Finite Group Databasebeta

← #16831310

Subgroups — #16831310

Order 3712.

462 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 139 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
1D4×D116C211yesmaximal, characteristic
2#14271511C211yesmaximal, characteristic, complemented
3#14271481C212yesmaximal, complemented
4#14271450C211yesmaximal, characteristic, complemented
5#14271394C212yesmaximal, complemented
6#14271363C211yesmaximal, characteristic, complemented
7#14271313C212yesmaximal, complemented
8C4×#90784C211yesmaximal, characteristic, complemented
9D4×#6597C211yesmaximal, characteristic, complemented
10C2×#11745157C212yesmaximal, complemented
11C29×#554C211yesmaximal, characteristic, complemented
12C2×D4×D29—21no—
13C22×D116C2212yes—
14C29×C4⋊Q8C2211yescharacteristic, complemented
15C29×#125—21no—
16C29×C22⋊Q8—22no—
17C29×C22⋊Q8C2212yescomplemented
18C4×Q8×C29C2211yescharacteristic, complemented
19C4×D4×C29—21no—
20C4×D4×C29C2211yescharacteristic
21C2×C4⋊C4×C29C2212yescomplemented
22C2×#90784C2211yescharacteristic
23C2×#90784C2212yescomplemented
24#11745292C2212yes—
25C22×#6597C2212yescomplemented
26#11745257—21no—
27#11745244—22no—
28#11745236C2211yescharacteristic
29C4×D116C2211yescharacteristic
30#11745183C2211yescharacteristic, complemented
31C22⋊#6597C2212yescomplemented
32#11745157C2211yescharacteristic
33#11745157C2212yes—
34#11745157C2212yescomplemented
35#11745157C2212yescomplemented
36#11745157C2212yescomplemented
37#11745156C2211yescharacteristic, complemented
38#11745156C2212yescomplemented
39C4⋊#6597C2211yescharacteristic, complemented
40C4×#6597C2211yescharacteristic, complemented
41C23×D29—22no—
42C2×Q8×C29—21no—
43C2×Q8×C29C2311yescharacteristic
44C2×D4×C29D411yescharacteristic
45C22×C4×C29—22no—
46C22×C4×C29C2312yes—
47C2×C22⋊D29—22no—
48D4×D29—44no—
49C2×D116—24no—
50C2×D116—22no—
51C2×D116—21no—
52C2×D116—21no—
53C2×D116—42no—
54C2×D116C2312yes—
55C2×D116C2311yescharacteristic
56C2×C4×D29—21no—
57C4⋊C4×C29—22no—
58C4⋊C4×C29—21no—
59C4⋊C4×C29—21no—
60C4⋊C4×C29—22no—
61C4⋊C4×C29D411yescharacteristic
62C4⋊C4×C29C2311yescharacteristic
63C4⋊C4×C29C2312yes—
64C22⋊C4×C29—22no—
65C22⋊C4×C29D412yes—
66C42×C29—21no—
67C42×C29C2311yescharacteristic
68#90784—22no—
69#90784—42no—
70#90781—22no—
71#90781—21no—
72C4⋊Dic29C2311yescharacteristic
73C2×#6597—42no—
74C2×#6597C2311yescharacteristic
75C2×#6597C2312yes—
76C2×#6597C2311yescharacteristic
77C23×C29C2×D412yes—
78C22×D29—21no—
79C22×D29—22no—
80C22×D29—44no—
81C22×D29—44no—
82C22×D29—82no—
83Q8×C29—21no—
84Q8×C29—21no—
85Q8×C29—81no—
86D4×C29SD1614yescomplemented
87C2×C4×C29—22no—
88C2×C4×C29—22no—
89C2×C4×C29—21no—
90C2×C4×C29—21no—
91C2×C4×C29—22no—
92C2×C4×C29—82no—
93C2×C4×C29C2×D411yescharacteristic
94C2×C4×C29C2×D412yes—
95C2×C4×C29C2×D411yescharacteristic
96C2×C4×C29C2411yescharacteristic
97C22⋊D29—44no—
98C2×Dic29—21no—
99D116—22no—
100D116—21no—
101D116—22no—
102D116—44no—
103D116—42no—
104D116—81no—
105D116#4112yes—
106C4×D29—42no—
107#6597—21no—
108#6597—21no—
109#6597—22no—
110#3295—291nomaximal, Sylow
111C22×C29—22no—
112C22×C29—22no—
113C22×C29C2×SD1614yes—
114C22×C29C22×D411yescharacteristic
115C2×D29—22no—
116C2×D29—41no—
117C2×D29—41no—
118C2×D29—42no—
119C2×D29—42no—
120C2×D29—42no—
121C2×D29—88no—
122C4×C29—22no—
123C4×C29—21no—
124C4×C29—41no—
125C4×C29—41no—
126C4×C29—42no—
127C4×C29C2×SD1612yes—
128C4×C29C2×#4111yescharacteristic
129C4×C29H(5,𝔽2)11yescharacteristic
130Dic29—41no—
131#554D2911yescharacteristic, complemented
132D42—291no—
133#486—291no—
134#481—292no—
135#476—291no—
136#465—292no—
137#460—291no—
138#450—292no—
139C4×SD16—291no—
140C8×D4—291no—
141C2×#102—292no—
142C2×C29—24no—
143C2×C29#54611yescharacteristic
144C2×C29C22×SD1611yescharacteristic
145C2×C29#57611yescharacteristic
146D29—42no—
147D29—82no—
148C22×D4—292no—
149C22×D4—581no—
150C2×SD16—291no—
151C2×SD16—292no—
152C22×C8—292no—
153C4⋊Q8C2×D2911yescharacteristic, complemented
154#127—291no—
155#125—21no—
156C22⋊Q8—22no—
157C22⋊Q8C2×D2912yescomplemented
158#121—292no—
159#121—581no—
160#120—582no—
161C4×Q8C2×D2911yescharacteristic, complemented
162C4×D4—21no—
163C4×D4—291no—
164C4×D4C2×D2911yescharacteristic
165C2×C4⋊C4C2×D2912yescomplemented
166#106—291no—
167#106—292no—
168C4⋊C8—291no—
169#103—291no—
170#102—291no—
171#102—292no—
172#102—292no—
173#102—292no—
174#102—292no—
175C22⋊C8—292no—
176C4×C8—291no—
177C29#329511yescharacteristic, complemented, Sylow
178C24—582no—
179C2×Q8—21no—
180C2×Q8C22×D2911yescharacteristic
181C2×D4—291no—
182C2×D4—292no—
183C2×D4—584no—
184C2×D4—582no—
185C2×D4—582no—
186C2×D4—581no—
187C2×D4—581no—
188C2×D4—1162no—
189C2×D4—1164no—
190C2×D4C22⋊D2911yescharacteristic
191C22×C4—22no—
192C22×C4—581no—
193C22×C4C22×D2912yes—
194SD16—582no—
195SD16—1162no—
196C2×C8—291no—
197C2×C8—291no—
198C2×C8—292no—
199C2×C8—1162no—
200C4⋊C4—22no—
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—22no—
204C4⋊C4—291no—
205C4⋊C4C22⋊D2911yescharacteristic
206C4⋊C4C22×D2911yescharacteristic
207C4⋊C4C22×D2912yes—
208C22⋊C4—22no—
209C22⋊C4—581no—
210C22⋊C4—582no—
211C22⋊C4C22⋊D2912yes—
212C42—21no—
213C42C22×D2911yescharacteristic
214C23—581no—
215C23—582no—
216C23—1164no—
217C23—1164no—
218C23—2322no—
219C23C2×C22⋊D2912yes—
220Q8—21no—
221Q8—21no—
222Q8—81no—
223D4—292no—
224D4—582no—
225D4—581no—
226D4—582no—
227D4—1164no—
228D4—1164no—
229D4—1162no—
230D4—2321no—
231D4#9078414yescomplemented
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—82no—
238C2×C4—581no—
239C2×C4—1162no—
240C2×C4C2×C22⋊D2911yescharacteristic
241C2×C4C2×C22⋊D2912yes—
242C2×C4C2×C22⋊D2911yescharacteristic
243C2×C4C23×D2911yescharacteristic
244C8—581no—
245C8—581no—
246C8—582no—
247C22—22no—
248C22—22no—
249C22—582no—
250C22—1161no—
251C22—1161no—
252C22—1162no—
253C22—1162no—
254C22—1162no—
255C22—2328no—
256C22C2×#9078414yes—
257C22C22×C22⋊D2911yescharacteristic
258C4—22no—
259C4—21no—
260C4—41no—
261C4—41no—
262C4—42no—
263C4—1161no—
264C4C2×#9078412yes—
265C4#41×D2911yescharacteristic
266C4#1174536411yescharacteristic
267C2—24no—
268C2—1162no—
269C2—2322no—
270C2#1427180111yescharacteristic
271C2C22×#9078411yescharacteristic
272C2#1427196611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D29
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC29×#554C2
socleC22×C29C22×D4
prime radicalD4×D116C2
hypercenter#554D29
hypocenterC29#3295
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C4×D4 ◁ #554

