Finite Group Databasebeta

← #16828798

Subgroups — #16828798

Order 3712.

764 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 149 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11745364C211yesmaximal, characteristic, complemented
2C2×#11745270C211yesmaximal, characteristic, complemented
3C2×#11745312C211yesmaximal, characteristic, complemented
4#14271744C211yesmaximal, characteristic, complemented
5#14271328C212yesmaximal, complemented
6#14271315C212yesmaximal, complemented
7#14271311C212yesmaximal, complemented
8#14271298C212yesmaximal, complemented
9C2×#11745157C211yesmaximal, characteristic, complemented
10#14271131C211yesmaximal, characteristic, complemented
11C29×#417C211yesmaximal, characteristic, complemented
12C2×#41×C29C2211yescharacteristic
13#11745364—24no—
14#11745364—24no—
15C2×#90809C2212yes—
16C2×D4×D29—22no—
17C2×D4×D29C2212yes—
18C2×#90805—21no—
19C2×#90805C2211yescharacteristic
20C2×#90805C2211yescharacteristic, complemented
21C22×D116—21no—
22C22×D116C2211yescharacteristic
23C2×M16×C29C2211yescharacteristic, complemented
24C2×C4⋊C4×C29C2211yescharacteristic, complemented
25#11745312—24no—
26C2×#90785C2211yescharacteristic, complemented
27C2×#90784C2211yescharacteristic, complemented
28C2×#90783C2211yescharacteristic, complemented
29C2×#90782C2211yescharacteristic, complemented
30#11745292—22no—
31C22×#6597C2211yescharacteristic, complemented
32#11745270—24no—
33C2×D232C2212yescomplemented
34C2×#90773C2212yescomplemented
35#11745259—21no—
36#11745259C2212yescomplemented
37#11745257—21no—
38#11745257C2212yescomplemented
39C29×#103C2212yescomplemented
40C29×#102C2212yescomplemented
41#11745168C2212yescomplemented
42#11745168C2212yescomplemented
43#11745157C2212yescomplemented
44#11745157C2212yescomplemented
45C23×D29—21no—
46C23×D29—22no—
47#41×C29—22no—
48#41×C29D414yescomplemented
49C2×Q8×C29C2311yescharacteristic
50C2×D4×C29—21no—
51C2×D4×C29C2311yescharacteristic
52C22×C4×C29—21no—
53C22×C4×C29—21no—
54C22×C4×C29C2311yescharacteristic
55C2×C22⋊D29—21no—
56C2×C22⋊D29—22no—
57C2×C22⋊D29D412yes—
58#90809—24no—
59#90809—24no—
60D4×D29—216no—
61D4×D29—24no—
62D4×D29—24no—
63#90805—28no—
64#90805—22no—
65#90805—22no—
66#90805—22no—
67#90805—22no—
68C2×D116—28no—
69C2×D116—22no—
70C2×D116—21no—
71C2×D116—22no—
72C2×D116—22no—
73C2×D116D411yescharacteristic
74C2×D116D414yescomplemented
75C2×D116C2311yescharacteristic
76C2×D116C2312yes—
77C2×C4×D29—22no—
78C2×C4×D29—21no—
79C2×C4×D29C2312yes—
80C2×Dic58D411yescharacteristic
81C2×Dic58C2311yescharacteristic
82M16×C29—22no—
83C2×C8×C29C2312yes—
84C4⋊C4×C29—21no—
85C4⋊C4×C29C2312yes—
86#90785—22no—
87#90784—22no—
88#90783—22no—
89#90782—22no—
90#90781—22no—
91#90781—22no—
92C4⋊Dic29—22no—
93C2×#6597—22no—
94C2×#6597C2311yescharacteristic
95C2×#6597C2311yescharacteristic
96D232—24no—
97#90773—24no—
98C23×C29—21no—
99C23×C29C2×D411yescharacteristic
100C22×D29—22no—
101C22×D29—28no—
102C22×D29—24no—
103C22×D29—21no—
104C22×D29—82no—
105C22×D29—82no—
106C22×D29—81no—
