Finite Group Databasebeta

← #16855085

Subgroups — #16855085

Order 3968.

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

normal subgroups only

This group has 736 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
2C2×#11758485C212yesmaximal, complemented
3#14521613C211yesmaximal, characteristic, complemented
4#14521224C214yesmaximal, complemented
5#14521219C214yesmaximal, complemented
6#14521065C212yesmaximal, complemented
7C31×#413C211yesmaximal, characteristic, complemented
8C2×#41×C31C2211yescharacteristic
9#11758567—22no—
10#11758566—22no—
11#41×D31—24no—
12#11758564C2212yescomplemented
13#11758561—22no—
14#11758561C2212yescomplemented
15C2×#92495—22no—
16C2×#92495C2212yes—
17C22×D124C2211yescharacteristic, complemented
18C2×M16×C31C2212yescomplemented
19#11758524C2212yescomplemented
20#11758517C2212yescomplemented
21#11758514—22no—
22#11758507—22no—
23#11758503C2211yescharacteristic, complemented
24#11758485—24no—
25#11758485C2218yescomplemented
26C2×D248—22no—
27C2×#92470—22no—
28C31×#101C2212yescomplemented
29C31×#100C2212yescomplemented
30#11758413C2218yescomplemented
31C23×D31—21no—
32#41×C31—22no—
33#41×C31—22no—
34C2×Q8×C31C2311yescharacteristic
35C2×D4×C31D412yes—
36C2×D4×C31C2311yescharacteristic
37C22×C4×C31D412yes—
38C22×C4×C31C2311yescharacteristic
39C2×C22⋊D31—24no—
40C2×C22⋊D31—22no—
41C2×C22⋊D31—24no—
42#92499—24no—
43Q8×D31—24no—
44#92497—28no—
45#92497—24no—
46D4×D31—28no—
47D4×D31—24no—
48#92495—24no—
49#92495—24no—
50#92495—22no—
51#92495—24no—
52#92495—22no—
53#92495D414yescomplemented
54#92495C2314yes—
55C2×D124—22no—
56C2×D124—22no—
57C2×D124—22no—
58C2×D124—21no—
59C2×D124D412yescomplemented
60C2×D124C2312yes—
61C2×C4×D31—24no—
62C2×C4×D31—22no—
63C2×Dic62—22no—
64C2×Dic62D412yescomplemented
65M16×C31—22no—
66M16×C31C2314yes—
67C2×C8×C31—22no—
68C22⋊Dic31—21no—
69(C2×D31)⋊C4—24no—
70C4⋊Dic31—21no—
71C4×Dic31C2312yes—
72D248—28no—
73#92470—28no—
74C23×C31C2×D412yes—
75C23×C31C2×D411yescharacteristic
76C22×D31—24no—
77C22×D31—22no—
78C22×D31—22no—
79C22×D31—42no—
80C22×D31—82no—
81Q8×C31—22no—
82D4×C31—24no—
83D4×C31—22no—
84C2×C4×C31—22no—
85C2×C4×C31—22no—
86C2×C4×C31C2×D412yes—
87C2×C4×C31C2×D411yescharacteristic
88C2×C4×C31C2×D412yes—
89C2×C4×C31C2×D412yes—
90C2×C4×C31C2411yescharacteristic
91C22⋊D31—28no—
92C22⋊D31—24no—
93C22⋊D31—24no—
94C22⋊D31—84no—
95C2×Dic31—24no—
96C2×Dic31—22no—
97C2×Dic31—22no—
98D124—24no—
99D124—22no—
100D124—24no—
101D124—41no—
102D124—41no—
103D124C2×D414yes—
104C4×D31—28no—
105C4×D31—24no—
106C4×D31—24no—
107Dic62—24no—
108Dic62—22no—
109Dic62C2×D414yes—
110C8×C31—24no—
111#3021—311nomaximal, Sylow
112C22×C31—22no—
113C22×C31—41no—
114C22×C31—42no—
115C22×C31#12012yes—
116C22×C31C22×D411yescharacteristic
117C2×D31—24no—
118C2×D31—44no—
119C2×D31—42no—
120C2×D31—82no—
121C2×D31—84no—
122C2×D31—84no—
123C4×C31—22no—
124C4×C31#12011yescharacteristic
125C4×C31#12011yescharacteristic
126C4×C31C22×D412yes—
127Dic31—24no—
128Dic31—42no—
129#585—311no—
130C2×AGL(1,ℤ/8)—312no—
131#535—311no—
132#454—314no—
133#453—314no—
134#421—312no—
135#413D3111yescharacteristic, complemented
136C2×C31—21no—
137C2×C31—22no—
138C2×C31—42no—
139C2×C31C2×#12011yescharacteristic
140D31—44no—
141D31—82no—
1422−1+4—312no—
1432−1+4—622no—
144H(5,𝔽2)—312no—
145H(5,𝔽2)—622no—
146H(5,𝔽2)—622no—
147C2×#41—312no—
