Finite Group Databasebeta

← #16828746

Subgroups — #16828746

Order 3712.

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

normal subgroups only

This group has 806 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×#90805C211yesmaximal, characteristic, complemented
2C2×#11745270C212yesmaximal, complemented
3C2×#11745292C211yesmaximal, characteristic, complemented
4#14271310C214yesmaximal, complemented
5#14271304C212yesmaximal, complemented
6#14271299C212yesmaximal, complemented
7#14271165C212yesmaximal, complemented
8C29×#407C211yesmaximal, characteristic, complemented
9C23×C4×C29C2211yescharacteristic
10C22×C22⋊D29—21no—
11C22×C22⋊D29—21no—
12C2×#90805—21no—
13C2×#90805—21no—
14C2×#90805—28no—
15C2×#90805C2212yes—
16C2×#90805C2212yescomplemented
17C22×D116C2211yescharacteristic, complemented
18C22×D116C2211yescharacteristic, complemented
19C22×C4×D29—21no—
20C22×Dic58C2211yescharacteristic, complemented
21C2×M16×C29C2212yescomplemented
22#11745292—21no—
23#11745292—21no—
24#11745292C2212yes—
25#11745292C2212yescomplemented
26C2×#90781—21no—
27C2×C4⋊Dic29C2211yescharacteristic, complemented
28#11745270—28no—
29C2×D232C2214yescomplemented
30C2×#90773C2214yescomplemented
31C29×C22⋊C8C2212yescomplemented
32C29×C22⋊C8C2212yescomplemented
33#11745169C2214yescomplemented
34#11745164C2214yescomplemented
35C24×C29D411yescharacteristic
36C23×D29—21no—
37C23×D29—21no—
38C22×C4×C29—24no—
39C22×C4×C29D411yescharacteristic
40C22×C4×C29D411yescharacteristic
41C22×C4×C29D411yescharacteristic
42C22×C4×C29C2311yescharacteristic
43C22×C4×C29C2311yescharacteristic
44C22×C4×C29C2311yescharacteristic
45C2×C22⋊D29—28no—
46C2×C22⋊D29—22no—
47C2×C22⋊D29—22no—
48C2×C22⋊D29—22no—
49C2×C22⋊D29—22no—
50C2×C22⋊D29—44no—
51C22×Dic29—21no—
52C22×Dic29—21no—
53#90805—28no—
54#90805—28no—
55#90805—24no—
56#90805—24no—
57#90805—24no—
58#90805D418yescomplemented
59C2×D116—21no—
60C2×D116—24no—
61C2×D116—24no—
62C2×D116—21no—
63C2×D116C2312yes—
64C2×D116C2312yes—
65C2×C4×D29—28no—
66C2×C4×D29—22no—
67C2×C4×D29—22no—
68C2×Dic58—21no—
69C2×Dic58—24no—
70C2×Dic58C2312yes—
71M16×C29—24no—
72C2×C8×C29C2314yes—
73#90781—22no—
74#90781—22no—
75C4⋊Dic29—21no—
76C4⋊Dic29C2312yes—
77D232—28no—
78#90773—28no—
79C23×C29—22no—
80C23×C29—22no—
81C23×C29—41no—
82C23×C29C2×D411yescharacteristic
83C23×C29C2×D411yescharacteristic
84C23×C29C2×D411yescharacteristic
85C22×D29—24no—
86C22×D29—22no—
87C22×D29—22no—
88C22×D29—42no—
89C22×D29—42no—
90C22×D29—42no—
91C22×D29—42no—
92C22×D29—42no—
93C2×C4×C29—22no—
94C2×C4×C29—22no—
95C2×C4×C29—22no—
96C2×C4×C29—22no—
97C2×C4×C29—22no—
98C2×C4×C29C2×D411yescharacteristic
99C2×C4×C29C2×D411yescharacteristic
100C2×C4×C29C2×D411yescharacteristic
101C2×C4×C29C2×D414yes—
102C2×C4×C29C2411yescharacteristic
103C22⋊D29—28no—
104C22⋊D29—28no—
105C22⋊D29—44no—
106C22⋊D29—44no—
107C2×Dic29—24no—
108C2×Dic29—22no—
109C2×Dic29—22no—
110C2×Dic29—42no—
111D116—24no—
112D116—22no—
113D116—24no—
114D116—42no—
115D116C2×D414yescomplemented
116C4×D29—28no—
117C4×D29—28no—
118Dic58—24no—
119Dic58—22no—
120Dic58C2×D414yescomplemented
121C8×C29—24no—
122#3006—291nomaximal, Sylow
123C22×C29—21no—
124C22×C29—21no—
125C22×C29—21no—
126C22×C29—21no—
127C22×C29—21no—
128C22×C29—21no—
129C22×C29—21no—
130C22×C29—44no—
131C22×C29#12012yes—
132C22×C29AGL(1,ℤ/8)12yescomplemented
133C22×C29C22×D411yescharacteristic
134C2×D29—24no—
135C2×D29—48no—
136C2×D29—42no—
137C2×D29—42no—
138C2×D29—42no—
139C2×D29—42no—
140C2×D29—42no—
141C2×D29—84no—
142C4×C29—21no—
143C4×C29—21no—
144C4×C29#12012yes—
145C4×C29C22×D412yes—
146Dic29—24no—
147Dic29—42no—
148C22×#41—291no—
149C2×AGL(1,ℤ/8)—292no—
150C2×#121—291no—
151#449—294no—
152#448—292no—
153#447—292no—
154#417—292no—
155#407D2911yescharacteristic, complemented
156C2×C29—22no—
157C2×C29—22no—
158C2×C29—41no—
159C2×C29C2×#12011yescharacteristic
160C2×C29C2×AGL(1,ℤ/8)11yescharacteristic
161C2×C29#57511yescharacteristic