107C22×D29—81no—
108C22×D29C2×D412yes—
109Q8×C29—21no—
110Q8×C29C2×D412yes—
111D4×C29—24no—
112D4×C29—21no—
113D4×C29C2×D412yes—
114C2×C4×C29—21no—
115C2×C4×C29—24no—
116C2×C4×C29—22no—
117C2×C4×C29—81no—
118C2×C4×C29C2×D412yes—
119C2×C4×C29C2×D411yescharacteristic
120C2×C4×C29C2×D412yes—
121C2×C4×C29C2411yescharacteristic
122C22⋊D29—28no—
123C22⋊D29—24no—
124C22⋊D29—42no—
125C2×Dic29—21no—
126C2×Dic29C2×D412yes—
127D116—28no—
128D116—22no—
129D116—24no—
130D116—22no—
131D116—21no—
132D116—21no—
133D116—82no—
134D116C2×D414yes—
135C4×D29—28no—
136C4×D29—22no—
137C4×D29—22no—
138C4×D29—42no—
139Dic58—22no—
140Dic58—21no—
141Dic58—21no—
142C8×C29—22no—
143#6597—21no—
144#6597—21no—
145#3012—291nomaximal, Sylow
146C22×C29—21no—
147C22×C29—21no—
148C22×C29—22no—
149C22×C29—41no—
150C22×C29—41no—
151C22×C29#12012yes—
152C22×C29C22×D411yescharacteristic
153C2×D29—22no—
154C2×D29—24no—
155C2×D29—22no—
156C2×D29—22no—
157C2×D29—42no—
158C2×D29—42no—
159C2×D29—41no—
160C2×D29—41no—
161C2×D29—41no—
162C2×D29—84no—
163C2×D29—88no—
164C4×C29—22no—
165C4×C29—42no—
166C4×C29#12012yes—
167C4×C29C22×D411yescharacteristic
168C4×C29C22×D411yescharacteristic
169Dic29—22no—
170Dic29—41no—
171C2×H(5,𝔽2)—291no—
172C2×AGL(1,ℤ/8)—291no—
173C2×#135—291no—
174#537—291no—
175#452—292no—
176#450—292no—
177#449—292no—
178#447—292no—
179#417D2911yescharacteristic, complemented
180C2×#102—291no—
181#408—291no—
182C2×C29—22no—
183C2×C29—42no—
184C2×C29C2×#12011yescharacteristic
185C2×C29#57611yescharacteristic
186C2×C29#57711yescharacteristic
187D29—42no—
188D29—44no—
189D29—81no—
190H(5,𝔽2)—584no—
191H(5,𝔽2)—584no—
192C2×#41—291no—
193C2×#41—292no—
194C2×#41—291no—
195C2×#41—581no—
196C2×#41C2×D2911yescharacteristic
197C22×D4—292no—
198C22×D4—291no—
199C22×D4—581no—
200C22×D4—582no—
201AGL(1,ℤ/8)—584no—
202#135—584no—
203C2×Q16—291no—
204C2×SD16—291no—
205C2×SD16—291no—
206C2×SD16—292no—
207C2×D8—291no—
208C2×D8—292no—
209C2×M16C2×D2911yescharacteristic, complemented
210C22×C8—291no—
211C22⋊Q8—292no—
212C22⋊Q8—581no—
213#121—292no—
214#121—581no—
215#121—582no—
216C2×C4⋊C4C2×D2911yescharacteristic, complemented
217#103C2×D2912yescomplemented
218#102—292no—
219#102—292no—
220#102C2×D2912yescomplemented
221C22⋊C8—292no—
222C22⋊C8—292no—
223C29#301211yescharacteristic, complemented, Sylow
224C24—582no—
225C24—581no—
226#41—22no—
227#41—588no—
228#41—584no—
229#41—584no—
230#41—582no—
231#41—582no—
232#41—582no—
233#41—582no—
234#41D11614yescomplemented
235C2×Q8—291no—
236C2×Q8—291no—
237C2×Q8C22×D2911yescharacteristic
238C2×D4—21no—
239C2×D4—292no—
240C2×D4—294no—
241C2×D4—291no—
242C2×D4—292no—
243C2×D4—291no—
244C2×D4—582no—
245C2×D4—588no—
246C2×D4—582no—
247C2×D4—582no—
248C2×D4—584no—
249C2×D4—5816no—
250C2×D4—584no—
251C2×D4—581no—
252C2×D4—582no—
253C2×D4—581no—
254C2×D4C22×D2911yescharacteristic
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—292no—
258C22×C4—582no—
259C22×C4—581no—
260C22×C4C22×D2911yescharacteristic
261Q16—582no—
262SD16—584no—
