Finite Group Databasebeta

← #16831020

Subgroups — #16831020

Order 3712.

500 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11745311C211yesmaximal, characteristic, complemented
2C2×#11745278C211yesmaximal, characteristic, complemented
3#14271812C211yesmaximal, characteristic, complemented
4#14271804C211yesmaximal, characteristic, complemented
5#14271556C211yesmaximal, characteristic, complemented
6#14271419C211yesmaximal, characteristic, complemented
7#14271395C211yesmaximal, characteristic, complemented
8#14271380C211yesmaximal, characteristic, complemented
9#14271367C211yesmaximal, characteristic, complemented
10#14271324C211yesmaximal, characteristic, complemented
11#14271316C211yesmaximal, characteristic, complemented
12#14271309C211yesmaximal, characteristic, complemented
13#14271272C211yesmaximal, characteristic, complemented
14#14271237C211yesmaximal, characteristic, complemented
15C29×#452C211yesmaximal, characteristic, complemented
16C2×#41×C29C2211yescharacteristic
17C2×#90809C2211yescharacteristic
18C2×#90807C2211yescharacteristic, complemented
19C22×D116C2211yescharacteristic
20C2×Q16×C29C2211yescharacteristic, complemented
21C2×SD16×C29C2211yescharacteristic, complemented
22C29×C22⋊Q8C2211yescharacteristic
23#11745311—24no—
24#11745309C2211yescharacteristic
25#11745308C2211yescharacteristic, complemented
26#11745306C2211yescharacteristic, complemented
27C2×#90785C2211yescharacteristic, complemented
28C2×#90784C2211yescharacteristic
29C2×#90784C2211yescharacteristic
30#11745301—21no—
31#11745297—21no—
32D4×Dic29C2211yescharacteristic, complemented
33C2×#90782C2211yescharacteristic
34C2×#90782C2211yescharacteristic
35#11745291—21no—
36C4×C22⋊D29C2211yescharacteristic
37#11745286C2211yescharacteristic
38C2×#90779—21no—
39C2×#90777C2211yescharacteristic
40#11745278—24no—
41C2×#90773C2211yescharacteristic
42C2×#90773C2211yescharacteristic
43C2×C8×D29C2211yescharacteristic, complemented
44#11745257—21no—
45#11745257C2211yescharacteristic
46#11745256—21no—
47#11745255C2211yescharacteristic
48#11745244—21no—
49C22⋊C4×D29—21no—
50C29×#103C2211yescharacteristic, complemented
51C29×#102C2211yescharacteristic, complemented
52C29×C22⋊C8C2211yescharacteristic
53#11745183C2211yescharacteristic, complemented
54#11745169C2211yescharacteristic
55#11745169C2211yescharacteristic
56#11745168C2211yescharacteristic, complemented
57#11745163C2211yescharacteristic, complemented
58#11745162C2211yescharacteristic, complemented
59#11745158C2211yescharacteristic, complemented
60#11745156C2211yescharacteristic, complemented
61C23×D29—21no—
62#41×C29—22no—
63#41×C29—41no—
64C2×Q8×C29C2311yescharacteristic
65C2×Q8×C29C2311yescharacteristic
66C2×D4×C29—21no—
67C2×D4×C29C2311yescharacteristic
68C22×C4×C29—21no—
69C22×C4×C29C2311yescharacteristic
70C2×C22⋊D29—21no—
71C2×C22⋊D29D411yescharacteristic
72C22×Dic29—21no—
73C22×Dic29—21no—
74#90809—22no—
75#90809—41no—
76#90807—22no—
77#90807—41no—
78C2×D116—22no—
79C2×D116—21no—
80C2×D116—21no—
81C2×D116—21no—
82C2×D116—41no—
83C2×D116C2311yescharacteristic
84C2×D116C2311yescharacteristic
85C2×C4×D29—21no—
86C2×C4×D29—21no—
87C2×C4×D29C2311yescharacteristic
88C2×Dic58C2311yescharacteristic
89Q16×C29—41no—
90SD16×C29—22no—
91C2×C8×C29C2311yescharacteristic
92C2×C8×C29C2311yescharacteristic
93C4⋊C4×C29—21no—
94C4⋊C4×C29C2311yescharacteristic
95C22⋊C4×C29—21no—
96C22⋊Dic29—21no—
97C22⋊Dic29—21no—
98C22⋊Dic29D411yescharacteristic
99#90785—22no—
100#90784—22no—
101#90784—22no—
102#90782—22no—
103#90782D414yescomplemented
104#90781—21no—
105#90781—21no—
106#90781—21no—
107#90781—21no—
