Finite Group Databasebeta

← #16822706

Subgroups — #16822706

Order 3712.

944 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 944 conjugacy classes of subgroups — too many to draw. Only its 163 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, complemented
2C2×#11745255C211yesmaximal, characteristic, complemented
3C2×#11745243C212yesmaximal, complemented
4#14271083C211yesmaximal, characteristic, complemented
5#14271078C212yesmaximal, complemented
6#14271076C211yesmaximal, characteristic, complemented
7#14271072C212yesmaximal, complemented
8#14271071C211yesmaximal, characteristic, complemented
9C22⋊C4×Dic29C212yesmaximal, complemented
10#14271010C211yesmaximal, characteristic, complemented
11C29×#394C211yesmaximal, characteristic, complemented
12C22×D4×C29C2211yescharacteristic
13C22×C22⋊D29C2212yes—
14C22×C22⋊D29C2212yes—
15C23×Dic29C2212yes—
16C2×#90807—28no—
17C2×#90807—24no—
18C22×D116C2211yescharacteristic
19C22×C4×D29C2211yescharacteristic
20C22×C4×D29C2211yescharacteristic
21C22×Dic58C2211yescharacteristic
22C2×C4⋊C4×C29C2211yescharacteristic
23C2×C22⋊C4×C29C2212yes—
24C2×C22⋊C4×C29C2212yes—
25C2×C22⋊Dic29C2212yes—
26C2×#90781C2212yes—
27C2×#90781C2211yescharacteristic
28C2×#90781C2212yes—
29C2×#90781C2211yescharacteristic
30C2×#90779C2212yes—
31C2×C4×Dic29C2212yes—
32C2×C4×Dic29C2211yescharacteristic
33#11745255—24no—
34#11745243—28no—
35C29×#95C2211yescharacteristic
36#11745179C2211yescharacteristic
37#11745179C2212yes—
38#11745179C2212yes—
39C24×C29C2312yes—
40C23×D29C2311yescharacteristic
41C23×D29C2311yescharacteristic
42C2×D4×C29—24no—
43C2×D4×C29—22no—
44C22×C4×C29C2312yes—
45C22×C4×C29C2311yescharacteristic
46C22×C4×C29C2311yescharacteristic
47C22×C4×C29C2311yescharacteristic
48C2×C22⋊D29—24no—
49C2×C22⋊D29—24no—
50C2×C22⋊D29—24no—
51C2×C22⋊D29—24no—
52C2×C22⋊D29—24no—
53C2×C22⋊D29—24no—
54C22×Dic29—24no—
55C22×Dic29—24no—
56C22×Dic29D418yescomplemented
57C22×Dic29C2311yescharacteristic
58C22×Dic29C2312yes—
59C22×Dic29C2312yes—
60C22×Dic29C2311yescharacteristic
61#90807—232no—
62C2×D116—22no—
63C2×D116—22no—
64C2×D116—22no—
65C2×C4×D29—24no—
66C2×C4×D29—24no—
67C2×C4×D29—22no—
68C2×C4×D29D414yescomplemented
69C2×Dic58—22no—
70C2×Dic58—24no—
71C4⋊C4×C29—22no—
72C22⋊C4×C29—24no—
73C22⋊C4×C29—24no—
74C22⋊Dic29—24no—
75#90781—22no—
76#90781—22no—
77#90781—24no—
78#90781—24no—
79#90779—24no—
80C4×Dic29—24no—
81C4×Dic29—22no—
82C23×C29—22no—
83C23×C29—22no—
84C23×C29—22no—
85C23×C29—22no—
86C23×C29—22no—
87C23×C29—22no—
88C23×C29C2×D414yes—
89C23×C29C2411yescharacteristic
90C22×D29—21no—
91C22×D29—22no—
92C22×D29—22no—
93C22×D29—22no—
