Finite Group Databasebeta

← #16831062

Subgroups — #16831062

Order 3712.

472 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11745280C211yesmaximal, characteristic, complemented
2Q8×C22⋊D29C211yesmaximal, characteristic, complemented
3#14271824C211yesmaximal, characteristic
4#14271556C211yesmaximal, characteristic, complemented
5#14271418C211yesmaximal, characteristic, complemented
6#14271392C211yesmaximal, characteristic, complemented
7#14271387C211yesmaximal, characteristic, complemented
8#14271371C211yesmaximal, characteristic, complemented
9#14271324C211yesmaximal, characteristic, complemented
10#14271319C211yesmaximal, characteristic, complemented
11#14271317C211yesmaximal, characteristic, complemented
12#14271302C211yesmaximal, characteristic, complemented
13#14271270C211yesmaximal, characteristic, complemented
14#14271234C211yesmaximal, characteristic, complemented
15C22×Q16×C29C211yesmaximal, characteristic, complemented
16C22×Q8×C29C2211yescharacteristic
17C22×Q8×C29C2211yescharacteristic
18C2×#90809C2211yescharacteristic
19C2×Q8×D29C2211yescharacteristic
20C2×Q16×C29—24no—
21C2×Q16×C29C2211yescharacteristic
22C2×Q16×C29C2211yescharacteristic
23C2×Q16×C29C2211yescharacteristic
24C2×Q16×C29C2211yescharacteristic
25C22×C8×C29C2211yescharacteristic
26#11745309—21no—
27#11745309C2211yescharacteristic
28#11745308—21no—
29#11745308—21no—
30#11745308C2211yescharacteristic
31#11745308C2211yescharacteristic
32Q8×Dic29C2211yescharacteristic
33Q8×Dic29C2211yescharacteristic
34#11745306—21no—
35#11745306C2211yescharacteristic
36C2×#90785C2211yescharacteristic
37C2×#90785C2211yescharacteristic
38C2×#90784C2211yescharacteristic
39C2×#90784C2211yescharacteristic
40#11745292—21no—
41#11745292C2211yescharacteristic
42C4×C22⋊D29—21no—
43C4×C22⋊D29—21no—
44C4×C22⋊D29C2211yescharacteristic
45#11745286—21no—
46#11745286C2211yescharacteristic
47#11745280—24no—
48C2×#90773C2211yescharacteristic
49C2×#90772C2211yescharacteristic
50#11745183C2211yescharacteristic
51#11745183C2211yescharacteristic
52#11745183C2211yescharacteristic
53#11745183C2211yescharacteristic
54C22⋊#6597C2211yescharacteristic
55#11745169C2211yescharacteristic
56#11745168C2211yescharacteristic
57#11745165C2211yescharacteristic
58#11745164C2211yescharacteristic
59#11745163C2211yescharacteristic
60#11745162C2211yescharacteristic
61C2×Q8×C29—21no—
62C2×Q8×C29—21no—
63C2×Q8×C29—22no—
64C2×Q8×C29—22no—
65C2×Q8×C29—41no—
66C2×Q8×C29—41no—
67C2×Q8×C29C2311yescharacteristic
68C2×Q8×C29C2311yescharacteristic
69C2×Q8×C29C2311yescharacteristic
70C2×Q8×C29C2311yescharacteristic
71C22×C4×C29—21no—
72C22×C4×C29—21no—
73C22×C4×C29C2311yescharacteristic
74C2×C22⋊D29—21no—
75C2×C22⋊D29D411yescharacteristic
76#90809—22no—
77#90809—41no—
78Q8×D29—22no—
79Q8×D29—41no—
80C2×D116—21no—
81C2×D116C2311yescharacteristic
82C2×C4×D29—21no—
83C2×C4×D29—21no—
84C2×C4×D29C2311yescharacteristic
85C2×Dic58—21no—
86C2×Dic58C2311yescharacteristic
87Q16×C29—22no—
88Q16×C29—22no—
89Q16×C29—22no—
90Q16×C29D414yescomplemented
91C2×C8×C29—22no—
92C2×C8×C29C2311yescharacteristic
93C2×C8×C29C2311yescharacteristic
94C22⋊Dic29—21no—
95C22⋊Dic29D411yescharacteristic
96#90785—22no—
97#90785—41no—
98#90784—22no—
99#90784—41no—
100#90781—21no—
101#90781—21no—
102#90781—21no—
103#90781—21no—
