Finite Group Databasebeta

← #16828742

Subgroups — #16828742

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×#11745294C212yesmaximal, complemented
3#14271306C214yesmaximal, complemented
4#14271301C212yesmaximal, complemented
5#14271296C212yesmaximal, complemented
6#14271162C212yesmaximal, complemented
7#14271127C211yesmaximal, characteristic, complemented
8C2×C29×#121C211yesmaximal, characteristic, complemented
9C22×D4×C29—21no—
10C22×D4×C29C2211yescharacteristic, complemented
11C23×C4×C29C2211yescharacteristic
12C22×C22⋊D29—21no—
13C2×#90805—21no—
14C2×#90805—21no—
15C2×#90805—28no—
16C2×#90805C2212yes—
17C2×#90805C2212yescomplemented
18C22×D116C2211yescharacteristic, complemented
19C22×C4×D29—21no—
20C22×Dic58C2211yescharacteristic, complemented
21C29×#121—21no—
22C29×#121—21no—
23C29×#121C2212yes—
24C29×#121C2212yescomplemented
25C2×C4⋊C4×C29C2211yescharacteristic, complemented
26C2×C22⋊C4×C29—21no—
27C2×#90783C2214yescomplemented
28#11745294—28no—
29C2×#90782C2214yescomplemented
30C2×#90777C2212yescomplemented
31C22⋊#6597C2212yescomplemented
32C22⋊#6597C2212yescomplemented
33#11745158C2214yescomplemented
34#11745157C2214yescomplemented
35C24×C29—21no—
36C24×C29D411yescharacteristic
37C23×D29—21no—
38C2×D4×C29—22no—
39C2×D4×C29—22no—
40C2×D4×C29—24no—
41C2×D4×C29—21no—
42C2×D4×C29—44no—
43C2×D4×C29C2312yes—
44C22×C4×C29—24no—
45C22×C4×C29—21no—
46C22×C4×C29D411yescharacteristic
47C22×C4×C29D411yescharacteristic
48C22×C4×C29D411yescharacteristic
49C22×C4×C29C2311yescharacteristic
50C22×C4×C29C2311yescharacteristic
51C22×C4×C29C2311yescharacteristic
52C2×C22⋊D29—28no—
53C2×C22⋊D29—22no—
54C2×C22⋊D29—22no—
55C22×Dic29—21no—
56#90805—28no—
57#90805—28no—
58#90805—24no—
59#90805—24no—
60#90805—24no—
61#90805D418yescomplemented
62C2×D116—21no—
63C2×D116—24no—
64C2×D116C2312yes—
65C2×C4×D29—28no—
66C2×C4×D29—22no—
67C2×C4×D29—22no—
68C2×Dic58—21no—
69C2×Dic58—24no—
70C2×Dic58C2312yes—
71C4⋊C4×C29—21no—
72C4⋊C4×C29C2312yes—
73C22⋊C4×C29—22no—
74C22⋊C4×C29—22no—
75#90783—28no—
76#90782—28no—
77#90777—24no—
78C2×#6597C2314yes—
79C23×C29—22no—
80C23×C29—22no—
81C23×C29—22no—
82C23×C29—41no—
83C23×C29—42no—
84C23×C29—42no—
85C23×C29—42no—
86C23×C29C2×D411yescharacteristic
87C23×C29C2×D411yescharacteristic
88C23×C29C2×D411yescharacteristic
89C22×D29—24no—
90C22×D29—22no—
91C22×D29—42no—
92C22×D29—42no—
93D4×C29—24no—
94D4×C29—44no—
95D4×C29—44no—
96D4×C29—42no—
97C2×C4×C29—22no—
98C2×C4×C29—22no—
99C2×C4×C29—22no—
100C2×C4×C29—22no—
101C2×C4×C29—22no—
102C2×C4×C29—22no—
103C2×C4×C29—42no—
104C2×C4×C29C2×D414yes—
105C2×C4×C29C2×D411yescharacteristic
106C2×C4×C29C2×D411yescharacteristic
107C2×C4×C29C2×D411yescharacteristic
108C2×C4×C29C2411yescharacteristic
109C22⋊D29—28no—
110C22⋊D29—28no—
111C2×Dic29—24no—
112C2×Dic29—22no—
113D116—24no—
114D116—22no—
115D116C2×D414yescomplemented
116C4×D29—28no—
117C4×D29—28no—
118Dic58—24no—
119Dic58—22no—
120Dic58C2×D414yescomplemented
121#6597—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—42no—
132C22×C29—42no—
133C22×C29—42no—
134C22×C29—84no—
135C22×C29#12012yes—
136C22×C29AGL(1,ℤ/8)12yescomplemented
137C22×C29C22×D411yescharacteristic
138C2×D29—24no—
139C2×D29—48no—
140C2×D29—42no—
141C2×D29—42no—
142C4×C29—21no—
143C4×C29—21no—
144C4×C29—42no—
145C4×C29#12012yes—
146C4×C29C22×D412yes—
147Dic29—24no—
148C22×#41—291no—
149C2×AGL(1,ℤ/8)—292no—
150C2×#121D2911yescharacteristic, complemented
151#449—294no—
152#448—292no—
153#447—292no—
154#417—292no—
155#407—291no—
156C2×C29—22no—
157C2×C29—22no—
158C2×C29—41no—
159C2×C29—82no—
160C2×C29C2×#12011yescharacteristic
161C2×C29C2×AGL(1,ℤ/8)11yescharacteristic
162C2×C29#57511yescharacteristic
163D29—44no—
164C2×#41—292no—
165C2×#41—292no—
166C2×#41—588no—
167C2×#41—581no—
168C2×#41—581no—