108#90781—21no—
109#90781D411yescharacteristic
110C4⋊Dic29C2311yescharacteristic
111#90779—21no—
112#90779—21no—
113#90779—21no—
114#90779—21no—
115#90779D411yescharacteristic
116C4×Dic29C2311yescharacteristic
117#90777—22no—
118C2×#6597C2311yescharacteristic
119C2×#6597C2311yescharacteristic
120#90773—22no—
121#90773—41no—
122C8×D29—22no—
123C23×C29—21no—
124C23×C29C2×D411yescharacteristic
125C22×D29—21no—
126C22×D29—21no—
127C22×D29—42no—
128C22×D29—42no—
129C22×D29—81no—
130C22×D29C2×D411yescharacteristic
131Q8×C29—21no—
132Q8×C29—21no—
133Q8×C29—21no—
134Q8×C29—21no—
135D4×C29—21no—
136D4×C29—42no—
137D4×C29C2×D412yes—
138C2×C4×C29—21no—
139C2×C4×C29—21no—
140C2×C4×C29—21no—
141C2×C4×C29—21no—
142C2×C4×C29—21no—
143C2×C4×C29—42no—
144C2×C4×C29C2×D411yescharacteristic
145C2×C4×C29C2411yescharacteristic
146C22⋊D29—41no—
147C22⋊D29—42no—
148C2×Dic29—21no—
149C2×Dic29—21no—
150C2×Dic29—42no—
151C2×Dic29—42no—
152C2×Dic29C2×D411yescharacteristic
153C2×Dic29C2×D411yescharacteristic
154C2×Dic29C2×D411yescharacteristic
155D116—21no—
156D116—21no—
157D116—21no—
158D116—42no—
159D116—42no—
160D116—81no—
161D116C2×D412yes—
162C4×D29—21no—
163C4×D29—21no—
164C4×D29—42no—
165C4×D29—81no—
166Dic58—21no—
167Dic58—21no—
168C8×C29—21no—
169C8×C29—21no—
170#6597—21no—
171#6597C2×D412yes—
172#3267—291nomaximal, Sylow
173C22×C29—22no—
174C22×C29—21no—
175C22×C29—21no—
176C22×C29—21no—
177C22×C29—41no—
178C22×C29—41no—
179C22×C29C22×D411yescharacteristic
180C2×D29—21no—
181C2×D29—22no—
182C2×D29—21no—
183C2×D29—21no—
184C2×D29—41no—
185C2×D29—41no—
186C2×D29—41no—
187C2×D29—41no—
188C2×D29—41no—
189C2×D29—84no—
190C4×C29—21no—
191C4×C29—41no—
192C4×C29—41no—
193C4×C29—41no—
194C4×C29C22×D411yescharacteristic
195C4×C29H(5,𝔽2)11yescharacteristic
196Dic29—21no—
197Dic29—21no—
198Dic29—21no—
199Dic29—41no—
200Dic29—41no—
201C2×AGL(1,ℤ/8)—291no—
202C2×#135—291no—
203#547—291no—
204#546—291no—
205#497—291no—
206#470—291no—
207#465—291no—
208#463—291no—
209#461—291no—
210#452—291no—
211#452D2911yescharacteristic, complemented
212#450—291no—
213#449—291no—
214#442—291no—
215#435—291no—
216C2×C29—41no—
217C2×C29—42no—
218C2×C29D4211yescharacteristic
219C2×C29#57511yescharacteristic
220C2×C29#57711yescharacteristic
221D29—41no—
222D29—42no—
223D29—81no—
224C2×#41—291no—
225C2×#41—291no—
226C2×#41C2×D2911yescharacteristic
227C22×D4—291no—
228AGL(1,ℤ/8)—584no—
229#135—584no—
230C2×Q16—291no—
231C2×Q16C2×D2911yescharacteristic, complemented
232C2×SD16—291no—
233C2×SD16—291no—
234C2×SD16—291no—
235C2×SD16—291no—
236C2×SD16C2×D2911yescharacteristic, complemented
237C2×D8—291no—
238C2×D8—291no—
239C2×M16—291no—
240C22×C8—291no—
241C4⋊Q8—291no—
242#124—291no—
243#123—581no—
244#123—581no—
245#123—581no—
246C22⋊Q8—291no—
247C22⋊Q8—291no—
248C22⋊Q8C2×D2911yescharacteristic
249#121—291no—
250#121—581no—
251#121—581no—
252#120—581no—
253C4×D4—291no—
254C4×D4—291no—
255C4×D4—291no—
256C2×C4⋊C4—581no—
257C2×C22⋊C4—581no—
258#106—291no—
259C4⋊C8—291no—
260#103—291no—
261#103—291no—
262#103C2×D2911yescharacteristic, complemented
263#102—291no—
264#102—291no—
265#102C2×D2911yescharacteristic, complemented
266C22⋊C8—291no—
267C22⋊C8C2×D2911yescharacteristic
268#97—291no—
269C29#326711yescharacteristic, complemented, Sylow
270C24—581no—
271#41—22no—
272#41—41no—
273#41—582no—
274#41—582no—
275#41—1161no—
276#41—1161no—
277C2×Q8—291no—
278C2×Q8C22×D2911yescharacteristic
279C2×Q8C22×D2911yescharacteristic
280C2×D4—21no—
281C2×D4—291no—
282C2×D4—291no—
283C2×D4—291no—
284C2×D4—582no—
285C2×D4—581no—
286C2×D4—581no—
287C2×D4—581no—
288C2×D4—581no—
289C2×D4—1161no—