104#90781—21no—
105#90781D411yescharacteristic
106C4⋊Dic29—21no—
107C4⋊Dic29—21no—
108C4⋊Dic29C2311yescharacteristic
109C4⋊Dic29C2311yescharacteristic
110#90779—21no—
111#90779—21no—
112#90779—21no—
113#90779—21no—
114#90779—21no—
115#90779D411yescharacteristic
116C4×Dic29—21no—
117C4×Dic29—21no—
118C4×Dic29C2311yescharacteristic
119C2×#6597C2311yescharacteristic
120C2×#6597C2311yescharacteristic
121#90773—22no—
122#90772—22no—
123C23×C29C2×D411yescharacteristic
124C22×D29—21no—
125C22×D29C2×D411yescharacteristic
126Q8×C29—21no—
127Q8×C29—21no—
128Q8×C29—21no—
129Q8×C29—21no—
130Q8×C29—21no—
131Q8×C29—21no—
132Q8×C29—42no—
133Q8×C29—42no—
134Q8×C29C2×D412yes—
135Q8×C29C2×D412yes—
136C2×C4×C29—21no—
137C2×C4×C29—21no—
138C2×C4×C29—21no—
139C2×C4×C29—21no—
140C2×C4×C29—21no—
141C2×C4×C29—21no—
142C2×C4×C29—42no—
143C2×C4×C29—42no—
144C2×C4×C29C2×D411yescharacteristic
145C2×C4×C29C2411yescharacteristic
146C22⋊D29—22no—
147C22⋊D29—42no—
148C2×Dic29—21no—
149C2×Dic29—21no—
150C2×Dic29—21no—
151C2×Dic29C2×D411yescharacteristic
152C2×Dic29C2×D411yescharacteristic
153C2×Dic29C2×D411yescharacteristic
154D116—21no—
155D116—21no—
156D116—42no—
157C4×D29—21no—
158C4×D29—21no—
159C4×D29—42no—
160C4×D29—42no—
161Dic58—21no—
162Dic58—21no—
163Dic58—42no—
164C8×C29—21no—
165C8×C29C2×D412yes—
166#6597—21no—
167#6597—21no—
168#3270—291nomaximal, Sylow
169C22×C29—21no—
170C22×C29—21no—
171C22×C29#13712yescomplemented
172C22×C29C22×D411yescharacteristic
173C2×D29—21no—
174C2×D29—21no—
175C2×D29—21no—
176C2×D29—41no—
177C2×D29—41no—
178C2×D29—41no—
179C4×C29—22no—
180C4×C29—22no—
181C4×C29—21no—
182C4×C29—41no—
183C4×C29—41no—
184C4×C29C22×D411yescharacteristic
185C4×C29H(5,𝔽2)11yescharacteristic
186Dic29—21no—
187Dic29—21no—
188Dic29—21no—
189Dic29—41no—
190Dic29—41no—
191Dic29—41no—
192C2×#137—291no—
193C22×Q16D2911yescharacteristic, complemented
194D4×Q8—291no—
195#548—291no—
196#497—291no—
197#470—291no—
198#465—291no—
199#464—291no—
200#462—291no—
201#452—291no—
202#451—291no—
203#451—291no—
204#448—291no—
205#441—291no—
206#435—291no—
207C2×C29—22no—
208C2×C29D4211yescharacteristic
209C2×C29C2×#13711yescharacteristic
210C2×C29#57811yescharacteristic
211D29—41no—
212D29—81no—
213C2×#41—291no—
214C22×Q8—291no—
215C22×Q8C2×D2911yescharacteristic
216C22×Q8C2×D2911yescharacteristic
217#137—584no—
218C2×Q16—24no—
219C2×Q16—291no—
220C2×Q16—291no—
221C2×Q16C2×D2911yescharacteristic
222C2×Q16C2×D2911yescharacteristic
223C2×Q16C2×D2911yescharacteristic
224C2×Q16C2×D2911yescharacteristic
225C2×SD16—291no—
226C2×SD16—291no—
227C2×SD16—291no—
228C2×M16—291no—
229C22×C8C2×D2911yescharacteristic
230C4⋊Q8—291no—
231C4⋊Q8—581no—
232#124—291no—
233#124—581no—
234C22⋊Q8—291no—
235C22⋊Q8—291no—
236C22⋊Q8—291no—
237C22⋊Q8—581no—
238C22⋊Q8—581no—
239C22⋊Q8—581no—
240#121—291no—
241#121—581no—
242C4×Q8—291no—
243C4×Q8—291no—
244C4×D4—291no—
245C4×D4—581no—
246C4×D4—581no—
247#106—291no—
248C4⋊C8—291no—
249#103—291no—
250#103—291no—
251#103—291no—
252#103—291no—
253#103—291no—
254#102—291no—
255C22⋊C8—291no—
256C22⋊C8—291no—
257#97—291no—
258C29#327011yescharacteristic, complemented, Sylow
259#41—582no—
260#41—1161no—
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8—22no—
264C2×Q8—22no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8—291no—
