Finite Group Databasebeta

← #16829968

Subgroups — #16829968

Order 3712.

564 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 165 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×#11745271C211yesmaximal, characteristic, complemented
2C2×#11745270C211yesmaximal, characteristic, complemented
3C2×#11745266C211yesmaximal, characteristic, complemented
4C22×#90773C211yesmaximal, characteristic, complemented
5#14271739C211yesmaximal, characteristic, complemented
6#14271732C211yesmaximal, characteristic, complemented
7#14271557C212yesmaximal, complemented
8#14271552C212yesmaximal, complemented
9#14271547C212yesmaximal, complemented
10#14271536C212yesmaximal, complemented
11C29×#405C211yesmaximal, characteristic, complemented
12C2×#90805C2211yescharacteristic
13C2×#90805C2211yescharacteristic
14C22×D116C2211yescharacteristic
15C22×Dic58C2211yescharacteristic
16C2×M16×C29C2211yescharacteristic
17C22×C8×C29C2211yescharacteristic
18C29×#117C2211yescharacteristic
19#11745271—24no—
20#11745270—24no—
21C2×Dic116C2212yescomplemented
22C2×Dic116C2211yescharacteristic, complemented
23#11745266—24no—
24C2×D232C2212yes—
25C2×D232C2211yescharacteristic, complemented
26C2×#90773—24no—
27C2×#90773C2212yes—
28C2×#90773C2212yes—
29C2×#90773C2212yes—
30C2×#90773C2211yescharacteristic, complemented
31C2×#90773C2212yescomplemented
32C2×#90773C2211yescharacteristic, complemented
33#11745259—22no—
34#11745257—22no—
35#11745248—22no—
36#11745244—22no—
37#11745237C2212yes—
38#11745237C2212yescomplemented
39#11745236C2212yes—
40#11745231C2212yescomplemented
41C29×#97C2212yescomplemented
42C4×C8×C29C2212yescomplemented
43C23×D29—21no—
44C22×C4×C29C2311yescharacteristic
45C2×C22⋊D29—21no—
46C2×C22⋊D29—21no—
47C22×Dic29—21no—
48#90805—22no—
49#90805—22no—
50#90805—22no—
51#90805—22no—
52C2×D116—22no—
53C2×D116—22no—
54C2×D116—21no—
55C2×D116—24no—
56C2×D116C2312yes—
57C2×D116C2311yescharacteristic
58C2×D116C2311yescharacteristic
59C2×C4×D29—21no—
60C2×C4×D29—21no—
61C2×Dic58—22no—
62C2×Dic58—22no—
63C2×Dic58—21no—
64C2×Dic58C2312yes—
65C2×Dic58C2311yescharacteristic
66C2×Dic58C2311yescharacteristic
67M16×C29D414yescomplemented
68C2×C8×C29D414yescomplemented
69C2×C8×C29C2311yescharacteristic
70C2×C8×C29C2311yescharacteristic
71C2×C8×C29C2312yes—
72C4⋊C4×C29D412yes—
73C22⋊C4×C29D412yes—
74C42×C29C2312yes—
75#90781—24no—
76#90781—24no—
77C4⋊Dic29—24no—
78Dic116—22no—
79Dic116—24no—
80D232—24no—
81D232—22no—
82#90773—24no—
83#90773—24no—
84#90773—24no—
85#90773—22no—
86#90773—22no—
87#90773—24no—
88C23×C29C2×D411yescharacteristic
89C22×D29—21no—
90C22×D29—22no—
91C22×D29—21no—
92C22×D29—42no—
93C22×D29—82no—
94C2×C4×C29C2×D414yes—
95C2×C4×C29C2×D411yescharacteristic
96C2×C4×C29C2×D412yes—
97C2×C4×C29C2×D412yes—
98C2×C4×C29C2411yescharacteristic
99C22⋊D29—42no—
100C22⋊D29—42no—
101C2×Dic29—21no—
102C2×Dic29—22no—
103C2×Dic29—21no—
104C2×Dic29—42no—
105D116—22no—
106D116—21no—
107D116—22no—
108D116—21no—
109D116—21no—
110D116—21no—
111D116—42no—
112D116—84no—
113C4×D29—42no—
114C4×D29—42no—
115Dic58—22no—
116Dic58—21no—
117Dic58—22no—
118Dic58—21no—
119Dic58—21no—
120Dic58—21no—
121Dic58—42no—
122C8×C29C2×D412yes—
123C8×C29C2×D412yes—
124C8×C29C2×D414yes—
125#3154—291nomaximal, Sylow
126C22×C29—22no—
127C22×C29#12712yes—
128C22×C29C22×D411yescharacteristic
129C2×D29—41no—
130C2×D29—41no—
131C2×D29—42no—
132C2×D29—42no—
133C2×D29—44no—
134C2×D29—42no—
135C2×D29—84no—
136C4×C29—24no—
137C4×C29#12712yes—
138C4×C29C22×D411yescharacteristic
139C4×C29C22×D411yescharacteristic
140Dic29—42no—
141Dic29—41no—
142Dic29—41no—
143C2×#137—291no—
144C2×AGL(1,ℤ/8)—291no—
145C2×#135—291no—
146C22×SD16—291no—
147#536—291no—
148#535—291no—
149#497—292no—
150#496—292no—
151#495—292no—
152#492—292no—
153#405D2911yescharacteristic, complemented
154C2×C29—22no—
155C2×C29C2×#12711yescharacteristic
156C2×C29#57712yes—
157D29—81no—
158D29—81no—
159D29—82no—
160C2×#41—291no—
161C2×#41—291no—
162C22×Q8—291no—
163C22×D4—291no—
164#137—584no—
165AGL(1,ℤ/8)—584no—
166#135—584no—
167C2×Q16—291no—
168C2×Q16—292no—
169C2×SD16—292no—
170C2×SD16—292no—
171C2×SD16—291no—
172C2×SD16—291no—
173C2×SD16—292no—
174C2×SD16—292no—
175C2×SD16—584no—
176C2×D8—291no—
177C2×D8—292no—
178C2×M16C2×D2911yescharacteristic
179C22×C8C2×D2911yescharacteristic
180C4⋊Q8—292no—
181#127—292no—
182#124—292no—
183#124—292no—
184#123—582no—
185C22⋊Q8—582no—
186#121—582no—
187#120—582no—
188#117C2×D2911yescharacteristic
189#97C2×D2912yescomplemented
190C4×C8C2×D2912yescomplemented
191C29#315411yescharacteristic, complemented, Sylow
192C24—581no—
193#41—582no—
194#41—582no—
195#41—582no—
196#41—582no—
197C2×Q8—291no—
198C2×Q8—292no—
199C2×Q8—291no—
200C2×Q8—581no—
201C2×Q8—582no—
202C2×Q8—582no—
203C2×D4—291no—
204C2×D4—292no—
205C2×D4—291no—
206C2×D4—581no—
207C2×D4—581no—
208C2×D4—582no—
209C2×D4—582no—
210C2×D4—581no—
211C2×D4—584no—
212C22×C4—581no—
213C22×C4—581no—
214C22×C4—581no—
215C22×C4C22×D2911yescharacteristic
216Q16—582no—
217Q16—584no—
218SD16—584no—
219SD16—584no—
220SD16—582no—
221SD16—582no—
222SD16—584no—
223SD16—584no—
224D8—582no—
225D8—584no—
226M16D11614yescomplemented
227C2×C8D11614yescomplemented
228C2×C8C22×D2911yescharacteristic
229C2×C8C22×D2912yes—
230C2×C8C22×D2911yescharacteristic
231C4⋊C4—584no—
232C4⋊C4D11612yes—
233C22⋊C4—584no—
234C22⋊C4—584no—
235C22⋊C4D11612yes—
236C42C22×D2912yes—
237C23—581no—
238C23—582no—
239C23—581no—
240C23—1162no—
241C23—2322no—
242C23C2×D11611yescharacteristic
243Q8—582no—
244Q8—581no—
245Q8—582no—
246Q8—581no—
247Q8—581no—
248Q8—581no—
249Q8—1162no—
250D4—581no—
251D4—582no—
252D4—581no—
253D4—581no—
254D4—582no—
255D4—581no—
256D4—1162no—
257D4—1162no—
258D4—1162no—
259D4—2324no—
260C2×C4—581no—
261C2×C4—582no—
262C2×C4—581no—
263C2×C4—1162no—
264C2×C4—1162no—
265C2×C4—1162no—
266C2×C4C2×D11614yes—
267C2×C4C2×D11612yes—
268C2×C4C2×D11612yes—
269C2×C4C2×D11611yescharacteristic
270C2×C4C23×D2911yescharacteristic
271C8C2×D11612yes—
272C8C2×D11612yes—
273C8C2×D11614yes—
274C22—22no—
275C22—1161no—
276C22—1161no—
277C22—1162no—
278C22—1162no—
279C22—1164no—
280C22—1162no—
281C22—2324no—
282C22#1174523612yes—
283C22C22×D11611yescharacteristic
284C4—24no—
285C4—1161no—
286C4—1161no—
287C4—1162no—
288C4#1174523612yes—
289C4C22×D11611yescharacteristic
290C4C22×D11611yescharacteristic
291C2—22no—
292C2—2322no—
293C2—2321no—
294C2—2321no—
295C2C2×#1174523611yescharacteristic
296C2#1427197512yes—
subgroupG/H
centerC22C22×D116
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC29×#405C2
socleC22×C29C22×D4
hypercenter#405D29
hypocenterC29#3154
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ #117 ◁ #405

