Finite Group Databasebeta

← #16833148

Subgroups — #16833148

Order 3712.

1536 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11745365C211yesmaximal, characteristic, complemented
2C22×#90807C211yesmaximal, characteristic, complemented
3#14271833C214yesmaximal, complemented
4#14271831C214yesmaximal, complemented
5#14271821C214yesmaximal, complemented
6#14271819C214yesmaximal, complemented
7#14271730C211yesmaximal, characteristic, complemented
8#14271728C214yesmaximal, complemented
9#14271675C211yesmaximal, characteristic, complemented
10#14271670C211yesmaximal, characteristic, complemented
11#14271669C214yesmaximal, complemented
12#14271640C211yesmaximal, characteristic, complemented
13C29×#535C211yesmaximal, characteristic, complemented
14C2×#41×C29C2211yescharacteristic
15C22×D4×C29C2211yescharacteristic
16C22×C22⋊D29C2212yes—
17C23×Dic29C2212yes—
18#11745365—28no—
19C2×Q8×D29C2212yescomplemented
20C2×#90807—28no—
21C2×#90807C2214yes—
22C2×#90807C2214yes—
23C2×#90807C2214yes—
24C2×#90807C2212yescomplemented
25C2×#90805C2211yescharacteristic
26C2×#90805C2212yescomplemented
27C22×C4×D29C2211yescharacteristic
28C22×Dic58C2211yescharacteristic
29C22×Dic58C2212yescomplemented
30C29×#127C2212yescomplemented
31C29×#124C2212yescomplemented
32C29×#121C2214yescomplemented
33C29×#120C2214yes—
34C29×#117C2211yescharacteristic
35#11745315C2214yes—
36#11745308C2212yescomplemented
37#11745302C2212yescomplemented
38#11745302C2218yescomplemented
39#11745301C2218yescomplemented
40#11745301C2218yescomplemented
41#11745300C2214yescomplemented
42#11745300C2212yescomplemented
43#11745299C2212yes—
44#11745298C2212yescomplemented
45#11745297C2212yes—
46D4×Dic29C2218yescomplemented
47C4×C22⋊D29C2214yescomplemented
48#11745288C2211yescharacteristic
49#11745259C2212yescomplemented
50#11745257C2212yescomplemented
51#11745249C2214yescomplemented
52#11745248C2212yes—
53#11745247C2218yescomplemented
54#11745246C2218yescomplemented
55#11745244C2212yes—
56#11745243C2218yescomplemented
57#11745240C2218yescomplemented
58#11745239C2214yes—
59C4×D116C2214yescomplemented
60C4×Dic58C2214yescomplemented
61C24×C29C2312yes—
62C23×D29C2311yescharacteristic
63#41×C29—24no—
64C2×Q8×C29C2311yescharacteristic
65C2×D4×C29—24no—
66C2×D4×C29C2312yes—
67C2×D4×C29C2312yes—
68C2×D4×C29C2312yes—
69C2×D4×C29C2311yescharacteristic
70C2×D4×C29C2318yes—
71C22×C4×C29C2312yes—
72C22×C4×C29C2311yescharacteristic
73C2×C22⋊D29—28no—
74C2×C22⋊D29C2314yes—
75C2×C22⋊D29C2318yes—
76C2×C22⋊D29C2312yes—
77C2×C22⋊D29C2314yes—
78C2×C22⋊D29C2318yes—
79C22×Dic29—28no—
80C22×Dic29C2314yes—
81C22×Dic29C2318yes—
82C22×Dic29C2311yescharacteristic
83C22×Dic29C2312yes—
84Q8×D29—28no—
85#90807—216no—
86#90807—216no—
87#90807—216no—
88#90807—28no—
89#90805—24no—
90#90805—28no—
91C2×D116C2311yescharacteristic
92C2×D116C2314yes—
93C2×C4×D29—24no—
94C2×C4×D29C2312yes—
95C2×C4×D29C2312yes—
96C2×C4×D29C2312yes—
97C2×C4×D29C2314yes—
98C2×Dic58—28no—
99C2×Dic58D418yescomplemented
100C2×Dic58C2311yescharacteristic
101C2×Dic58C2314yes—
102C2×Dic58C2312yes—
103C2×Dic58C2312yes—
104C4⋊C4×C29C2312yes—
105C22⋊C4×C29C2312yes—
106C22⋊C4×C29C2318yes—
107C42×C29C2312yes—
108C22⋊Dic29C2312yes—
109C22⋊Dic29C2314yes—
110C22⋊Dic29C2318yes—
111#90781C2314yes—
112#90781C2314yes—
113#90781C2314yes—
