Finite Group Databasebeta

← #16828751

Subgroups — #16828751

Order 3712.

754 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 754 conjugacy classes of subgroups — too many to draw. Only its 141 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×#90809C211yesmaximal, characteristic
2C2×#11745280C212yesmaximal, complemented
3C22⋊Q8×D29C211yesmaximal, characteristic, complemented
4#14271328C212yesmaximal, complemented
5#14271324C212yesmaximal, complemented
6#14271319C211yesmaximal, characteristic, complemented
7#14271316C211yesmaximal, characteristic, complemented
8#14271313C211yesmaximal, characteristic, complemented
9#14271163C212yesmaximal, complemented
10#14271125C211yesmaximal, characteristic, complemented
11C29×#451C211yesmaximal, characteristic, complemented
12C22×Q8×C29C2211yescharacteristic
13C2×#90809—28no—
14C2×#90809—21no—
15C2×#90809—21no—
16C2×#90809C2212yes—
17C2×#90809C2212yes—
18C2×Q8×D29C2211yescharacteristic, complemented
19C22×D116—21no—
20C22×D116C2211yescharacteristic
21C22×C4×D29—21no—
22C22×C4×D29C2211yescharacteristic
23C2×Q16×C29C2212yescomplemented
24C29×C22⋊Q8C2211yescharacteristic
25#11745308C2211yescharacteristic, complemented
26C2×#90785C2212yescomplemented
27C2×#90784C2212yescomplemented
28#11745286C2211yescharacteristic
29#11745280—28no—
30C2×#90773C2212yescomplemented
31C2×#90772C2212yescomplemented
32#11745259C2211yescharacteristic, complemented
33#11745258—21no—
34C4⋊C4×D29—21no—
35C4⋊C4×D29C2211yescharacteristic, complemented
36C22⋊C4×D29—21no—
37#11745240—21no—
38C29×#103C2212yescomplemented
39C29×C22⋊C8C2211yescharacteristic
40#11745183C2212yescomplemented
41C22⋊#6597C2211yescharacteristic
42#11745169C2212yescomplemented
43#11745168C2212yescomplemented
44#11745157C2212yescomplemented
45C23×D29—21no—
46C23×D29D411yescharacteristic
47C2×Q8×C29—24no—
48C2×Q8×C29—21no—
49C2×Q8×C29C2311yescharacteristic
50C2×Q8×C29C2312yes—
51C22×C4×C29—21no—
52C22×C4×C29C2311yescharacteristic
53C22×Dic29D411yescharacteristic
54#90809—28no—
55#90809—28no—
56#90809—24no—
57#90809—24no—
58#90809—24no—
59#90809D418yes—
60Q8×D29—24no—
61C2×D116—28no—
62C2×D116—22no—
63C2×D116—22no—
64C2×D116—21no—
65C2×D116—24no—
66C2×D116C2312yes—
67C2×C4×D29—28no—
68C2×C4×D29—22no—
69C2×C4×D29—22no—
70C2×C4×D29—24no—
71C2×C4×D29—21no—
72C2×C4×D29—21no—
73C2×C4×D29D411yescharacteristic
74C2×C4×D29D411yescharacteristic
75C2×C4×D29C2311yescharacteristic
76C2×C4×D29C2311yescharacteristic
77C2×Dic58—21no—
78C2×Dic58C2311yescharacteristic
79Q16×C29—24no—
80C2×C8×C29C2312yes—
81C4⋊C4×C29—21no—
82C4⋊C4×C29C2311yescharacteristic
83C22⋊C4×C29—21no—
84C22⋊Dic29—21no—
85#90785—24no—
86#90784—24no—
87#90781—21no—
88#90781—21no—
89C4⋊Dic29—21no—
90C4⋊Dic29C2311yescharacteristic
91#90779—21no—
92#90779—21no—
93#90779—21no—
94C2×#6597C2312yes—
95#90773—24no—
96#90772—24no—
97C23×C29C2×D411yescharacteristic
