Finite Group Databasebeta

← #16855112

Subgroups — #16855112

Order 3968.

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#14521885C211yesmaximal, characteristic, complemented
2#14521616C212yesmaximal, complemented
3#14521235C218yesmaximal, complemented
4#14521023C211yesmaximal, characteristic, complemented
5#14521022C212yesmaximal, complemented
6C31×#535C211yesmaximal, characteristic, complemented
7C2×#41×C31C2211yescharacteristic
8C22×D4×C31C2211yescharacteristic, complemented
9#11758567—22no—
10#11758566—22no—
11#41×D31—24no—
12#11758564—21no—
13#11758561—21no—
14#11758561C2214yes—
15C2×#92495—22no—
16C2×#92495C2212yes—
17C31×#127—21no—
18C31×#124—21no—
19C31×#121—22no—
20C31×#120C2214yescomplemented
21C31×#117C2211yescharacteristic, complemented
22#11758530C2218yescomplemented
23#11758517—22no—
24#11758515—24no—
25#11758513—22no—
26#11758503C2212yescomplemented
27#11758422C2214yescomplemented
28#11758394C2218yescomplemented
29C24×C31C2312yes—
30#41×C31—22no—
31#41×C31—22no—
32C2×Q8×C31D411yescharacteristic, complemented
33C2×D4×C31—24no—
34C2×D4×C31—22no—
35C2×D4×C31—22no—
36C2×D4×C31—21no—
37C2×D4×C31—21no—
38C2×D4×C31D411yescharacteristic, complemented
39C2×D4×C31C2312yes—
40C22×C4×C31D412yescomplemented
41C22×C4×C31C2311yescharacteristic
42C2×C22⋊D31—24no—
43C2×C22⋊D31—28no—
44C2×C22⋊D31C2314yes—
45#92499—24no—
46Q8×D31—24no—
47#92497—24no—
48#92497—28no—
49D4×D31—24no—
50D4×D31—28no—
51#92495—24no—
52#92495—24no—
53#92495—22no—
54#92495—24no—
55#92495—22no—
56#92495—22no—
57#92495—22no—
58C2×D124—22no—
59C2×D124D412yescomplemented
60C2×C4×D31—24no—
61C2×C4×D31D414yescomplemented
62C2×Dic62—22no—
63C2×Dic62D412yescomplemented
64C4⋊C4×C31—21no—
65C22⋊C4×C31—24no—
66C22⋊C4×C31C2312yes—
67C42×C31—21no—
68C22⋊Dic31—28no—
69C22⋊Dic31C2314yes—
70C4⋊Dic31—22no—
71C4×Dic31—22no—
72C23×C31—24no—
73C23×C31—22no—
74C23×C31—82no—
75C23×C31C2×D412yes—
76C23×C31C2411yescharacteristic
77C22×D31—24no—
78C22×D31C2×D414yes—
79Q8×C31—22no—
80D4×C31—24no—
81D4×C31—22no—
82D4×C31—41no—
83D4×C31—42no—
84D4×C31—41no—
85D4×C31—84no—
86C2×C4×C31—22no—
87C2×C4×C31—22no—
88C2×C4×C31—22no—
89C2×C4×C31C2×D412yes—
90C2×C4×C31C2×D412yes—
91C2×C4×C31C2×D412yes—
92C2×C4×C31C2×D411yescharacteristic
93C2×C4×C31C2×D411yescharacteristic
94C22⋊D31—28no—
95C22⋊D31—24no—
96C22⋊D31—24no—
97C22⋊D31—88no—
98C2×Dic31—24no—
99C2×Dic31—24no—
100C2×Dic31C2×D414yes—
101D124—24no—
102D124—22no—
103D124—22no—
104C4×D31—28no—
105C4×D31—24no—
106C4×D31—24no—
107Dic62—24no—
108Dic62—22no—
109Dic62—22no—
110#3027—311nomaximal, Sylow
111C22×C31—22no—
112C22×C31—41no—
113C22×C31—42no—
114C22×C31—48no—
115C22×C31—44no—
116C22×C31—42no—
117C22×C31C22×D412yes—
118C22×C31C22×D411yescharacteristic
119C2×D31—24no—
120C2×D31—44no—
121C2×D31—84no—
122C4×C31—22no—
123C4×C31—42no—
124C4×C31#12011yescharacteristic
125C4×C31#12012yes—
126C4×C31#12011yescharacteristic
127Dic31—24no—
128Dic31—44no—
129#585—311no—
130#535—312no—
131#535D3111yescharacteristic, complemented
132B(4,𝔽2)—318no—
133#410—311no—
134#410—312no—
135C2×C31—22no—
136C2×C31—21no—
137C2×C31—42no—
138C2×C31—82no—
139C2×C31C2×#12011yescharacteristic
140D31—44no—
1412−1+4—622no—
1422−1+4—621no—
143H(5,𝔽2)—314no—
144H(5,𝔽2)—622no—
145H(5,𝔽2)—621no—
146C2×#41—312no—
147C2×#41—624no—
148C2×#41—622no—
149C2×#41C2×D3111yescharacteristic