114C4⋊Dic29C2312yes—
115C4⋊Dic29C2314yes—
116#90779C2314yes—
117#90779C2318yes—
118C4×Dic29C2312yes—
119C4×Dic29C2318yes—
120C23×C29—24no—
121C23×C29—44no—
122C23×C29C2414yes—
123C23×C29C2412yes—
124C23×C29C2411yescharacteristic
125C22×D29—22no—
126C22×D29—42no—
127C22×D29C2412yes—
128C22×D29C2412yes—
129Q8×C29—22no—
130D4×C29—24no—
131D4×C29—24no—
132D4×C29—24no—
133D4×C29—22no—
134D4×C29—48no—
135C2×C4×C29—24no—
136C2×C4×C29C2×D414yes—
137C2×C4×C29C2414yes—
138C2×C4×C29C2412yes—
139C2×C4×C29C2411yescharacteristic
140C2×C4×C29C2411yescharacteristic
141C22⋊D29—24no—
142C22⋊D29—28no—
143C22⋊D29—216no—
144C22⋊D29—48no—
145C22⋊D29—44no—
146C2×Dic29—22no—
147C2×Dic29—28no—
148C2×Dic29—216no—
149C2×Dic29C2×D418yes—
150C2×Dic29C2414yes—
151C2×Dic29C2412yes—
152C2×Dic29C2412yes—
153C2×Dic29C2414yes—
154D116—22no—
155D116—44no—
156C4×D29—24no—
157C4×D29—24no—
158C4×D29—24no—
159C4×D29—28no—
160Dic58—22no—
161Dic58—28no—
162Dic58C2×D418yes—
163Dic58C2×D418yes—
164#3487—291nomaximal, Sylow
165C22×C29—22no—
166C22×C29—22no—
167C22×C29—24no—
168C22×C29—24no—
169C22×C29—28no—
170C22×C29—48no—
171C22×C29C22×D412yes—
172C22×C29C2511yescharacteristic
173C2×D29—22no—
174C2×D29—22no—
175C2×D29—22no—
176C2×D29—24no—
177C2×D29—22no—
178C2×D29—44no—
179C4×C29—22no—
180C4×C29—24no—
181C4×C29C22×D412yes—
182C4×C29C22×D412yes—
183Dic29—22no—
184Dic29—22no—
185Dic29—24no—
186Dic29C22×D418yes—
187C2×2−1+4—291no—
188C22×#41—291no—
189#550—294no—
190#550—294no—
191#548—294no—
192#548—294no—
193#535—291no—
194#535—294no—
195#535D2911yescharacteristic, complemented
196#525—291no—
197#525—294no—
198#525—291no—
199#519—291no—
200C2×C29—22no—
201C2×C29—42no—
202C2×C29—44no—
203C2×C29C23×D411yescharacteristic
204C2×C29#58512yes—
205D29—42no—
206D29—42no—
2072−1+4—588no—
208C2×#41—294no—
209C2×#41—294no—
210C2×#41—294no—
211C2×#41—292no—
212C2×#41—291no—
213C2×#41—292no—
214C2×#41—588no—
215C2×#41C2×D2911yescharacteristic
216C22×Q8—292no—
217C22×Q8—291no—
218C22×Q8—292no—
219C22×D4—292no—
220C22×D4C2×D2911yescharacteristic
221C23×C4—292no—
222C23×C4—291no—
223#127—298no—
224#127—292no—
225#127C2×D2912yescomplemented
226#124—292no—
227#124—298no—
228#124C2×D2912yescomplemented
229#123—292no—
230#123—292no—
231C22⋊Q8—292no—
232C22⋊Q8—292no—
233C22⋊Q8—298no—
234#121—292no—
235#121—298no—
236#121—294no—
237#121—298no—
238#121—292no—
239#121—298no—
240#121C2×D2914yescomplemented
241#120—294no—
242#120—292no—
243#120—292no—
244#120C2×D2914yes—
245C4×Q8—294no—
246C4×Q8—294no—
247C4×D4—298no—
248C4×D4—294no—
249C4×D4—298no—
250C4×D4—294no—
251#117—291no—
252#117—294no—
253#117C2×D2911yescharacteristic
254C29#348711yescharacteristic, complemented, Sylow
255C24—291no—
256C24C22×D2912yes—
257#41—24no—
258#41—588no—
259#41—5816no—
260#41—5816no—
261#41—5816no—
262#41—588no—
263#41—584no—
264C2×Q8—298no—
265C2×Q8—292no—
266C2×Q8—292no—
267C2×Q8—294no—
268C2×Q8—291no—
269C2×Q8—588no—
270C2×Q8—588no—
271C2×Q8C22×D2911yescharacteristic
272C2×D4—24no—
273C2×D4—292no—
274C2×D4—298no—
275C2×D4—294no—
276C2×D4—294no—
277C2×D4—298no—
278C2×D4—291no—
279C2×D4—294no—
280C2×D4—588no—
281C2×D4C22×D2912yes—
282C2×D4C22×D2912yes—
283C2×D4C22×D2911yescharacteristic
284C2×D4C22×D2912yes—
285C2×D4C22×D2918yes—
286C22×C4—292no—
287C22×C4—294no—
288C22×C4—298no—