290C2×D4C22×D2911yescharacteristic
291C22×C4—21no—
292C22×C4—291no—
293C22×C4—581no—
294C22×C4—581no—
295C22×C4—581no—
296C22×C4—581no—
297C22×C4C22×D2911yescharacteristic
298Q16—41no—
299Q16—582no—
300SD16—22no—
301SD16—582no—
302SD16—582no—
303SD16—582no—
304SD16—1161no—
305D8—294no—
306D8—582no—
307M16—582no—
308C2×C8—291no—
309C2×C8—291no—
310C2×C8—582no—
311C2×C8C22×D2911yescharacteristic
312C2×C8C22×D2911yescharacteristic
313C4⋊C4—21no—
314C4⋊C4—291no—
315C4⋊C4—291no—
316C4⋊C4—581no—
317C4⋊C4—581no—
318C4⋊C4—581no—
319C4⋊C4—581no—
320C4⋊C4C22×D2911yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—291no—
323C22⋊C4—291no—
324C22⋊C4—581no—
325C22⋊C4—581no—
326C22⋊C4—581no—
327C22⋊C4—581no—
328C22⋊C4—581no—
329C22⋊C4—581no—
330C22⋊C4—581no—
331C42—291no—
332C23—21no—
333C23—291no—
334C23—581no—
335C23—581no—
336C23—1162no—
337C23—1162no—
338C23—2321no—
339C23D4×D2911yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—21no—
344Q8—581no—
345Q8—581no—
346D4—21no—
347D4—42no—
348D4—292no—
349D4—581no—
350D4—581no—
351D4—581no—
352D4—1162no—
353D4—1162no—
354D4—1161no—
355D4—1162no—
356D4—2321no—
357D4D4×D2912yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—42no—
364C2×C4—291no—
365C2×C4—291no—
366C2×C4—291no—
367C2×C4—581no—
368C2×C4—581no—
369C2×C4—581no—
370C2×C4—581no—
371C2×C4—1162no—
372C2×C4—1162no—
373C2×C4—1162no—
374C2×C4—2321no—
375C2×C4D4×D2911yescharacteristic
376C2×C4C23×D2911yescharacteristic
377C8—21no—
378C8—21no—
379C8—292no—
380C8—581no—
381C22—21no—
382C22—22no—
383C22—21no—
384C22—21no—
385C22—41no—
386C22—41no—
387C22—581no—
388C22—582no—
389C22—581no—
390C22—581no—
391C22—1161no—
392C22—1161no—
393C22—1161no—
394C22—1161no—
395C22—1161no—
396C22—2324no—
397C22C2×D4×D2911yescharacteristic
398C4—21no—
399C4—41no—
400C4—41no—
401C4—41no—
402C4—581no—
403C4—581no—
404C4—581no—
405C4—1161no—
406C4—1161no—
407C4C2×D4×D2911yescharacteristic
408C4#1174535911yescharacteristic
409C2—41no—
410C2—42no—
411C2—1161no—
412C2—1162no—
413C2—2321no—
414C2#1427179511yescharacteristic
415C2#1427196211yescharacteristic
416C2#1427197011yescharacteristic
subgroupG/H
centerC22C2×D4×D29
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC29×#452C2
socleC22×C29C22×D4
hypercenter#452D29
hypocenterC29#3267
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. C2×#11745311C2semidirect
2. C2×#11745278C2semidirect
3. #14271812C2semidirect
4. #14271804C2semidirect
5. #14271556C2semidirect
6. #14271419C2semidirect
7. #14271395C2semidirect
8. #14271380C2semidirect
9. #14271367C2semidirect
10. #14271324C2semidirect
11. #14271316C2semidirect
12. #14271309C2semidirect
13. #14271272C2semidirect
14. #14271237C2semidirect
15. C29×#452C2semidirect
18. C2×#90807C22semidirect
20. C2×Q16×C29C22semidirect
21. C2×SD16×C29C22semidirect
25. #11745308C22semidirect
26. #11745306C22semidirect
27. C2×#90785C22semidirect
32. D4×Dic29C22semidirect
43. C2×C8×D29C22semidirect
50. C29×#103C22semidirect
51. C29×#102C22semidirect
53. #11745183C22semidirect
56. #11745168C22semidirect
57. #11745163C22semidirect
58. #11745162C22semidirect
59. #11745158C22semidirect
60. #11745156C22semidirect
103. #90782D4semidirect
211. #452D29semidirect
231. C2×Q16C2×D29semidirect
236. C2×SD16C2×D29semidirect
262. #103C2×D29semidirect
265. #102C2×D29semidirect
269. C29#3267semidirect

Maximal subgroups

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

subgroupindex
C2×#117453112
C2×#117452782
#142718122
#142718042
#142715562
#142714192
#142713952
#142713802
#142713672
#142713242
#142713162
#142713092
#142712722
#142712372
C29×#4522
#326729