Finite Group Databasebeta

← #16828721

Subgroups — #16828721

Order 3712.

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

normal subgroups only

This group has 994 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
1C22×D4×D29C211yesmaximal, characteristic, complemented
2C2×#11745270C211yesmaximal, characteristic, complemented
3C2×#11745294C211yesmaximal, characteristic, complemented
4#14271675C211yesmaximal, characteristic, complemented
5#14271315C214yesmaximal, complemented
6#14271298C214yesmaximal, complemented
7#14271167C211yesmaximal, characteristic, complemented
8#14271126C211yesmaximal, characteristic, complemented
9C29×#418C211yesmaximal, characteristic, complemented
10C24×D29—21no—
11C22×D4×C29C2211yescharacteristic
12C22×C22⋊D29—21no—
13C2×D4×D29—24no—
14C2×D4×D29—24no—
15C2×D4×D29—22no—
16C2×D4×D29C2214yes—
17C2×#90805C2211yescharacteristic, complemented
18C22×D116C2211yescharacteristic
19C22×C4×D29C2211yescharacteristic
20C2×M16×C29C2211yescharacteristic, complemented
21C29×#117C2211yescharacteristic, complemented
22C2×#90783C2212yescomplemented
23#11745294—24no—
24C2×#90782C2212yescomplemented
25C2×#90777C2211yescharacteristic, complemented
26#11745270—24no—
27C2×D232C2212yescomplemented
28C2×#90773C2212yescomplemented
29#11745259C2212yescomplemented
30#11745257C2212yescomplemented
31#11745248—21no—
32#11745244—21no—
33C4×D116—22no—
34C29×#102C2214yescomplemented
35#11745168C2214yescomplemented
36#11745157C2214yescomplemented
37C24×C29—21no—
38C23×D29—28no—
39C23×D29—21no—
40C23×D29—24no—
41C23×D29—48no—
42C23×D29D411yescharacteristic
43C2×D4×C29—21no—
44C2×D4×C29—22no—
45C2×D4×C29D414yescomplemented
46C2×D4×C29C2312yes—
47C22×C4×C29C2311yescharacteristic
48C2×C22⋊D29—28no—
49C2×C22⋊D29—21no—
50C2×C22⋊D29—24no—
51C22×Dic29D411yescharacteristic
52D4×D29—216no—
53D4×D29—28no—
54D4×D29—28no—
55#90805—22no—
56#90805—22no—
57C2×D116—22no—
58C2×D116—21no—
59C2×D116—22no—
60C2×D116D414yescomplemented
61C2×D116C2312yes—
62C2×D116C2311yescharacteristic, complemented
63C2×C4×D29—22no—
64C2×C4×D29—22no—
65C2×C4×D29—21no—
66C2×C4×D29D412yes—
67C2×C4×D29C2312yes—
68C2×Dic58C2311yescharacteristic, complemented
69M16×C29—22no—
70C2×C8×C29C2312yescomplemented
71C4⋊C4×C29C2312yescomplemented
72C22⋊C4×C29—21no—
73C42×C29—21no—
74#90783—24no—
75#90782—24no—
76#90781—22no—
77#90781—22no—
78C4⋊Dic29—22no—
79#90777—22no—
80C2×#6597C2312yescomplemented
81D232—24no—
82#90773—24no—
83C23×C29—24no—
84C23×C29—22no—
85C23×C29—44no—
86C23×C29C2×D411yescharacteristic
87C22×D29—22no—
88C22×D29—24no—
89C22×D29—216no—
90C22×D29—24no—
91C22×D29—22no—
92C22×D29—21no—
93C22×D29—432no—
94C22×D29—48no—
