Finite Group Databasebeta

← #16831029

Subgroups — #16831029

Order 3712.

510 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11745273C211yesmaximal, characteristic, complemented
2C2×#11745266C211yesmaximal, characteristic, complemented
3#14271831C211yesmaximal, characteristic, complemented
4#14271810C211yesmaximal, characteristic, complemented
5#14271551C211yesmaximal, characteristic, complemented
6#14271410C211yesmaximal, characteristic, complemented
7#14271384C211yesmaximal, characteristic, complemented
8#14271357C211yesmaximal, characteristic, complemented
9#14271326C211yesmaximal, characteristic, complemented
10#14271306C211yesmaximal, characteristic, complemented
11#14271306C211yesmaximal, characteristic, complemented
12#14271298C211yesmaximal, characteristic, complemented
13#14271262C211yesmaximal, characteristic, complemented
14#14271234C211yesmaximal, characteristic, complemented
15C29×#466C211yesmaximal, characteristic, complemented
16C2×#90807C2211yescharacteristic
17C2×D4×D29C2211yescharacteristic
18C2×#90805C2211yescharacteristic
19C2×#90805C2211yescharacteristic, complemented
20C2×D8×C29C2211yescharacteristic
21C22×C8×C29C2211yescharacteristic, complemented
22C29×#121C2211yescharacteristic
23C29×#121C2211yescharacteristic, complemented
24#11745302—21no—
25#11745302C2211yescharacteristic, complemented
26#11745301—21no—
27#11745300—21no—
28#11745299—21no—
29#11745298C2211yescharacteristic
30C2×#90783C2211yescharacteristic
31C2×#90783C2211yescharacteristic, complemented
32C2×#90782C2211yescharacteristic
33C2×#90782C2211yescharacteristic
34C4×C22⋊D29C2211yescharacteristic
35#11745273—24no—
36C2×Dic116C2211yescharacteristic, complemented
37#11745266—24no—
38C2×D232C2211yescharacteristic
39C2×#90773C2211yescharacteristic
40C2×#90773C2211yescharacteristic, complemented
41C2×#90772C2211yescharacteristic
42#11745259C2211yescharacteristic
43#11745257C2211yescharacteristic
44#11745255C2211yescharacteristic
45#11745249C2211yescharacteristic, complemented
46#11745246—21no—
47#11745245—21no—
48#11745243—21no—
49C22⋊C4×D29—21no—
50C29×#107C2211yescharacteristic, complemented
51C29×#102C2211yescharacteristic
52C29×#102C2211yescharacteristic, complemented
53C22⋊#6597C2211yescharacteristic, complemented
54#11745168C2211yescharacteristic
55#11745163C2211yescharacteristic, complemented
56#11745162C2211yescharacteristic, complemented
57#11745158C2211yescharacteristic, complemented
58#11745158C2211yescharacteristic, complemented
59#11745157C2211yescharacteristic
60#11745157C2211yescharacteristic, complemented
61C23×D29—21no—
62C2×D4×C29—21no—
63C2×D4×C29—21no—
64C2×D4×C29C2311yescharacteristic
65C2×D4×C29C2311yescharacteristic
66C22×C4×C29C2311yescharacteristic
67C2×C22⋊D29—21no—
68C2×C22⋊D29—21no—
69C2×C22⋊D29—21no—
70C2×C22⋊D29—21no—
71C2×C22⋊D29—21no—
72C2×C22⋊D29—21no—
73C2×C22⋊D29D411yescharacteristic
74C22×Dic29—21no—
75#90807—22no—
76#90807—41no—
77D4×D29—22no—
78D4×D29—41no—
79#90805—22no—
80#90805—22no—
81#90805—41no—
82#90805—41no—
83C2×D116—21no—
84C2×D116C2311yescharacteristic
85C2×D116C2311yescharacteristic
86C2×C4×D29—21no—
87C2×C4×D29—21no—
88C2×C4×D29—21no—
89C2×C4×D29C2311yescharacteristic
90C2×Dic58C2311yescharacteristic
91C2×Dic58C2311yescharacteristic
92D8×C29—22no—
93C2×C8×C29—22no—
94C2×C8×C29C2311yescharacteristic
95C2×C8×C29C2311yescharacteristic
96C4⋊C4×C29C2311yescharacteristic
97C4⋊C4×C29C2311yescharacteristic
98C22⋊C4×C29—21no—
99C22⋊C4×C29—21no—
100C22⋊Dic29—21no—
101C22⋊Dic29—21no—
102C22⋊Dic29D411yescharacteristic
103#90783—22no—
104#90783—41no—
105#90782—22no—
106#90782—41no—
107#90781—21no—
108#90781—21no—
109#90781—21no—
