Finite Group Databasebeta

← #16822822

Subgroups — #16822822

Order 3712.

864 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 864 conjugacy classes of subgroups — too many to draw. Only its 159 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×#90807C211yesmaximal, characteristic
2C2×#11745245C212yesmaximal, complemented
3C2×C4⋊C4×D29C211yesmaximal, characteristic, complemented
4#14271084C211yesmaximal, characteristic, complemented
5#14271083C211yesmaximal, characteristic, complemented
6#14271077C211yesmaximal, characteristic, complemented
7#14271039C212yesmaximal, complemented
8#14271038C212yesmaximal, complemented
9#14271035C212yesmaximal, complemented
10#14271025C211yesmaximal, characteristic, complemented
11C29×#394C211yesmaximal, characteristic, complemented
12C22×D4×C29C2211yescharacteristic
13C22×C22⋊D29C2212yes—
14C23×Dic29C2212yes—
15C2×#90807—28no—
16C2×#90807—24no—
17C22×C4×D29C2211yescharacteristic
18C22×C4×D29C2211yescharacteristic
19C22×C4×D29C2211yescharacteristic
20C22×Dic58C2211yescharacteristic
21C2×C4⋊C4×C29C2211yescharacteristic
22C2×C22⋊C4×C29C2212yes—
23C2×C22⋊C4×C29C2212yes—
24C2×C22⋊Dic29C2212yes—
25C2×C22⋊Dic29C2212yes—
26C2×#90781C2212yes—
27C2×C4⋊Dic29C2211yescharacteristic
28C2×C4⋊Dic29C2212yes—
29C2×#90779C2211yescharacteristic
30C2×#90779C2212yes—
31C2×#90779C2211yescharacteristic
32C4⋊C4×D29—24no—
33#11745245—28no—
34C29×#95C2211yescharacteristic
35#11745179C2212yes—
36#11745179C2211yescharacteristic
37#11745179C2212yes—
38#11745179C2212yes—
39C24×C29C2312yes—
40C23×D29C2311yescharacteristic
41C2×D4×C29—24no—
42C2×D4×C29—22no—
43C22×C4×C29C2311yescharacteristic
44C22×C4×C29C2312yes—
45C22×C4×C29C2311yescharacteristic
46C22×C4×C29C2311yescharacteristic
47C2×C22⋊D29—24no—
48C2×C22⋊D29—24no—
49C2×C22⋊D29D418yes—
50C22×Dic29—24no—
51C22×Dic29—24no—
52C22×Dic29—24no—
53C22×Dic29C2311yescharacteristic
54C22×Dic29C2312yes—
55C22×Dic29C2312yes—
56C22×Dic29C2311yescharacteristic
57C22×Dic29C2311yescharacteristic
58#90807—232no—
59C2×C4×D29—24no—
60C2×C4×D29—22no—
61C2×C4×D29—22no—
62C2×C4×D29—22no—
63C2×C4×D29—24no—
64C2×C4×D29—22no—
65C2×C4×D29D414yes—
66C2×Dic58—22no—
67C2×Dic58—24no—
68C4⋊C4×C29—22no—
69C22⋊C4×C29—24no—
70C22⋊C4×C29—24no—
71C22⋊Dic29—24no—
72C22⋊Dic29—24no—
73#90781—24no—
74C4⋊Dic29—24no—
75C4⋊Dic29—22no—
76#90779—22no—
77#90779—24no—
78#90779—22no—
79C23×C29—22no—
80C23×C29—22no—
81C23×C29—22no—
82C23×C29—22no—
83C23×C29—22no—
84C23×C29—22no—
85C23×C29C2×D414yes—
86C23×C29C2411yescharacteristic
87C22×D29—21no—
88C22×D29—21no—
89C22×D29—22no—
90C22×D29C2×D414yes—
91C22×D29C2×D412yes—
92D4×C29—28no—
93C2×C4×C29—21no—
94C2×C4×C29—22no—
95C2×C4×C29—21no—
96C2×C4×C29—22no—
97C2×C4×C29—21no—
98C2×C4×C29—22no—
99C2×C4×C29—22no—
100C2×C4×C29—22no—
101C2×C4×C29C2×D412yes—
102C2×C4×C29#4112yescomplemented
103C22⋊D29—216no—
104C2×Dic29—216no—
105C2×Dic29—21no—
106C2×Dic29—22no—
107C2×Dic29—22no—
108C2×Dic29—22no—
109C2×Dic29—21no—
110C2×Dic29—21no—
111C2×Dic29—22no—
112C2×Dic29—22no—
113C2×Dic29—22no—
114C2×Dic29—22no—
115C2×Dic29C2×D414yes—
116C2×Dic29C2×D412yes—
117C2×Dic29#4112yescomplemented
118C4×D29—28no—
119C4×D29—28no—
120C4×D29—44no—
121Dic58—28no—
122#2406—291nomaximal, Sylow
123C22×C29—24no—
124C22×C29—24no—
125C22×C29—24no—
126C22×C29—42no—
127C22×C29—42no—
128C22×C29—42no—
129C22×C29—42no—
130C22×C29C22×D411yescharacteristic
131C22×C29C22×D412yes—
132C22×C29C2×#4111yescharacteristic
133C22×C29C2×#4111yescharacteristic
134C22×C292−1+412yes—
135C2×D29—22no—
136C2×D29—22no—
137C2×D29—24no—
138C2×D29—24no—
139C2×D29#12014yes—
140C4×C29—22no—
141C4×C29—22no—
142C4×C29—41no—
143C4×C29—42no—
144Dic29—24no—
145Dic29—22no—
146Dic29—22no—
147Dic29—41no—
148Dic29—42no—
149C22×#41—291no—
