Finite Group Databasebeta

← #16828720

Subgroups — #16828720

Order 3712.

994 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 143 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
1C22×D4×D29C211yesmaximal, characteristic, complemented
2C2×#11745276C212yesmaximal, complemented
3#14271676C211yesmaximal, characteristic, complemented
4#14271313C211yesmaximal, characteristic, complemented
5#14271312C212yesmaximal, complemented
6#14271299C211yesmaximal, characteristic, complemented
7#14271298C212yesmaximal, complemented
8#14271297C211yesmaximal, characteristic, complemented
9#14271166C212yesmaximal, complemented
10#14271125C211yesmaximal, characteristic, complemented
11C29×#450C211yesmaximal, characteristic, complemented
12C24×D29—21no—
13C22×D4×C29C2211yescharacteristic
14C22×C22⋊D29—21no—
15C2×#90809C2211yescharacteristic, complemented
16C2×D4×D29—28no—
17C2×D4×D29—21no—
18C2×D4×D29—21no—
19C2×D4×D29C2212yes—
20C2×D4×D29C2212yes—
21C22×D116C2211yescharacteristic
22C22×C4×D29C2211yescharacteristic
23C2×SD16×C29C2212yescomplemented
24C29×C22⋊Q8C2211yescharacteristic, complemented
25#11745308C2211yescharacteristic, complemented
26C2×#90784C2212yescomplemented
27C2×#90782C2212yescomplemented
28#11745292C2211yescharacteristic, complemented
29#11745276—28no—
30C2×D232C2212yescomplemented
31C2×#90772C2212yescomplemented
32#11745257C2211yescharacteristic, complemented
33#11745256—21no—
34#11745255—21no—
35#11745254C2211yescharacteristic, complemented
36#11745244—21no—
37#11745243—21no—
38C29×#102C2212yescomplemented
39C29×C22⋊C8C2211yescharacteristic, complemented
40#11745180C2212yescomplemented
41C22⋊#6597C2211yescharacteristic, complemented
42#11745169C2212yescomplemented
43#11745168C2212yescomplemented
44#11745157C2212yescomplemented
45C24×C29—21no—
46C23×D29—28no—
47C23×D29—21no—
48C23×D29—22no—
49C23×D29—22no—
50C23×D29—44no—
51C23×D29—44no—
52C23×D29D411yescharacteristic
53C2×Q8×C29C2311yescharacteristic, complemented
54C2×D4×C29—24no—
55C2×D4×C29—21no—
56C2×D4×C29C2312yes—
57C22×C4×C29C2311yescharacteristic
58C2×C22⋊D29—28no—
59C2×C22⋊D29—21no—
60C2×C22⋊D29—22no—
61C2×C22⋊D29—22no—
62C22×Dic29D411yescharacteristic
63#90809—24no—
64D4×D29—28no—
65D4×D29—28no—
66D4×D29—24no—
67D4×D29—24no—
68D4×D29—24no—
69D4×D29D418yescomplemented
70C2×D116—21no—
71C2×D116—24no—
72C2×D116—21no—
73C2×D116—21no—
74C2×D116C2312yes—
75C2×D116C2311yescharacteristic, complemented
76C2×C4×D29—24no—
77C2×C4×D29—21no—
78C2×C4×D29D411yescharacteristic
79C2×C4×D29D411yescharacteristic
80C2×C4×D29C2311yescharacteristic
81C2×C4×D29C2311yescharacteristic
82SD16×C29—24no—
83C2×C8×C29C2312yescomplemented
84C4⋊C4×C29—21no—
85C4⋊C4×C29C2311yescharacteristic, complemented
86C22⋊C4×C29—21no—
87#90784—24no—
88#90782—24no—
89#90781—21no—
90#90781—21no—
91#90781—21no—
92#90781—21no—
93C4⋊Dic29C2311yescharacteristic, complemented
94#90779—21no—
95C4×Dic29—21no—
96C2×#6597C2312yescomplemented
97D232—24no—
98#90772—24no—
99C23×C29—24no—
100C23×C29—22no—
101C23×C29—42no—
102C23×C29—42no—
103C23×C29C2×D411yescharacteristic
104C22×D29—22no—
105C22×D29—22no—
106C22×D29—22no—
107C22×D29—24no—
108C22×D29—28no—
109C22×D29—22no—
110C22×D29—28no—
111C22×D29—21no—
112C22×D29—432no—
113C22×D29—44no—
114C22×D29—44no—