150C22×D4C2×D3111yescharacteristic, complemented
151#127—21no—
152#127—622no—
153#124—21no—
154#124—622no—
155#121—22no—
156#121—624no—
157#120—318no—
158#120C2×D3114yescomplemented
159#117—312no—
160#117C2×D3111yescharacteristic, complemented
161#99—314no—
162#99—318no—
163C31#302711yescharacteristic, complemented, Sylow
164C24C22×D3112yes—
165#41—22no—
166#41—22no—
167#41—624no—
168#41—624no—
169#41—628no—
170#41—624no—
171#41—624no—
172#41—622no—
173#41—622no—
174#41—622no—
175#41—622no—
176#41—624no—
177C2×Q8—312no—
178C2×Q8—622no—
179C2×Q8—624no—
180C2×Q8C22⋊D3111yescharacteristic, complemented
181C2×D4—24no—
182C2×D4—22no—
183C2×D4—22no—
184C2×D4—21no—
185C2×D4—21no—
186C2×D4—314no—
187C2×D4—312no—
188C2×D4—622no—
189C2×D4—628no—
190C2×D4—624no—
191C2×D4—624no—
192C2×D4—628no—
193C2×D4C22⋊D3111yescharacteristic, complemented
194C2×D4C22×D3112yes—
195C22×C4—314no—
196C22×C4—624no—
197C22×C4C22⋊D3112yescomplemented
198C22×C4C22×D3111yescharacteristic
199C4⋊C4—21no—
200C4⋊C4—622no—
201C22⋊C4—24no—
202C22⋊C4—314no—
203C22⋊C4—628no—
204C22⋊C4C22×D3112yes—
205C42—21no—
206C42—622no—
207C23—24no—
208C23—22no—
209C23—82no—
210C23—314no—
211C23—624no—
212C23C2×C22⋊D3112yes—
213C23C23×D3111yescharacteristic
214Q8—22no—
215Q8—624no—
216Q8—622no—
217Q8—622no—
218D4—24no—
219D4—22no—
220D4—41no—
221D4—41no—
222D4—42no—
223D4—84no—
224D4—624no—
225D4—628no—
226D4—624no—
227D4—624no—
228D4—622no—
229D4—622no—
230D4—2488no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—314no—
235C2×C4—628no—
236C2×C4—624no—
237C2×C4—624no—
238C2×C4—624no—
239C2×C4—624no—
240C2×C4D4×D3112yes—
241C2×C4D4×D3112yes—
242C2×C4C2×C22⋊D3112yes—
243C2×C4C2×C22⋊D3111yescharacteristic
244C2×C4C2×C22⋊D3111yescharacteristic
245C22—22no—
246C22—41no—
247C22—42no—
248C22—44no—
249C22—48no—
250C22—42no—
251C22—624no—
252C22—1244no—
253C22—2484no—
254C22C2×D4×D3112yes—
255C22C22×C22⋊D3111yescharacteristic
256C4—22no—
257C4—42no—
258C4—624no—
259C4—1244no—
260C4#1175851411yescharacteristic
261C4#1175851411yescharacteristic
262C4#1175851412yes—
263C2—21no—
264C2—22no—
265C2—42no—
266C2—82no—
267C2—1244no—
268C2C2×#1175851411yescharacteristic
subgroupG/H
centerC2C2×#11758514
commutator, cosocleC23×C31C24
FrattiniC23C23×D31
FittingC31×#535C2
socleC2×C31C2×#120
hypercenter#535D31
hypocenterC31#3027
LCSG ▷ C23×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C22×C4 ◁ #535

Complements

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

NH ≅ G/Ntype
1. #14521885C2semidirect
2. #14521616C2semidirect
3. #14521235C2semidirect
4. #14521023C2semidirect
5. #14521022C2semidirect
6. C31×#535C2semidirect
8. C22×D4×C31C22semidirect
20. C31×#120C22semidirect
21. C31×#117C22semidirect
22. #11758530C22semidirect
26. #11758503C22semidirect
27. #11758422C22semidirect
28. #11758394C22semidirect
32. C2×Q8×C31D4semidirect
38. C2×D4×C31D4semidirect
40. C22×C4×C31D4semidirect
59. C2×D124D4semidirect
61. C2×C4×D31D4semidirect
63. C2×Dic62D4semidirect
131. #535D31semidirect
150. C22×D4C2×D31semidirect
158. #120C2×D31semidirect
160. #117C2×D31semidirect
163. C31#3027semidirect
180. C2×Q8C22⋊D31semidirect
193. C2×D4C22⋊D31semidirect
197. C22×C4C22⋊D31semidirect

Maximal subgroups

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

subgroupindex
#145218852
#145216162
#145212352
#145210232
#145210222
C31×#5352
#302731