Finite Group Databasebeta

← #16828967

Subgroups — #16828967

Order 3712.

584 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 133 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×#11745273C212yesmaximal, complemented
2#127×D29C211yesmaximal, characteristic, complemented
3C4×#90807C211yesmaximal, characteristic, complemented
4#14271364C211yesmaximal, characteristic, complemented
5#14271362C212yesmaximal, complemented
6#14271360C211yesmaximal, characteristic, complemented
7#14271358C211yesmaximal, characteristic, complemented
8#14271356C212yesmaximal, complemented
9#14271185C211yesmaximal, characteristic, complemented
10#14271166C212yesmaximal, complemented
11C29×#459C211yesmaximal, characteristic, complemented
12C2×#90807C2211yescharacteristic
13C2×D4×D29—21no—
14C2×D4×D29—21no—
15C2×D4×D29C2212yes—
16C2×D8×C29C2212yescomplemented
17C29×#127C2211yescharacteristic
18C4×D4×C29C2211yescharacteristic, complemented
19#11745302—21no—
20#11745302—21no—
21#11745302C2212yes—
22D4×Dic29C2211yescharacteristic, complemented
23C2×#90783C2212yescomplemented
24C2×#90782C2212yescomplemented
25C4×C22⋊D29—21no—
26C2×C4×Dic29—21no—
27#11745273—28no—
28C2×#90773C2212yescomplemented
29C2×#90772C2212yes—
30#11745254C2211yescharacteristic
31#11745249C2211yescharacteristic, complemented
32#11745243—21no—
33#11745239—21no—
34#11745236C2211yescharacteristic
35C42×D29C2211yescharacteristic
36C4×Dic58C2211yescharacteristic, complemented
37C29×C4⋊C8C2211yescharacteristic, complemented
38C29×#102C2212yescomplemented
39#11745180C2212yescomplemented
40#11745169C2212yescomplemented
41#11745162C2212yescomplemented
42#11745157C2212yescomplemented
43C4⋊#6597C2211yescharacteristic, complemented
44C23×D29—22no—
45C2×D4×C29—21no—
46C2×D4×C29—21no—
47C2×D4×C29C2312yes—
48C2×D4×C29C2311yescharacteristic
49C22×C4×C29—21no—
50C2×C22⋊D29—21no—
51C2×C22⋊D29—22no—
52C2×C22⋊D29—22no—
53C2×C22⋊D29—22no—
54C22×Dic29—21no—
55C22×Dic29—21no—
56#90807—22no—
57#90807D414yescomplemented
58D4×D29—24no—
59D4×D29—24no—
60D4×D29—44no—
61D4×D29—44no—
62C2×D116—21no—
63C2×D116—21no—
64C2×D116C2312yes—
65C2×C4×D29D411yescharacteristic
66C2×C4×D29D411yescharacteristic
67C2×C4×D29C2311yescharacteristic
68C2×Dic58C2311yescharacteristic
69D8×C29—24no—
70C2×C8×C29C2312yes—
71C4⋊C4×C29C2311yescharacteristic
72C22⋊C4×C29—21no—
73C42×C29C2311yescharacteristic
74C22⋊Dic29—21no—
75#90783—24no—
76#90782—24no—
77#90781—21no—
78C4⋊Dic29C2311yescharacteristic
79#90779—21no—
80#90779—21no—
81C4×Dic29—21no—
82C4×Dic29—21no—
83C4×Dic29—21no—
84C4×Dic29D411yescharacteristic
85C4×Dic29D411yescharacteristic
86C4×Dic29C2311yescharacteristic
87C2×#6597C2312yes—
88#90773—24no—
89#90772—24no—
90C23×C29—21no—
91C23×C29—22no—
92C22×D29—22no—
93C22×D29—44no—
94C22×D29—44no—
95C22×D29—44no—
96C22×D29C2×D411yescharacteristic
97D4×C29—22no—
98D4×C29—21no—
99D4×C29—22no—
