Finite Group Databasebeta

← #16829940

Subgroups — #16829940

Order 3712.

656 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 153 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×#11745270C211yesmaximal, characteristic, complemented
2C2×#11745294C211yesmaximal, characteristic, complemented
3C2×#11745311C211yesmaximal, characteristic, complemented
4#14271709C211yesmaximal, characteristic, complemented
5C2×#11745302C211yesmaximal, characteristic, complemented
6#14271551C212yesmaximal, complemented
7#14271549C212yesmaximal, complemented
8#14271538C212yesmaximal, complemented
9#14271534C212yesmaximal, complemented
10M16×Dic29C211yesmaximal, characteristic, complemented
11C2×C29×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
12C2×#41×C29C2211yescharacteristic
13C22×D4×C29C2211yescharacteristic
14C22×C22⋊D29—22no—
15C2×#90805C2211yescharacteristic
16C22×D116C2211yescharacteristic
17C29×AGL(1,ℤ/8)—24no—
18C2×SD16×C29C2212yescomplemented
19C2×D8×C29C2212yescomplemented
20C2×M16×C29C2211yescharacteristic
21#11745311—24no—
22#11745309C2211yescharacteristic, complemented
23#11745306C2211yescharacteristic, complemented
24C2×#90784C2212yescomplemented
25#11745302—22no—
26#11745302C2212yes—
27#11745302C2212yes—
28#11745302C2211yescharacteristic, complemented
29#11745301—22no—
30#11745298C2211yescharacteristic, complemented
31C2×#90783C2212yescomplemented
32#11745294—24no—
33C2×#90782C2212yescomplemented
34C2×#90782C2212yescomplemented
35C4×C22⋊D29—22no—
36C2×C4×Dic29C2211yescharacteristic
37C2×#90777C2211yescharacteristic
38#11745270—24no—
39C2×D232C2212yescomplemented
40C2×#90773C2212yescomplemented
41#11745163C2212yescomplemented
42C8×Dic29C2212yescomplemented
43C24×C29—21no—
44C23×D29—21no—
45#41×C29—22no—
46#41×C29—22no—
47C2×Q8×C29C2311yescharacteristic
48C2×D4×C29—21no—
49C2×D4×C29—22no—
50C2×D4×C29—22no—
51C2×D4×C29—21no—
52C2×D4×C29C2311yescharacteristic
53C2×D4×C29C2312yes—
54C22×C4×C29—21no—
55C22×C4×C29C2311yescharacteristic
56C2×C22⋊D29—21no—
57C2×C22⋊D29—22no—
58C2×C22⋊D29—24no—
59C2×C22⋊D29—24no—
60C2×C22⋊D29—48no—
61C22×Dic29D412yes—
62#90805—22no—
63#90805—22no—
64C2×D116—22no—
65C2×D116—22no—
66C2×D116—21no—
67C2×D116C2311yescharacteristic
68C2×D116C2312yes—
69C2×C4×D29—21no—
70C2×Dic58C2311yescharacteristic
71SD16×C29—24no—
72D8×C29—24no—
73M16×C29D414yescomplemented
74C2×C8×C29C2312yes—
75C22⋊Dic29—22no—
76#90784—24no—
77#90783—24no—
78#90782—24no—
79#90782—24no—
80#90781—22no—
81#90779—22no—
82C4×Dic29D412yes—
83C4×Dic29C2311yescharacteristic
84C4×Dic29C2311yescharacteristic
85#90777D414yescomplemented
86C2×#6597C2312yes—
87D232—24no—
88#90773—24no—
89C23×C29—21no—
90C23×C29—22no—
91C23×C29—44no—
92C23×C29—42no—
93C23×C29C2×D411yescharacteristic
94C22×D29—21no—
