Finite Group Databasebeta

← #16829115

Subgroups — #16829115

Order 3712.

496 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11745277C211yesmaximal, characteristic, complemented
2C22×#90772C211yesmaximal, characteristic, complemented
3#14271673C211yesmaximal, characteristic
4C22⋊Q8×D29C211yesmaximal, characteristic, complemented
5#14271422C211yesmaximal, characteristic, complemented
6#14271420C211yesmaximal, characteristic, complemented
7#14271408C211yesmaximal, characteristic, complemented
8#14271406C211yesmaximal, characteristic, complemented
9#14271404C211yesmaximal, characteristic, complemented
10#14271402C211yesmaximal, characteristic, complemented
11#14271390C211yesmaximal, characteristic, complemented
12#14271202C211yesmaximal, characteristic, complemented
13#14271170C211yesmaximal, characteristic, complemented
14#14271161C211yesmaximal, characteristic, complemented
15C29×#465C211yesmaximal, characteristic, complemented
16C2×Q8×D29C2211yescharacteristic
17C2×#90807C2211yescharacteristic
18C22×C4×D29C2211yescharacteristic
19C2×SD16×C29C2211yescharacteristic, complemented
20C22×C8×C29C2211yescharacteristic
21C29×C22⋊Q8C2211yescharacteristic
22C29×#121C2211yescharacteristic
23#11745308C2211yescharacteristic
24C2×#90785C2211yescharacteristic, complemented
25#11745300C2211yescharacteristic
26#11745299—21no—
27D4×Dic29—21no—
28C2×#90783C2211yescharacteristic, complemented
29#11745286C2211yescharacteristic
30#11745286C2211yescharacteristic
31C22×#6597C2211yescharacteristic
32#11745277—24no—
33C2×Dic116C2211yescharacteristic, complemented
34C2×#90772—24no—
35C2×#90772C2211yescharacteristic
36C2×#90772C2211yescharacteristic
37C2×#90772C2211yescharacteristic
38C2×#90772C2211yescharacteristic
39#11745259C2211yescharacteristic
40#11745259C2211yescharacteristic
41#11745258—21no—
42#11745254C2211yescharacteristic
43C4⋊C4×D29—21no—
44C4⋊C4×D29C2211yescharacteristic
45#11745245—21no—
46#11745243—21no—
47C22⋊C4×D29—21no—
48#11745240—21no—
49C29×#106C2211yescharacteristic, complemented
50C29×#103C2211yescharacteristic, complemented
51C29×#102C2211yescharacteristic, complemented
52#11745183C2211yescharacteristic, complemented
53#11745180C2211yescharacteristic, complemented
54#11745166C2211yescharacteristic, complemented
55#11745164C2211yescharacteristic, complemented
56#11745164C2211yescharacteristic, complemented
57#11745158C2211yescharacteristic, complemented
58#11745158C2211yescharacteristic, complemented
59#11745156C2211yescharacteristic, complemented
60#11745155C2211yescharacteristic, complemented
61C23×D29D411yescharacteristic
62C2×Q8×C29C2311yescharacteristic
63C2×D4×C29—21no—
64C2×D4×C29C2311yescharacteristic
65C22×C4×C29C2311yescharacteristic
66C2×C22⋊D29—21no—
67C2×C22⋊D29—21no—
68C22×Dic29—21no—
69C22×Dic29D411yescharacteristic
70Q8×D29—22no—
71Q8×D29—22no—
72#90807—22no—
73#90807—22no—
74C2×C4×D29—22no—
75C2×C4×D29—22no—
76C2×C4×D29—21no—
77C2×C4×D29—21no—
78C2×C4×D29D411yescharacteristic
79C2×C4×D29D411yescharacteristic
80C2×C4×D29C2311yescharacteristic
81C2×C4×D29C2311yescharacteristic
82C2×Dic58—21no—
83C2×Dic58C2311yescharacteristic
84C2×Dic58C2311yescharacteristic
85SD16×C29—22no—
86C2×C8×C29—22no—
87C2×C8×C29C2311yescharacteristic
88C2×C8×C29C2311yescharacteristic
89C4⋊C4×C29—21no—
90C4⋊C4×C29C2311yescharacteristic
91C4⋊C4×C29C2311yescharacteristic
92C22⋊C4×C29—21no—
93C22⋊C4×C29—21no—
94C22⋊Dic29—21no—
95C22⋊Dic29—21no—
96C22⋊Dic29—21no—
97#90785—22no—
98#90783—22no—
99#90781—21no—
100#90781—21no—
101#90781—21no—
102#90781—21no—
103C4⋊Dic29—21no—
