2312yes—
49#16864401—22no—
50#16864401—22no—
51C2×D1076—22no—
52C2×D1076C2312yes—
53C2×C22⋊D269—42no—
54C2×C22⋊D269—42no—
55D4×D269—22no—
56D4×D269—22no—
57D4×D269—22no—
58D4×D269—42no—
59Dic1076D411yescharacteristic, complemented
60#16864392D411yescharacteristic, complemented
61#16864391—22no—
62#16864391—22no—
63#16864391D412yescomplemented
64#16864390—22no—
65#16864390D411yescharacteristic, complemented
66#16864389D411yescharacteristic, complemented
67D2152—22no—
68D2152—22no—
69D2152—21no—
70D2152—21no—
71D2152D411yescharacteristic, complemented
72#16864387—22no—
73#16864387—22no—
74#16864387—22no—
75#16864387D411yescharacteristic, complemented
76#16864386C2311yescharacteristic
77#16864385—22no—
78#16864385—21no—
79C2×#14524130C2311yescharacteristic
80C8×D269—22no—
81C8×D269—21no—
82C4×Dic269C2311yescharacteristic
83#41×C269C2311yescharacteristic
84C2×D4×C269C2312yes—
85SD16×C269—22no—
86D8×C269—22no—
87D8×C269—21no—
88D8×C269—21no—
89M16×C269—21no—
90M16×C269C2312yes—
91M16×C269C2311yescharacteristic
92C2×C8×C269—21no—
93C2×C8×C269C2311yescharacteristic
94C22×D269—22no—
95C22×D269—42no—
96C22×D269—42no—
97Dic538C2×D412yes—
98Dic538C2×D411yescharacteristic
99D1076—22no—
100D1076—22no—
101D1076—42no—
102D1076C2×D412yes—
103D1076C2×D411yescharacteristic
104C22⋊D269—22no—
105C22⋊D269—21no—
106C22⋊D269—42no—
107C22⋊D269—42no—
108C22⋊D269—42no—
109C22⋊D269—84no—
110C2×Dic269—21no—
111C2×Dic269—42no—
112C4×D269—21no—
113C4×D269—22no—
114C4×D269—42no—
115#14524130—21no—
116#14524130C2×D411yescharacteristic
117#14524130C2×D411yescharacteristic
118C23×C269—22no—
119Q8×C269C2×D411yescharacteristic
120D4×C269—21no—
121D4×C269—22no—
122D4×C269—22no—
123D4×C269C2×D411yescharacteristic
124C2×C4×C269—21no—
125C2×C4×C269C2411yescharacteristic
126C8×C269—21no—
127C8×C269—22no—
128C8×C269C2×D411yescharacteristic
129C8×C269C2×D411yescharacteristic
130C22×C269—21no—
131C22×C269—42no—
132C22×C269—82no—
133C22×C269H(5,𝔽2)11yescharacteristic
134C2×D269—21no—
135C2×D269—22no—
136C2×D269—42no—
137C2×D269—82no—
138C2×D269—162no—
139C2×D269—162no—
140C4×C269—21no—
141C4×C269C22×D411yescharacteristic
142C4×C269C22×D411yescharacteristic
143Dic269—22no—
144Dic269—22no—
145Dic269—21no—
146C2×C269—21no—
147C2×C269—41no—
148C2×C269—82no—
149C2×C269D4211yescharacteristic
150D269—41no—
151D269—42no—
152D269—82no—
153C269#329211yescharacteristic, complemented, Sylow
154#3292—2691nomaximal, Sylow
155#576—2692no—
156#576—2691no—
157#576—2691no—
158#493—2691no—
159#472—2691no—
160#472—2692no—
161#472D26911yescharacteristic, complemented
162#453—2692no—
163#453—2692no—
164#443—2691no—
165#443—2691no—
166H(5,𝔽2)—2692no—
167H(5,𝔽2)—2692no—
168AGL(1,ℤ/8)—2692no—
169AGL(1,ℤ/8)—2692no—
170AGL(1,ℤ/8)—2692no—
171AGL(1,ℤ/8)—5382no—
172AGL(1,ℤ/8)—5382no—
173AGL(1,ℤ/8)—5382no—
174AGL(1,ℤ/8)—5382no—
175AGL(1,ℤ/8)C2×D26912yescomplemented
176#135—2691no—
177#135—2691no—
178#135—5381no—
179#135—5381no—
180#135—5382no—
181C2×D8—2692no—
182C2×D8—2691no—
183C2×D8—5381no—
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#108C2×D26911yescharacteristic, complemented
193#104—2691no—
194#104—2692no—
195#104—2691no—
196#100—2692no—
197#100C2×D26912yescomplemented
198C4×C8—2691no—
199#41—2692no—
200#41—2691no—
201#41—5382no—
202#41—5382no—
203#41—5382no—
204#41—5382no—
205#41C22×D26911yescharacteristic
206C2×D4—2692no—
207C2×D4—5382no—
208C2×D4—5382no—
209C2×D4—5382no—
210C2×D4—5382no—
211C2×D4—10762no—
212C2×D4—10762no—
213C2×D4—10762no—
214C2×D4C22×D26912yes—
215Q16—2691no—
216Q16—2691no—
217SD16—22no—
218SD16—2691no—
219SD16—2691no—
220SD16—2692no—
221SD16—5382no—
222SD16—5382no—
223SD16—5382no—
224D8—22no—
225D8—21no—
226D8—21no—
227D8—2691no—
228D8—2691no—
229D8—5382no—
230D8—5382no—
231D8—5382no—
232D8—5382no—
233D8—5382no—
234D8—5381no—
235D8—5381no—
236M16—21no—
237M16—2691no—
238M16—5382no—
239M16—5381no—
240M16C22×D26912yes—
241M16C22×D26911yescharacteristic
242C2×C8—21no—
243C2×C8—2691no—
244C2×C8—5382no—
245C2×C8—5381no—
246C2×C8C22×D26911yescharacteristic
247C42—2691no—
248C23—22no—
249C23—5382no—
250C23—10762no—
251C23—10762no—
252Q8—2691no—
253Q8—2692no—
254Q8D4×D26911yescharacteristic
255D4—21no—
256D4—22no—
257D4—22no—
258D4—2691no—
259D4—2692no—
260D4—5382no—
261D4—5381no—
262D4—5382no—
263D4—5382no—
264D4—10762no—
265D4—10762no—
266D4—10762no—
267D4—10762no—
268D4—21524no—
269D4D4×D26911yescharacteristic
270C2×C4—21no—
271C2×C4—5381no—
272C2×C4—5382no—
273C2×C4—5381no—
274C2×C4—10762no—
275C2×C4—10762no—
276C2×C4C23×D26911yescharacteristic
277C8—21no—
278C8—22no—
279C8—2691no—
280C8—2691no—
281C8—5381no—
282C8D4×D26911yescharacteristic
283C8D4×D26911yescharacteristic
284C22—21no—
285C22—42no—
286C22—82no—
287C22—5382no—
288C22—5381no—
289C22—10762no—
290C22—21522no—
291C22—43042no—
292C22—43042no—
293C22#2257549911yescharacteristic
294C4—21no—
295C4—5382no—
296C4—5382no—
297C4—5381no—
298C4C2×D4×D26911yescharacteristic
299C4C2×D4×D26911yescharacteristic
300C2—21no—
301C2—41no—
302C2—82no—
303C2—10762no—
304C2—10761no—
305C2—21522no—
306C2#3177123911yescharacteristic
subgroupG/H
centerC2#31771239
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#472×C269C2
socleC2×C269D42
hypercenter#472D269
hypocenterC269#3292
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C2 ◁ C2×C4 ◁ #472