98C22×D29—24no—
99C22×D29—22no—
100C22×D29—22no—
101C22×D29—22no—
102C22×D29—41no—
103C22×D29—42no—
104C22×D29—42no—
105C22×D29C2×D411yescharacteristic
106C22×D29C2×D411yescharacteristic
107Q8×C29—24no—
108Q8×C29—22no—
109Q8×C29—22no—
110Q8×C29C2×D414yes—
111C2×C4×C29—24no—
112C2×C4×C29—22no—
113C2×C4×C29—22no—
114C2×C4×C29—21no—
115C2×C4×C29—21no—
116C2×C4×C29C2×D411yescharacteristic
117C2×C4×C29C2411yescharacteristic
118C2×Dic29—22no—
119C2×Dic29—21no—
120C2×Dic29—21no—
121C2×Dic29C2×D411yescharacteristic
122C2×Dic29C2×D411yescharacteristic
123D116—28no—
124D116—24no—
125D116—28no—
126D116—22no—
127D116C2×D414yes—
128C4×D29—28no—
129C4×D29—22no—
130C4×D29—22no—
131C4×D29—28no—
132C4×D29—22no—
133C4×D29—42no—
134C4×D29—42no—
135C4×D29C2×D414yes—
136Dic58—22no—
137Dic58—42no—
138C8×C29—22no—
139#6597—22no—
140#3007—291nomaximal, Sylow
141C22×C29—21no—
142C22×C29—21no—
143C22×C29#13712yescomplemented
144C22×C29C22×D411yescharacteristic
145C2×D29—21no—
146C2×D29—21no—
147C2×D29—21no—
148C2×D29—24no—
149C2×D29—21no—
150C2×D29—21no—
151C2×D29—44no—
152C2×D29—48no—
153C2×D29—42no—
154C2×D29—42no—
155C2×D29#12012yes—
156C4×C29—24no—
157C4×C29—21no—
158C4×C29—41no—
159C4×C29—41no—
160C4×C29C22×D412yes—
161Dic29—21no—
162Dic29—41no—
163Dic29—41no—
164Dic29#12012yes—
165C22×#41—291no—
166C2×#137—292no—
167C2×C22⋊Q8—291no—
168#452—292no—
169#452—292no—
170#451—291no—
171#451D2911yescharacteristic, complemented
172#450—291no—
173#450—291no—
174#417—292no—
175#407—291no—
176C2×C29—22no—
177C2×C29C2×#12011yescharacteristic
178C2×C29C2×#13711yescharacteristic
179C2×C29#57511yescharacteristic
180D29—22no—
181D29—41no—
182D29—44no—
183C2×#41—292no—
184C2×#41—292no—
185C2×#41—581no—
186C2×#41—581no—
187C2×#41—588no—
188C22×Q8—291no—
189C22×Q8C2×D2911yescharacteristic
190C22×D4—291no—
191C22×D4—581no—
192C23×C4—291no—
193C23×C4—581no—
194#137—588no—
195C2×Q16—292no—
196C2×Q16C2×D2912yescomplemented
197C2×SD16—292no—
198C2×SD16—292no—
199C2×M16—292no—
200C22⋊Q8—291no—
201C22⋊Q8—291no—
202C22⋊Q8—291no—
203C22⋊Q8—581no—
204C22⋊Q8—581no—
205C22⋊Q8C2×D2911yescharacteristic
206C2×C4⋊C4—291no—
207C2×C4⋊C4—581no—
208C2×C22⋊C4—581no—
209#103—292no—
210#103C2×D2912yescomplemented
211#102—292no—
212#102—292no—
213C22⋊C8—291no—
214C22⋊C8—292no—
215C22⋊C8C2×D2911yescharacteristic
216C29#300711yescharacteristic, complemented, Sylow
217C24—291no—
218C24—581no—
219#41—298no—
220#41—584no—
221#41—584no—
222#41—588no—
223#41—588no—
224#41—584no—
225C2×Q8—24no—
226C2×Q8—21no—
227C2×Q8—291no—
228C2×Q8—581no—
229C2×Q8—584no—
230C2×Q8C22×D2911yescharacteristic
231C2×Q8C22×D2912yes—
232C2×D4—292no—
233C2×D4—584no—
234C2×D4—582no—
235C2×D4—582no—
236C2×D4—588no—
237C2×D4—581no—
238C22×C4—21no—
239C22×C4—291no—
240C22×C4—291no—
241C22×C4—291no—
242C22×C4—291no—
243C22×C4—291no—
244C22×C4—584no—
245C22×C4—582no—
246C22×C4—582no—