94C22×D29—21no—
95C22×D29—21no—
96C22×D29—21no—
97C22×D29—22no—
98C22×D29C2×D412yes—
99C22×D29#4112yescomplemented
100D4×C29—28no—
101C2×C4×C29—21no—
102C2×C4×C29—22no—
103C2×C4×C29—21no—
104C2×C4×C29—22no—
105C2×C4×C29—22no—
106C2×C4×C29—21no—
107C2×C4×C29—22no—
108C2×C4×C29—22no—
109C2×C4×C29C2×D412yes—
110C2×C4×C29#4112yescomplemented
111C22⋊D29—216no—
112C22⋊D29—216no—
113C2×Dic29—22no—
114C2×Dic29—22no—
115C2×Dic29—22no—
116C2×Dic29—22no—
117C2×Dic29—216no—
118C2×Dic29—21no—
119C2×Dic29—22no—
120C2×Dic29C2×D414yes—
121C2×Dic29C2×D414yes—
122C2×Dic29C2×D412yes—
123C2×Dic29#4114yescomplemented
124D116—28no—
125C4×D29—28no—
126C4×D29—44no—
127Dic58—28no—
128#2388—291nomaximal, Sylow
129C22×C29—24no—
130C22×C29—24no—
131C22×C29—24no—
132C22×C29—42no—
133C22×C29—42no—
134C22×C29—42no—
135C22×C29—42no—
136C22×C29C22×D411yescharacteristic
137C22×C29C22×D412yes—
138C22×C29C2×#4111yescharacteristic
139C22×C29C2×#4111yescharacteristic
140C22×C29C2×#4112yes—
141C2×D29—22no—
142C2×D29—22no—
143C2×D29—22no—
144C2×D29—24no—
145C2×D29—22no—
146C2×D29—41no—
147C2×D29—42no—
148C2×D29—42no—
149C2×D29—42no—
150C2×D29—41no—
151C4×C29—22no—
152C4×C29—22no—
153C4×C29—41no—
154C4×C29—42no—
155Dic29—24no—
156Dic29—24no—
157Dic29—41no—
158Dic29#12014yes—
159C22×#41—291no—
160C2×C4×D4—291no—
161C2×C4×D4—292no—
162#395—291no—
163#394—292no—
164#394D2911yescharacteristic, complemented
165#393—291no—
166#392—291no—
167#392—292no—
168C4×C22⋊C4—292no—
169C4×C22⋊C4—291no—
170C2×C29—44no—
171C2×C29C2×#12011yescharacteristic
172C2×C29#52512yes—
173C2×C29#52511yescharacteristic
174C2×C29#53212yes—
175C2×C29#53211yescharacteristic
176D29—42no—
177D29—42no—
178C2×#41—588no—
179C2×#41—584no—
180C22×Q8—291no—
181C22×D4—292no—
182C22×D4—292no—
183C22×D4—291no—
184C22×D4C2×D2911yescharacteristic
185C23×C4—292no—
186C23×C4—291no—
187C23×C4—291no—
188C4×D4—588no—
189C4×D4—584no—
190C2×C4⋊C4—292no—
191C2×C4⋊C4C2×D2911yescharacteristic
192C2×C22⋊C4—292no—
193C2×C22⋊C4—292no—
194C2×C22⋊C4—291no—
195C2×C22⋊C4—291no—
196C2×C22⋊C4—292no—
197C2×C22⋊C4C2×D2912yes—
198C2×C22⋊C4C2×D2912yes—
199C2×C42—292no—
200C2×C42—291no—
201#95—291no—
202#95—292no—
203#95—292no—
204#95C2×D2911yescharacteristic
205C29#238811yescharacteristic, complemented, Sylow
206C24—291no—
207C24—291no—
208C24C22×D2912yes—
209#41—5832no—
210C2×Q8—584no—
211C2×Q8—582no—
212C2×D4—24no—
213C2×D4—22no—
214C2×D4—582no—
215C2×D4—582no—
216C2×D4—584no—
217C2×D4—584no—
218C2×D4—584no—
219C2×D4—584no—
220C2×D4—584no—
221C2×D4—584no—
222C2×D4—582no—
223C22×C4—291no—
224C22×C4—298no—
225C22×C4—292no—
226C22×C4—292no—
227C22×C4—294no—