95C22×D29—44no—
96C22×D29—48no—
97C22×D29C2×D412yes—
98D4×C29—24no—
99D4×C29—22no—
100D4×C29C2×D414yes—
101C2×C4×C29—22no—
102C2×C4×C29C2×D412yes—
103C2×C4×C29C2×D411yescharacteristic
104C2×C4×C29C2×D412yes—
105C2×C4×C29C2411yescharacteristic
106C22⋊D29—216no—
107C22⋊D29—42no—
108C2×Dic29—22no—
109C2×Dic29—21no—
110C2×Dic29C2×D412yes—
111D116—24no—
112D116—22no—
113D116—21no—
114D116—21no—
115D116—44no—
116D116C2×D414yes—
117C4×D29—24no—
118C4×D29—22no—
119C4×D29—22no—
120C4×D29—42no—
121Dic58—21no—
122Dic58—21no—
123C8×C29—22no—
124#6597—22no—
125#3003—291nomaximal, Sylow
126C22×C29—22no—
127C22×C29—24no—
128C22×C29—48no—
129C22×C29—44no—
130C22×C29#12012yes—
131C22×C29C22×D411yescharacteristic
132C2×D29—28no—
133C2×D29—22no—
134C2×D29—22no—
135C2×D29—24no—
136C2×D29—432no—
137C2×D29—48no—
138C2×D29—44no—
139C2×D29—41no—
140C2×D29—42no—
141C2×D29AGL(1,ℤ/8)14yescomplemented
142C4×C29—42no—
143C4×C29#12012yes—
144C4×C29C22×D411yescharacteristic
145C4×C29C22×D411yescharacteristic
146Dic29—22no—
147Dic29—41no—
148C23×D4—291no—
149C2×AGL(1,ℤ/8)—291no—
150C2×AGL(1,ℤ/8)—291no—
151#525—291no—
152#450—294no—
153#447—294no—
154#418—291no—
155#418D2911yescharacteristic, complemented
156#407—291no—
157C2×C29—22no—
158C2×C29—44no—
159C2×C29C2×#12011yescharacteristic
160C2×C29C2×AGL(1,ℤ/8)12yes—
161D29—24no—
162D29—44no—
163D29—81no—
164C25—581no—
165C2×#41—291no—
166C22×D4—294no—
167C22×D4—291no—
168C22×D4—582no—
169C22×D4—581no—
170C22×D4—584no—
171C22×D4—584no—
172C22×D4C2×D2911yescharacteristic
173C23×C4—291no—
174AGL(1,ℤ/8)—584no—
175AGL(1,ℤ/8)—584no—
176C2×SD16—292no—
177C2×SD16—292no—
178C2×D8—292no—
179C2×D8—292no—
180C2×M16—291no—
181C2×M16C2×D2911yescharacteristic, complemented
182#123—581no—
183C22⋊Q8—292no—
184#121—292no—
185#120—581no—
186C4×D4—582no—
187#117C2×D2911yescharacteristic, complemented
188#102—294no—
189#102C2×D2914yescomplemented
190C22⋊C8—294no—
191C29#300311yescharacteristic, complemented, Sylow
192C24—21no—
193C24—291no—
194C24—584no—
195C24—588no—
196C24—581no—
197C24—1168no—
198#41—582no—
199#41—582no—
200C2×Q8—291no—
201C2×D4—22no—
202C2×D4—21no—
203C2×D4—294no—
204C2×D4—292no—
205C2×D4—291no—
206C2×D4—582no—
207C2×D4—584no—
208C2×D4—588no—
209C2×D4—5816no—
210C2×D4—588no—
211C2×D4—588no—
212C2×D4—581no—
213C2×D4—582no—
214C2×D4—581no—
215C2×D4D11614yescomplemented
216C2×D4C22×D2912yes—
217C22×C4—291no—
218C22×C4—292no—
219C22×C4—292no—
220C22×C4—582no—
221C22×C4—582no—
222C22×C4—581no—
223C22×C4C22×D2911yescharacteristic
224SD16—584no—
225SD16—584no—
226D8—584no—
227D8—584no—
228M16—22no—
229M16—582no—
230C2×C8—292no—
231C2×C8C22×D2912yescomplemented
232C4⋊C4—582no—