115C22×D29—44no—
116C22×D29—42no—
117C22×D29—42no—
118C22×D29—44no—
119C22×D29C2×D411yescharacteristic
120C22×D29C2×D411yescharacteristic
121Q8×C29—22no—
122D4×C29—24no—
123D4×C29—22no—
124D4×C29C2×D414yes—
125C2×C4×C29—22no—
126C2×C4×C29—21no—
127C2×C4×C29—21no—
128C2×C4×C29C2×D411yescharacteristic
129C2×C4×C29C2411yescharacteristic
130C22⋊D29—28no—
131C22⋊D29—28no—
132C22⋊D29—42no—
133C2×Dic29—22no—
134C2×Dic29—21no—
135C2×Dic29C2×D411yescharacteristic
136C2×Dic29C2×D411yescharacteristic
137D116—24no—
138D116—22no—
139D116—22no—
140D116—42no—
141D116—42no—
142D116C2×D414yes—
143C4×D29—22no—
144C4×D29—22no—
145C4×D29—22no—
146C4×D29—42no—
147C4×D29C2×D414yes—
148C8×C29—22no—
149#6597—22no—
150#3003—291nomaximal, Sylow
151C22×C29—21no—
152C22×C29—21no—
153C22×C29—24no—
154C22×C29—48no—
155C22×C29—42no—
156C22×C29—42no—
157C22×C29AGL(1,ℤ/8)12yescomplemented
158C22×C29C22×D411yescharacteristic
159C2×D29—28no—
160C2×D29—21no—
161C2×D29—21no—
162C2×D29—21no—
163C2×D29—21no—
164C2×D29—24no—
165C2×D29—416no—
166C2×D29—416no—
167C2×D29—48no—
168C2×D29—42no—
169C2×D29—42no—
170C2×D29—41no—
171C2×D29—41no—
172C2×D29—41no—
173C2×D29#12012yes—
174C2×D29AGL(1,ℤ/8)12yescomplemented
175C4×C29—21no—
176C4×C29—41no—
177C4×C29—41no—
178C4×C29C22×D412yes—
179Dic29—21no—
180Dic29—41no—
181Dic29#12012yes—
182C23×D4—291no—
183C2×AGL(1,ℤ/8)—292no—
184#525—291no—
185#450—292no—
186#450—291no—
187#450D2911yescharacteristic, complemented
188#447—291no—
189#447—292no—
190#447—291no—
191#418—292no—
192#407—291no—
193C2×C29—22no—
194C2×C29—44no—
195C2×C29C2×#12011yescharacteristic
196C2×C29C2×AGL(1,ℤ/8)11yescharacteristic
197C2×C29C2×AGL(1,ℤ/8)11yescharacteristic
198D29—22no—
199D29—22no—
200D29—44no—
201D29—81no—
202C25—581no—
203C2×#41—291no—
204C22×D4—292no—
205C22×D4—292no—
206C22×D4—291no—
207C22×D4—581no—
208C22×D4—581no—
209C22×D4—581no—
210C22×D4—588no—
211C22×D4C2×D2911yescharacteristic
212C23×C4—291no—
213AGL(1,ℤ/8)—588no—
214C2×SD16—292no—
215C2×SD16C2×D2912yescomplemented
216C2×D8—292no—
217C2×D8—292no—
218C2×M16—292no—
219#123—581no—
220C22⋊Q8—291no—
221C22⋊Q8C2×D2911yescharacteristic, complemented
222#121—291no—
223#121—291no—
224#120—581no—
225C4×D4—581no—
226C4×D4—581no—
227#117—291no—
228#102—292no—
229#102—292no—
230#102—292no—
231#102C2×D2912yescomplemented
232C22⋊C8—291no—
233C22⋊C8—292no—
234C22⋊C8C2×D2911yescharacteristic, complemented
235C29#300311yescharacteristic, complemented, Sylow
236C24—21no—
237C24—291no—
238C24—582no—
239C24—582no—
240C24—588no—
241C24—581no—
242C24—1164no—
243C24—1164no—
244#41—584no—
245C2×Q8C22×D2911yescharacteristic, complemented
246C2×D4—24no—
247C2×D4—21no—
248C2×D4—298no—
249C2×D4—292no—
250C2×D4—291no—
251C2×D4—584no—
252C2×D4—581no—
253C2×D4—581no—
254C2×D4—584no—
255C2×D4—582no—
256C2×D4—582no—
257C2×D4—584no—
258C2×D4—588no—
259C2×D4—588no—
260C2×D4—588no—
261C2×D4—584no—
262C2×D4—581no—
263C2×D4—581no—
264C2×D4C22×D2912yes—
265C22×C4—291no—
266C22×C4—291no—
267C22×C4—291no—
268C22×C4—291no—
269C22×C4—291no—
270C22×C4—584no—
271C22×C4—581no—
272C22×C4C22×D2911yescharacteristic
273SD16—24no—
274SD16—584no—
275D8—584no—
276D8—584no—
277M16—584no—
278C2×C8—292no—
279C2×C8C22×D2912yescomplemented
280C4⋊C4—21no—
281C4⋊C4—291no—
282C4⋊C4—581no—
