Finite Group Databasebeta

← #16828895

Subgroups — #16828895

Order 3712.

772 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 772 conjugacy classes of subgroups — too many to draw. Only its 145 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
1#14271999C211yesmaximal, characteristic, complemented
2#14271732C212yesmaximal, complemented
3#14271727C211yesmaximal, characteristic, complemented
4#14271351C218yesmaximal, complemented
5#14271136C212yesmaximal, complemented
6C29×#410C211yesmaximal, characteristic, complemented
7C2×#41×C29C2211yescharacteristic
8#11745365—22no—
9#11745364—22no—
10#41×D29—24no—
11#11745362—21no—
12#11745359—21no—
13#11745359C2214yes—
14C2×#90805—22no—
15C2×#90805C2212yes—
16C22×D116C2211yescharacteristic, complemented
17C29×#117C2212yescomplemented
18#11745309—21no—
19#11745302—21no—
20#11745299C2214yescomplemented
21#11745292—22no—
22#11745288C2211yescharacteristic, complemented
23#11745257—24no—
24#11745244C2218yescomplemented
25#11745237—22no—
26#11745236—22no—
27C29×#99C2214yescomplemented
28#11745154C2218yescomplemented
29C23×D29C2312yes—
30#41×C29—22no—
31#41×C29—22no—
32C2×Q8×C29D411yescharacteristic, complemented
33C2×D4×C29D411yescharacteristic, complemented
34C2×D4×C29C2312yes—
35C22×C4×C29D412yescomplemented
36C22×C4×C29C2311yescharacteristic
37C2×C22⋊D29—24no—
38C2×C22⋊D29—24no—
39C2×C22⋊D29C2314yes—
40#90809—24no—
41Q8×D29—24no—
42#90807—24no—
43#90807—28no—
44D4×D29—24no—
45D4×D29—28no—
46#90805—24no—
47#90805—24no—
48#90805—22no—
49#90805—24no—
50#90805—22no—
51#90805—22no—
52#90805—22no—
53C2×D116—22no—
54C2×D116—21no—
55C2×D116—21no—
56C2×D116—22no—
57C2×D116—22no—
58C2×D116—28no—
59C2×D116D412yescomplemented
60C2×C4×D29—24no—
61C2×C4×D29D414yescomplemented
62C2×Dic58—22no—
63C2×Dic58D412yescomplemented
64C4⋊C4×C29—22no—
65C22⋊C4×C29C2314yes—
66C42×C29—22no—
67C22⋊Dic29C2312yes—
68#90781—24no—
69#90781—28no—
70C4⋊Dic29—21no—
71C4×Dic29—21no—
72C23×C29C2×D412yes—
73C23×C29C2411yescharacteristic
74C22×D29—24no—
75C22×D29—24no—
76C22×D29—22no—
77C22×D29—82no—
78C22×D29C2×D414yes—
79Q8×C29—22no—
80D4×C29—24no—
81D4×C29—22no—
82C2×C4×C29—22no—
83C2×C4×C29—22no—
84C2×C4×C29—24no—
85C2×C4×C29C2×D412yes—
86C2×C4×C29C2×D411yescharacteristic
87C2×C4×C29C2×D411yescharacteristic
88C2×C4×C29C2×D412yes—
89C2×C4×C29C2×D412yes—
90C22⋊D29—28no—
91C22⋊D29—24no—
92C22⋊D29—24no—
93C22⋊D29—84no—
94C2×Dic29—24no—
95C2×Dic29—22no—
96C2×Dic29C2×D414yes—
97D116—24no—
98D116—22no—
99D116—22no—
100D116—41no—
101D116—42no—
102D116—41no—
103D116—88no—
104C4×D29—28no—
105C4×D29—24no—
106C4×D29—24no—
107Dic58—24no—
108Dic58—22no—
109Dic58—22no—
110#3027—291nomaximal, Sylow
111C22×C29—22no—
112C22×C29—41no—
113C22×C29—42no—
114C22×C29C22×D411yescharacteristic
115C22×C29C22×D412yes—
116C2×D29—24no—
117C2×D29—44no—
118C2×D29—48no—
119C2×D29—44no—
120C2×D29—42no—
121C2×D29—84no—
122C4×C29—22no—
123C4×C29—44no—
124C4×C29#12011yescharacteristic
125C4×C29#12012yes—
126C4×C29#12011yescharacteristic
127Dic29—24no—
128Dic29—42no—
129#585—291no—
130#535—291no—
131#535—292no—
132B(4,𝔽2)—298no—
133#410—292no—
134#410D2911yescharacteristic, complemented
135C2×C29—21no—
136C2×C29—22no—
137C2×C29—42no—
138C2×C29C2×#12011yescharacteristic
139D29—44no—
140D29—82no—
1412−1+4—582no—
1422−1+4—581no—
143H(5,𝔽2)—294no—
