Finite Group Databasebeta

← #16828747

Subgroups — #16828747

Order 3712.

754 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 754 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C22×#90807C211yesmaximal, characteristic
2C2×#11745277C212yesmaximal, complemented
3C22⋊Q8×D29C211yesmaximal, characteristic, complemented
4#14271326C212yesmaximal, complemented
5#14271322C212yesmaximal, complemented
6#14271321C211yesmaximal, characteristic, complemented
7#14271318C211yesmaximal, characteristic, complemented
8#14271314C211yesmaximal, characteristic, complemented
9#14271161C212yesmaximal, complemented
10#14271125C211yesmaximal, characteristic, complemented
11C29×#450C211yesmaximal, characteristic, complemented
12C22×D4×C29C2211yescharacteristic
13C22×C22⋊D29—21no—
14C23×Dic29—21no—
15C2×Q8×D29C2211yescharacteristic, complemented
16C2×#90807—28no—
17C2×#90807—21no—
18C2×#90807—21no—
19C2×#90807C2212yes—
20C2×#90807C2212yes—
21C22×C4×D29C2211yescharacteristic
22C22×Dic58C2211yescharacteristic
23C2×SD16×C29C2212yescomplemented
24C29×C22⋊Q8C2211yescharacteristic
25#11745308C2211yescharacteristic, complemented
26C2×#90785C2212yescomplemented
27C2×#90783C2212yescomplemented
28#11745286C2211yescharacteristic
29#11745277—28no—
30C2×Dic116C2212yescomplemented
31C2×#90772C2212yescomplemented
32#11745259C2211yescharacteristic, complemented
33#11745258—21no—
34C4⋊C4×D29—21no—
35C4⋊C4×D29C2211yescharacteristic, complemented
36C22⋊C4×D29—21no—
37#11745240—21no—
38C29×#102C2212yescomplemented
39C29×C22⋊C8C2211yescharacteristic
40#11745180C2212yescomplemented
41C22⋊#6597C2211yescharacteristic
42#11745168C2212yescomplemented
43#11745164C2212yescomplemented
44#11745158C2212yescomplemented
45C24×C29—21no—
46C23×D29D411yescharacteristic
47C2×Q8×C29C2311yescharacteristic
48C2×D4×C29—24no—
49C2×D4×C29—21no—
50C2×D4×C29C2312yes—
51C22×C4×C29C2311yescharacteristic
52C2×C22⋊D29—28no—
53C2×C22⋊D29—22no—
54C2×C22⋊D29—22no—
55C22×Dic29—28no—
56C22×Dic29—21no—
57C22×Dic29—22no—
58C22×Dic29—22no—
59C22×Dic29D411yescharacteristic
60Q8×D29—24no—
61#90807—28no—
62#90807—28no—
63#90807—24no—
64#90807—24no—
65#90807—24no—
66#90807D418yes—
67C2×C4×D29—24no—
68C2×C4×D29—21no—
69C2×C4×D29—21no—
70C2×C4×D29D411yescharacteristic
71C2×C4×D29D411yescharacteristic
72C2×C4×D29C2311yescharacteristic
73C2×C4×D29C2311yescharacteristic
74C2×Dic58—21no—
75C2×Dic58—24no—
76C2×Dic58—21no—
77C2×Dic58C2311yescharacteristic
78C2×Dic58C2312yes—
79SD16×C29—24no—
80C2×C8×C29C2312yes—
81C4⋊C4×C29—21no—
82C4⋊C4×C29C2311yescharacteristic
83C22⋊C4×C29—21no—
84C22⋊Dic29—21no—
85#90785—24no—
86#90783—24no—
87#90781—21no—
88#90781—21no—
89C4⋊Dic29—21no—
90C4⋊Dic29C2311yescharacteristic
91#90779—21no—
92#90779—21no—
93#90779—21no—
94C2×#6597C2312yes—
95Dic116—24no—
96#90772—24no—
97C23×C29—24no—
98C23×C29—22no—
99C23×C29—42no—
100C23×C29—42no—