283C4⋊C4C22×D2911yescharacteristic, complemented
284C22⋊C4—21no—
285C22⋊C4—581no—
286C22⋊C4—581no—
287C22⋊C4—581no—
288C22⋊C4—581no—
289C42—581no—
290C23—24no—
291C23—22no—
292C23—42no—
293C23—42no—
294C23—291no—
295C23—291no—
296C23—582no—
297C23—582no—
298C23—588no—
299C23—584no—
300C23—588no—
301C23—582no—
302C23—582no—
303C23—581no—
304C23—1162no—
305C23—1164no—
306C23—11632no—
307C23—1164no—
308C23—1164no—
309C23—1164no—
310C23—1162no—
311C23D4×D2911yescharacteristic
312Q8—22no—
313D4—24no—
314D4—22no—
315D4—294no—
316D4—584no—
317D4—588no—
318D4—588no—
319D4—582no—
320D4—582no—
321D4—1162no—
322D4—1162no—
323D4—1162no—
324D4D4×D2914yes—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—291no—
329C2×C4—294no—
330C2×C4—291no—
331C2×C4—582no—
332C2×C4—582no—
333C2×C4—582no—
334C2×C4—582no—
335C2×C4—581no—
336C2×C4—1162no—
337C2×C4D4×D2911yescharacteristic
338C2×C4C23×D2911yescharacteristic
339C8—22no—
340C8—582no—
341C22—21no—
342C22—21no—
343C22—24no—
344C22—48no—
345C22—42no—
346C22—42no—
347C22—292no—
348C22—292no—
349C22—588no—
350C22—581no—
351C22—581no—
352C22—581no—
353C22—581no—
354C22—584no—
355C22—1168no—
356C22—11616no—
357C22—11616no—
358C22—1162no—
359C22—1162no—
360C22—1161no—
361C22—1161no—
362C22—1161no—
363C22#1174527612yescomplemented
364C22C2×D4×D2911yescharacteristic
365C4—21no—
366C4—41no—
367C4—41no—
368C4—292no—
369C4—581no—
370C4—1161no—
371C4C2×D4×D2912yes—
372C2—22no—
373C2—44no—
374C2—582no—
375C2—582no—
376C2—1164no—
377C2—2321no—
378C2#120×D2911yescharacteristic
379C2AGL(1,ℤ/8)×D2911yescharacteristic
380C2C2×#1174527611yescharacteristic
subgroupG/H
centerC22C2×D4×D29
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC29×#450C2
socleC22×C29C22×D4
hypercenter#450D29
hypocenterC29#3003
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C22×D4×D29C2semidirect
2. C2×#11745276C2semidirect
3. #14271676C2semidirect
4. #14271313C2semidirect
5. #14271312C2semidirect
6. #14271299C2semidirect
7. #14271298C2semidirect
8. #14271297C2semidirect
9. #14271166C2semidirect
10. #14271125C2semidirect
11. C29×#450C2semidirect
15. C2×#90809C22semidirect
23. C2×SD16×C29C22semidirect
24. C29×C22⋊Q8C22semidirect
25. #11745308C22semidirect
26. C2×#90784C22semidirect
27. C2×#90782C22semidirect
28. #11745292C22semidirect
30. C2×D232C22semidirect
31. C2×#90772C22semidirect
32. #11745257C22semidirect
35. #11745254C22semidirect
38. C29×#102C22semidirect
39. C29×C22⋊C8C22semidirect
40. #11745180C22semidirect
41. C22⋊#6597C22semidirect
42. #11745169C22semidirect
43. #11745168C22semidirect
44. #11745157C22semidirect
53. C2×Q8×C29C23semidirect
69. D4×D29D4semidirect
75. C2×D116C23semidirect
83. C2×C8×C29C23semidirect
85. C4⋊C4×C29C23semidirect
93. C4⋊Dic29C23semidirect
96. C2×#6597C23semidirect
157. C22×C29AGL(1,ℤ/8)semidirect
174. C2×D29AGL(1,ℤ/8)semidirect
187. #450D29semidirect
215. C2×SD16C2×D29semidirect
221. C22⋊Q8C2×D29semidirect
231. #102C2×D29semidirect
234. C22⋊C8C2×D29semidirect
235. C29#3003semidirect
245. C2×Q8C22×D29semidirect
279. C2×C8C22×D29semidirect
283. C4⋊C4C22×D29semidirect
363. C22#11745276semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D292
C2×#117452762
#142716762
#142713132
#142713122
#142712992
#142712982
#142712972
#142711662
#142711252
C29×#4502
#300329