95C22×D29—22no—
96C22×D29—44no—
97C22×D29—42no—
98Q8×C29—21no—
99Q8×C29—21no—
100D4×C29—21no—
101D4×C29—22no—
102D4×C29—21no—
103D4×C29—22no—
104D4×C29—42no—
105D4×C29—42no—
106C2×C4×C29—21no—
107C2×C4×C29—42no—
108C2×C4×C29C2×D411yescharacteristic
109C2×C4×C29C2×D412yes—
110C2×C4×C29C2×D412yes—
111C2×C4×C29C2411yescharacteristic
112C22⋊D29—48no—
113C22⋊D29—42no—
114C22⋊D29—44no—
115C22⋊D29—48no—
116C2×Dic29—24no—
117C2×Dic29—21no—
118C2×Dic29C2×D412yes—
119C2×Dic29C2×D412yes—
120D116—22no—
121D116—21no—
122D116—21no—
123D116—22no—
124D116—42no—
125C4×D29—42no—
126Dic58—21no—
127Dic58—21no—
128C8×C29C2×D414yescomplemented
129#6597C2×D414yescomplemented
130#3151—291nomaximal, Sylow
131C22×C29—22no—
132C22×C29—41no—
133C22×C29—42no—
134C22×C29—44no—
135C22×C29—42no—
136C22×C29—84no—
137C22×C29#12712yes—
138C22×C29C22×D411yescharacteristic
139C2×D29—41no—
140C2×D29—42no—
141C2×D29—44no—
142C2×D29—42no—
143C2×D29—84no—
144C4×C29—41no—
145C4×C29#12712yes—
146C4×C29C22×D411yescharacteristic
147C4×C29C22×D411yescharacteristic
148Dic29—22no—
149Dic29—41no—
150Dic29AGL(1,ℤ/8)14yescomplemented
151C2×AGL(1,ℤ/8)—291no—
152C2×AGL(1,ℤ/8)—291no—
153C2×AGL(1,ℤ/8)—291no—
154C2×AGL(1,ℤ/8)D2911yescharacteristic, complemented
155#532—291no—
156C2×#127—291no—
157#496—292no—
158#496—292no—
159#493—292no—
160#492—292no—
161C4×M16—291no—
162C2×C29—22no—
163C2×C29—81no—
164C2×C29—82no—
165C2×C29C2×#12711yescharacteristic
166C2×C29C2×AGL(1,ℤ/8)12yes—
167D29—81no—
168D29—82no—
169C2×#41—291no—
170C2×#41C2×D2911yescharacteristic
171C22×D4—291no—
172C22×D4—582no—
173C22×D4C2×D2911yescharacteristic
174AGL(1,ℤ/8)—24no—
175AGL(1,ℤ/8)—584no—
176AGL(1,ℤ/8)—584no—
177AGL(1,ℤ/8)—584no—
178C2×SD16—292no—
179C2×SD16—292no—
180C2×SD16—292no—
181C2×SD16C2×D2912yescomplemented
182C2×D8—292no—
183C2×D8—292no—
184C2×D8—292no—
185C2×D8C2×D2912yescomplemented
186C2×M16—291no—
187C2×M16C2×D2911yescharacteristic
188C4⋊Q8—291no—
189#127—292no—
190#127—292no—
191#127—291no—
192#127—582no—
193#124—291no—
194#124—291no—
195#121—582no—
196C4×D4—582no—
197C2×C42—291no—
198#97—292no—
199C4×C8—292no—
200C29#315111yescharacteristic, complemented, Sylow
201C24—21no—
202C24—581no—
203#41—22no—
204#41—22no—
205#41—582no—
206#41—582no—
207C2×Q8—291no—
208C2×Q8C22×D2911yescharacteristic
209C2×D4—21no—
210C2×D4—22no—
211C2×D4—21no—
212C2×D4—22no—
213C2×D4—291no—
214C2×D4—292no—
215C2×D4—582no—
216C2×D4—584no—
217C2×D4—584no—
218C2×D4—581no—
219C2×D4—582no—
220C2×D4—581no—
221C2×D4—582no—
222C2×D4—1168no—
223C2×D4C22×D2911yescharacteristic
224C2×D4C22×D2912yes—
225C22×C4—21no—
226C22×C4—292no—
227C22×C4—581no—
228C22×C4C22×D2911yescharacteristic
229SD16—24no—
230SD16—584no—
231SD16—584no—
232SD16—584no—
233D8—24no—
234D8—584no—
235D8—584no—
236D8—584no—
