Finite Group Databasebeta

← #16855089

Subgroups — #16855089

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×#11758519C212yesmaximal, complemented
3#14521231C214yesmaximal, complemented
4#14521226C214yesmaximal, complemented
5#14521067C212yesmaximal, complemented
6#14521033C211yesmaximal, characteristic, complemented
7C31×#536C211yesmaximal, characteristic, complemented
8C2×#41×C31C2211yescharacteristic
9C22×Q8×C31C2211yescharacteristic, complemented
10#11758567—22no—
11#11758566—22no—
12#41×D31—24no—
13#11758564C2212yes—
14#11758561—22no—
15#11758561C2212yes—
16C2×#92495—22no—
17C2×#92495C2212yes—
18C31×C4⋊Q8C2212yescomplemented
19C31×#124C2212yescomplemented
20C31×#123—22no—
21C31×C22⋊Q8—22no—
22C31×#117C2211yescharacteristic, complemented
23C2×#92482—22no—
24#11758519—24no—
25#11758519C2218yescomplemented
26C2×#92481—22no—
27C2×#92474C2212yescomplemented
28#11758424C2212yescomplemented
29#11758421C2212yescomplemented
30#11758393C2218yescomplemented
31#41×C31—22no—
32#41×C31—22no—
33C2×Q8×C31—21no—
34C2×Q8×C31—22no—
35C2×Q8×C31—22no—
36C2×Q8×C31C2312yes—
37C2×Q8×C31C2311yescharacteristic
38C2×D4×C31D412yes—
39C2×D4×C31C2311yescharacteristic
40C22×C4×C31—21no—
41C22×C4×C31D412yes—
42C22×C4×C31C2311yescharacteristic
43C2×C22⋊D31—24no—
44C2×C22⋊D31—22no—
45#92499—24no—
46Q8×D31—24no—
47#92497—28no—
48#92497—24no—
49D4×D31—28no—
50D4×D31—24no—
51#92495—24no—
52#92495—24no—
53#92495—22no—
54#92495—24no—
55#92495—22no—
56#92495D414yes—
57#92495C2314yes—
58C2×D124—22no—
59C2×D124D412yes—
60C2×C4×D31—24no—
61C2×C4×D31—22no—
62C2×Dic62—22no—
63C2×Dic62D412yes—
64C4⋊C4×C31—21no—
65C4⋊C4×C31—24no—
66C22⋊C4×C31—21no—
67C22⋊C4×C31—24no—
68C42×C31C2312yes—
69#92482—28no—
70#92481—28no—
71#92474—22no—
72#92474C2314yes—
73C2×C31⋊C8—22no—
74C23×C31C2×D412yes—
75C23×C31C2×D411yescharacteristic
76C22×D31—24no—
77C22×D31—22no—
78Q8×C31—22no—
79Q8×C31—24no—
80Q8×C31—41no—
81Q8×C31—41no—
82D4×C31—24no—
83D4×C31—22no—
84C2×C4×C31—22no—
85C2×C4×C31—22no—
86C2×C4×C31—22no—
87C2×C4×C31—22no—
88C2×C4×C31—42no—
89C2×C4×C31C2×D412yes—
90C2×C4×C31C2×D412yes—
91C2×C4×C31C2×D412yes—
92C2×C4×C31C2×D411yescharacteristic
93C2×C4×C31C2411yescharacteristic
94C22⋊D31—28no—
95C22⋊D31—24no—
96C22⋊D31—24no—
97C2×Dic31—24no—
98C2×Dic31—22no—
99D124—24no—
100D124—22no—
101D124C2×D414yes—
102C4×D31—28no—
103C4×D31—24no—
104C4×D31—24no—
105Dic62—24no—
106Dic62—22no—
107Dic62C2×D414yes—
108C31⋊C8—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—42no—
120C4×C31—42no—
121C4×C31#12011yescharacteristic
122C4×C31#12011yescharacteristic
123C4×C31C22×D412yes—
124Dic31—24no—
125#585—311no—
126C2×#137—312no—
127#536D3111yescharacteristic, complemented
128#456—314no—
129#455—314no—
130#421—312no—
131#413—311no—
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×Q8C2×D3111yescharacteristic, complemented
