Finite Group Databasebeta

← #16855114

Subgroups — #16855114

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#14521618C212yesmaximal, complemented
3#14521613C211yesmaximal, characteristic, complemented
4#14521237C218yesmaximal, complemented
5#14521022C212yesmaximal, complemented
6C31×#410C211yesmaximal, characteristic, complemented
7C2×#41×C31C2211yescharacteristic
8#11758567—22no—
9#11758566—22no—
10#41×D31—24no—
11#11758564—21no—
12#11758561—21no—
13#11758561C2214yes—
14C2×#92495—22no—
15C2×#92495C2212yes—
16C22×D124C2211yescharacteristic, complemented
17C31×#117C2212yescomplemented
18#11758524—21no—
19#11758517—21no—
20#11758514C2214yescomplemented
21#11758507—22no—
22#11758503C2211yescharacteristic, complemented
23#11758472—24no—
24#11758459C2218yescomplemented
25#11758452—22no—
26#11758451—22no—
27C31×#99C2214yescomplemented
28#11758394C2218yescomplemented
29C23×D31C2312yes—
30#41×C31—22no—
31#41×C31—22no—
32C2×Q8×C31D411yescharacteristic, complemented
33C2×D4×C31D411yescharacteristic, complemented
34C2×D4×C31C2312yes—
35C22×C4×C31D412yescomplemented
36C22×C4×C31C2311yescharacteristic
37C2×C22⋊D31—24no—
38C2×C22⋊D31—24no—
39C2×C22⋊D31C2314yes—
40#92499—24no—
41Q8×D31—24no—
42#92497—24no—
43#92497—28no—
44D4×D31—24no—
45D4×D31—28no—
46#92495—24no—
47#92495—24no—
48#92495—22no—
49#92495—24no—
50#92495—22no—
51#92495—22no—
52#92495—22no—
53C2×D124—22no—
54C2×D124—22no—
55C2×D124—22no—
56C2×D124—21no—
57C2×D124—21no—
58C2×D124—28no—
59C2×D124D412yescomplemented
60C2×C4×D31—24no—
61C2×C4×D31D414yescomplemented
62C2×Dic62—22no—
63C2×Dic62D412yescomplemented
64C4⋊C4×C31—22no—
65C22⋊C4×C31C2314yes—
66C42×C31—22no—
67C22⋊Dic31C2312yes—
68(C2×D31)⋊C4—24no—
69(C2×D31)⋊C4—28no—
70C4⋊Dic31—21no—
71C4×Dic31—21no—
72C23×C31C2×D412yes—
73C23×C31C2411yescharacteristic
74C22×D31—24no—
75C22×D31—24no—
76C22×D31—22no—
77C22×D31—82no—
78C22×D31C2×D414yes—
79Q8×C31—22no—
80D4×C31—24no—
81D4×C31—22no—
82C2×C4×C31—22no—
83C2×C4×C31—22no—
84C2×C4×C31—24no—
85C2×C4×C31C2×D412yes—
86C2×C4×C31C2×D411yescharacteristic
87C2×C4×C31C2×D411yescharacteristic
88C2×C4×C31C2×D412yes—
89C2×C4×C31C2×D412yes—
90C22⋊D31—28no—
91C22⋊D31—24no—
92C22⋊D31—24no—
93C22⋊D31—84no—
94C2×Dic31—24no—
95C2×Dic31—22no—
96C2×Dic31C2×D414yes—
97D124—24no—
98D124—22no—
99D124—22no—
100D124—41no—
101D124—42no—
102D124—41no—
103D124—88no—
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×C31C22×D411yescharacteristic
115C22×C31C22×D412yes—
116C2×D31—24no—
117C2×D31—44no—
118C2×D31—48no—
119C2×D31—44no—
120C2×D31—42no—
121C2×D31—84no—
122C4×C31—22no—
123C4×C31—44no—
124C4×C31#12011yescharacteristic
125C4×C31#12012yes—
126C4×C31#12011yescharacteristic
127Dic31—24no—
128Dic31—42no—
129#585—311no—
130#535—311no—
131#535—312no—
132B(4,𝔽2)—318no—
133#410—312no—
134#410D3111yescharacteristic, complemented
135C2×C31—21no—
136C2×C31—22no—
137C2×C31—42no—
138C2×C31C2×#12011yescharacteristic
139D31—44no—
140D31—82no—
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×D4—311no—
151#127—622no—
152#127—621no—
153#124—622no—