101C23×C29C2×D411yescharacteristic
102C22×D29—22no—
103C22×D29—22no—
104C22×D29—41no—
105C22×D29C2×D411yescharacteristic
106C22×D29C2×D411yescharacteristic
107Q8×C29—22no—
108D4×C29—24no—
109D4×C29—22no—
110D4×C29C2×D414yes—
111C2×C4×C29—22no—
112C2×C4×C29—21no—
113C2×C4×C29—21no—
114C2×C4×C29C2×D411yescharacteristic
115C2×C4×C29C2411yescharacteristic
116C22⋊D29—28no—
117C22⋊D29—28no—
118C2×Dic29—24no—
119C2×Dic29—28no—
120C2×Dic29—22no—
121C2×Dic29—22no—
122C2×Dic29—28no—
123C2×Dic29—21no—
124C2×Dic29—21no—
125C2×Dic29C2×D411yescharacteristic
126C2×Dic29C2×D411yescharacteristic
127C4×D29—22no—
128C4×D29—22no—
129C4×D29—22no—
130C4×D29—42no—
131C4×D29—42no—
132C4×D29C2×D414yes—
133Dic58—24no—
134Dic58—22no—
135Dic58—22no—
136Dic58—42no—
137Dic58C2×D414yes—
138C8×C29—22no—
139#6597—22no—
140#3007—291nomaximal, Sylow
141C22×C29—21no—
142C22×C29—21no—
143C22×C29—24no—
144C22×C29—48no—
145C22×C29—42no—
146C22×C29—42no—
147C22×C29#13712yescomplemented
148C22×C29C22×D411yescharacteristic
149C2×D29—21no—
150C2×D29—21no—
151C2×D29—21no—
152C2×D29—21no—
153C2×D29—21no—
154C2×D29—44no—
155C2×D29#12012yes—
156C4×C29—21no—
157C4×C29—41no—
158C4×C29—41no—
159C4×C29C22×D412yes—
160Dic29—24no—
161Dic29—21no—
162Dic29—41no—
163Dic29—41no—
164Dic29#12012yes—
165C22×#41—291no—
166C2×#137—292no—
167C2×C22⋊Q8—291no—
168#452—292no—
169#452—292no—
170#451—291no—
171#451—291no—
172#450—291no—
173#450D2911yescharacteristic, complemented
174#417—292no—
175#407—291no—
176C2×C29—22no—
177C2×C29—44no—
178C2×C29C2×#12011yescharacteristic
179C2×C29C2×#13711yescharacteristic
180C2×C29#57511yescharacteristic
181D29—22no—
182D29—41no—
183C2×#41—292no—
184C2×#41—292no—
185C2×#41—581no—
186C2×#41—581no—
187C2×#41—588no—
188C22×Q8—291no—
189C22×Q8—291no—
190C22×D4—581no—
191C22×D4C2×D2911yescharacteristic
192C23×C4—291no—
193C23×C4—581no—
194#137—588no—
195C2×Q16—292no—
196C2×Q16—292no—
197C2×SD16—292no—
198C2×SD16C2×D2912yescomplemented
199C2×M16—292no—
200C22⋊Q8—291no—
201C22⋊Q8—291no—
202C22⋊Q8—291no—
203C22⋊Q8—581no—
204C22⋊Q8—581no—
205C22⋊Q8C2×D2911yescharacteristic
206C2×C4⋊C4—291no—
207C2×C4⋊C4—581no—
208C2×C22⋊C4—581no—
209#103—292no—
210#103—292no—
211#102—292no—
212#102C2×D2912yescomplemented
213C22⋊C8—291no—
214C22⋊C8—292no—
215C22⋊C8C2×D2911yescharacteristic
216C29#300711yescharacteristic, complemented, Sylow
217C24—21no—
218C24—291no—
219#41—298no—
220#41—584no—
221#41—584no—
222#41—588no—
223#41—588no—
224#41—584no—
225C2×Q8—292no—
226C2×Q8—291no—
227C2×Q8—581no—
228C2×Q8—584no—
229C2×Q8—584no—
230C2×Q8—581no—
231C2×Q8C22×D2911yescharacteristic
232C2×D4—24no—
233C2×D4—21no—
234C2×D4—582no—
235C2×D4—582no—
236C2×D4—588no—
237C2×D4C22×D2912yes—
238C22×C4—291no—
239C22×C4—291no—
240C22×C4—291no—
241C22×C4—291no—
242C22×C4—291no—
243C22×C4—584no—
244C22×C4—581no—
245C22×C4—581no—
246C22×C4—582no—
247C22×C4—582no—