263SD16—582no—
264SD16—582no—
265D8—584no—
266D8—582no—
267M16—22no—
268C2×C8—291no—
269C2×C8—291no—
270C2×C8—582no—
271C2×C8C22×D2912yes—
272C4⋊C4—21no—
273C4⋊C4—582no—
274C4⋊C4C22×D2912yes—
275C22⋊C4—582no—
276C22⋊C4—582no—
277C23—21no—
278C23—292no—
279C23—588no—
280C23—582no—
281C23—584no—
282C23—581no—
283C23—2321no—
284C23—2321no—
285C23—2322no—
286C23—2322no—
287C23D4×D2911yescharacteristic
288Q8—21no—
289Q8—582no—
290Q8—581no—
291Q8—581no—
292Q8C2×D11612yes—
293D4—24no—
294D4—21no—
295D4—294no—
296D4—588no—
297D4—588no—
298D4—584no—
299D4—582no—
300D4—581no—
301D4—581no—
302D4—584no—
303D4—582no—
304D4—1162no—
305D4—2322no—
306D4C2×D11612yes—
307C2×C4—21no—
308C2×C4—24no—
309C2×C4—22no—
310C2×C4—81no—
311C2×C4—292no—
312C2×C4—588no—
313C2×C4—582no—
314C2×C4—582no—
315C2×C4—581no—
316C2×C4—1162no—
317C2×C4C2×D11612yes—
318C2×C4D4×D2912yes—
319C2×C4D4×D2911yescharacteristic
320C2×C4C23×D2911yescharacteristic
321C8—22no—
322C8—581no—
323C8—581no—
324C22—21no—
325C22—21no—
326C22—22no—
327C22—41no—
328C22—41no—
329C22—584no—
330C22—582no—
331C22—582no—
332C22—582no—
333C22—1162no—
334C22—1162no—
335C22—1161no—
336C22—1161no—
337C22—1161no—
338C22—2324no—
339C22—2328no—
340C22#1174524412yes—
341C22C2×D4×D2911yescharacteristic
342C4—22no—
343C4—42no—
344C4—582no—
345C4—1161no—
346C4#1174524412yes—
347C4C22×D11611yescharacteristic
348C4C2×D4×D2911yescharacteristic
349C2—22no—
350C2—42no—
351C2—1162no—
352C2—1164no—
353C2—2321no—
354C2C2×#1174524411yescharacteristic
355C2#1427196711yescharacteristic
356C2#1427197211yescharacteristic
subgroupG/H
centerC22C2×D4×D29
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC29×#417C2
socleC22×C29C22×D4
hypercenter#417D29
hypocenterC29#3012
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. C2×#11745364C2semidirect
2. C2×#11745270C2semidirect
3. C2×#11745312C2semidirect
4. #14271744C2semidirect
5. #14271328C2semidirect
6. #14271315C2semidirect
7. #14271311C2semidirect
8. #14271298C2semidirect
9. C2×#11745157C2semidirect
10. #14271131C2semidirect
11. C29×#417C2semidirect
20. C2×#90805C22semidirect
23. C2×M16×C29C22semidirect
24. C2×C4⋊C4×C29C22semidirect
26. C2×#90785C22semidirect
27. C2×#90784C22semidirect
28. C2×#90783C22semidirect
29. C2×#90782C22semidirect
31. C22×#6597C22semidirect
33. C2×D232C22semidirect
34. C2×#90773C22semidirect
36. #11745259C22semidirect
38. #11745257C22semidirect
39. C29×#103C22semidirect
40. C29×#102C22semidirect
41. #11745168C22semidirect
42. #11745168C22semidirect
43. #11745157C22semidirect
44. #11745157C22semidirect
48. #41×C29D4semidirect
74. C2×D116D4semidirect
179. #417D29semidirect
209. C2×M16C2×D29semidirect
216. C2×C4⋊C4C2×D29semidirect
217. #103C2×D29semidirect
220. #102C2×D29semidirect
223. C29#3012semidirect
234. #41D116semidirect

Maximal subgroups

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

subgroupindex
C2×#117453642
C2×#117452702
C2×#117453122
#142717442
#142713282
#142713152
#142713112
#142712982
C2×#117451572
#142711312
C29×#4172
#301229