148C2×#41—624no—
149C2×#41—622no—
150C2×#41C2×D3111yescharacteristic
151C22×D4—311no—
152AGL(1,ℤ/8)—318no—
153AGL(1,ℤ/8)—624no—
154C2×SD16—622no—
155C2×D8—622no—
156C2×M16C2×D3112yescomplemented
157#127—312no—
158#124—312no—
159#121—622no—
160#120—622no—
161#117—311no—
162#104—318no—
163#101C2×D3112yescomplemented
164#100C2×D3112yescomplemented
165C31#302111yescharacteristic, complemented, Sylow
166C24—621no—
167#41—22no—
168#41—22no—
169#41—314no—
170#41—314no—
171#41—624no—
172#41—624no—
173#41—624no—
174#41—628no—
175#41—624no—
176#41—622no—
177#41—624no—
178#41—622no—
179C2×Q8—312no—
180C2×Q8—622no—
181C2×Q8—624no—
182C2×Q8C22×D3111yescharacteristic
183C2×D4—312no—
184C2×D4—312no—
185C2×D4—622no—
186C2×D4—624no—
187C2×D4—628no—
188C2×D4—622no—
189C2×D4—624no—
190C2×D4—624no—
191C2×D4—621no—
192C2×D4—622no—
193C2×D4—622no—
194C2×D4D12412yes—
195C2×D4C22×D3111yescharacteristic
196C22×C4—624no—
197C22×C4—622no—
198C22×C4D12412yes—
199C22×C4C22×D3111yescharacteristic
200SD16—628no—
201D8—628no—
202M16—22no—
203M16C22×D3114yes—
204C2×C8—22no—
205C4⋊C4—621no—
206C22⋊C4—621no—
207C22⋊C4—624no—
208C42—312no—
209C23—624no—
210C23—622no—
211C23—622no—
212C23—1242no—
213C23—2482no—
214C23C2×D12412yes—
215C23C2×D12411yescharacteristic
216Q8—22no—
217Q8—314no—
218Q8—624no—
219Q8—622no—
220D4—24no—
221D4—22no—
222D4—314no—
223D4—624no—
224D4—628no—
225D4—624no—
226D4—624no—
227D4—622no—
228D4—624no—
229D4—1241no—
230D4—1241no—
231D4—2484no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—628no—
235C2×C4—624no—
236C2×C4—622no—
237C2×C4—624no—
238C2×C4—624no—
239C2×C4—622no—
240C2×C4C2×D12412yes—
241C2×C4C2×D12411yescharacteristic
242C2×C4D4×D3112yes—
243C2×C4D4×D3112yes—
244C2×C4C23×D3111yescharacteristic
245C8—24no—
246C22—22no—
247C22—41no—
248C22—42no—
249C22—624no—
250C22—1244no—
251C22—1242no—
252C22—2482no—
253C22—2484no—
254C22—2484no—
255C22#1175845912yes—
256C22C22×D12411yescharacteristic
257C4—22no—
258C4—624no—
259C4—1242no—
260C4#1175845911yescharacteristic
261C4#1175845911yescharacteristic
262C4C2×D4×D3112yes—
263C2—21no—
264C2—22no—
265C2—42no—
266C2—1244no—
267C2—2482no—
268C2C2×#1175845911yescharacteristic
subgroupG/H
centerC2C2×#11758459
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#413C2
socleC2×C31C2×#120
hypercenter#413D31
hypocenterC31#3021
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C22×C4 ◁ #413

Complements

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

NH ≅ G/Ntype
1. #14521885C2semidirect
2. C2×#11758485C2semidirect
3. #14521613C2semidirect
4. #14521224C2semidirect
5. #14521219C2semidirect
6. #14521065C2semidirect
7. C31×#413C2semidirect
12. #11758564C22semidirect
14. #11758561C22semidirect
17. C22×D124C22semidirect
18. C2×M16×C31C22semidirect
19. #11758524C22semidirect
20. #11758517C22semidirect
23. #11758503C22semidirect
25. #11758485C22semidirect
28. C31×#101C22semidirect
29. C31×#100C22semidirect
30. #11758413C22semidirect
53. #92495D4semidirect
59. C2×D124D4semidirect
64. C2×Dic62D4semidirect
135. #413D31semidirect
156. C2×M16C2×D31semidirect
163. #101C2×D31semidirect
164. #100C2×D31semidirect
165. C31#3021semidirect

Maximal subgroups

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

subgroupindex
#145218852
C2×#117584852
#145216132
#145212242
#145212192
#145210652
C31×#4132
#302131