Complements

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

NH ≅ G/Ntype
1. #31771410C2semidirect
2. #31771409C2semidirect
3. #31771406C2semidirect
4. #31770980C2semidirect
5. #31770876C2semidirect
6. #31770874C2semidirect
7. #31770775C2semidirect
8. #31770771C2semidirect
9. #31770721C2semidirect
10. #31770720C2semidirect
11. #472×C269C2semidirect
12. #22575499C22semidirect
13. #22575498C22semidirect
15. #22575481C22semidirect
16. #22575481C22semidirect
18. #22575479C22semidirect
27. C2×D2152C22semidirect
28. C2×#16864387C22semidirect
31. #22575461C22semidirect
32. #22575459C22semidirect
33. #22575449C22semidirect
34. #22575388C22semidirect
35. #22575380C22semidirect
36. #22575379C22semidirect
37. #22575379C22semidirect
38. #22575369C22semidirect
39. C8×Dic269C22semidirect
40. AGL(1,ℤ/8)×C269C22semidirect
41. C2×D8×C269C22semidirect
42. #131×C269C22semidirect
43. #108×C269C22semidirect
44. #100×C269C22semidirect
59. Dic1076D4semidirect
60. #16864392D4semidirect
63. #16864391D4semidirect
65. #16864390D4semidirect
66. #16864389D4semidirect
71. D2152D4semidirect
75. #16864387D4semidirect
153. C269#3292semidirect
161. #472D269semidirect
175. AGL(1,ℤ/8)C2×D269semidirect
186. C2×D8C2×D269semidirect
189. #131C2×D269semidirect
192. #108C2×D269semidirect
197. #100C2×D269semidirect

Maximal subgroups

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

subgroupindex
#317714102
#317714092
#317714062
#317709802
#317708762
#317708742
#317707752
#317707712
#317707212
#317707202
#472×C2692
#3292269