Finite Group Databasebeta

← #16822810

Subgroups — #16822810

Order 3712.

1056 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#90807C211yesmaximal, characteristic, complemented
2C2×#11745244C211yesmaximal, characteristic, complemented
3C2×#11745299C211yesmaximal, characteristic, complemented
4C22×#90781C211yesmaximal, characteristic, complemented
5#14271076C212yesmaximal, complemented
6#14271072C212yesmaximal, complemented
7#14271042C211yesmaximal, characteristic, complemented
8#14271041C212yesmaximal, complemented
9#14271040C212yesmaximal, complemented
10#14271024C211yesmaximal, characteristic, complemented
11C29×#386C211yesmaximal, characteristic, complemented
12C24×D29C2211yescharacteristic
13C22×D4×C29C2211yescharacteristic
14C23×C4×C29C2211yescharacteristic
15C22×C22⋊D29C2212yes—
16C22×C22⋊D29C2211yescharacteristic
17C22×C22⋊D29C2211yescharacteristic, complemented
18C23×Dic29C2211yescharacteristic
19C23×Dic29C2211yescharacteristic, complemented
20C2×#90807—24no—
21C2×#90807—24no—
22C2×#90807—24no—
23C22×D116C2212yes—
24C22×C4×D29C2211yescharacteristic, complemented
25C22×Dic58C2211yescharacteristic, complemented
26C2×C22⋊C4×C29C2211yescharacteristic
27C2×C22⋊C4×C29C2212yescomplemented
28C2×C22⋊Dic29C2211yescharacteristic
29#11745299—24no—
30C2×#90781—24no—
31C2×#90781C2212yes—
32C2×#90781C2212yes—
33C2×#90781C2212yes—
34C2×#90781C2212yes—
35C2×#90781C2212yescomplemented
36#11745244—24no—
37C29×#95C2212yescomplemented
38#11745179C2212yescomplemented
39#11745179C2212yescomplemented
40#11745179C2212yescomplemented
41C24×C29C2311yescharacteristic
42C24×C29C2311yescharacteristic
43C23×D29—22no—
44C23×D29—24no—
45C23×D29—22no—
46C23×D29—22no—
47C23×D29—42no—
48C23×D29C2311yescharacteristic
49C23×D29C2312yes—
50C2×D4×C29—22no—
51C2×D4×C29—22no—
52C2×D4×C29D414yescomplemented
53C22×C4×C29—22no—
54C22×C4×C29—42no—
55C22×C4×C29C2312yes—
56C22×C4×C29C2312yes—
57C22×C4×C29C2311yescharacteristic
58C2×C22⋊D29—22no—
59C2×C22⋊D29—22no—
60C2×C22⋊D29—22no—
61C2×C22⋊D29—22no—
62C2×C22⋊D29—24no—
63C2×C22⋊D29—24no—
64C2×C22⋊D29—22no—
65C2×C22⋊D29—42no—
66C2×C22⋊D29D414yescomplemented
67C22×Dic29—22no—
68C22×Dic29—22no—
69C22×Dic29—22no—
70C22×Dic29—22no—
71C22×Dic29—22no—
72C22×Dic29D414yescomplemented
73C22×Dic29C2312yes—
74C22×Dic29C2311yescharacteristic
75C22×Dic29C2311yescharacteristic
76C22×Dic29C2311yescharacteristic
77#90807—232no—
78C2×D116—24no—
79C2×D116—24no—
80C2×D116—42no—
81C2×C4×D29—22no—
82C2×C4×D29—22no—
83C2×C4×D29—22no—
84C2×Dic58—22no—
85C2×Dic58—22no—
86C2×Dic58—22no—
87C22⋊C4×C29—22no—
88C22⋊C4×C29—42no—
89C22⋊Dic29—22no—
90#90781—24no—
91#90781—24no—
92#90781—24no—
93#90781—24no—
94#90781—42no—
95C23×C29—22no—
96C23×C29—22no—
97C23×C29—21no—
98C23×C29—21no—
99C23×C29—22no—
100C23×C29—22no—
101C23×C29C2×D412yes—
102C23×C29C2×D412yes—
103C23×C29C2×D412yes—
104C23×C29C2×D412yes—
105C23×C29C2411yescharacteristic
106C22×D29—28no—
107C22×D29—22no—
108C22×D29—22no—
109C22×D29—22no—
110C22×D29—21no—
111C22×D29—21no—
112C22×D29—22no—
113C22×D29—22no—
114C22×D29—22no—
115C22×D29—22no—
116C22×D29—22no—
117C22×D29—48no—
