Finite Group Databasebeta

← #16828780

Subgroups — #16828780

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×#11745365C211yesmaximal, characteristic, complemented
2C2×#11745270C211yesmaximal, characteristic, complemented
3C22×#90773C211yesmaximal, characteristic, complemented
4#14271727C211yesmaximal, characteristic, complemented
5#14271310C214yesmaximal, complemented
6#14271304C214yesmaximal, complemented
7#14271173C211yesmaximal, characteristic, complemented
8#14271160C211yesmaximal, characteristic, complemented
9C29×#408C211yesmaximal, characteristic, complemented
10C2×#41×C29C2211yescharacteristic
11#11745365—24no—
12#11745365—24no—
13C2×Q8×D29—21no—
14C2×#90807—21no—
15C2×#90807—22no—
16C2×#90805C2211yescharacteristic
17C2×#90805C2212yescomplemented
18C22×D116C2211yescharacteristic
19C22×Dic58C2211yescharacteristic
20C22×Dic58C2212yescomplemented
21C2×M16×C29C2211yescharacteristic
22C22×C8×C29C2211yescharacteristic
23#11745309—21no—
24#11745302—21no—
25#11745299—22no—
26#11745292C2214yes—
27#11745288C2211yescharacteristic
28#11745270—24no—
29C2×D232C2212yes—
30C2×#90773—24no—
31C2×#90773C2212yes—
32C2×#90773C2214yes—
33C29×C22⋊C8C2214yescomplemented
34#11745169C2212yescomplemented
35#11745164C2212yescomplemented
36#11745164C2214yescomplemented
37C23×D29—21no—
38#41×C29—22no—
39#41×C29—22no—
40C2×Q8×C29D411yescharacteristic
41C2×D4×C29D412yes—
42C2×D4×C29D411yescharacteristic
43C22×C4×C29C2312yes—
44C22×C4×C29C2311yescharacteristic
45C2×C22⋊D29—21no—
46C2×C22⋊D29—22no—
47C2×C22⋊D29—24no—
48C22×Dic29—21no—
49C22×Dic29—22no—
50Q8×D29—28no—
51#90807—216no—
52#90807—28no—
53#90805—22no—
54#90805—24no—
55#90805—24no—
56#90805D414yescomplemented
57C2×D116—22no—
58C2×D116—22no—
59C2×D116—21no—
60C2×D116C2312yes—
61C2×D116C2311yescharacteristic
62C2×C4×D29—21no—
63C2×C4×D29—22no—
64C2×Dic58—21no—
65C2×Dic58—22no—
66C2×Dic58—22no—
67C2×Dic58—24no—
68C2×Dic58—24no—
69C2×Dic58D414yescomplemented
70C2×Dic58C2312yes—
71C2×Dic58C2311yescharacteristic
72M16×C29—22no—
73C2×C8×C29—22no—
74C2×C8×C29C2312yes—
75C2×C8×C29C2312yes—
76C22⋊Dic29—21no—
77#90781—24no—
78C4⋊Dic29C2312yes—
79C4×Dic29—21no—
80D232—24no—
81#90773—28no—
82#90773—24no—
83C23×C29C2×D412yes—
84C23×C29C2×D411yescharacteristic
85C22×D29—21no—
86C22×D29—22no—
87C22×D29—42no—
88C22×D29—82no—
89Q8×C29—22no—
90D4×C29—24no—
91D4×C29—22no—
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—28no—
100C22⋊D29—24no—
101C22⋊D29—84no—
102C2×Dic29—28no—
103C2×Dic29—24no—
104C2×Dic29—22no—
105C2×Dic29—21no—
106C2×Dic29—22no—
107D116—21no—
108D116—22no—
109D116—22no—
110D116—42no—
111D116C2×D412yes—
112C4×D29—28no—
113C4×D29—24no—
114Dic58—28no—
115Dic58—24no—
116Dic58—22no—
117Dic58—21no—
118Dic58C2×D412yes—
119Dic58C2×D414yes—
120C8×C29—22no—
121C8×C29—22no—
122#3010—291nomaximal, Sylow
123C22×C29—22no—
124C22×C29—22no—
125C22×C29—24no—
126C22×C29#12012yes—
127C22×C29C22×D411yescharacteristic
128C2×D29—22no—
129C2×D29—42no—
130C2×D29—42no—
131C2×D29—44no—
132C2×D29—84no—
133C4×C29—22no—
134C4×C29#12012yes—
135C4×C29C22×D411yescharacteristic
136C4×C29C22×D411yescharacteristic
137Dic29—22no—
138Dic29—24no—
139Dic29—42no—
140C2×2−1+4—291no—
141C2×AGL(1,ℤ/8)—291no—
142C22×SD16—291no—
143#535—291no—
144#449—294no—
145#448—294no—
146#419—291no—
147#416—291no—
148#408D2911yescharacteristic, complemented
149C2×C29—22no—
150C2×C29—42no—
151C2×C29C2×#12011yescharacteristic
