Finite Group Databasebeta

← #16828779

Subgroups — #16828779

Order 3712.

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

normal subgroups only

This group has 732 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×#11745362C211yesmaximal, characteristic, complemented
2C2×#11745270C211yesmaximal, characteristic, complemented
3C22×#90784C211yesmaximal, characteristic, complemented
4#14271732C211yesmaximal, characteristic, complemented
5#14271311C214yesmaximal, complemented
6#14271303C214yesmaximal, complemented
7#14271157C211yesmaximal, characteristic, complemented
8#14271131C211yesmaximal, characteristic, complemented
9C29×#419C211yesmaximal, characteristic, complemented
10C22×Q8×C29C2211yescharacteristic
11#11745362—24no—
12#11745362—24no—
13C2×#90809—21no—
14C2×#90809C2212yescomplemented
15C2×Q8×D29—21no—
16C2×Q8×D29C2212yescomplemented
17C2×#90805—22no—
18C2×#90805C2211yescharacteristic
19C2×#90805C2211yescharacteristic
20C22×D116C2211yescharacteristic
21C2×M16×C29C2211yescharacteristic
22C29×#117C2211yescharacteristic
23C2×#90784—24no—
24C2×#90784C2214yes—
25C22×#6597C2211yescharacteristic
26#11745270—24no—
27C2×D232C2212yes—
28C2×#90773C2212yes—
29#11745257C2214yes—
30#11745244—22no—
31#11745237—21no—
32#11745236—21no—
33C29×#103C2214yescomplemented
34#11745168C2214yescomplemented
35#11745158C2212yescomplemented
36#11745157C2212yescomplemented
37C23×D29—21no—
38C2×Q8×C29—21no—
39C2×Q8×C29—22no—
40C2×Q8×C29D414yescomplemented
41C2×Q8×C29C2312yes—
42C22×C4×C29—21no—
43C22×C4×C29C2311yescharacteristic
44C2×C22⋊D29—21no—
45C2×C22⋊D29D412yes—
46#90809—28no—
47#90809—24no—
48#90809—24no—
49Q8×D29—28no—
50Q8×D29—24no—
51Q8×D29—24no—
52#90805—216no—
53#90805—22no—
54#90805—22no—
55#90805—22no—
56#90805D414yescomplemented
57C2×D116—22no—
58C2×D116—22no—
59C2×D116—22no—
60C2×D116—21no—
61C2×D116—24no—
62C2×D116D411yescharacteristic
63C2×D116C2312yes—
64C2×D116C2311yescharacteristic
65C2×C4×D29—22no—
66C2×C4×D29—21no—
67C2×C4×D29—22no—
68C2×C4×D29C2312yes—
69C2×Dic58—22no—
70C2×Dic58D411yescharacteristic
71C2×Dic58C2311yescharacteristic
72M16×C29—22no—
73C2×C8×C29C2312yes—
74C4⋊C4×C29C2312yes—
75C22⋊C4×C29—21no—
76C42×C29—21no—
77#90784—28no—
78#90781—24no—
79C2×#6597—22no—
80C2×#6597C2312yes—
81D232—24no—
82#90773—24no—
83C23×C29C2×D411yescharacteristic
84C22×D29—21no—
85C22×D29—22no—
86C22×D29—42no—
87C22×D29—82no—
88C22×D29C2×D412yes—
89Q8×C29—24no—
90Q8×C29—22no—
91Q8×C29C2×D414yes—
92C2×C4×C29—24no—
93C2×C4×C29—22no—
94C2×C4×C29—22no—
95C2×C4×C29C2×D412yes—
96C2×C4×C29C2×D411yescharacteristic
97C2×C4×C29C2×D412yes—
98C2×C4×C29C2411yescharacteristic
99C22⋊D29—24no—
100C22⋊D29—24no—
101C2×Dic29—21no—
102C2×Dic29C2×D412yes—
103D116—28no—
104D116—22no—
105D116—21no—
106D116—22no—
107D116—22no—
108D116—42no—
109D116—84no—
110D116C2×D412yes—
111C4×D29—28no—
112C4×D29—28no—
113C4×D29—24no—
114C4×D29—22no—
115C4×D29—22no—
116Dic58—28no—
117Dic58—22no—
118Dic58—21no—
119Dic58C2×D412yes—
120C8×C29—22no—
121#6597—22no—
122#3010—291nomaximal, Sylow
123C22×C29—22no—
124C22×C29#12012yes—
125C22×C29C22×D411yescharacteristic
126C2×D29—22no—
127C2×D29—22no—
128C2×D29—24no—
129C2×D29—42no—
130C2×D29—42no—
131C2×D29—44no—
132C2×D29—84no—
133C4×C29—24no—
134C4×C29—42no—
135C4×C29#12012yes—
136C4×C29C22×D411yescharacteristic
137C4×C29C22×D411yescharacteristic
138Dic29—22no—
139Dic29—22no—
140C2×2−1+4—291no—
141C2×AGL(1,ℤ/8)—291no—
142C22×SD16—291no—
143#535—291no—
144#449—294no—
145#448—294no—
146#419D2911yescharacteristic, complemented