Complements

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

NH ≅ G/Ntype
2. #14271511C2semidirect
3. #14271481C2semidirect
4. #14271450C2semidirect
5. #14271394C2semidirect
6. #14271363C2semidirect
7. #14271313C2semidirect
8. C4×#90784C2semidirect
9. D4×#6597C2semidirect
10. C2×#11745157C2semidirect
11. C29×#554C2semidirect
14. C29×C4⋊Q8C22semidirect
17. C29×C22⋊Q8C22semidirect
18. C4×Q8×C29C22semidirect
21. C2×C4⋊C4×C29C22semidirect
23. C2×#90784C22semidirect
25. C22×#6597C22semidirect
30. #11745183C22semidirect
31. C22⋊#6597C22semidirect
34. #11745157C22semidirect
35. #11745157C22semidirect
36. #11745157C22semidirect
37. #11745156C22semidirect
38. #11745156C22semidirect
39. C4⋊#6597C22semidirect
40. C4×#6597C22semidirect
86. D4×C29SD16semidirect
131. #554D29semidirect
153. C4⋊Q8C2×D29semidirect
157. C22⋊Q8C2×D29semidirect
161. C4×Q8C2×D29semidirect
165. C2×C4⋊C4C2×D29semidirect
177. C29#3295semidirect
231. D4#90784semidirect

Maximal subgroups

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

subgroupindex
D4×D1162
#142715112
#142714812
#142714502
#142713942
#142713632
#142713132
C4×#907842
D4×#65972
C2×#117451572
C29×#5542
#329529