233C4⋊C4C22×D2912yescomplemented
234C22⋊C4—21no—
235C22⋊C4—582no—
236C22⋊C4—582no—
237C42—21no—
238C23—22no—
239C23—24no—
240C23—44no—
241C23—292no—
242C23—584no—
243C23—5816no—
244C23—582no—
245C23—582no—
246C23—584no—
247C23—581no—
248C23—1164no—
249C23—1168no—
250C23—11632no—
251C23—1168no—
252C23D4×D2911yescharacteristic
253Q8—581no—
254Q8—581no—
255D4—22no—
256D4—24no—
257D4—294no—
258D4—5816no—
259D4—582no—
260D4—581no—
261D4—581no—
262D4—584no—
263D4—1164no—
264D4—1162no—
265D4C2×D11614yes—
266C2×C4—22no—
267C2×C4—292no—
268C2×C4—582no—
269C2×C4—582no—
270C2×C4—582no—
271C2×C4—584no—
272C2×C4—581no—
273C2×C4—1162no—
274C2×C4C2×D11612yes—
275C2×C4D4×D2912yes—
276C2×C4D4×D2911yescharacteristic
277C2×C4C23×D2911yescharacteristic
278C8—22no—
279C8—582no—
280C22—22no—
281C22—24no—
282C22—48no—
283C22—44no—
284C22—294no—
285C22—588no—
286C22—582no—
287C22—582no—
288C22—584no—
289C22—1168no—
290C22—11632no—
291C22—1164no—
292C22—1161no—
293C22—1162no—
294C22#1174524412yes—
295C22C2×D4×D2911yescharacteristic
296C4—42no—
297C4—582no—
298C4—1161no—
299C4#1174524412yes—
300C4C22×D11611yescharacteristic
301C4C2×D4×D2911yescharacteristic
302C2—22no—
303C2—44no—
304C2—584no—
305C2—1164no—
306C2—2321no—
307C2C2×#1174524411yescharacteristic
308C2AGL(1,ℤ/8)×D2912yes—
subgroupG/H
centerC22C2×D4×D29
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC29×#418C2
socleC22×C29C22×D4
hypercenter#418D29
hypocenterC29#3003
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. C22×D4×D29C2semidirect
2. C2×#11745270C2semidirect
3. C2×#11745294C2semidirect
4. #14271675C2semidirect
5. #14271315C2semidirect
6. #14271298C2semidirect
7. #14271167C2semidirect
8. #14271126C2semidirect
9. C29×#418C2semidirect
17. C2×#90805C22semidirect
20. C2×M16×C29C22semidirect
21. C29×#117C22semidirect
22. C2×#90783C22semidirect
24. C2×#90782C22semidirect
25. C2×#90777C22semidirect
27. C2×D232C22semidirect
28. C2×#90773C22semidirect
29. #11745259C22semidirect
30. #11745257C22semidirect
34. C29×#102C22semidirect
35. #11745168C22semidirect
36. #11745157C22semidirect
45. C2×D4×C29D4semidirect
60. C2×D116D4semidirect
62. C2×D116C23semidirect
68. C2×Dic58C23semidirect
70. C2×C8×C29C23semidirect
71. C4⋊C4×C29C23semidirect
80. C2×#6597C23semidirect
141. C2×D29AGL(1,ℤ/8)semidirect
155. #418D29semidirect
181. C2×M16C2×D29semidirect
187. #117C2×D29semidirect
189. #102C2×D29semidirect
191. C29#3003semidirect
215. C2×D4D116semidirect
231. C2×C8C22×D29semidirect
233. C4⋊C4C22×D29semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D292
C2×#117452702
C2×#117452942
#142716752
#142713152
#142712982
#142711672
#142711262
C29×#4182
#300329