162D29—44no—
163D29—82no—
164C2×#41—292no—
165C2×#41—292no—
166C2×#41—588no—
167C2×#41—581no—
168C2×#41—581no—
169C22×Q8—291no—
170C22×D4—291no—
171C22×D4—291no—
172C22×D4—581no—
173C22×D4—581no—
174C23×C4—581no—
175C23×C4C2×D2911yescharacteristic
176AGL(1,ℤ/8)—588no—
177C2×SD16—294no—
178C2×D8—294no—
179C2×M16C2×D2912yescomplemented
180#121—292no—
181#121—292no—
182#121—581no—
183#121—581no—
184C2×C4⋊C4—291no—
185C2×C22⋊C4—581no—
186#103—294no—
187#102—294no—
188C22⋊C8C2×D2912yescomplemented
189C22⋊C8C2×D2912yescomplemented
190C29#300611yescharacteristic, complemented, Sylow
191C24—581no—
192C24—581no—
193C24D11611yescharacteristic
194#41—298no—
195#41—584no—
196#41—584no—
197#41—588no—
198#41—588no—
199#41—584no—
200C2×Q8—292no—
201C2×Q8—584no—
202C2×Q8—581no—
203C2×D4—292no—
204C2×D4—292no—
205C2×D4—584no—
206C2×D4—588no—
207C2×D4—582no—
208C2×D4—582no—
209C2×D4—582no—
210C2×D4—582no—
211C2×D4—581no—
212C2×D4—581no—
213C2×D4—584no—
214C2×D4—1164no—
215C22×C4—24no—
216C22×C4—588no—
217C22×C4—581no—
218C22×C4—582no—
219C22×C4—582no—
220C22×C4—581no—
221C22×C4D11611yescharacteristic
222C22×C4D11611yescharacteristic
223C22×C4D11611yescharacteristic
224C22×C4C22×D2911yescharacteristic
225C22×C4C22×D2911yescharacteristic
226C22×C4C22×D2911yescharacteristic
227SD16—588no—
228D8—588no—
229M16—24no—
230C2×C8C22×D2914yes—
231C4⋊C4—292no—
232C4⋊C4—581no—
233C22⋊C4—582no—
234C22⋊C4—582no—
235C23—22no—
236C23—22no—
237C23—41no—
238C23—584no—
239C23—582no—
240C23—582no—
241C23—1162no—
242C23—1162no—
243C23—1162no—
244C23—1162no—
245C23—1162no—
246C23C2×D11611yescharacteristic
247C23C2×D11611yescharacteristic
248C23C2×D11611yescharacteristic
249Q8—294no—
250Q8—584no—
251Q8—582no—
252D4—294no—
253D4—584no—
254D4—588no—
255D4—588no—
256D4—582no—
257D4—584no—
258D4—1164no—
259D4—1164no—
260D4—1162no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—588no—
267C2×C4—584no—
268C2×C4—582no—
269C2×C4—588no—
270C2×C4—582no—
271C2×C4—1162no—
272C2×C4C2×D11611yescharacteristic
273C2×C4C2×D11611yescharacteristic
274C2×C4C2×D11611yescharacteristic
275C2×C4D4×D2914yes—
276C2×C4C23×D2911yescharacteristic
277C8—24no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22—21no—
285C22—44no—
286C22—584no—
287C22—1168no—
288C22—1162no—
289C22—1162no—
290C22—1162no—
291C22—1162no—
292C22—1162no—
293C22—2324no—
294C22#1174524412yes—
295C22#1174527012yescomplemented
296C22C22×D11611yescharacteristic
297C4—21no—
298C4—21no—
299C4—584no—
300C4—1162no—
301C4#1174524412yes—
302C4C2×D4×D2912yes—
303C2—22no—
304C2—22no—
305C2—41no—
306C2—1164no—
307C2—2322no—
308C2C2×#1174524411yescharacteristic
309C2C2×#1174527011yescharacteristic
310C2#1427196311yescharacteristic
subgroupG/H
centerC22C22×D116
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC29×#407C2
socleC22×C29C22×D4
hypercenter#407D29
hypocenterC29#3006
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
1. C22×#90805C2semidirect
2. C2×#11745270C2semidirect
3. C2×#11745292C2semidirect
4. #14271310C2semidirect
5. #14271304C2semidirect
6. #14271299C2semidirect
7. #14271165C2semidirect
8. C29×#407C2semidirect
16. C2×#90805C22semidirect
17. C22×D116C22semidirect
18. C22×D116C22semidirect
20. C22×Dic58C22semidirect
21. C2×M16×C29C22semidirect
25. #11745292C22semidirect
27. C2×C4⋊Dic29C22semidirect
29. C2×D232C22semidirect
30. C2×#90773C22semidirect
31. C29×C22⋊C8C22semidirect
32. C29×C22⋊C8C22semidirect
33. #11745169C22semidirect
34. #11745164C22semidirect
58. #90805D4semidirect
115. D116C2×D4semidirect
120. Dic58C2×D4semidirect
132. C22×C29AGL(1,ℤ/8)semidirect
155. #407D29semidirect
179. C2×M16C2×D29semidirect
188. C22⋊C8C2×D29semidirect
189. C22⋊C8C2×D29semidirect
190. C29#3006semidirect
295. C22#11745270semidirect

Maximal subgroups

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

subgroupindex
C22×#908052
C2×#117452702
C2×#117452922
#142713102
#142713042
#142712992
#142711652
C29×#4072
#300629