152C2×C29#57712yes—
153D29—42no—
154D29—82no—
1552−1+4—584no—
1562−1+4—584no—
157C2×#41—291no—
158C2×#41—292no—
159C2×#41—582no—
160C2×#41—581no—
161C2×#41C2×D2911yescharacteristic
162C22×Q8—291no—
163C22×Q8—292no—
164C22×Q8—581no—
165C22×D4—291no—
166AGL(1,ℤ/8)—584no—
167C2×SD16—294no—
168C2×SD16—292no—
169C2×SD16—584no—
170C2×D8—292no—
171C2×M16C2×D2911yescharacteristic
172C22×C8C2×D2911yescharacteristic
173#127—581no—
174#124—581no—
175#121—294no—
176#120—582no—
177#117—291no—
178#103—292no—
179#103—294no—
180#102—292no—
181C22⋊C8C2×D2914yescomplemented
182C29#301011yescharacteristic, complemented, Sylow
183C24—581no—
184#41—22no—
185#41—22no—
186#41—294no—
187#41—584no—
188#41—584no—
189#41—5816no—
190#41—588no—
191#41—582no—
192C2×Q8—294no—
193C2×Q8—292no—
194C2×Q8—291no—
195C2×Q8—584no—
196C2×Q8—584no—
197C2×Q8—588no—
198C2×Q8—582no—
199C2×Q8—581no—
200C2×Q8—582no—
201C2×Q8D11611yescharacteristic
202C2×D4—291no—
203C2×D4—292no—
204C2×D4—582no—
205C2×D4—581no—
206C2×D4—584no—
207C2×D4—581no—
208C2×D4—582no—
209C2×D4—582no—
210C2×D4D11612yes—
211C2×D4D11611yescharacteristic
212C22×C4—582no—
213C22×C4—581no—
214C22×C4—582no—
215C22×C4—581no—
216C22×C4C22×D2912yes—
217C22×C4C22×D2911yescharacteristic
218SD16—588no—
219SD16—584no—
220D8—584no—
221M16—22no—
222C2×C8—22no—
223C2×C8C22×D2912yes—
224C2×C8C22×D2912yes—
225C4⋊C4—292no—
226C22⋊C4—581no—
227C22⋊C4—584no—
228C42—581no—
229C23—581no—
230C23—582no—
231C23—1162no—
232C23—2322no—
233C23C2×D11612yes—
234C23C2×D11611yescharacteristic
235Q8—22no—
236Q8—294no—
237Q8—292no—
238Q8—588no—
239Q8—582no—
240Q8—581no—
241Q8—584no—
242D4—24no—
243D4—22no—
244D4—292no—
245D4—588no—
246D4—584no—
247D4—581no—
248D4—582no—
249D4—582no—
250D4—1162no—
251D4—2324no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—588no—
255C2×C4—588no—
256C2×C4—582no—
257C2×C4—581no—
258C2×C4—584no—
259C2×C4—584no—
260C2×C4—582no—
261C2×C4C2×D11612yes—
262C2×C4C2×D11611yescharacteristic
263C2×C4D4×D2912yes—
264C2×C4D4×D2912yes—
265C2×C4C23×D2911yescharacteristic
266C8—22no—
267C8—22no—
268C22—22no—
269C22—22no—
270C22—24no—
271C22—582no—
272C22—1162no—
273C22—1162no—
274C22—1164no—
275C22—2324no—
276C22#1174524412yes—
277C22C22×D11611yescharacteristic
278C4—22no—
279C4—584no—
280C4—582no—
281C4—1162no—
282C4#1174524412yes—
283C4C2×D4×D2911yescharacteristic
284C4C2×D4×D2911yescharacteristic
285C2—22no—
286C2—42no—
287C2—1162no—
288C2—2322no—
289C2C2×#1174524411yescharacteristic
290C2#1427197512yes—
subgroupG/H
centerC22C22×D116
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC29×#408C2
socleC22×C29C22×D4
hypercenter#408D29
hypocenterC29#3010
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
1. C2×#11745365C2semidirect
2. C2×#11745270C2semidirect
3. C22×#90773C2semidirect
4. #14271727C2semidirect
5. #14271310C2semidirect
6. #14271304C2semidirect
7. #14271173C2semidirect
8. #14271160C2semidirect
9. C29×#408C2semidirect
17. C2×#90805C22semidirect
20. C22×Dic58C22semidirect
33. C29×C22⋊C8C22semidirect
34. #11745169C22semidirect
35. #11745164C22semidirect
36. #11745164C22semidirect
56. #90805D4semidirect
69. C2×Dic58D4semidirect
148. #408D29semidirect
181. C22⋊C8C2×D29semidirect
182. C29#3010semidirect

Maximal subgroups

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

subgroupindex
C2×#117453652
C2×#117452702
C22×#907732
#142717272
#142713102
#142713042
#142711732
#142711602
C29×#4082
#301029