237M16—294no—
238M16C22⋊D2914yescomplemented
239C2×C8—292no—
240C2×C8C22×D2912yes—
241C4⋊C4—582no—
242C22⋊C4—582no—
243C22⋊C4—582no—
244C42—292no—
245C42—291no—
246C42—291no—
247C23—21no—
248C23—22no—
249C23—44no—
250C23—42no—
251C23—581no—
252C23—582no—
253C23—1164no—
254C23—1162no—
255C23D4×D2911yescharacteristic
256Q8—21no—
257Q8—21no—
258Q8—581no—
259Q8—581no—
260D4—21no—
261D4—22no—
262D4—21no—
263D4—22no—
264D4—42no—
265D4—42no—
266D4—581no—
267D4—582no—
268D4—581no—
269D4—582no—
270D4—1168no—
271D4—1164no—
272D4—1168no—
273D4—1162no—
274D4—1162no—
275C2×C4—21no—
276C2×C4—42no—
277C2×C4—292no—
278C2×C4—292no—
279C2×C4—584no—
280C2×C4—581no—
281C2×C4—1162no—
282C2×C4D4×D2911yescharacteristic
283C2×C4D4×D2912yes—
284C2×C4C2×C22⋊D2912yes—
285C2×C4C23×D2911yescharacteristic
286C8—294no—
287C8C2×C22⋊D2914yescomplemented
288C22—22no—
289C22—44no—
290C22—42no—
291C22—41no—
292C22—42no—
293C22—84no—
294C22—1164no—
295C22—1162no—
296C22—1161no—
297C22—1162no—
298C22—2324no—
299C22#1174530212yes—
300C22C2×D4×D2911yescharacteristic
301C4—41no—
302C4—294no—
303C4—582no—
304C4—1161no—
305C4#1174530212yes—
306C4C2×D4×D2911yescharacteristic
307C4C22×C22⋊D2911yescharacteristic
308C2—22no—
309C2—82no—
310C2—81no—
311C2—2322no—
312C2—2321no—
313C2C2×#1174530211yescharacteristic
314C2AGL(1,ℤ/8)×D2912yes—
subgroupG/H
centerC22C2×D4×D29
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC2×C29×AGL(1,ℤ/8)C2
socleC22×C29C22×D4
hypercenterC2×AGL(1,ℤ/8)D29
hypocenterC29#3151
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#11745270C2semidirect
2. C2×#11745294C2semidirect
3. C2×#11745311C2semidirect
4. #14271709C2semidirect
5. C2×#11745302C2semidirect
6. #14271551C2semidirect
7. #14271549C2semidirect
8. #14271538C2semidirect
9. #14271534C2semidirect
10. M16×Dic29C2semidirect
11. C2×C29×AGL(1,ℤ/8)C2semidirect
18. C2×SD16×C29C22semidirect
19. C2×D8×C29C22semidirect
22. #11745309C22semidirect
23. #11745306C22semidirect
24. C2×#90784C22semidirect
28. #11745302C22semidirect
30. #11745298C22semidirect
31. C2×#90783C22semidirect
33. C2×#90782C22semidirect
34. C2×#90782C22semidirect
39. C2×D232C22semidirect
40. C2×#90773C22semidirect
41. #11745163C22semidirect
42. C8×Dic29C22semidirect
73. M16×C29D4semidirect
85. #90777D4semidirect
128. C8×C29C2×D4semidirect
129. #6597C2×D4semidirect
150. Dic29AGL(1,ℤ/8)semidirect
154. C2×AGL(1,ℤ/8)D29semidirect
181. C2×SD16C2×D29semidirect
185. C2×D8C2×D29semidirect
200. C29#3151semidirect
238. M16C22⋊D29semidirect
287. C8C2×C22⋊D29semidirect

Maximal subgroups

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

subgroupindex
C2×#117452702
C2×#117452942
C2×#117453112
#142717092
C2×#117453022
#142715512
#142715492
#142715382
#142715342
M16×Dic292
C2×C29×AGL(1,ℤ/8)2
#315129