118C22×D29—48no—
119C22×D29—44no—
120C22×D29—44no—
121C22×D29C2×D412yes—
122C22×D29#4114yescomplemented
123D4×C29—28no—
124C2×C4×C29—21no—
125C2×C4×C29—21no—
126C2×C4×C29—22no—
127C2×C4×C29—22no—
128C2×C4×C29—22no—
129C2×C4×C29—22no—
130C2×C4×C29—22no—
131C2×C4×C29—22no—
132C2×C4×C29—44no—
133C2×C4×C29C2×D412yes—
134C22⋊D29—28no—
135C22⋊D29—28no—
136C22⋊D29—48no—
137C2×Dic29—28no—
138C2×Dic29—28no—
139C2×Dic29—21no—
140C2×Dic29—21no—
141C2×Dic29—21no—
142C2×Dic29—21no—
143C2×Dic29—21no—
144C2×Dic29—21no—
145C2×Dic29—22no—
146C2×Dic29—22no—
147C2×Dic29—22no—
148C2×Dic29C2×D412yes—
149C2×Dic29C2×D412yes—
150C2×Dic29C2×D412yes—
151D116—48no—
152C4×D29—28no—
153Dic58—28no—
154#2404—291nomaximal, Sylow
155C22×C29—22no—
156C22×C29—22no—
157C22×C29—24no—
158C22×C29—24no—
159C22×C29—22no—
160C22×C29—24no—
161C22×C29—41no—
162C22×C29—41no—
163C22×C29—42no—
164C22×C29#12014yes—
165C22×C29C22×D411yescharacteristic
166C22×C29C22×D411yescharacteristic
167C22×C29C22×D411yescharacteristic
168C22×C29C2×#4112yes—
169C22×C29H(5,𝔽2)12yes—
170C2×D29—22no—
171C2×D29—24no—
172C2×D29—24no—
173C2×D29—24no—
174C2×D29—24no—
175C2×D29—416no—
176C2×D29—42no—
177C2×D29—41no—
178C2×D29—41no—
179C2×D29—42no—
180C2×D29—42no—
181C2×D29—42no—
182C2×D29—42no—
183C4×C29—22no—
184C4×C29—42no—
185C4×C29—42no—
186Dic29—22no—
187Dic29—22no—
188Dic29—22no—
189Dic29—42no—
190C22×#41—291no—
191C2×#120—291no—
192C2×#120—291no—
193C22×C22⋊C4—291no—
194#393—292no—
195#392—292no—
196#386—291no—
197#386—292no—
198#386—292no—
199#386D2911yescharacteristic, complemented
200#381—291no—
201C2×C29—24no—
202C2×C29—42no—
203C2×C29C2×#12011yescharacteristic
204C2×C29#52511yescharacteristic
205C2×C29#53511yescharacteristic
206C2×C29#54612yes—
207C2×C29#54612yes—
208D29—42no—
209D29—44no—
210C25—291no—
211C2×#41—584no—
212C2×#41—584no—
213C2×#41—584no—
214C22×Q8—291no—
215C22×D4—291no—
216C22×D4—291no—
217C22×D4—292no—
218C22×D4—292no—
219C22×D4C2×D2911yescharacteristic
220C23×C4—291no—
221C23×C4—291no—
222C23×C4—291no—
223C23×C4C2×D2911yescharacteristic
224#120—584no—
225#120—584no—
226C2×C22⋊C4—291no—
227C2×C22⋊C4—292no—
228C2×C22⋊C4—292no—
229C2×C22⋊C4—292no—
230C2×C22⋊C4—292no—
231C2×C22⋊C4—292no—
232C2×C22⋊C4—584no—
233C2×C22⋊C4C2×D2911yescharacteristic
234C2×C22⋊C4C2×D2912yescomplemented
235#95—292no—
236#95—292no—
237#95—292no—
238#95C2×D2912yescomplemented
239C29#240411yescharacteristic, complemented, Sylow
240C24—292no—
241C24—291no—
242C24—584no—
243C24—582no—
244C24—582no—
245C24—582no—
246C24—1162no—
247C24C22×D2911yescharacteristic
248C24C22×D2911yescharacteristic
249#41—5832no—
250C2×Q8—582no—
251C2×Q8—582no—
252C2×Q8—582no—
253C2×D4—22no—
254C2×D4—22no—
255C2×D4—294no—
256C2×D4—584no—
257C2×D4—584no—
258C2×D4—582no—
259C2×D4—582no—
260C2×D4—582no—
261C2×D4—582no—
262C2×D4—582no—
263C2×D4—584no—
264C2×D4—584no—
265C2×D4—1162no—
266C2×D4—1162no—
267C2×D4D11614yescomplemented
268C22×C4—22no—
269C22×C4—42no—
270C22×C4—294no—
271C22×C4—291no—
272C22×C4—291no—
273C22×C4—291no—
274C22×C4—292no—
275C22×C4—582no—
276C22×C4—582no—
277C22×C4—582no—
278C22×C4—582no—
279C22×C4—582no—
