Finite Group Databasebeta

← #16828718

Subgroups — #16828718

Order 3712.

994 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 157 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
1C23×D116C211yesmaximal, characteristic, complemented
2C2×#11745311C212yesmaximal, complemented
3#14271313C214yesmaximal, complemented
4#14271296C214yesmaximal, complemented
5#14271167C212yesmaximal, complemented
6#14271127C211yesmaximal, characteristic, complemented
7C29×#525C211yesmaximal, characteristic, complemented
8C24×D29—21no—
9C2×#41×C29C2211yescharacteristic, complemented
10C23×C4×C29C2211yescharacteristic
11C22×D116—22no—
12C22×D116—21no—
13C22×D116—28no—
14C22×D116C2214yes—
15C22×D116C2212yes—
16C29×#123—21no—
17C29×C22⋊Q8C2212yescomplemented
18C29×#121C2212yescomplemented
19C29×#120—21no—
20C4×D4×C29—22no—
21C29×#117C2211yescharacteristic, complemented
22#11745311—28no—
23C2×#90784C2214yescomplemented
24C2×#90782C2214yescomplemented
25C2×#90777C2212yescomplemented
26C22⋊#6597C2214yescomplemented
27#11745157C2218yescomplemented
28C24×C29D411yescharacteristic
29C23×D29—28no—
30C23×D29—24no—
31C23×D29—22no—
32C23×D29—48no—
33#41×C29—24no—
34C2×Q8×C29C2311yescharacteristic, complemented
35C2×D4×C29—22no—
36C2×D4×C29—21no—
37C2×D4×C29C2311yescharacteristic, complemented
38C22×C4×C29—24no—
39C22×C4×C29—21no—
40C22×C4×C29D412yes—
41C22×C4×C29D411yescharacteristic
42C22×C4×C29C2312yes—
43C22×C4×C29C2311yescharacteristic
44C2×D116—28no—
45C2×D116—216no—
46C2×D116—24no—
47C2×D116—24no—
48C2×D116—22no—
49C2×D116—216no—
50C2×D116D418yescomplemented
51C2×D116C2314yes—
52C4⋊C4×C29—22no—
53C4⋊C4×C29C2312yescomplemented
54C22⋊C4×C29—21no—
55C22⋊C4×C29—22no—
56C22⋊C4×C29—22no—
57C42×C29—21no—
58#90784—28no—
59#90782—28no—
60#90777—24no—
61C2×#6597C2314yescomplemented
62C23×C29—22no—
63C23×C29—24no—
64C23×C29—21no—
65C23×C29C2×D411yescharacteristic
66C23×C29C2×D412yes—
67C22×D29—216no—
68C22×D29—28no—
69C22×D29—24no—
70C22×D29—432no—
71C22×D29—48no—
72C22×D29—48no—
73C22×D29—48no—
74Q8×C29—22no—
75D4×C29—22no—
76D4×C29—42no—
77D4×C29—44no—
78C2×C4×C29—22no—
79C2×C4×C29—24no—
80C2×C4×C29—24no—
81C2×C4×C29—22no—
82C2×C4×C29—21no—
83C2×C4×C29—42no—
84C2×C4×C29C2×D414yes—
85C2×C4×C29C2×D412yes—
86C2×C4×C29C2×D411yescharacteristic
87C2×C4×C29C2411yescharacteristic
88D116—216no—
89D116—28no—
90D116—24no—
91D116C2×D418yes—
92#6597—24no—
93#3003—291nomaximal, Sylow
94C22×C29—28no—
95C22×C29—22no—
96C22×C29—22no—
97C22×C29—22no—
98C22×C29—41no—
99C22×C29—42no—
100C22×C29#12012yes—
101C22×C29AGL(1,ℤ/8)14yescomplemented
102C22×C29C22×D411yescharacteristic
103C2×D29—28no—
104C2×D29—416no—
105C2×D29—432no—
106C2×D29—48no—
107C4×C29—22no—
108C4×C29—41no—
109C4×C29—42no—
110C4×C29#12012yes—
111C4×C29C22×D412yes—
112C23×D4—291no—
113C2×AGL(1,ℤ/8)—292no—
114#525D2911yescharacteristic, complemented
115#450—294no—
116#447—294no—
117#418—292no—
118#407—291no—
119C2×C29—22no—
120C2×C29—24no—
121C2×C29—81no—
122C2×C29C2×#12011yescharacteristic
123C2×C29C2×AGL(1,ℤ/8)12yes—
124D29—48no—
125C25—581no—
126C2×#41C2×D2911yescharacteristic, complemented
127C22×D4—292no—
128C22×D4—294no—
129C22×D4—588no—
130C22×D4—582no—
131C22×D4—581no—
132C23×C4C2×D2911yescharacteristic