147#416—291no—
148#408—291no—
149C2×C29—22no—
150C2×C29C2×#12011yescharacteristic
151C2×C29#57712yes—
152D29—42no—
153D29—42no—
154D29—82no—
1552−1+4—584no—
1562−1+4—584no—
157C2×#41—291no—
158C2×#41—292no—
159C2×#41—291no—
160C2×#41—582no—
161C2×#41—581no—
162C22×Q8—292no—
163C22×Q8—581no—
164C22×Q8C2×D2911yescharacteristic
165C22×D4—291no—
166AGL(1,ℤ/8)—584no—
167C2×SD16—294no—
168C2×SD16—292no—
169C2×SD16—584no—
170C2×D8—292no—
171C2×M16C2×D2911yescharacteristic
172C22×C8—291no—
173#127—581no—
174#124—581no—
175#121—294no—
176#120—582no—
177#117C2×D2911yescharacteristic
178#103—292no—
179#103C2×D2914yescomplemented
180#102—292no—
181C22⋊C8—294no—
182C29#301011yescharacteristic, complemented, Sylow
183C24—581no—
184#41—294no—
185#41—5816no—
186#41—584no—
187#41—588no—
188#41—584no—
189#41—582no—
190#41—582no—
191#41—582no—
192C2×Q8—21no—
193C2×Q8—22no—
194C2×Q8—291no—
195C2×Q8—291no—
196C2×Q8—582no—
197C2×Q8—584no—
198C2×Q8—588no—
199C2×Q8—584no—
200C2×Q8D11614yescomplemented
201C2×Q8C22×D2912yes—
202C2×D4—292no—
203C2×D4—291no—
204C2×D4—291no—
205C2×D4—292no—
206C2×D4—582no—
207C2×D4—584no—
208C2×D4—581no—
209C2×D4—582no—
210C2×D4—582no—
211C2×D4—581no—
212C22×C4—21no—
213C22×C4—292no—
214C22×C4—582no—
215C22×C4—582no—
216C22×C4—581no—
217C22×C4C22×D2911yescharacteristic
218SD16—584no—
219SD16—588no—
220D8—584no—
221M16—22no—
222C2×C8—292no—
223C2×C8—582no—
224C2×C8C22×D2912yes—
225C4⋊C4C22×D2912yes—
226C22⋊C4—21no—
227C22⋊C4—584no—
228C42—21no—
229C23—292no—
230C23—581no—
231C23—582no—
232C23—1162no—
233C23—2322no—
234C23D4×D2911yescharacteristic
235Q8—22no—
236Q8—24no—
237Q8—292no—
238Q8—588no—
239Q8—581no—
240Q8—582no—
241Q8C2×D11614yes—
242D4—292no—
243D4—588no—
244D4—584no—
245D4—584no—
246D4—581no—
247D4—582no—
248D4—582no—
249D4—582no—
250D4—1162no—
251D4—2324no—
252C2×C4—22no—
253C2×C4—24no—
254C2×C4—22no—
255C2×C4—292no—
256C2×C4—588no—
257C2×C4—588no—
258C2×C4—581no—
259C2×C4—582no—
260C2×C4—582no—
261C2×C4—584no—
262C2×C4C2×D11612yes—
263C2×C4D4×D2912yes—
264C2×C4D4×D2911yescharacteristic
265C2×C4C23×D2911yescharacteristic
266C8—22no—
267C8—582no—
268C22—22no—
269C22—582no—
270C22—582no—
271C22—584no—
272C22—1162no—
273C22—1162no—
274C22—1164no—
275C22—2324no—
276C22#1174524412yes—
277C22C2×D4×D2911yescharacteristic
278C4—24no—
279C4—42no—
280C4—582no—
281C4—582no—
282C4#1174524412yes—
283C4C22×D11611yescharacteristic
284C4C2×D4×D2911yescharacteristic
285C2—22no—
286C2—1162no—
287C2—1162no—
288C2—2322no—
289C2C2×#1174524411yescharacteristic
290C2#1427197212yes—
subgroupG/H
centerC22C2×D4×D29
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC29×#419C2
socleC22×C29C22×D4
hypercenter#419D29
hypocenterC29#3010
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
1. C2×#11745362C2semidirect
2. C2×#11745270C2semidirect
3. C22×#90784C2semidirect
4. #14271732C2semidirect
5. #14271311C2semidirect
6. #14271303C2semidirect
7. #14271157C2semidirect
8. #14271131C2semidirect
9. C29×#419C2semidirect
14. C2×#90809C22semidirect
16. C2×Q8×D29C22semidirect
33. C29×#103C22semidirect
34. #11745168C22semidirect
35. #11745158C22semidirect
36. #11745157C22semidirect
40. C2×Q8×C29D4semidirect
56. #90805D4semidirect
146. #419D29semidirect
179. #103C2×D29semidirect
182. C29#3010semidirect
200. C2×Q8D116semidirect

Maximal subgroups

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

subgroupindex
C2×#117453622
C2×#117452702
C22×#907842
#142717322
#142713112
#142713032
#142711572
#142711312
C29×#4192
#301029