154#124—621no—
155#121—624no—
156#121—622no—
157#120—314no—
158#120—318no—
159#117—311no—
160#117C2×D3112yescomplemented
161#99—318no—
162#99C2×D3114yescomplemented
163C31#302711yescharacteristic, complemented, Sylow
164C24—312no—
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—624no—
175#41—622no—
176#41—622no—
177C2×Q8—312no—
178C2×Q8—622no—
179C2×Q8—624no—
180C2×Q8D12411yescharacteristic, complemented
181C2×D4—314no—
182C2×D4—312no—
183C2×D4—622no—
184C2×D4—628no—
185C2×D4—628no—
186C2×D4—624no—
187C2×D4—624no—
188C2×D4—624no—
189C2×D4—621no—
190C2×D4—621no—
191C2×D4—622no—
192C2×D4—622no—
193C2×D4D12411yescharacteristic, complemented
194C2×D4C22×D3112yes—
195C22×C4—314no—
196C22×C4—624no—
197C22×C4D12412yescomplemented
198C22×C4C22×D3111yescharacteristic
199C4⋊C4—22no—
200C4⋊C4—621no—
201C22⋊C4—312no—
202C22⋊C4—628no—
203C22⋊C4—624no—
204C22⋊C4C22×D3114yes—
205C42—22no—
206C42—621no—
207C23—314no—
208C23—624no—
209C23—622no—
210C23—624no—
211C23—2482no—
212C23C2×D12412yes—
213C23C23×D3111yescharacteristic
214Q8—22no—
215Q8—624no—
216Q8—622no—
217Q8—622no—
218D4—24no—
219D4—22no—
220D4—624no—
221D4—628no—
222D4—624no—
223D4—624no—
224D4—622no—
225D4—622no—
226D4—1241no—
227D4—1241no—
228D4—1242no—
229D4—2488no—
230D4—2484no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—24no—
234C2×C4—314no—
235C2×C4—628no—
236C2×C4—624no—
237C2×C4—624no—
238C2×C4—624no—
239C2×C4—622no—
240C2×C4C2×D12412yes—
241C2×C4C2×D12411yescharacteristic
242C2×C4C2×D12411yescharacteristic
243C2×C4D4×D3112yes—
244C2×C4D4×D3112yes—
245C22—22no—
246C22—41no—
247C22—42no—
248C22—624no—
249C22—1244no—
250C22—1248no—
251C22—1244no—
252C22—1242no—
253C22—2484no—
254C22C22×D12411yescharacteristic
255C22C2×D4×D3112yes—
256C4—22no—
257C4—44no—
258C4—624no—
259C4—1242no—
260C4#1175845911yescharacteristic
261C4#1175845911yescharacteristic
262C4#1175845912yes—
263C2—22no—
264C2—21no—
265C2—42no—
266C2—1244no—
267C2—2482no—
268C2C2×#1175845911yescharacteristic
subgroupG/H
centerC2C2×#11758459
commutator, cosocleC23×C31C24
FrattiniC23C23×D31
FittingC31×#410C2
socleC2×C31C2×#120
hypercenter#410D31
hypocenterC31#3027
LCSG ▷ C23×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
1. #14521885C2semidirect
2. #14521618C2semidirect
3. #14521613C2semidirect
4. #14521237C2semidirect
5. #14521022C2semidirect
6. C31×#410C2semidirect
16. C22×D124C22semidirect
17. C31×#117C22semidirect
20. #11758514C22semidirect
22. #11758503C22semidirect
24. #11758459C22semidirect
27. C31×#99C22semidirect
28. #11758394C22semidirect
32. C2×Q8×C31D4semidirect
33. C2×D4×C31D4semidirect
35. C22×C4×C31D4semidirect
59. C2×D124D4semidirect
61. C2×C4×D31D4semidirect
63. C2×Dic62D4semidirect
134. #410D31semidirect
160. #117C2×D31semidirect
162. #99C2×D31semidirect
163. C31#3027semidirect
180. C2×Q8D124semidirect
193. C2×D4D124semidirect
197. C22×C4D124semidirect

Maximal subgroups

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

subgroupindex
#145218852
#145216182
#145216132
#145212372
#145210222
C31×#4102
#302731