100D4×C29—42no—
101D4×C29—42no—
102D4×C29C2×D412yescomplemented
103C2×C4×C29—21no—
104C2×C4×C29—42no—
105C2×C4×C29C2×D411yescharacteristic
106C2×C4×C29C2×D411yescharacteristic
107C2×C4×C29C2411yescharacteristic
108C22⋊D29—24no—
109C22⋊D29—44no—
110C22⋊D29—44no—
111C22⋊D29—44no—
112C2×Dic29—21no—
113C2×Dic29—24no—
114C2×Dic29—21no—
115C2×Dic29—42no—
116C2×Dic29C2×D411yescharacteristic
117C2×Dic29C2×D411yescharacteristic
118C2×Dic29C2×D411yescharacteristic
119D116—22no—
120D116—22no—
121D116—42no—
122D116—42no—
123C4×D29—22no—
124C4×D29—22no—
125C4×D29C2×D412yes—
126C4×D29#4112yes—
127Dic58—21no—
128Dic58C2×D412yescomplemented
129C8×C29—22no—
130#6597—22no—
131#3038—291nomaximal, Sylow
132C22×C29—22no—
133C22×C29—41no—
134C22×C29—41no—
135C22×C29—42no—
136C22×C29—42no—
137C22×C29—42no—
138C22×C29C22×D411yescharacteristic
139C2×D29—21no—
140C2×D29—21no—
141C2×D29—42no—
142C2×D29—42no—
143C2×D29—42no—
144C2×D29—88no—
145C2×D29#12112yes—
146C4×C29—21no—
147C4×C29—41no—
148C4×C29AGL(1,ℤ/8)12yescomplemented
149C4×C29C22×D411yescharacteristic
150C4×C29C2×#4111yescharacteristic
151Dic29—22no—
152Dic29—21no—
153Dic29—41no—
154Dic29#12112yes—
155Dic29AGL(1,ℤ/8)12yescomplemented
156C2×AGL(1,ℤ/8)—292no—
157C2×#127—291no—
158C4×#41—291no—
159#460—291no—
160#460—291no—
161#460—292no—
162#459—292no—
163#459—291no—
164#459D2911yescharacteristic, complemented
165#423—291no—
166#418—292no—
167C2×C29—42no—
168C2×C29—82no—
169C2×C29C2×#12111yescharacteristic
170C2×C29C2×AGL(1,ℤ/8)11yescharacteristic
171C2×C29C2×AGL(1,ℤ/8)11yescharacteristic
172D29—22no—
173D29—82no—
174C2×#41—291no—
175C22×D4—292no—
176C22×D4—581no—
177C22×D4—581no—
178AGL(1,ℤ/8)—588no—
179C2×SD16—292no—
180C2×SD16—292no—
181C2×D8—292no—
182C2×D8C2×D2912yescomplemented
183C2×M16—292no—
184#127—292no—
185#127—291no—
186#127—581no—
187#127—581no—
188#127C2×D2911yescharacteristic
189C4×Q8—291no—
190C4×Q8—291no—
191C4×D4—291no—
192C4×D4—581no—
193C4×D4—581no—
194C4×D4C2×D2911yescharacteristic, complemented
195#117—291no—
196#117—581no—
197C2×C42—291no—
198C2×C42—581no—
199C4⋊C8—292no—
200C4⋊C8—291no—
201C4⋊C8C2×D2911yescharacteristic, complemented
202#102—292no—
203#102—292no—
204#102—292no—
205#102C2×D2912yescomplemented
206C29#303811yescharacteristic, complemented, Sylow
207C24—582no—
208#41—294no—
209#41—582no—
210C2×Q8—291no—
211C2×D4—21no—
212C2×D4—21no—
213C2×D4—292no—
214C2×D4—584no—
215C2×D4—584no—
216C2×D4—582no—
217C2×D4—582no—
218C2×D4—582no—
219C2×D4—581no—
220C2×D4—581no—
221C2×D4—581no—
222C2×D4—1164no—
223C2×D4—1164no—
224C2×D4C22×D2912yes—
225C2×D4C22×D2911yescharacteristic
226C22×C4—21no—
227C22×C4—291no—
228C22×C4—291no—
229C22×C4—291no—
230C22×C4—581no—
231C22×C4—581no—
232SD16—584no—
233SD16—584no—
234D8—24no—
235D8—584no—
236M16—584no—
237C2×C8—292no—
238C2×C8C22×D2912yes—
239C4⋊C4—291no—
240C4⋊C4—581no—
241C4⋊C4—581no—