144H(5,𝔽2)—582no—
145H(5,𝔽2)—581no—
146C2×#41—292no—
147C2×#41—584no—
148C2×#41—582no—
149C2×#41C2×D2911yescharacteristic
150C22×D4—291no—
151#127—582no—
152#127—581no—
153#124—582no—
154#124—581no—
155#121—584no—
156#121—582no—
157#120—294no—
158#120—298no—
159#117—291no—
160#117C2×D2912yescomplemented
161#99—298no—
162#99C2×D2914yescomplemented
163C29#302711yescharacteristic, complemented, Sylow
164C24—292no—
165#41—22no—
166#41—22no—
167#41—584no—
168#41—584no—
169#41—588no—
170#41—584no—
171#41—584no—
172#41—582no—
173#41—582no—
174#41—584no—
175#41—582no—
176#41—582no—
177C2×Q8—292no—
178C2×Q8—582no—
179C2×Q8—584no—
180C2×Q8D11611yescharacteristic, complemented
181C2×D4—294no—
182C2×D4—292no—
183C2×D4—582no—
184C2×D4—588no—
185C2×D4—588no—
186C2×D4—584no—
187C2×D4—584no—
188C2×D4—584no—
189C2×D4—581no—
190C2×D4—581no—
191C2×D4—582no—
192C2×D4—582no—
193C2×D4D11611yescharacteristic, complemented
194C2×D4C22×D2912yes—
195C22×C4—294no—
196C22×C4—584no—
197C22×C4D11612yescomplemented
198C22×C4C22×D2911yescharacteristic
199C4⋊C4—22no—
200C4⋊C4—581no—
201C22⋊C4—292no—
202C22⋊C4—588no—
203C22⋊C4—584no—
204C22⋊C4C22×D2914yes—
205C42—22no—
206C42—581no—
207C23—294no—
208C23—584no—
209C23—582no—
210C23—584no—
211C23—2322no—
212C23C2×D11612yes—
213C23C23×D2911yescharacteristic
214Q8—22no—
215Q8—584no—
216Q8—582no—
217Q8—582no—
218D4—24no—
219D4—22no—
220D4—584no—
221D4—588no—
222D4—584no—
223D4—584no—
224D4—582no—
225D4—582no—
226D4—1161no—
227D4—1161no—
228D4—1162no—
229D4—2328no—
230D4—2324no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—24no—
234C2×C4—294no—
235C2×C4—588no—
236C2×C4—584no—
237C2×C4—584no—
238C2×C4—584no—
239C2×C4—582no—
240C2×C4C2×D11612yes—
241C2×C4C2×D11611yescharacteristic
242C2×C4C2×D11611yescharacteristic
243C2×C4D4×D2912yes—
244C2×C4D4×D2912yes—
245C22—22no—
246C22—41no—
247C22—42no—
248C22—584no—
249C22—1164no—
250C22—1168no—
251C22—1164no—
252C22—1162no—
253C22—2324no—
254C22C22×D11611yescharacteristic
255C22C2×D4×D2912yes—
256C4—22no—
257C4—44no—
258C4—584no—
259C4—1162no—
260C4#1174524411yescharacteristic
261C4#1174524411yescharacteristic
262C4#1174524412yes—
263C2—22no—
264C2—21no—
265C2—42no—
266C2—1164no—
267C2—2322no—
268C2C2×#1174524411yescharacteristic
subgroupG/H
centerC2C2×#11745244
commutator, cosocleC23×C29C24
FrattiniC23C23×D29
FittingC29×#410C2
socleC2×C29C2×#120
hypercenter#410D29
hypocenterC29#3027
LCSG ▷ C23×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
1. #14271999C2semidirect
2. #14271732C2semidirect
3. #14271727C2semidirect
4. #14271351C2semidirect
5. #14271136C2semidirect
6. C29×#410C2semidirect
16. C22×D116C22semidirect
17. C29×#117C22semidirect
20. #11745299C22semidirect
22. #11745288C22semidirect
24. #11745244C22semidirect
27. C29×#99C22semidirect
28. #11745154C22semidirect
32. C2×Q8×C29D4semidirect
33. C2×D4×C29D4semidirect
35. C22×C4×C29D4semidirect
59. C2×D116D4semidirect
61. C2×C4×D29D4semidirect
63. C2×Dic58D4semidirect
134. #410D29semidirect
160. #117C2×D29semidirect
162. #99C2×D29semidirect
163. C29#3027semidirect
180. C2×Q8D116semidirect
193. C2×D4D116semidirect
197. C22×C4D116semidirect

Maximal subgroups

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

subgroupindex
#142719992
#142717322
#142717272
#142713512
#142711362
C29×#4102
#302729