228C22×C4—291no—
229C22×C4—584no—
230C22×C4—584no—
231C22×C4—582no—
232C22×C4—584no—
233C22×C4—584no—
234C22×C4C22×D2912yes—
235C22×C4C22×D2911yescharacteristic
236C22×C4C22×D2911yescharacteristic
237C22×C4C22×D2911yescharacteristic
238C4⋊C4—22no—
239C4⋊C4—584no—
240C22⋊C4—24no—
241C22⋊C4—24no—
242C22⋊C4—584no—
243C22⋊C4—584no—
244C22⋊C4—582no—
245C22⋊C4—584no—
246C22⋊C4—582no—
247C42—582no—
248C42—584no—
249C23—22no—
250C23—22no—
251C23—22no—
252C23—22no—
253C23—22no—
254C23—22no—
255C23—292no—
256C23—292no—
257C23—582no—
258C23—581no—
259C23—582no—
260C23—581no—
261C23—581no—
262C23—582no—
263C23—582no—
264C23—581no—
265C23D4×D2914yes—
266C23C23×D2911yescharacteristic
267Q8—588no—
268D4—28no—
269D4—588no—
270D4—5816no—
271D4—5816no—
272C2×C4—21no—
273C2×C4—22no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—294no—
281C2×C4—292no—
282C2×C4—294no—
283C2×C4—294no—
284C2×C4—588no—
285C2×C4—5816no—
286C2×C4—582no—
287C2×C4—582no—
288C2×C4—582no—
289C2×C4—582no—
290C2×C4—581no—
291C2×C4—582no—
292C2×C4—1164no—
293C2×C4D4×D2912yes—
294C2×C4#9080712yescomplemented
295C22—24no—
296C22—24no—
297C22—24no—
298C22—42no—
299C22—42no—
300C22—42no—
301C22—42no—
302C22—582no—
303C22—584no—
304C22—582no—
305C22—582no—
306C22—582no—
307C22—1161no—
308C22—1162no—
309C22—1162no—
310C22—1162no—
311C22—1161no—
312C22C2×D4×D2912yes—
313C22C2×D4×D2911yescharacteristic
314C22C2×#9080711yescharacteristic
315C22#41×D2911yescharacteristic
316C22#41×D2912yes—
317C4—22no—
318C4—22no—
319C4—41no—
320C4—42no—
321C4—294no—
322C4—584no—
323C4—584no—
324C4—1161no—
325C2—44no—
326C2—1162no—
327C2—1162no—
328C2#120×D2911yescharacteristic
329C2#1427167112yes—
330C2#1427167311yescharacteristic
331C2#1427170812yes—
332C2#1427171211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D29
commutator, socle, cosocleC23×C29C24
FittingC29×#394C2
hypercenter#394D29
hypocenterC29#2388
LCSG ▷ C23×C29 ▷ C29
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C22×#90807C2semidirect
2. C2×#11745255C2semidirect
3. C2×#11745243C2semidirect
4. #14271083C2semidirect
5. #14271078C2semidirect
6. #14271076C2semidirect
7. #14271072C2semidirect
8. #14271071C2semidirect
9. C22⋊C4×Dic29C2semidirect
10. #14271010C2semidirect
11. C29×#394C2semidirect
56. C22×Dic29D4semidirect
68. C2×C4×D29D4semidirect
99. C22×D29#41semidirect
110. C2×C4×C29#41semidirect
123. C2×Dic29#41semidirect
164. #394D29semidirect
205. C29#2388semidirect
294. 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×#117452552
C2×#117452432
#142710832
#142710782
#142710762
#142710722
#142710712
C22⋊C4×Dic292
#142710102
C29×#3942
#238829