280C22×C4—582no—
281C22×C4—582no—
282C22×C4—582no—
283C22×C4C22×D2912yes—
284C22×C4C22×D2912yes—
285C22×C4C22×D2911yescharacteristic
286C22⋊C4—22no—
287C22⋊C4—42no—
288C22⋊C4—584no—
289C22⋊C4—584no—
290C22⋊C4—584no—
291C22⋊C4—582no—
292C22⋊C4—584no—
293C22⋊C4—1162no—
294C23—22no—
295C23—22no—
296C23—21no—
297C23—21no—
298C23—22no—
299C23—22no—
300C23—292no—
301C23—294no—
302C23—588no—
303C23—582no—
304C23—581no—
305C23—582no—
306C23—582no—
307C23—581no—
308C23—582no—
309C23—582no—
310C23—582no—
311C23—582no—
312C23—582no—
313C23—1168no—
314C23—1168no—
315C23—1164no—
316C23—1164no—
317C23C2×D11612yes—
318C23C2×D11612yes—
319C23D4×D2912yes—
320C23D4×D2912yes—
321C23C23×D2911yescharacteristic
322Q8—588no—
323D4—28no—
324D4—588no—
325D4—588no—
326D4—1168no—
327D4—1168no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—44no—
337C2×C4—292no—
338C2×C4—292no—
339C2×C4—292no—
340C2×C4—588no—
341C2×C4—588no—
342C2×C4—581no—
343C2×C4—581no—
344C2×C4—581no—
345C2×C4—581no—
346C2×C4—581no—
347C2×C4—581no—
348C2×C4—588no—
349C2×C4—582no—
350C2×C4—582no—
351C2×C4—582no—
352C2×C4C2×D11612yes—
353C22—24no—
354C22—24no—
355C22—22no—
356C22—22no—
357C22—24no—
358C22—22no—
359C22—42no—
360C22—41no—
361C22—41no—
362C22—584no—
363C22—584no—
364C22—584no—
365C22—582no—
366C22—584no—
367C22—11616no—
368C22—1162no—
369C22—1162no—
370C22—1162no—
371C22—1162no—
372C22—1161no—
373C22—1162no—
374C22—1161no—
375C22#1174524414yes—
376C22C22×D11611yescharacteristic
377C22C2×D4×D2911yescharacteristic
378C22C2×D4×D2911yescharacteristic
379C22#1174535912yes—
380C22#41×D2912yes—
381C4—22no—
382C4—42no—
383C4—42no—
384C4—582no—
385C4—582no—
386C4—582no—
387C4—1162no—
388C2—24no—
389C2—42no—
390C2—1164no—
391C2—1162no—
392C2C2×#1174524411yescharacteristic
393C2#1427167111yescharacteristic
394C2#1427172811yescharacteristic
395C2#1427180212yes—
396C2#1427180812yes—
subgroupG/H
center, FrattiniC23C23×D29
commutator, socle, cosocleC23×C29C24
FittingC29×#386C2
hypercenter#386D29
hypocenterC29#2404
LCSG ▷ C23×C29 ▷ C29
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. C22×#90807C2semidirect
2. C2×#11745244C2semidirect
3. C2×#11745299C2semidirect
4. C22×#90781C2semidirect
5. #14271076C2semidirect
6. #14271072C2semidirect
7. #14271042C2semidirect
8. #14271041C2semidirect
9. #14271040C2semidirect
10. #14271024C2semidirect
11. C29×#386C2semidirect
17. C22×C22⋊D29C22semidirect
19. C23×Dic29C22semidirect
24. C22×C4×D29C22semidirect
25. C22×Dic58C22semidirect
27. C2×C22⋊C4×C29C22semidirect
35. C2×#90781C22semidirect
37. C29×#95C22semidirect
38. #11745179C22semidirect
39. #11745179C22semidirect
40. #11745179C22semidirect
52. C2×D4×C29D4semidirect
66. C2×C22⋊D29D4semidirect
72. C22×Dic29D4semidirect
122. C22×D29#41semidirect
199. #386D29semidirect
234. C2×C22⋊C4C2×D29semidirect
238. #95C2×D29semidirect
239. C29#2404semidirect
267. C2×D4D116semidirect

Maximal subgroups

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

subgroupindex
C22×#908072
C2×#117452442
C2×#117452992
C22×#907812
#142710762
#142710722
#142710422
#142710412
#142710402
#142710242
C29×#3862
#240429