Finite Group Databasebeta

← #16855092

Subgroups — #16855092

Order 3968.

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

normal subgroups only

This group has 704 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
2C2×#11758486C212yesmaximal, complemented
3#14521620C211yesmaximal, characteristic, complemented
4#14521234C214yesmaximal, complemented
5#14521229C214yesmaximal, complemented
6#14521065C212yesmaximal, complemented
7C31×#413C211yesmaximal, characteristic, complemented
8C2×#41×C31C2211yescharacteristic
9#11758567—22no—
10#11758566—22no—
11#41×D31—24no—
12#11758564C2212yes—
13#11758561—22no—
14#11758561C2212yes—
15C2×#92495—22no—
16C2×#92495C2212yes—
17C22×Dic62C2211yescharacteristic, complemented
18C2×M16×C31C2212yescomplemented
19#11758521C2212yescomplemented
20#11758513C2212yescomplemented
21#11758512—22no—
22#11758503C2211yescharacteristic, complemented
23#11758501—22no—
24#11758486—24no—
25#11758486C2218yescomplemented
26C2×Dic124—22no—
27C2×#92470—22no—
28C31×#101C2212yescomplemented
29C31×#100C2212yescomplemented
30#11758413C2218yescomplemented
31#41×C31—22no—
32#41×C31—22no—
33C2×Q8×C31C2311yescharacteristic
34C2×D4×C31D412yes—
35C2×D4×C31C2311yescharacteristic
36C22×C4×C31D412yes—
37C22×C4×C31C2311yescharacteristic
38C2×C22⋊D31—24no—
39C2×C22⋊D31—22no—
40C22×Dic31—21no—
41#92499—24no—
42Q8×D31—24no—
43#92497—28no—
44#92497—24no—
45D4×D31—28no—
46D4×D31—24no—
47#92495—24no—
48#92495—24no—
49#92495—22no—
50#92495—24no—
51#92495—22no—
52#92495D414yes—
53#92495C2314yes—
54C2×D124—22no—
55C2×D124D412yes—
56C2×C4×D31—24no—
57C2×C4×D31—22no—
58C2×Dic62—22no—
59C2×Dic62—22no—
60C2×Dic62—22no—
61C2×Dic62—21no—
62C2×Dic62D412yes—
63C2×Dic62C2312yes—
64M16×C31—22no—
65M16×C31C2314yes—
66C2×C8×C31—22no—
67C22⋊Dic31—24no—
68C22⋊Dic31—21no—
69C4⋊Dic31—21no—
70Dic31⋊C4—24no—
71C4×Dic31C2312yes—
72Dic124—28no—
73#92470—28no—
74C23×C31C2×D412yes—
75C23×C31C2×D411yescharacteristic
76C22×D31—24no—
77C22×D31—22no—
78Q8×C31—22no—
79D4×C31—24no—
80D4×C31—22no—
81C2×C4×C31—22no—
82C2×C4×C31—22no—
83C2×C4×C31C2×D412yes—
84C2×C4×C31C2×D411yescharacteristic
85C2×C4×C31C2×D412yes—
86C2×C4×C31C2×D412yes—
87C2×C4×C31C2411yescharacteristic
88C22⋊D31—28no—
89C22⋊D31—24no—
90C22⋊D31—24no—
91C2×Dic31—24no—
92C2×Dic31—22no—
93C2×Dic31—22no—
94C2×Dic31—22no—
95C2×Dic31—42no—
96D124—24no—
97D124—22no—
98D124C2×D414yes—
99C4×D31—28no—
100C4×D31—24no—
101C4×D31—24no—
102Dic62—24no—
103Dic62—22no—
104Dic62—24no—
105Dic62—41no—
106Dic62—41no—
107Dic62C2×D414yes—
108C8×C31—24no—
109#3022—311nomaximal, Sylow
110C22×C31—22no—
111C22×C31—41no—
112C22×C31—42no—
113C22×C31#12012yes—
114C22×C31C22×D411yescharacteristic
115C2×D31—24no—
116C2×D31—82no—
117C2×D31—84no—
118C4×C31—22no—
119C4×C31#12011yescharacteristic
120C4×C31#12011yescharacteristic
121C4×C31C22×D412yes—
122Dic31—24no—
123Dic31—42no—
124Dic31—42no—
125#585—311no—
126C2×#137—312no—
127#536—311no—
128#456—314no—
129#455—314no—
130#421—312no—
131#413D3111yescharacteristic, complemented
132C2×C31—21no—
133C2×C31—22no—
134C2×C31—42no—
135C2×C31C2×#12011yescharacteristic
136D31—44no—
1372−1+4—312no—