110#90781D411yescharacteristic
111C4⋊Dic29—21no—
112#90779—21no—
113#90779D411yescharacteristic
114C4×Dic29—21no—
115C4×Dic29C2311yescharacteristic
116C2×#6597C2311yescharacteristic
117C2×#6597C2311yescharacteristic
118Dic116—22no—
119D232—22no—
120#90773—22no—
121#90773D414yescomplemented
122#90772—22no—
123C23×C29—21no—
124C23×C29—21no—
125C23×C29C2×D411yescharacteristic
126C22×D29—21no—
127C22×D29—21no—
128C22×D29—42no—
129C22×D29—42no—
130C22×D29—81no—
131C22×D29C2×D411yescharacteristic
132D4×C29—21no—
133D4×C29—21no—
134D4×C29—21no—
135D4×C29—21no—
136D4×C29—81no—
137D4×C29—81no—
138C2×C4×C29—21no—
139C2×C4×C29—21no—
140C2×C4×C29—21no—
141C2×C4×C29—21no—
142C2×C4×C29C2×D411yescharacteristic
143C2×C4×C29C2411yescharacteristic
144C22⋊D29—41no—
145C22⋊D29—42no—
146C22⋊D29—42no—
147C22⋊D29—42no—
148C22⋊D29—42no—
149C22⋊D29—42no—
150C22⋊D29—81no—
151C2×Dic29—21no—
152C2×Dic29—21no—
153C2×Dic29—42no—
154C2×Dic29C2×D411yescharacteristic
155C2×Dic29C2×D411yescharacteristic
156C2×Dic29C2×D411yescharacteristic
157D116—21no—
158D116—21no—
159D116—21no—
160D116—81no—
161D116C2×D412yes—
162C4×D29—21no—
163C4×D29—21no—
164C4×D29—42no—
165C4×D29—42no—
166C4×D29—81no—
167Dic58—21no—
168Dic58—21no—
169Dic58—21no—
170Dic58C2×D412yes—
171C8×C29—21no—
172C8×C29C2×D412yes—
173#6597—21no—
174#6597—21no—
175#3268—291nomaximal, Sylow
176C22×C29—21no—
177C22×C29—21no—
178C22×C29—21no—
179C22×C29—41no—
180C22×C29—41no—
181C22×C29—41no—
182C22×C29—41no—
183C22×C29—41no—
184C22×C29—41no—
185C22×C29C22×D411yescharacteristic
186C2×D29—21no—
187C2×D29—22no—
188C2×D29—21no—
189C2×D29—21no—
190C2×D29—41no—
191C2×D29—41no—
192C2×D29—41no—
193C2×D29—41no—
194C2×D29—41no—
195C2×D29—84no—
196C4×C29—21no—
197C4×C29—41no—
198C4×C29—41no—
199C4×C29C22×D411yescharacteristic
200C4×C29H(5,𝔽2)11yescharacteristic
201Dic29—22no—
202Dic29—21no—
203Dic29—21no—
204Dic29—21no—
205Dic29—41no—
206C2×AGL(1,ℤ/8)—291no—
207C2×#135—291no—
208#550—291no—
209#546—291no—
210#496—291no—
211#468—291no—
212#466D2911yescharacteristic, complemented
213#464—291no—
214#459—291no—
215#452—291no—
216#449—291no—
217#449—291no—
218#447—291no—
219#440—291no—
220#435—291no—
221C2×C29—41no—
222C2×C29—81no—
223C2×C29—81no—
224C2×C29D4211yescharacteristic
225C2×C29#57511yescharacteristic
226C2×C29#57611yescharacteristic
227D29—41no—
228D29—42no—
229D29—81no—
230C2×#41—291no—
231C2×#41—291no—
232C2×#41—291no—
233C22×D4—291no—
234AGL(1,ℤ/8)—584no—
235#135—584no—
236C2×Q16—291no—
237C2×SD16—291no—
238C2×SD16—291no—
239C2×SD16—291no—
240C2×SD16—291no—
241C2×D8—291no—
242C2×D8—291no—
243C2×D8—291no—
244C2×D8C2×D2911yescharacteristic
245C2×M16—291no—
246C22×C8C2×D2911yescharacteristic, complemented
247#127—291no—
248#127—581no—
249#124—291no—
250#123—581no—
251C22⋊Q8—291no—
252#121—291no—
253#121—581no—
254#121—581no—
255#121—581no—
256#121C2×D2911yescharacteristic
257#121C2×D2911yescharacteristic, complemented
258#120—581no—
259C4×Q8—291no—
260C4×D4—291no—
261C4×D4—291no—
262C4×D4—581no—
263C2×C22⋊C4—581no—
264#107C2×D2911yescharacteristic, complemented
265C4⋊C8—291no—
266#103—291no—
267#103—291no—
268#102—291no—
269#102—291no—
270#102C2×D2911yescharacteristic
271#102C2×D2911yescharacteristic, complemented
272C22⋊C8—291no—
273C22⋊C8—291no—
274#97—291no—
275C29#326811yescharacteristic, complemented, Sylow
276C24—581no—
277#41—582no—
278#41—582no—
279#41—582no—
280#41—1161no—
281#41—1161no—
282#41—1161no—
283C2×Q8—291no—
284C2×Q8—291no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—291no—
288C2×D4—291no—
289C2×D4—291no—
290C2×D4—581no—
291C2×D4—581no—
292C2×D4—581no—
293C2×D4—581no—
294C2×D4—582no—
295C2×D4—581no—