289C22×C4—291no—
290C22×C4—294no—
291C22×C4—292no—
292C22×C4—292no—
293C22×C4—292no—
294C22×C4—588no—
295C22×C4—584no—
296C22×C4C22×D2912yes—
297C22×C4C22×D2911yescharacteristic
298C4⋊C4—298no—
299C4⋊C4—292no—
300C4⋊C4—294no—
301C4⋊C4—294no—
302C4⋊C4C22×D2912yes—
303C22⋊C4—292no—
304C22⋊C4—294no—
305C22⋊C4—298no—
306C22⋊C4—294no—
307C22⋊C4—294no—
308C22⋊C4—294no—
309C22⋊C4C22×D2912yes—
310C22⋊C4C22×D2918yes—
311C42—298no—
312C42—292no—
313C42C22×D2912yes—
314C23—24no—
315C23—44no—
316C23—292no—
317C23—292no—
318C23—582no—
319C23—1162no—
320C23C23×D2914yes—
321C23C23×D2912yes—
322C23C23×D2911yescharacteristic
323Q8—22no—
324Q8—298no—
325Q8—298no—
326Q8—588no—
327Q8—582no—
328D4—24no—
329D4—24no—
330D4—24no—
331D4—22no—
332D4—48no—
333D4—5816no—
334D4—588no—
335D4—584no—
336D4—582no—
337D4—1164no—
338D4—1164no—
339D4—1168no—
340C2×C4—24no—
341C2×C4—298no—
342C2×C4—294no—
343C2×C4—294no—
344C2×C4—292no—
345C2×C4—292no—
346C2×C4—588no—
347C2×C4—5816no—
348C2×C4—588no—
349C2×C4—582no—
350C2×C4—584no—
351C2×C4—584no—
352C2×C4—584no—
353C2×C4D4×D2914yes—
354C2×C4C23×D2914yes—
355C2×C4C23×D2912yes—
356C2×C4C23×D2911yescharacteristic
357C2×C4C23×D2911yescharacteristic
358C22—22no—
359C22—22no—
360C22—24no—
361C22—24no—
362C22—28no—
363C22—48no—
364C22—582no—
365C22—582no—
366C22—584no—
367C22—582no—
368C22—582no—
369C22—1164no—
370C22C2×D4×D2912yes—
371C22C24×D2911yescharacteristic
372C4—22no—
373C4—24no—
374C4—298no—
375C4—582no—
376C4—582no—
377C4—584no—
378C4C2×D4×D2912yes—
379C4C2×D4×D2912yes—
380C2—22no—
381C2—44no—
382C2—42no—
383C2—1162no—
384C2—1162no—
385C2C22×D4×D2911yescharacteristic
386C2#1427199812yes—
subgroupG/H
center, FrattiniC22C24×D29
commutator, socle, cosocleC22×C29C25
FittingC29×#535C2
hypercenter#535D29
hypocenterC29#3487
LCSG ▷ C22×C29 ▷ C29
UCSC1 ◁ C22 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#11745365C2semidirect
2. C22×#90807C2semidirect
3. #14271833C2semidirect
4. #14271831C2semidirect
5. #14271821C2semidirect
6. #14271819C2semidirect
7. #14271730C2semidirect
8. #14271728C2semidirect
9. #14271675C2semidirect
10. #14271670C2semidirect
11. #14271669C2semidirect
12. #14271640C2semidirect
13. C29×#535C2semidirect
19. C2×Q8×D29C22semidirect
24. C2×#90807C22semidirect
26. C2×#90805C22semidirect
29. C22×Dic58C22semidirect
30. C29×#127C22semidirect
31. C29×#124C22semidirect
32. C29×#121C22semidirect
36. #11745308C22semidirect
37. #11745302C22semidirect
38. #11745302C22semidirect
39. #11745301C22semidirect
40. #11745301C22semidirect
41. #11745300C22semidirect
42. #11745300C22semidirect
44. #11745298C22semidirect
46. D4×Dic29C22semidirect
47. C4×C22⋊D29C22semidirect
49. #11745259C22semidirect
50. #11745257C22semidirect
51. #11745249C22semidirect
53. #11745247C22semidirect
54. #11745246C22semidirect
56. #11745243C22semidirect
57. #11745240C22semidirect
59. C4×D116C22semidirect
60. C4×Dic58C22semidirect
99. C2×Dic58D4semidirect
195. #535D29semidirect
225. #127C2×D29semidirect
228. #124C2×D29semidirect
240. #121C2×D29semidirect
254. C29#3487semidirect

Maximal subgroups

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

subgroupindex
C2×#117453652
C22×#908072
#142718332
#142718312
#142718212
#142718192
#142717302
#142717282
#142716752
#142716702
#142716692
#142716402
C29×#5352
#348729