169C22×Q8—291no—
170C22×D4—21no—
171C22×D4—291no—
172C22×D4—581no—
173C22×D4C2×D2911yescharacteristic, complemented
174C23×C4—581no—
175C23×C4C2×D2911yescharacteristic
176AGL(1,ℤ/8)—588no—
177C2×SD16—294no—
178C2×D8—294no—
179C2×M16—292no—
180#121—21no—
181#121—21no—
182#121C2×D2912yes—
183#121C2×D2912yescomplemented
184C2×C4⋊C4C2×D2911yescharacteristic, complemented
185C2×C22⋊C4—21no—
186#103—294no—
187#102—294no—
188C22⋊C8—292no—
189C22⋊C8—292no—
190C29#300611yescharacteristic, complemented, Sylow
191C24—21no—
192C24—581no—
193C24C22⋊D2911yescharacteristic
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—22no—
204C2×D4—22no—
205C2×D4—24no—
206C2×D4—21no—
207C2×D4—44no—
208C2×D4—292no—
209C2×D4—584no—
210C2×D4—588no—
211C2×D4—582no—
212C2×D4—582no—
213C2×D4—581no—
214C2×D4C22×D2912yes—
215C22×C4—24no—
216C22×C4—21no—
217C22×C4—588no—
218C22×C4—581no—
219C22×C4—582no—
220C22×C4—582no—
221C22×C4C22⋊D2911yescharacteristic
222C22×C4C22⋊D2911yescharacteristic
223C22×C4C22⋊D2911yescharacteristic
224C22×C4C22×D2911yescharacteristic
225C22×C4C22×D2911yescharacteristic
226C22×C4C22×D2911yescharacteristic
227SD16—588no—
228D8—588no—
229M16—584no—
230C2×C8—294no—
231C4⋊C4—21no—
232C4⋊C4C22×D2912yes—
233C22⋊C4—22no—
234C22⋊C4—22no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—41no—
239C23—42no—
240C23—42no—
241C23—42no—
242C23—584no—
243C23—582no—
244C23—1162no—
245C23—1162no—
246C23C2×C22⋊D2911yescharacteristic
247C23C2×C22⋊D2911yescharacteristic
248C23C2×C22⋊D2911yescharacteristic
249Q8—294no—
250Q8—584no—
251Q8—582no—
252D4—24no—
253D4—44no—
254D4—44no—
255D4—42no—
256D4—294no—
257D4—584no—
258D4—588no—
259D4—588no—
260D4—582no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—42no—
268C2×C4—588no—
269C2×C4—584no—
270C2×C4—582no—
271C2×C4—588no—
272C2×C4D4×D2914yes—
273C2×C4C2×C22⋊D2911yescharacteristic
274C2×C4C2×C22⋊D2911yescharacteristic
275C2×C4C2×C22⋊D2911yescharacteristic
276C2×C4C23×D2911yescharacteristic
277C8—584no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22—21no—
285C22—44no—
286C22—42no—
287C22—42no—
288C22—42no—
289C22—84no—
290C22—584no—
291C22—1168no—
292C22—1162no—
293C22—1162no—
294C22#1174529412yescomplemented
295C22#1174529912yes—
296C22C22×C22⋊D2911yescharacteristic
297C4—21no—
298C4—21no—
299C4—42no—
300C4—584no—
301C4#1174529912yes—
302C4C2×D4×D2912yes—
303C2—22no—
304C2—22no—
305C2—41no—
306C2—82no—
307C2—1164no—
308C2C2×#1174529911yescharacteristic
309C2C2×#1174529411yescharacteristic
310C2#1427196111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D29
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC2×C29×#121C2
socleC22×C29C22×D4
hypercenterC2×#121D29
hypocenterC29#3006
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C22×#90805C2semidirect
2. C2×#11745294C2semidirect
3. #14271306C2semidirect
4. #14271301C2semidirect
5. #14271296C2semidirect
6. #14271162C2semidirect
7. #14271127C2semidirect
8. C2×C29×#121C2semidirect
10. C22×D4×C29C22semidirect
17. C2×#90805C22semidirect
18. C22×D116C22semidirect
20. C22×Dic58C22semidirect
24. C29×#121C22semidirect
25. C2×C4⋊C4×C29C22semidirect
27. C2×#90783C22semidirect
29. C2×#90782C22semidirect
30. C2×#90777C22semidirect
31. C22⋊#6597C22semidirect
32. C22⋊#6597C22semidirect
33. #11745158C22semidirect
34. #11745157C22semidirect
61. #90805D4semidirect
115. D116C2×D4semidirect
120. Dic58C2×D4semidirect
136. C22×C29AGL(1,ℤ/8)semidirect
150. C2×#121D29semidirect
173. C22×D4C2×D29semidirect
183. #121C2×D29semidirect
184. C2×C4⋊C4C2×D29semidirect
190. C29#3006semidirect
294. C22#11745294semidirect

Maximal subgroups

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

subgroupindex
C22×#908052
C2×#117452942
#142713062
#142713012
#142712962
#142711622
#142711272
C2×C29×#1212
#300629