104C4⋊Dic29—21no—
105C4⋊Dic29C2311yescharacteristic
106C4⋊Dic29C2311yescharacteristic
107#90779—21no—
108#90779—21no—
109#90779—21no—
110#90779—21no—
111C4×Dic29—21no—
112C2×#6597—22no—
113C2×#6597C2311yescharacteristic
114C2×#6597C2311yescharacteristic
115Dic116—22no—
116#90772—22no—
117#90772—22no—
118#90772—22no—
119#90772D414yescomplemented
120C23×C29—21no—
121C23×C29C2×D411yescharacteristic
122C22×D29—22no—
123C22×D29—22no—
124C22×D29—41no—
125C22×D29C2×D411yescharacteristic
126C22×D29C2×D411yescharacteristic
127Q8×C29—21no—
128Q8×C29—21no—
129D4×C29—21no—
130D4×C29—21no—
131D4×C29—42no—
132C2×C4×C29—21no—
133C2×C4×C29—21no—
134C2×C4×C29—21no—
135C2×C4×C29—21no—
136C2×C4×C29—21no—
137C2×C4×C29C2×D411yescharacteristic
138C2×C4×C29C2411yescharacteristic
139C22⋊D29—42no—
140C22⋊D29—42no—
141C2×Dic29—22no—
142C2×Dic29—21no—
143C2×Dic29—21no—
144C2×Dic29—21no—
145C2×Dic29—21no—
146C2×Dic29—42no—
147C2×Dic29C2×D411yescharacteristic
148C2×Dic29C2×D411yescharacteristic
149C4×D29—22no—
150C4×D29—21no—
151C4×D29—21no—
152C4×D29—22no—
153C4×D29—42no—
154C4×D29—42no—
155C4×D29C2×D412yes—
156C4×D29#4112yes—
157Dic58—21no—
158Dic58—21no—
159Dic58—21no—
160Dic58—21no—
161Dic58—42no—
162C8×C29—21no—
163C8×C29C2×D412yes—
164#6597—21no—
165#6597C2×D412yes—
166#3056—291nomaximal, Sylow
167C22×C29—21no—
168C22×C29—21no—
169C22×C29—41no—
170C22×C29—41no—
171C22×C29—41no—
172C22×C29#13712yescomplemented
173C22×C29C22×D411yescharacteristic
174C2×D29—21no—
175C2×D29—21no—
176C2×D29—21no—
177C2×D29—21no—
178C2×D29—21no—
179C2×D29—44no—
180C2×D29#12112yes—
181C4×C29—21no—
182C4×C29—41no—
183C4×C29—41no—
184C4×C29—41no—
185C4×C29C22×D411yescharacteristic
186C4×C29C2×#4111yescharacteristic
187Dic29—21no—
188Dic29—41no—
189Dic29—41no—
190Dic29—41no—
191Dic29—41no—
192Dic29#12112yes—
193C2×#137—291no—
194C22×M16—291no—
195#525—291no—
196C2×C22⋊Q8—291no—
197#470—291no—
198#470—291no—
199#468—291no—
200#468—291no—
201#467—291no—
202#467—291no—
203#465—291no—
204#465D2911yescharacteristic, complemented
205#428—291no—
206#419—291no—
207#417—291no—
208C2×C29—22no—
209C2×C29—81no—
210C2×C29C2×#12111yescharacteristic
211C2×C29C2×#13711yescharacteristic
212C2×C29#57511yescharacteristic
213D29—22no—
214D29—41no—
215C2×#41—291no—
216C22×Q8—291no—
217C23×C4—291no—
218#137—584no—
219C2×Q16—291no—
220C2×Q16—291no—
221C2×SD16—291no—
222C2×SD16C2×D2911yescharacteristic, complemented
223C2×M16—291no—
224C2×M16—291no—
225C2×M16—291no—
226C2×M16—291no—
227C2×M16—584no—
228C22×C8—291no—
229C22×C8C2×D2911yescharacteristic
230#123—581no—
231C22⋊Q8—291no—
232C22⋊Q8—291no—
233C22⋊Q8—291no—
234C22⋊Q8—291no—
235C22⋊Q8—291no—
236C22⋊Q8—581no—
237C22⋊Q8—581no—
238C22⋊Q8C2×D2911yescharacteristic
239#121—291no—
240#121C2×D2911yescharacteristic
241#120—581no—
242C4×D4—581no—
243C4×D4—581no—
244#117—291no—
245C2×C4⋊C4—291no—
246C2×C4⋊C4—581no—
247C2×C22⋊C4—581no—
248#107—291no—
249#107—291no—
250#106—291no—
251#106C2×D2911yescharacteristic, complemented
252#103—291no—
253#103—291no—
254#103—291no—
255#103—291no—
256#103—291no—
257#103C2×D2911yescharacteristic, complemented
258#102—291no—
259#102C2×D2911yescharacteristic, complemented
260C29#305611yescharacteristic, complemented, Sylow
261C24—291no—
262#41—582no—
263#41—582no—
264C2×Q8—291no—
265C2×Q8—291no—
266C2×Q8—581no—
267C2×Q8—582no—
268C2×Q8—582no—
269C2×Q8C22×D2911yescharacteristic
270C2×D4—21no—
271C2×D4—581no—
272C2×D4—581no—
273C2×D4C22×D2911yescharacteristic
274C22×C4—291no—
275C22×C4—291no—
276C22×C4—291no—
277C22×C4—291no—
278C22×C4—291no—
279C22×C4—582no—
280C22×C4—582no—