Complements

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

NH ≅ G/Ntype
1. C2×#11745271C2semidirect
2. C2×#11745270C2semidirect
3. C2×#11745266C2semidirect
4. C22×#90773C2semidirect
5. #14271739C2semidirect
6. #14271732C2semidirect
7. #14271557C2semidirect
8. #14271552C2semidirect
9. #14271547C2semidirect
10. #14271536C2semidirect
11. C29×#405C2semidirect
21. C2×Dic116C22semidirect
22. C2×Dic116C22semidirect
25. C2×D232C22semidirect
30. C2×#90773C22semidirect
31. C2×#90773C22semidirect
32. C2×#90773C22semidirect
38. #11745237C22semidirect
40. #11745231C22semidirect
41. C29×#97C22semidirect
42. C4×C8×C29C22semidirect
67. M16×C29D4semidirect
68. C2×C8×C29D4semidirect
153. #405D29semidirect
189. #97C2×D29semidirect
190. C4×C8C2×D29semidirect
191. C29#3154semidirect
226. M16D116semidirect
227. C2×C8D116semidirect

Maximal subgroups

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

subgroupindex
C2×#117452712
C2×#117452702
C2×#117452662
C22×#907732
#142717392
#142717322
#142715572
#142715522
#142715472
#142715362
C29×#4052
#315429