268C2×Q8—581no—
269C2×Q8—582no—
270C2×Q8—1161no—
271C2×Q8C22×D2911yescharacteristic
272C2×Q8C22×D2911yescharacteristic
273C2×Q8C22×D2911yescharacteristic
274C2×Q8C22×D2911yescharacteristic
275C2×D4—291no—
276C2×D4—291no—
277C2×D4—581no—
278C2×D4—581no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—291no—
282C22×C4—581no—
283C22×C4—581no—
284C22×C4C22×D2911yescharacteristic
285Q16—22no—
286Q16—22no—
287Q16—22no—
288Q16—582no—
289Q16—1161no—
290Q16C22⋊D2914yescomplemented
291SD16—582no—
292SD16—582no—
293SD16—1161no—
294M16—582no—
295C2×C8—22no—
296C2×C8—291no—
297C2×C8—291no—
298C2×C8C22×D2911yescharacteristic
299C2×C8C22×D2911yescharacteristic
300C4⋊C4—291no—
301C4⋊C4—291no—
302C4⋊C4—291no—
303C4⋊C4—581no—
304C4⋊C4—581no—
305C4⋊C4—581no—
306C4⋊C4—581no—
307C4⋊C4—581no—
308C4⋊C4—581no—
309C4⋊C4—581no—
310C22⋊C4—291no—
311C22⋊C4—291no—
312C22⋊C4—581no—
313C22⋊C4—581no—
314C22⋊C4—581no—
315C22⋊C4—581no—
316C22⋊C4—581no—
317C22⋊C4—581no—
318C42—291no—
319C42—581no—
320C42—581no—
321C23—291no—
322C23—581no—
323C23D4×D2911yescharacteristic
324Q8—21no—
325Q8—21no—
326Q8—21no—
327Q8—21no—
328Q8—21no—
329Q8—21no—
330Q8—42no—
331Q8—42no—
332Q8—581no—
333Q8—581no—
334Q8—1162no—
335Q8C2×C22⋊D2912yes—
336Q8C2×C22⋊D2912yes—
337D4—582no—
338D4—581no—
339D4—581no—
340D4—1162no—
341D4—1162no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—291no—
351C2×C4—291no—
352C2×C4—291no—
353C2×C4—581no—
354C2×C4—581no—
355C2×C4—581no—
356C2×C4—581no—
357C2×C4—581no—
358C2×C4—1162no—
359C2×C4—1162no—
360C2×C4D4×D2911yescharacteristic
361C2×C4C23×D2911yescharacteristic
362C8—21no—
363C8—581no—
364C8—581no—
365C8C2×C22⋊D2912yes—
366C22—21no—
367C22—21no—
368C22—581no—
369C22—581no—
370C22—581no—
371C22—1161no—
372C22—1161no—
373C22—1161no—
374C22#1174528012yescomplemented
375C22C2×D4×D2911yescharacteristic
376C4—21no—
377C4—22no—
378C4—22no—
379C4—41no—
380C4—41no—
381C4—581no—
382C4—581no—
383C4—581no—
384C4—1161no—
385C4—1161no—
386C4—1161no—
387C4#1174535911yescharacteristic
388C4C22×C22⋊D2911yescharacteristic
389C2—22no—
390C2—1161no—
391C2—2321no—
392C2D4×C22⋊D2911yescharacteristic
393C2C2×#1174528011yescharacteristic
394C2#1427197611yescharacteristic
subgroupG/H
centerC22C2×D4×D29
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC22×Q16×C29C2
socleC22×C29C22×D4
prime radical#14271824C2
hypercenterC22×Q16D29
hypocenterC29#3270
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q16

Complements

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

NH ≅ G/Ntype
1. C2×#11745280C2semidirect
2. Q8×C22⋊D29C2semidirect
4. #14271556C2semidirect
5. #14271418C2semidirect
6. #14271392C2semidirect
7. #14271387C2semidirect
8. #14271371C2semidirect
9. #14271324C2semidirect
10. #14271319C2semidirect
11. #14271317C2semidirect
12. #14271302C2semidirect
13. #14271270C2semidirect
14. #14271234C2semidirect
15. C22×Q16×C29C2semidirect
90. Q16×C29D4semidirect
171. C22×C29#137semidirect
193. C22×Q16D29semidirect
258. C29#3270semidirect
290. Q16C22⋊D29semidirect
374. C22#11745280semidirect

Maximal subgroups

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

subgroupindex
C2×#117452802
Q8×C22⋊D292
#142718242
#142715562
#142714182
#142713922
#142713872
#142713712
#142713242
#142713192
#142713172
#142713022
#142712702
#142712342
C22×Q16×C292
#327029