133AGL(1,ℤ/8)—588no—
134C2×SD16—294no—
135C2×D8—294no—
136C2×M16—292no—
137#123—21no—
138C22⋊Q8C2×D2912yescomplemented
139#121C2×D2912yescomplemented
140#120—21no—
141C4×D4—22no—
142#117C2×D2911yescharacteristic, complemented
143#102—298no—
144C22⋊C8—294no—
145C29#300311yescharacteristic, complemented, Sylow
146C24—588no—
147C24—584no—
148C24—582no—
149C24—1168no—
150C24C22⋊D2911yescharacteristic
151#41—24no—
152C2×Q8C22×D2911yescharacteristic, complemented
153C2×D4—22no—
154C2×D4—21no—
155C2×D4—298no—
156C2×D4—294no—
157C2×D4—5816no—
158C2×D4—588no—
159C2×D4—5816no—
160C2×D4—584no—
161C2×D4—584no—
162C2×D4—582no—
163C2×D4C22×D2911yescharacteristic, complemented
164C22×C4—24no—
165C22×C4—21no—
166C22×C4C22⋊D2912yes—
167C22×C4C22⋊D2911yescharacteristic
168C22×C4C22×D2912yes—
169C22×C4C22×D2911yescharacteristic
170SD16—588no—
171D8—588no—
172M16—584no—
173C2×C8—294no—
174C4⋊C4—22no—
175C4⋊C4C22×D2912yescomplemented
176C22⋊C4—21no—
177C22⋊C4—22no—
178C22⋊C4—22no—
179C42—21no—
180C23—22no—
181C23—24no—
182C23—21no—
183C23—5816no—
184C23—588no—
185C23—584no—
186C23—11632no—
187C23—1168no—
188C23—1168no—
189C23—1168no—
190C23C2×C22⋊D2911yescharacteristic
191C23C2×C22⋊D2912yes—
192Q8—22no—
193D4—22no—
194D4—42no—
195D4—44no—
196D4—298no—
197D4—5816no—
198D4—588no—
199D4—584no—
200C2×C4—22no—
201C2×C4—24no—
202C2×C4—24no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—42no—
206C2×C4D4×D2914yes—
207C2×C4C2×C22⋊D2912yes—
208C2×C4C2×C22⋊D2911yescharacteristic
209C2×C4C23×D2911yescharacteristic
210C8—584no—
211C22—28no—
212C22—22no—
213C22—22no—
214C22—22no—
215C22—41no—
216C22—42no—
217C22—588no—
218C22—11616no—
219C22—11632no—
220C22—1168no—
221C22#1174529912yes—
222C22#1174531114yescomplemented
223C22C22×C22⋊D2911yescharacteristic
224C4—22no—
225C4—41no—
226C4—42no—
227C4#1174529912yes—
228C4C2×D4×D2912yes—
229C2—22no—
230C2—24no—
231C2—81no—
232C2—1168no—
233C2C2×#1174529911yescharacteristic
234C2C2×#1174531112yes—
subgroupG/H
centerC22C22×C22⋊D29
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC29×#525C2
socleC22×C29C22×D4
hypercenter#525D29
hypocenterC29#3003
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C23×D116C2semidirect
2. C2×#11745311C2semidirect
3. #14271313C2semidirect
4. #14271296C2semidirect
5. #14271167C2semidirect
6. #14271127C2semidirect
7. C29×#525C2semidirect
9. C2×#41×C29C22semidirect
17. C29×C22⋊Q8C22semidirect
18. C29×#121C22semidirect
21. C29×#117C22semidirect
23. C2×#90784C22semidirect
24. C2×#90782C22semidirect
25. C2×#90777C22semidirect
26. C22⋊#6597C22semidirect
27. #11745157C22semidirect
34. C2×Q8×C29C23semidirect
37. C2×D4×C29C23semidirect
50. C2×D116D4semidirect
53. C4⋊C4×C29C23semidirect
61. C2×#6597C23semidirect
101. C22×C29AGL(1,ℤ/8)semidirect
114. #525D29semidirect
126. C2×#41C2×D29semidirect
138. C22⋊Q8C2×D29semidirect
139. #121C2×D29semidirect
142. #117C2×D29semidirect
145. C29#3003semidirect
152. C2×Q8C22×D29semidirect
163. C2×D4C22×D29semidirect
175. C4⋊C4C22×D29semidirect
222. C22#11745311semidirect

Maximal subgroups

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

subgroupindex
C23×D1162
C2×#117453112
#142713132
#142712962
#142711672
#142711272
C29×#5252
#300329