147#137—318no—
148#137—624no—
149C2×Q16—622no—
150C2×SD16—622no—
151C2×M16—312no—
152C4⋊Q8C2×D3112yescomplemented
153#124C2×D3112yescomplemented
154#123—22no—
155C22⋊Q8—22no—
156#117C2×D3111yescharacteristic, complemented
157#104—318no—
158#101—312no—
159#100—312no—
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—622no—
172#41—624no—
173C2×Q8—21no—
174C2×Q8—22no—
175C2×Q8—22no—
176C2×Q8—312no—
177C2×Q8—622no—
178C2×Q8—624no—
179C2×Q8C22×D3112yes—
180C2×Q8C22×D3111yescharacteristic
181C2×D4—312no—
182C2×D4—622no—
183C2×D4—624no—
184C2×D4—628no—
185C2×D4—622no—
186C2×D4—624no—
187C2×D4C22⋊D3112yes—
188C2×D4C22×D3111yescharacteristic
189C22×C4—21no—
190C22×C4—624no—
191C22×C4—622no—
192C22×C4C22⋊D3112yes—
193C22×C4C22×D3111yescharacteristic
194Q16—628no—
195SD16—628no—
196M16—314no—
197M16—622no—
198C2×C8—622no—
199C4⋊C4—21no—
200C4⋊C4—24no—
201C22⋊C4—21no—
202C22⋊C4—24no—
203C42C22×D3112yes—
204C23—624no—
205C23—622no—
206C23C2×C22⋊D3112yes—
207C23C2×C22⋊D3111yescharacteristic
208Q8—22no—
209Q8—24no—
210Q8—41no—
211Q8—41no—
212Q8—314no—
213Q8—624no—
214Q8—622no—
215D4—24no—
216D4—22no—
217D4—314no—
218D4—624no—
219D4—628no—
220D4—624no—
221D4—624no—
222D4—622no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—42no—
228C2×C4—628no—
229C2×C4—624no—
230C2×C4—622no—
231C2×C4—624no—
232C2×C4—624no—
233C2×C4D4×D3112yes—
234C2×C4D4×D3112yes—
235C2×C4C2×C22⋊D3112yes—
236C2×C4C2×C22⋊D3111yescharacteristic
237C2×C4C23×D3111yescharacteristic
238C8—624no—
239C22—22no—
240C22—41no—
241C22—42no—
242C22—624no—
243C22—2482no—
244C22—2484no—
245C22#1175851412yes—
246C22C22×C22⋊D3111yescharacteristic
247C4—22no—
248C4—42no—
249C4—42no—
250C4—624no—
251C4#1175851411yescharacteristic
252C4#1175851411yescharacteristic
253C4C2×D4×D3112yes—
254C2—21no—
255C2—22no—
256C2—42no—
257C2—1244no—
258C2C2×#1175851411yescharacteristic
subgroupG/H
centerC2C2×#11758514
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#536C2
socleC2×C31C2×#120
prime radical#14521885C2
hypercenter#536D31
hypocenterC31#3022
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C22×C4 ◁ #536

Complements

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

NH ≅ G/Ntype
2. C2×#11758519C2semidirect
3. #14521231C2semidirect
4. #14521226C2semidirect
5. #14521067C2semidirect
6. #14521033C2semidirect
7. C31×#536C2semidirect
9. C22×Q8×C31C22semidirect
18. C31×C4⋊Q8C22semidirect
19. C31×#124C22semidirect
22. C31×#117C22semidirect
25. #11758519C22semidirect
27. C2×#92474C22semidirect
28. #11758424C22semidirect
29. #11758421C22semidirect
30. #11758393C22semidirect
127. #536D31semidirect
146. C22×Q8C2×D31semidirect
152. C4⋊Q8C2×D31semidirect
153. #124C2×D31semidirect
156. #117C2×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×#117585192
#145212312
#145212262
#145210672
#145210332
C31×#5362
#302231