248C22×C4—588no—
249C22×C4—581no—
250C22×C4C22×D2911yescharacteristic
251Q16—584no—
252Q16—584no—
253SD16—24no—
254SD16—584no—
255M16—584no—
256C2×C8—292no—
257C2×C8C22×D2912yes—
258C4⋊C4—21no—
259C4⋊C4—291no—
260C4⋊C4—581no—
261C4⋊C4—581no—
262C4⋊C4—581no—
263C4⋊C4—581no—
264C4⋊C4C22×D2911yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—581no—
267C22⋊C4—581no—
268C22⋊C4—581no—
269C23—24no—
270C23—22no—
271C23—42no—
272C23—42no—
273C23—291no—
274C23—291no—
275C23—582no—
276C23—582no—
277C23—1161no—
278C23D4×D2911yescharacteristic
279Q8—22no—
280Q8—294no—
281Q8—584no—
282Q8—582no—
283Q8—582no—
284Q8—1162no—
285D4—24no—
286D4—22no—
287D4—588no—
288D4—588no—
289D4D4×D2914yes—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—291no—
294C2×C4—294no—
295C2×C4—291no—
296C2×C4—582no—
297C2×C4—582no—
298C2×C4—582no—
299C2×C4—588no—
300C2×C4—584no—
301C2×C4—588no—
302C2×C4—582no—
303C2×C4—582no—
304C2×C4—581no—
305C2×C4—581no—
306C2×C4—1162no—
307C2×C4—1162no—
308C2×C4D4×D2911yescharacteristic
309C2×C4C23×D2911yescharacteristic
310C8—22no—
311C8—582no—
312C22—21no—
313C22—21no—
314C22—24no—
315C22—48no—
316C22—42no—
317C22—42no—
318C22—292no—
319C22—581no—
320C22—581no—
321C22—581no—
322C22—581no—
323C22—581no—
324C22—1164no—
325C22#1174527712yescomplemented
326C22C2×D4×D2911yescharacteristic
327C4—21no—
328C4—41no—
329C4—41no—
330C4—292no—
331C4—581no—
332C4—584no—
333C4—1161no—
334C4—1161no—
335C4C2×D4×D2912yes—
336C2—22no—
337C2—44no—
338C2—582no—
339C2—1161no—
340C2#120×D2911yescharacteristic
341C2C2×#1174527711yescharacteristic
342C2#1427195911yescharacteristic
subgroupG/H
centerC22C2×D4×D29
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC29×#450C2
socleC22×C29C22×D4
prime radicalC22×#90807C2
hypercenter#450D29
hypocenterC29#3007
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
2. C2×#11745277C2semidirect
3. C22⋊Q8×D29C2semidirect
4. #14271326C2semidirect
5. #14271322C2semidirect
6. #14271321C2semidirect
7. #14271318C2semidirect
8. #14271314C2semidirect
9. #14271161C2semidirect
10. #14271125C2semidirect
11. C29×#450C2semidirect
15. C2×Q8×D29C22semidirect
23. C2×SD16×C29C22semidirect
25. #11745308C22semidirect
26. C2×#90785C22semidirect
27. C2×#90783C22semidirect
30. C2×Dic116C22semidirect
31. C2×#90772C22semidirect
32. #11745259C22semidirect
35. C4⋊C4×D29C22semidirect
38. C29×#102C22semidirect
40. #11745180C22semidirect
42. #11745168C22semidirect
43. #11745164C22semidirect
44. #11745158C22semidirect
147. C22×C29#137semidirect
173. #450D29semidirect
198. C2×SD16C2×D29semidirect
212. #102C2×D29semidirect
216. C29#3007semidirect
325. C22#11745277semidirect

Maximal subgroups

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

subgroupindex
C22×#908072
C2×#117452772
C22⋊Q8×D292
#142713262
#142713222
#142713212
#142713182
#142713142
#142711612
#142711252
C29×#4502
#300729