Finite Group Databasebeta

← #16828737

Subgroups — #16828737

Order 3712.

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

normal subgroups only

This group has 758 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
1C22×#90805C211yesmaximal, characteristic, complemented
2C2×#11745271C211yesmaximal, characteristic, complemented
3C2×#11745270C211yesmaximal, characteristic, complemented
4#14271674C211yesmaximal, characteristic, complemented
5#14271327C214yesmaximal, complemented
6#14271310C214yesmaximal, complemented
7#14271173C211yesmaximal, characteristic, complemented
8#14271169C211yesmaximal, characteristic, complemented
9C29×#407C211yesmaximal, characteristic, complemented
10C23×C4×C29C2211yescharacteristic
11C22×C22⋊D29—21no—
12C2×#90805—22no—
13C2×#90805—24no—
14C2×#90805—24no—
15C2×#90805C2214yes—
16C2×#90805C2211yescharacteristic
17C22×D116C2211yescharacteristic
18C22×C4×D29—21no—
19C22×Dic58C2211yescharacteristic
20C2×M16×C29C2211yescharacteristic
21C2×M16×C29C2211yescharacteristic
22#11745315—21no—
23#11745292C2212yescomplemented
24#11745291—21no—
25C4×C22⋊D29—22no—
26#11745288C2211yescharacteristic
27#11745286C2212yescomplemented
28#11745271—24no—
29#11745270—24no—
30C2×Dic116C2212yescomplemented
31C2×D232C2212yescomplemented
32C2×#90773C2212yescomplemented
33C2×#90773C2212yescomplemented
34C29×C22⋊C8C2214yescomplemented
35#11745169C2214yescomplemented
36#11745164C2214yescomplemented
37C24×C29D411yescharacteristic
38C23×D29—21no—
39C22×C4×C29—22no—
40C22×C4×C29—22no—
41C22×C4×C29D412yes—
42C22×C4×C29D411yescharacteristic
43C22×C4×C29C2312yes—
44C22×C4×C29C2311yescharacteristic
45C2×C22⋊D29—28no—
46C2×C22⋊D29—24no—
47C2×C22⋊D29—22no—
48C2×C22⋊D29—21no—
49C22×Dic29—21no—
50#90805—216no—
51#90805—28no—
52#90805—22no—
53#90805—22no—
54#90805—28no—
55C2×D116—21no—
56C2×D116—22no—
57C2×D116D414yescomplemented
58C2×D116C2311yescharacteristic
59C2×D116C2312yes—
60C2×C4×D29—28no—
61C2×C4×D29—24no—
62C2×C4×D29—21no—
63C2×Dic58—21no—
64C2×Dic58—22no—
65C2×Dic58D414yescomplemented
66C2×Dic58C2311yescharacteristic
67C2×Dic58C2312yes—
68M16×C29—22no—
69M16×C29—22no—
70C2×C8×C29C2312yes—
71C2×C8×C29C2312yes—
72C22⋊Dic29—22no—
73C22⋊Dic29—21no—
74#90781—22no—
75C4⋊Dic29C2312yes—
76#90779—22no—
77C4×Dic29—21no—
78Dic116—24no—
79D232—24no—
80#90773—24no—
81#90773—24no—
82C23×C29—22no—
83C23×C29—42no—
84C23×C29C2×D412yes—
85C23×C29C2×D411yescharacteristic
86C22×D29—24no—
87C22×D29—22no—
88C22×D29—21no—
89C22×D29—44no—
90C2×C4×C29—22no—
91C2×C4×C29—24no—
92C2×C4×C29—22no—
93C2×C4×C29—22no—
94C2×C4×C29C2×D412yes—
95C2×C4×C29C2×D411yescharacteristic
96C2×C4×C29C2×D412yes—
97C2×C4×C29C2×D412yes—
98C2×C4×C29C2411yescharacteristic
99C22⋊D29—216no—
100C22⋊D29—44no—
101C22⋊D29—42no—
102C2×Dic29—24no—
103C2×Dic29—22no—
104C2×Dic29—22no—
105C2×Dic29—21no—
106D116—24no—
107D116—22no—
108D116—21no—
109D116—21no—
110D116C2×D414yes—
111C4×D29—216no—
112C4×D29—42no—
113Dic58—24no—
114Dic58—22no—
115Dic58—21no—
116Dic58—21no—
117Dic58C2×D414yes—
118C8×C29—22no—
119C8×C29—22no—
120#3005—291nomaximal, Sylow
121C22×C29—22no—
122C22×C29—22no—
123C22×C29—22no—
124C22×C29—22no—
125C22×C29—44no—
126C22×C29#12012yes—
127C22×C29C22×D411yescharacteristic
128C2×D29—24no—
129C2×D29—48no—
130C2×D29—44no—
131C2×D29—41no—
132C2×D29—42no—
133C4×C29—22no—
134C4×C29#12012yes—
135C4×C29C22×D411yescharacteristic
136C4×C29C22×D411yescharacteristic
137Dic29—24no—
138Dic29—41no—
139Dic29—42no—
140C22×#41—291no—
141C2×#137—291no—
142C2×AGL(1,ℤ/8)—291no—
143#525—291no—
144#452—294no—
145#449—294no—
146#419—291no—
147#418—291no—