247C22×C4—588no—
248C22×C4—581no—
249C22×C4—581no—
250C22×C4C22×D2911yescharacteristic
251Q16—24no—
252Q16—584no—
253SD16—584no—
254SD16—584no—
255M16—584no—
256C2×C8—292no—
257C2×C8C22×D2912yes—
258C4⋊C4—21no—
259C4⋊C4—291no—
260C4⋊C4—581no—
261C4⋊C4—581no—
262C4⋊C4—581no—
263C4⋊C4—581no—
264C4⋊C4C22×D2911yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—581no—
267C22⋊C4—581no—
268C22⋊C4—581no—
269C23—291no—
270C23—291no—
271C23—582no—
272C23—584no—
273C23—582no—
274C23—582no—
275C23—1161no—
276C23—1162no—
277C23—1162no—
278C23D4×D2911yescharacteristic
279Q8—24no—
280Q8—22no—
281Q8—22no—
282Q8—582no—
283Q8—1162no—
284Q8D4×D2914yes—
285D4—294no—
286D4—588no—
287D4—588no—
288D4—584no—
289D4—582no—
290C2×C4—24no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—291no—
296C2×C4—294no—
297C2×C4—291no—
298C2×C4—582no—
299C2×C4—588no—
300C2×C4—582no—
301C2×C4—588no—
302C2×C4—582no—
303C2×C4—582no—
304C2×C4—581no—
305C2×C4—581no—
306C2×C4—1162no—
307C2×C4—1162no—
308C2×C4D4×D2911yescharacteristic
309C2×C4C23×D2911yescharacteristic
310C8—22no—
311C8—582no—
312C22—21no—
313C22—21no—
314C22—292no—
315C22—581no—
316C22—581no—
317C22—584no—
318C22—581no—
319C22—581no—
320C22—581no—
321C22—1164no—
322C22—1168no—
323C22—1162no—
324C22—1162no—
325C22#1174528012yescomplemented
326C22C2×D4×D2911yescharacteristic
327C4—24no—
328C4—21no—
329C4—41no—
330C4—41no—
331C4—292no—
332C4—581no—
333C4—1161no—
334C4—1161no—
335C4C2×D4×D2912yes—
336C2—22no—
337C2—582no—
338C2—1161no—
339C2—1164no—
340C2#120×D2911yescharacteristic
341C2C2×#1174528011yescharacteristic
342C2#1427196211yescharacteristic
subgroupG/H
centerC22C2×D4×D29
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC29×#451C2
socleC22×C29C22×D4
prime radicalC22×#90809C2
hypercenter#451D29
hypocenterC29#3007
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
2. C2×#11745280C2semidirect
3. C22⋊Q8×D29C2semidirect
4. #14271328C2semidirect
5. #14271324C2semidirect
6. #14271319C2semidirect
7. #14271316C2semidirect
8. #14271313C2semidirect
9. #14271163C2semidirect
10. #14271125C2semidirect
11. C29×#451C2semidirect
18. C2×Q8×D29C22semidirect
23. C2×Q16×C29C22semidirect
25. #11745308C22semidirect
26. C2×#90785C22semidirect
27. C2×#90784C22semidirect
30. C2×#90773C22semidirect
31. C2×#90772C22semidirect
32. #11745259C22semidirect
35. C4⋊C4×D29C22semidirect
38. C29×#103C22semidirect
40. #11745183C22semidirect
42. #11745169C22semidirect
43. #11745168C22semidirect
44. #11745157C22semidirect
143. C22×C29#137semidirect
171. #451D29semidirect
196. C2×Q16C2×D29semidirect
210. #103C2×D29semidirect
216. C29#3007semidirect
325. C22#11745280semidirect

Maximal subgroups

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

subgroupindex
C22×#908092
C2×#117452802
C22⋊Q8×D292
#142713282
#142713242
#142713192
#142713162
#142713132
#142711632
#142711252
C29×#4512
#300729