Finite Group Databasebeta

← #16828763

Subgroups — #16828763

Order 3712.

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

normal subgroups only

This group has 784 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×#11745311C211yesmaximal, characteristic, complemented
3C22×D232C211yesmaximal, characteristic, complemented
4#14271732C211yesmaximal, characteristic, complemented
5#14271311C214yesmaximal, complemented
6#14271298C214yesmaximal, complemented
7#14271167C211yesmaximal, characteristic, complemented
8#14271129C211yesmaximal, characteristic, complemented
9C29×#416C211yesmaximal, characteristic, complemented
10C2×#41×C29C2211yescharacteristic
11#11745364—24no—
12#11745364—24no—
13C2×#90809C2212yescomplemented
14C2×D4×D29—22no—
15C2×D4×D29C2212yescomplemented
16C2×#90805—21no—
17C2×#90805C2211yescharacteristic
18C22×D116—21no—
19C22×D116C2211yescharacteristic
20C22×D116C2211yescharacteristic, complemented
21C22×C8×C29C2211yescharacteristic, complemented
22C29×#117C2211yescharacteristic, complemented
23#11745311—24no—
24C2×#90784C2212yescomplemented
25C2×#90782C2212yescomplemented
26C2×#90777C2211yescharacteristic, complemented
27C2×D232—24no—
28C2×D232C2214yescomplemented
29#11745257C2214yescomplemented
30#11745244—22no—
31#11745237—21no—
32#11745236—21no—
33C29×#103C2212yescomplemented
34C29×#102C2212yescomplemented
35#11745168C2214yescomplemented
36#11745157C2214yescomplemented
37C23×D29—21no—
38C23×D29—22no—
39C23×D29—21no—
40#41×C29—22no—
41#41×C29D414yescomplemented
42C2×Q8×C29C2311yescharacteristic
43C2×D4×C29—21no—
44C2×D4×C29C2311yescharacteristic
45C22×C4×C29—21no—
46C22×C4×C29C2311yescharacteristic
47C2×C22⋊D29—22no—
48C2×C22⋊D29D412yes—
49#90809—24no—
50#90809—24no—
51D4×D29—216no—
52D4×D29—24no—
53D4×D29—24no—
54#90805—28no—
55#90805—22no—
56#90805—22no—
57C2×D116—28no—
58C2×D116—22no—
59C2×D116—21no—
60C2×D116—22no—
61C2×D116—22no—
62C2×D116—22no—
63C2×D116—21no—
64C2×D116—24no—
65C2×D116D411yescharacteristic
66C2×D116D414yescomplemented
67C2×D116C2312yes—
68C2×D116C2312yes—
69C2×C4×D29—22no—
70C2×C4×D29C2312yes—
71C2×Dic58D411yescharacteristic
72C2×C8×C29—22no—
73C2×C8×C29C2312yes—
74C4⋊C4×C29C2312yes—
75C22⋊C4×C29—21no—
76C42×C29—21no—
77#90784—24no—
78#90782—24no—
79#90781—24no—
80#90777—22no—
81C2×#6597C2312yes—
82D232—28no—
83C23×C29—21no—
84C23×C29C2×D411yescharacteristic
85C22×D29—22no—
86C22×D29—28no—
87C22×D29—24no—
88C22×D29—22no—
89C22×D29—42no—
90C22×D29—84no—
91C22×D29—82no—
92C22×D29—82no—
93C22×D29C2×D412yes—
94Q8×C29—21no—
95Q8×C29C2×D412yes—
96D4×C29—24no—
97D4×C29—21no—
98D4×C29C2×D412yes—
99C2×C4×C29—21no—
100C2×C4×C29—24no—
101C2×C4×C29—22no—
102C2×C4×C29C2×D412yes—
103C2×C4×C29C2×D411yescharacteristic
104C2×C4×C29C2×D412yes—
105C2×C4×C29C2411yescharacteristic
106C22⋊D29—28no—
107C22⋊D29—24no—
108C2×Dic29C2×D412yes—
109D116—28no—
110D116—22no—
111D116—24no—
112D116—22no—
113D116—22no—
114D116—22no—
115D116—42no—
116D116—84no—
117D116C2×D414yes—
118C4×D29—28no—
119C4×D29—22no—
120C4×D29—22no—
121Dic58—22no—
122C8×C29—22no—
123#6597—22no—
124#3008—291nomaximal, Sylow
125C22×C29—22no—
126C22×C29—22no—
127C22×C29—42no—
128C22×C29#12012yes—
129C22×C29C22×D411yescharacteristic
130C2×D29—22no—
131C2×D29—24no—
132C2×D29—24no—
133C2×D29—44no—
134C2×D29—42no—
135C2×D29—44no—
136C2×D29—84no—
137C2×D29—88no—
138C2×D29—84no—
139C4×C29—22no—
140C4×C29—42no—
141C4×C29#12012yes—
142C4×C29C22×D411yescharacteristic
143C4×C29C22×D411yescharacteristic
144Dic29—22no—
145C2×H(5,𝔽2)—291no—
146C2×AGL(1,ℤ/8)—291no—
147C22×D8—291no—
148#535—291no—
149#449—294no—
150#447—294no—
151#418—291no—
152#416D2911yescharacteristic, complemented
153#408—291no—
154C2×C29—22no—
155C2×C29—42no—
156C2×C29C2×#12011yescharacteristic
157C2×C29#57612yes—
158D29—42no—
159D29—44no—
160D29—82no—