148#407D2911yescharacteristic, complemented
149C2×C29—22no—
150C2×C29—42no—
151C2×C29C2×#12011yescharacteristic
152C2×C29#57512yes—
153D29—44no—
154D29—81no—
155C2×#41—291no—
156C2×#41—294no—
157C2×#41—584no—
158C2×#41—584no—
159C2×#41—582no—
160C22×Q8—291no—
161C22×D4—291no—
162C22×D4—581no—
163C23×C4—581no—
164C23×C4C2×D2911yescharacteristic
165#137—584no—
166AGL(1,ℤ/8)—584no—
167C2×Q16—292no—
168C2×SD16—292no—
169C2×SD16—292no—
170C2×D8—292no—
171C2×M16C2×D2911yescharacteristic
172C2×M16C2×D2911yescharacteristic
173#123—581no—
174C22⋊Q8—292no—
175#121—292no—
176#120—581no—
177C4×D4—582no—
178#117—291no—
179#103—294no—
180#102—294no—
181C22⋊C8C2×D2914yescomplemented
182C29#300511yescharacteristic, complemented, Sylow
183C24—581no—
184C24D11611yescharacteristic
185#41—588no—
186#41—588no—
187#41—5816no—
188#41—582no—
189#41—582no—
190C2×Q8—294no—
191C2×Q8—292no—
192C2×Q8—291no—
193C2×Q8—581no—
194C2×Q8—582no—
195C2×D4—294no—
196C2×D4—292no—
197C2×D4—291no—
198C2×D4—588no—
199C2×D4—584no—
200C2×D4—582no—
201C2×D4—581no—
202C2×D4—581no—
203C2×D4—582no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—588no—
207C22×C4—581no—
208C22×C4—584no—
209C22×C4—581no—
210C22×C4D11612yes—
211C22×C4D11611yescharacteristic
212C22×C4C22×D2912yes—
213C22×C4C22×D2911yescharacteristic
214Q16—584no—
215SD16—584no—
216SD16—584no—
217D8—584no—
218M16—22no—
219M16—22no—
220C2×C8C22×D2912yes—
221C2×C8C22×D2912yes—
222C4⋊C4—292no—
223C4⋊C4—582no—
224C22⋊C4—581no—
225C22⋊C4—582no—
226C22⋊C4—582no—
227C42—581no—
228C23—22no—
229C23—42no—
230C23—582no—
231C23—584no—
232C23—581no—
233C23—1164no—
234C23C2×D11612yes—
235C23C2×D11611yescharacteristic
236Q8—294no—
237Q8—582no—
238Q8—584no—
239Q8—581no—
240Q8—581no—
241D4—294no—
242D4—5816no—
243D4—582no—
244D4—581no—
245D4—581no—
246D4—584no—
247D4—1164no—
248D4—1162no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—24no—
253C2×C4—5816no—
254C2×C4—582no—
255C2×C4—584no—
256C2×C4—582no—
257C2×C4—581no—
258C2×C4—1162no—
259C2×C4C2×D11612yes—
260C2×C4C2×D11611yescharacteristic
261C2×C4D4×D2912yes—
262C2×C4D4×D2912yes—
263C2×C4C23×D2911yescharacteristic
264C8—22no—
265C8—22no—
266C22—22no—
267C22—22no—
268C22—22no—
269C22—22no—
270C22—44no—
271C22—584no—
272C22—1168no—
273C22—1164no—
274C22—1161no—
275C22—1162no—
276C22#1174524412yes—
277C22C22×D11611yescharacteristic
278C4—22no—
279C4—584no—
280C4—1161no—
281C4—1162no—
282C4#1174524412yes—
283C4C2×D4×D2911yescharacteristic
284C4C2×D4×D2911yescharacteristic
285C2—22no—
286C2—42no—
287C2—1164no—
288C2—2321no—
289C2C2×#1174524411yescharacteristic
290C2#1427196312yes—
subgroupG/H
centerC22C22×D116
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC29×#407C2
socleC22×C29C22×D4
hypercenter#407D29
hypocenterC29#3005
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×#11745271C2semidirect
3. C2×#11745270C2semidirect
4. #14271674C2semidirect
5. #14271327C2semidirect
6. #14271310C2semidirect
7. #14271173C2semidirect
8. #14271169C2semidirect
9. C29×#407C2semidirect
23. #11745292C22semidirect
27. #11745286C22semidirect
30. C2×Dic116C22semidirect
31. C2×D232C22semidirect
32. C2×#90773C22semidirect
33. C2×#90773C22semidirect
34. C29×C22⋊C8C22semidirect
35. #11745169C22semidirect
36. #11745164C22semidirect
57. C2×D116D4semidirect
65. C2×Dic58D4semidirect
148. #407D29semidirect
181. C22⋊C8C2×D29semidirect
182. C29#3005semidirect

Maximal subgroups

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

subgroupindex
C22×#908052
C2×#117452712
C2×#117452702
#142716742
#142713272
#142713102
#142711732
#142711692
C29×#4072
#300529