1382−1+4—622no—
139H(5,𝔽2)—312no—
140H(5,𝔽2)—622no—
141H(5,𝔽2)—622no—
142C2×#41—312no—
143C2×#41—624no—
144C2×#41—622no—
145C2×#41C2×D3111yescharacteristic
146C22×Q8—311no—
147#137—318no—
148#137—624no—
149C2×Q16—622no—
150C2×SD16—622no—
151C2×M16C2×D3112yescomplemented
152C4⋊Q8—312no—
153#124—312no—
154#123—622no—
155C22⋊Q8—622no—
156#117—311no—
157#104—318no—
158#101C2×D3112yescomplemented
159#100C2×D3112yescomplemented
160C31#302211yescharacteristic, complemented, Sylow
161#41—22no—
162#41—22no—
163#41—314no—
164#41—314no—
165#41—624no—
166#41—624no—
167#41—624no—
168#41—628no—
169#41—624no—
170#41—622no—
171#41—624no—
172#41—622no—
173C2×Q8—312no—
174C2×Q8—312no—
175C2×Q8—622no—
176C2×Q8—624no—
177C2×Q8—621no—
178C2×Q8—622no—
179C2×Q8—622no—
180C2×Q8C22×D3111yescharacteristic
181C2×D4—312no—
182C2×D4—622no—
183C2×D4—624no—
184C2×D4—628no—
185C2×D4—622no—
186C2×D4—624no—
187C2×D4D12412yes—
188C2×D4C22×D3111yescharacteristic
189C22×C4—624no—
190C22×C4—622no—
191C22×C4—621no—
192C22×C4D12412yes—
193C22×C4C22×D3111yescharacteristic
194Q16—628no—
195SD16—628no—
196M16—22no—
197M16C22×D3114yes—
198C2×C8—22no—
199C4⋊C4—624no—
200C4⋊C4—621no—
201C22⋊C4—621no—
202C22⋊C4—624no—
203C42—312no—
204C23—624no—
205C23—622no—
206C23C2×D12412yes—
207C23C2×D12411yescharacteristic
208Q8—22no—
209Q8—314no—
210Q8—624no—
211Q8—622no—
212Q8—624no—
213Q8—1241no—
214Q8—1241no—
215D4—24no—
216D4—22no—
217D4—314no—
218D4—624no—
219D4—628no—
220D4—624no—
221D4—624no—
222D4—622no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—628no—
226C2×C4—624no—
227C2×C4—622no—
228C2×C4—624no—
229C2×C4—624no—
230C2×C4—622no—
231C2×C4—622no—
232C2×C4—1242no—
233C2×C4C2×D12412yes—
234C2×C4C2×D12411yescharacteristic
235C2×C4D4×D3112yes—
236C2×C4D4×D3112yes—
237C2×C4C23×D3111yescharacteristic
238C8—24no—
239C22—22no—
240C22—41no—
241C22—42no—
242C22—624no—
243C22—2482no—
244C22—2484no—
245C22#1175845912yes—
246C22C22×D12411yescharacteristic
247C4—22no—
248C4—624no—
249C4—1242no—
250C4—1242no—
251C4#1175845911yescharacteristic
252C4#1175845911yescharacteristic
253C4C2×D4×D3112yes—
254C2—21no—
255C2—22no—
256C2—42no—
257C2—1244no—
258C2C2×#1175845911yescharacteristic
subgroupG/H
centerC2C2×#11758459
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#413C2
socleC2×C31C2×#120
prime radical#14521885C2
hypercenter#413D31
hypocenterC31#3022
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
2. C2×#11758486C2semidirect
3. #14521620C2semidirect
4. #14521234C2semidirect
5. #14521229C2semidirect
6. #14521065C2semidirect
7. C31×#413C2semidirect
17. C22×Dic62C22semidirect
18. C2×M16×C31C22semidirect
19. #11758521C22semidirect
20. #11758513C22semidirect
22. #11758503C22semidirect
25. #11758486C22semidirect
28. C31×#101C22semidirect
29. C31×#100C22semidirect
30. #11758413C22semidirect
131. #413D31semidirect
151. C2×M16C2×D31semidirect
158. #101C2×D31semidirect
159. #100C2×D31semidirect
160. C31#3022semidirect

Maximal subgroups

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

subgroupindex
#145218852
C2×#117584862
#145216202
#145212342
#145212292
#145210652
C31×#4132
#302231