161H(5,𝔽2)—584no—
162H(5,𝔽2)—584no—
163C2×#41—291no—
164C2×#41—292no—
165C2×#41—581no—
166C2×#41C2×D2911yescharacteristic
167C22×D4—292no—
168C22×D4—291no—
169C22×D4—291no—
170C22×D4—581no—
171C22×D4—582no—
172AGL(1,ℤ/8)—584no—
173C2×SD16—292no—
174C2×D8—292no—
175C2×D8—294no—
176C2×D8—584no—
177C2×M16—291no—
178C22×C8C2×D2911yescharacteristic, complemented
179#127—581no—
180#124—581no—
181#121—294no—
182#120—582no—
183#117C2×D2911yescharacteristic, complemented
184#103C2×D2912yescomplemented
185#102—294no—
186#102C2×D2912yescomplemented
187C22⋊C8—294no—
188C29#300811yescharacteristic, complemented, Sylow
189C24—582no—
190C24—581no—
191C24—581no—
192#41—22no—
193#41—588no—
194#41—584no—
195#41—584no—
196#41—582no—
197#41—582no—
198#41D11614yescomplemented
199C2×Q8—291no—
200C2×Q8C22×D2911yescharacteristic
201C2×D4—21no—
202C2×D4—292no—
203C2×D4—294no—
204C2×D4—291no—
205C2×D4—292no—
206C2×D4—292no—
207C2×D4—582no—
208C2×D4—588no—
209C2×D4—584no—
210C2×D4—582no—
211C2×D4—584no—
212C2×D4—5816no—
213C2×D4—584no—
214C2×D4—582no—
215C2×D4—582no—
216C2×D4—581no—
217C2×D4—582no—
218C2×D4—581no—
219C2×D4C22×D2911yescharacteristic
220C22×C4—21no—
221C22×C4—292no—
222C22×C4—582no—
223C22×C4C22×D2911yescharacteristic
224SD16—584no—
225D8—588no—
226D8—584no—
227M16—582no—
228C2×C8—22no—
229C2×C8—292no—
230C2×C8C22×D2912yes—
231C4⋊C4C22×D2912yes—
232C22⋊C4—21no—
233C22⋊C4—584no—
234C42—21no—
235C23—21no—
236C23—292no—
237C23—588no—
238C23—582no—
239C23—584no—
240C23—582no—
241C23—1162no—
242C23—2322no—
243C23—2322no—
244C23—2324no—
245C23D4×D2911yescharacteristic
246Q8—21no—
247Q8—582no—
248Q8C2×D11612yes—
249D4—24no—
250D4—21no—
251D4—294no—
252D4—588no—
253D4—588no—
254D4—584no—
255D4—582no—
256D4—582no—
257D4—582no—
258D4—584no—
259D4—582no—
260D4—1162no—
261D4—2324no—
262D4C2×D11612yes—
263C2×C4—21no—
264C2×C4—24no—
265C2×C4—22no—
266C2×C4—292no—
267C2×C4—588no—
268C2×C4—582no—
269C2×C4—582no—
270C2×C4C2×D11612yes—
271C2×C4D4×D2912yes—
272C2×C4D4×D2911yescharacteristic
273C2×C4C23×D2911yescharacteristic
274C8—22no—
275C8—582no—
276C22—22no—
277C22—22no—
278C22—42no—
279C22—584no—
280C22—582no—
281C22—584no—
282C22—1164no—
283C22—1162no—
284C22—1164no—
285C22—2324no—
286C22—2324no—
287C22—2328no—
288C22#1174524412yes—
289C22C2×D4×D2911yescharacteristic
290C4—22no—
291C4—42no—
292C4—582no—
293C4#1174524412yes—
294C4C22×D11611yescharacteristic
295C4C2×D4×D2911yescharacteristic
296C2—22no—
297C2—42no—
298C2—1162no—
299C2—1164no—
300C2—2322no—
301C2C2×#1174524411yescharacteristic
302C2#1427196512yes—
subgroupG/H
centerC22C2×D4×D29
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC29×#416C2
socleC22×C29C22×D4
hypercenter#416D29
hypocenterC29#3008
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. C2×#11745364C2semidirect
2. C2×#11745311C2semidirect
3. C22×D232C2semidirect
4. #14271732C2semidirect
5. #14271311C2semidirect
6. #14271298C2semidirect
7. #14271167C2semidirect
8. #14271129C2semidirect
9. C29×#416C2semidirect
13. C2×#90809C22semidirect
15. C2×D4×D29C22semidirect
20. C22×D116C22semidirect
21. C22×C8×C29C22semidirect
22. C29×#117C22semidirect
24. C2×#90784C22semidirect
25. C2×#90782C22semidirect
26. C2×#90777C22semidirect
28. C2×D232C22semidirect
29. #11745257C22semidirect
33. C29×#103C22semidirect
34. C29×#102C22semidirect
35. #11745168C22semidirect
36. #11745157C22semidirect
41. #41×C29D4semidirect
66. C2×D116D4semidirect
152. #416D29semidirect
178. C22×C8C2×D29semidirect
183. #117C2×D29semidirect
184. #103C2×D29semidirect
186. #102C2×D29semidirect
188. C29#3008semidirect
198. #41D116semidirect

Maximal subgroups

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

subgroupindex
C2×#117453642
C2×#117453112
C22×D2322
#142717322
#142713112
#142712982
#142711672
#142711292
C29×#4162
#300829