150C2×#123—292no—
151C22×C4⋊C4—291no—
152#395—291no—
153#395—291no—
154#394—291no—
155#394D2911yescharacteristic, complemented
156#386—292no—
157#385—292no—
158#385—292no—
159#381—291no—
160C2×C29—44no—
161C2×C29C2×#12011yescharacteristic
162C2×C29#52511yescharacteristic
163C2×C29#53611yescharacteristic
164C2×C29#54712yes—
165C2×C29#54712yes—
166D29—24no—
167C2×#41—584no—
168C2×#41—588no—
169C22×Q8—291no—
170C22×D4—292no—
171C22×D4C2×D2911yescharacteristic
172C23×C4—291no—
173C23×C4—292no—
174C23×C4—291no—
175C23×C4—291no—
176#123—588no—
177C2×C4⋊C4—291no—
178C2×C4⋊C4—291no—
179C2×C4⋊C4—291no—
180C2×C4⋊C4—292no—
181C2×C4⋊C4—292no—
182C2×C4⋊C4—584no—
183C2×C4⋊C4C2×D2911yescharacteristic
184C2×C22⋊C4—292no—
185C2×C22⋊C4—292no—
186C2×C22⋊C4—292no—
187C2×C22⋊C4C2×D2912yes—
188C2×C22⋊C4C2×D2912yes—
189#95—292no—
190#95—291no—
191#95—292no—
192#95—292no—
193#95C2×D2911yescharacteristic
194C29#240611yescharacteristic, complemented, Sylow
195C24—291no—
196C24C22×D2912yes—
197#41—5832no—
198C2×Q8—584no—
199C2×Q8—582no—
200C2×D4—24no—
201C2×D4—22no—
202C2×D4—298no—
203C2×D4—584no—
204C2×D4—584no—
205C22×C4—291no—
206C22×C4—291no—
207C22×C4—292no—
208C22×C4—294no—
209C22×C4—292no—
210C22×C4—291no—
211C22×C4—582no—
212C22×C4—582no—
213C22×C4—584no—
214C22×C4—582no—
215C22×C4—584no—
216C22×C4—584no—
217C22×C4—584no—
218C22×C4—584no—
219C22×C4—582no—
220C22×C4C22×D2912yes—
221C22×C4C22×D2911yescharacteristic
222C22×C4C22×D2911yescharacteristic
223C22×C4C22×D2911yescharacteristic
224C4⋊C4—22no—
225C4⋊C4—582no—
226C4⋊C4—582no—
227C4⋊C4—584no—
228C4⋊C4—584no—
229C4⋊C4—582no—
230C22⋊C4—24no—
231C22⋊C4—24no—
232C22⋊C4—584no—
233C22⋊C4—584no—
234C22⋊C4—584no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—22no—
239C23—22no—
240C23—22no—
241C23—294no—
242C23—292no—
243C23—581no—
244C23—581no—
245C23—582no—
246C23D4×D2914yes—
247C23C23×D2911yescharacteristic
248Q8—588no—
249D4—28no—
250D4—5816no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—292no—
260C2×C4—294no—
261C2×C4—292no—
262C2×C4—588no—
263C2×C4—588no—
264C2×C4—5816no—
265C2×C4—581no—
266C2×C4—582no—
267C2×C4—582no—
268C2×C4—582no—
269C2×C4—581no—
270C2×C4—582no—
271C2×C4—582no—
272C2×C4—581no—
273C2×C4—582no—
274C2×C4—582no—
275C2×C4—1164no—
276C2×C4D4×D2912yes—
277C2×C4#9080712yescomplemented
278C22—24no—
279C22—24no—
280C22—24no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—294no—
286C22—584no—
287C22—584no—
288C22—582no—
289C22—582no—
290C22C2×D4×D2912yes—
291C22C2×D4×D2911yescharacteristic
292C22C2×#9080711yescharacteristic
293C22#41×D2911yescharacteristic
294C22#1174536512yes—
295C4—22no—
296C4—22no—
297C4—42no—
298C4—41no—
299C4—582no—
300C4—584no—
301C4—582no—
302C4—1162no—
303C4—1161no—
304C2—44no—
305C2—584no—
306C2#120×D2911yescharacteristic
307C2#1427167311yescharacteristic
308C2#1427173311yescharacteristic
309C2#1427181112yes—
310C2#1427181312yes—
subgroupG/H
center, FrattiniC23C23×D29
commutator, socle, cosocleC23×C29C24
FittingC29×#394C2
prime radicalC22×#90807C2
hypercenter#394D29
hypocenterC29#2406
LCSG ▷ C23×C29 ▷ C29
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#11745245C2semidirect
3. C2×C4⋊C4×D29C2semidirect
4. #14271084C2semidirect
5. #14271083C2semidirect
6. #14271077C2semidirect
7. #14271039C2semidirect
8. #14271038C2semidirect
9. #14271035C2semidirect
10. #14271025C2semidirect
11. C29×#394C2semidirect
102. C2×C4×C29#41semidirect
117. C2×Dic29#41semidirect
155. #394D29semidirect
194. C29#2406semidirect
277. C2×C4#90807semidirect

Maximal subgroups

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

subgroupindex
C22×#908072
C2×#117452452
C2×C4⋊C4×D292
#142710842
#142710832
#142710772
#142710392
#142710382
#142710352
#142710252
C29×#3942
#240629