296C2×D4—581no—
297C2×D4—581no—
298C2×D4—1161no—
299C2×D4C22×D2911yescharacteristic
300C2×D4C22×D2911yescharacteristic
301C22×C4—291no—
302C22×C4—581no—
303C22×C4—581no—
304C22×C4—581no—
305C22×C4—581no—
306C22×C4C22×D2911yescharacteristic
307Q16—582no—
308SD16—294no—
309SD16—582no—
310SD16—582no—
311SD16—1161no—
312D8—22no—
313D8—582no—
314D8—582no—
315D8—1161no—
316M16—582no—
317C2×C8—22no—
318C2×C8—291no—
319C2×C8—291no—
320C2×C8C22×D2911yescharacteristic
321C2×C8C22×D2911yescharacteristic
322C4⋊C4—291no—
323C4⋊C4—581no—
324C4⋊C4—581no—
325C4⋊C4C22×D2911yescharacteristic
326C4⋊C4C22×D2911yescharacteristic
327C22⋊C4—21no—
328C22⋊C4—21no—
329C22⋊C4—291no—
330C22⋊C4—291no—
331C22⋊C4—581no—
332C22⋊C4—581no—
333C22⋊C4—581no—
334C22⋊C4—581no—
335C22⋊C4—581no—
336C42—291no—
337C42—581no—
338C23—21no—
339C23—21no—
340C23—291no—
341C23—581no—
342C23—581no—
343C23—1162no—
344C23—1162no—
345C23—2321no—
346C23D4×D2911yescharacteristic
347Q8—292no—
348Q8—581no—
349Q8—581no—
350Q8—581no—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—21no—
355D4—81no—
356D4—81no—
357D4—292no—
358D4—581no—
359D4—581no—
360D4—581no—
361D4—1161no—
362D4—1162no—
363D4—1162no—
364D4—1162no—
365D4—1162no—
366D4—1162no—
367D4—2321no—
368D4—2321no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—291no—
374C2×C4—291no—
375C2×C4—291no—
376C2×C4—581no—
377C2×C4—581no—
378C2×C4—581no—
379C2×C4—581no—
380C2×C4—1162no—
381C2×C4—1162no—
382C2×C4—1162no—
383C2×C4—2321no—
384C2×C4D4×D2911yescharacteristic
385C2×C4C23×D2911yescharacteristic
386C8—21no—
387C8—581no—
388C8—581no—
389C8D4×D2912yes—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—581no—
400C22—582no—
401C22—581no—
402C22—581no—
403C22—1161no—
404C22—1161no—
405C22—1161no—
406C22—1161no—
407C22—1161no—
408C22—2324no—
409C22C2×D4×D2911yescharacteristic
410C4—21no—
411C4—41no—
412C4—41no—
413C4—581no—
414C4—582no—
415C4—581no—
416C4—581no—
417C4—1161no—
418C4C2×D4×D2911yescharacteristic
419C4#1174536411yescharacteristic
420C2—41no—
421C2—81no—
422C2—81no—
423C2—1161no—
424C2—1162no—
425C2—2321no—
426C2#1427179711yescharacteristic
427C2#1427196011yescharacteristic
428C2#1427196411yescharacteristic
subgroupG/H
centerC22C2×D4×D29
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC29×#466C2
socleC22×C29C22×D4
hypercenter#466D29
hypocenterC29#3268
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. C2×#11745273C2semidirect
2. C2×#11745266C2semidirect
3. #14271831C2semidirect
4. #14271810C2semidirect
5. #14271551C2semidirect
6. #14271410C2semidirect
7. #14271384C2semidirect
8. #14271357C2semidirect
9. #14271326C2semidirect
10. #14271306C2semidirect
11. #14271306C2semidirect
12. #14271298C2semidirect
13. #14271262C2semidirect
14. #14271234C2semidirect
15. C29×#466C2semidirect
19. C2×#90805C22semidirect
21. C22×C8×C29C22semidirect
23. C29×#121C22semidirect
25. #11745302C22semidirect
31. C2×#90783C22semidirect
36. C2×Dic116C22semidirect
40. C2×#90773C22semidirect
45. #11745249C22semidirect
50. C29×#107C22semidirect
52. C29×#102C22semidirect
53. C22⋊#6597C22semidirect
55. #11745163C22semidirect
56. #11745162C22semidirect
57. #11745158C22semidirect
58. #11745158C22semidirect
60. #11745157C22semidirect
121. #90773D4semidirect
212. #466D29semidirect
246. C22×C8C2×D29semidirect
257. #121C2×D29semidirect
264. #107C2×D29semidirect
271. #102C2×D29semidirect
275. C29#3268semidirect

Maximal subgroups

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

subgroupindex
C2×#117452732
C2×#117452662
#142718312
#142718102
#142715512
#142714102
#142713842
#142713572
#142713262
#142713062
#142712982
#142712622
#142712342
C29×#4662
#326829