281C22×C4—581no—
282C22×C4—581no—
283C22×C4—581no—
284C22×C4C22×D2911yescharacteristic
285Q16—582no—
286Q16—582no—
287SD16—22no—
288SD16—582no—
289M16—294no—
290M16—582no—
291M16—582no—
292M16—582no—
293C2×C8—22no—
294C2×C8—291no—
295C2×C8—291no—
296C2×C8—582no—
297C2×C8C22×D2911yescharacteristic
298C2×C8C22×D2911yescharacteristic
299C4⋊C4—21no—
300C4⋊C4—291no—
301C4⋊C4—291no—
302C4⋊C4—581no—
303C4⋊C4—581no—
304C4⋊C4—581no—
305C4⋊C4—581no—
306C4⋊C4—581no—
307C4⋊C4—581no—
308C4⋊C4C22×D2911yescharacteristic
309C4⋊C4C22×D2911yescharacteristic
310C22⋊C4—21no—
311C22⋊C4—21no—
312C22⋊C4—581no—
313C22⋊C4—581no—
314C22⋊C4—581no—
315C22⋊C4—581no—
316C22⋊C4—581no—
317C22⋊C4—581no—
318C22⋊C4—581no—
319C42—581no—
320C23—21no—
321C23—291no—
322C23—291no—
323C23—582no—
324C23—582no—
325C23—1161no—
326C23D4×D2911yescharacteristic
327Q8—21no—
328Q8—21no—
329Q8—581no—
330Q8—581no—
331Q8—581no—
332Q8—581no—
333Q8—1162no—
334D4—21no—
335D4—21no—
336D4—42no—
337D4—1162no—
338D4—1162no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—291no—
345C2×C4—292no—
346C2×C4—292no—
347C2×C4—291no—
348C2×C4—582no—
349C2×C4—582no—
350C2×C4—582no—
351C2×C4—581no—
352C2×C4—581no—
353C2×C4—581no—
354C2×C4—581no—
355C2×C4—581no—
356C2×C4—581no—
357C2×C4—1162no—
358C2×C4—1162no—
359C2×C4—1162no—
360C2×C4D4×D2911yescharacteristic
361C2×C4C23×D2911yescharacteristic
362C8—21no—
363C8—292no—
364C8—581no—
365C8D4×D2912yes—
366C22—21no—
367C22—21no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—292no—
372C22—581no—
373C22—581no—
374C22—581no—
375C22—581no—
376C22—581no—
377C22—1164no—
378C22#1174527712yescomplemented
379C22C2×D4×D2911yescharacteristic
380C4—21no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—292no—
385C4—581no—
386C4—1161no—
387C4—1161no—
388C4—1161no—
389C4—1161no—
390C4C2×D4×D2911yescharacteristic
391C4#41×D2911yescharacteristic
392C2—22no—
393C2—81no—
394C2—582no—
395C2—1161no—
396C2#121×D2911yescharacteristic
397C2C2×#1174527711yescharacteristic
398C2#1427196011yescharacteristic
subgroupG/H
centerC22C2×D4×D29
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC29×#465C2
socleC22×C29C22×D4
prime radical#14271673C2
hypercenter#465D29
hypocenterC29#3056
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. C2×#11745277C2semidirect
2. C22×#90772C2semidirect
4. C22⋊Q8×D29C2semidirect
5. #14271422C2semidirect
6. #14271420C2semidirect
7. #14271408C2semidirect
8. #14271406C2semidirect
9. #14271404C2semidirect
10. #14271402C2semidirect
11. #14271390C2semidirect
12. #14271202C2semidirect
13. #14271170C2semidirect
14. #14271161C2semidirect
15. C29×#465C2semidirect
19. C2×SD16×C29C22semidirect
24. C2×#90785C22semidirect
28. C2×#90783C22semidirect
33. C2×Dic116C22semidirect
49. C29×#106C22semidirect
50. C29×#103C22semidirect
51. C29×#102C22semidirect
52. #11745183C22semidirect
53. #11745180C22semidirect
54. #11745166C22semidirect
55. #11745164C22semidirect
56. #11745164C22semidirect
57. #11745158C22semidirect
58. #11745158C22semidirect
59. #11745156C22semidirect
60. #11745155C22semidirect
119. #90772D4semidirect
172. C22×C29#137semidirect
204. #465D29semidirect
222. C2×SD16C2×D29semidirect
251. #106C2×D29semidirect
257. #103C2×D29semidirect
259. #102C2×D29semidirect
260. C29#3056semidirect
378. C22#11745277semidirect

Maximal subgroups

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

subgroupindex
C2×#117452772
C22×#907722
#142716732
C22⋊Q8×D292
#142714222
#142714202
#142714082
#142714062
#142714042
#142714022
#142713902
#142712022
#142711702
#142711612
C29×#4652
#305629