242C4⋊C4C22×D2911yescharacteristic
243C22⋊C4—21no—
244C22⋊C4—581no—
245C22⋊C4—581no—
246C42—291no—
247C42—291no—
248C42—291no—
249C42—581no—
250C42—581no—
251C42—581no—
252C42C22×D2911yescharacteristic
253C23—21no—
254C23—22no—
255C23—291no—
256C23—582no—
257C23—1164no—
258C23—1164no—
259C23—1164no—
260Q8—292no—
261Q8—581no—
262D4—22no—
263D4—21no—
264D4—22no—
265D4—42no—
266D4—42no—
267D4—584no—
268D4—582no—
269D4—582no—
270D4—1162no—
271D4—1164no—
272D4—1164no—
273D4—1164no—
274D4—1162no—
275D4D4×D2912yescomplemented
276C2×C4—21no—
277C2×C4—42no—
278C2×C4—291no—
279C2×C4—291no—
280C2×C4—292no—
281C2×C4—292no—
282C2×C4—291no—
283C2×C4—582no—
284C2×C4—584no—
285C2×C4—581no—
286C2×C4—582no—
287C2×C4—581no—
288C2×C4—1162no—
289C2×C4D4×D2911yescharacteristic
290C2×C4D4×D2911yescharacteristic
291C2×C4C23×D2911yescharacteristic
292C8—22no—
293C8—582no—
294C22—22no—
295C22—41no—
296C22—41no—
297C22—42no—
298C22—42no—
299C22—42no—
300C22—292no—
301C22—581no—
302C22—581no—
303C22—1162no—
304C22—1162no—
305C22—1162no—
306C22—2328no—
307C22C2×D4×D2911yescharacteristic
308C4—21no—
309C4—41no—
310C4—292no—
311C4—292no—
312C4—582no—
313C4—581no—
314C4—1161no—
315C4#1174527312yescomplemented
316C4C2×D4×D2911yescharacteristic
317C4#41×D2911yescharacteristic
318C2—42no—
319C2—82no—
320C2—582no—
321C2—2322no—
322C2#121×D2911yescharacteristic
323C2AGL(1,ℤ/8)×D2911yescharacteristic
324C2C2×#1174527311yescharacteristic
subgroupG/H
centerC22C2×D4×D29
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC29×#459C2
socleC22×C29C22×D4
hypercenter#459D29
hypocenterC29#3038
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#11745273C2semidirect
2. #127×D29C2semidirect
3. C4×#90807C2semidirect
4. #14271364C2semidirect
5. #14271362C2semidirect
6. #14271360C2semidirect
7. #14271358C2semidirect
8. #14271356C2semidirect
9. #14271185C2semidirect
10. #14271166C2semidirect
11. C29×#459C2semidirect
16. C2×D8×C29C22semidirect
18. C4×D4×C29C22semidirect
22. D4×Dic29C22semidirect
23. C2×#90783C22semidirect
24. C2×#90782C22semidirect
28. C2×#90773C22semidirect
31. #11745249C22semidirect
36. C4×Dic58C22semidirect
37. C29×C4⋊C8C22semidirect
38. C29×#102C22semidirect
39. #11745180C22semidirect
40. #11745169C22semidirect
41. #11745162C22semidirect
42. #11745157C22semidirect
43. C4⋊#6597C22semidirect
57. #90807D4semidirect
102. D4×C29C2×D4semidirect
128. Dic58C2×D4semidirect
148. C4×C29AGL(1,ℤ/8)semidirect
155. Dic29AGL(1,ℤ/8)semidirect
164. #459D29semidirect
182. C2×D8C2×D29semidirect
194. C4×D4C2×D29semidirect
201. C4⋊C8C2×D29semidirect
205. #102C2×D29semidirect
206. C29#3038semidirect
275. D4D4×D29semidirect
315. C4#11745273semidirect

Maximal subgroups

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

subgroupindex
C2×#117452732
#127×D292
C4×#908072
#142713642
#142713622
#142713602
#142713582
#142713562
#142711852
#142711662
C29×#4592
#303829