Finite Group Databasebeta

← #16855090

Subgroups — #16855090

Order 3968.

704 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 161 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
1C2×#11758528C212yesmaximal, complemented
2#14521620C211yesmaximal, characteristic, complemented
3#14521232C214yesmaximal, complemented
4#14521227C214yesmaximal, complemented
5#14521066C212yesmaximal, complemented
6#14521033C211yesmaximal, characteristic, complemented
7C31×#585C211yesmaximal, characteristic
8C31×2−1+4—22no—
9C31×2−1+4C2212yes—
10C31×H(5,𝔽2)—22no—
11C31×H(5,𝔽2)—22no—
12C31×H(5,𝔽2)C2212yes—
13C2×#41×C31—24no—
14C2×#41×C31—22no—
15C2×#41×C31C2212yes—
16C2×#41×C31C2211yescharacteristic
17C22×Dic62C2211yescharacteristic, complemented
18#11758528—24no—
19#11758528C2218yescomplemented
20#11758521C2212yescomplemented
21C2×#92482—22no—
22#11758513C2212yescomplemented
23#11758512—22no—
24C2×#92480—22no—
25#11758503C2211yescharacteristic, complemented
26#11758501—22no—
27C2×#92474C2212yescomplemented
28#11758425C2218yescomplemented
29#11758424C2212yescomplemented
30#11758421C2212yescomplemented
31#41×C31—24no—
32#41×C31—28no—
33#41×C31—24no—
34#41×C31—24no—
35#41×C31—24no—
36#41×C31—22no—
37#41×C31—22no—
38#41×C31—24no—
39#41×C31—22no—
40#41×C31—22no—
41#41×C31D414yes—
42#41×C31C2314yes—
43C2×Q8×C31—22no—
44C2×Q8×C31—24no—
45C2×Q8×C31D412yes—
46C2×Q8×C31C2311yescharacteristic
47C2×D4×C31—28no—
48C2×D4×C31—22no—
49C2×D4×C31—24no—
50C2×D4×C31—22no—
51C2×D4×C31—24no—
52C2×D4×C31D412yes—
53C2×D4×C31D412yes—
54C2×D4×C31C2311yescharacteristic
55C22×C4×C31—24no—
56C22×C4×C31—22no—
57C22×C4×C31D412yes—
58C22×C4×C31C2311yescharacteristic
59C22×Dic31—21no—
60C2×Dic62—22no—
61C2×Dic62—22no—
62C2×Dic62—21no—
63C2×Dic62C2312yes—
64C22⋊Dic31—24no—
65C22⋊Dic31—21no—
66#92482—28no—
67#92480—28no—
68C4⋊Dic31—21no—
69Dic31⋊C4—24no—
70C4×Dic31C2312yes—
71#92474—22no—
72#92474C2314yes—
73C2×C31⋊C8—22no—
74C23×C31—24no—
75C23×C31—22no—
76C23×C31C2×D412yes—
77C23×C31C2×D411yescharacteristic
78Q8×C31—24no—
79Q8×C31—22no—
80Q8×C31—22no—
81Q8×C31C2×D414yes—
82D4×C31—28no—
83D4×C31—24no—
84D4×C31—24no—
85D4×C31—24no—
86D4×C31—24no—
87D4×C31—22no—
88D4×C31—22no—
89D4×C31C2×D414yes—
90C2×C4×C31—24no—
91C2×C4×C31—28no—
92C2×C4×C31—22no—
93C2×C4×C31—24no—
94C2×C4×C31—22no—
95C2×C4×C31—22no—
96C2×C4×C31—24no—
97C2×C4×C31C2×D412yes—
98C2×C4×C31C2×D411yescharacteristic
99C2×C4×C31C2×D412yes—
100C2×C4×C31C2×D412yes—
101C2×C4×C31C2411yescharacteristic
102C2×Dic31—22no—
103C2×Dic31—22no—
104C2×Dic31—42no—
105Dic62—24no—
106Dic62—41no—
107Dic62—41no—
108C31⋊C8—24no—
109#3022—311nomaximal, Sylow
110C22×C31—22no—
111C22×C31—24no—
112C22×C31—41no—
113C22×C31—42no—
114C22×C31—82no—
115C22×C31—84no—
116C22×C31#12012yes—
117C22×C31C22×D411yescharacteristic
118C4×C31—24no—
119C4×C31—22no—
120C4×C31#12011yescharacteristic
121C4×C31#12011yescharacteristic
122C4×C31C22×D412yes—
123Dic31—42no—
124Dic31—42no—
125#585D3111yescharacteristic
126C2×#137—312no—
127#536—311no—
128#456—314no—
129#455—314no—
130#421—312no—
131#413—311no—
132C2×C31—21no—
133C2×C31—22no—
134C2×C31—42no—
135C2×C31—44no—
136C2×C31C2×#12011yescharacteristic
1372−1+4—22no—
1382−1+4C2×D3112yes—
139H(5,𝔽2)—22no—
140H(5,𝔽2)—22no—
141H(5,𝔽2)C2×D3112yes—
142C2×#41—24no—
143C2×#41—22no—
144C2×#41C2×D3112yes—
145C2×#41C2×D3111yescharacteristic
146C22×Q8—311no—
147#137—318no—
148#137—624no—
149C2×Q16—622no—
150C2×SD16—622no—
151C2×M16—312no—
152C4⋊Q8—312no—
153#124—312no—
154#123—622no—
155C22⋊Q8—622no—
156#117—311no—
157#104—318no—
158#101—312no—
159#100—312no—
160C31#302211yescharacteristic, complemented, Sylow
161#41—24no—
162#41—28no—
163#41—24no—
164#41—24no—
165#41—24no—
166#41—22no—
167#41—22no—
168#41—24no—
169#41—22no—
170#41—22no—
171#41C22⋊D3114yes—
172#41C22×D3114yes—
173C2×Q8—22no—
174C2×Q8—24no—
175C2×Q8—312no—
176C2×Q8—621no—
177C2×Q8—622no—
178C2×Q8—622no—
179C2×Q8C22⋊D3112yes—
180C2×Q8C22×D3111yescharacteristic
181C2×D4—28no—
182C2×D4—22no—
183C2×D4—24no—
184C2×D4—22no—
185C2×D4—24no—
186C2×D4C22⋊D3112yes—
187C2×D4C22⋊D3112yes—
188C2×D4C22×D3111yescharacteristic
189C22×C4—24no—
190C22×C4—22no—
191C22×C4—621no—
192C22×C4C22⋊D3112yes—
193C22×C4C22×D3111yescharacteristic
194Q16—628no—
195SD16—628no—
196M16—314no—
197M16—622no—
198C2×C8—622no—
199C4⋊C4—624no—
200C4⋊C4—621no—
201C22⋊C4—621no—
202C22⋊C4—624no—
203C42—312no—
204C23—24no—
205C23—22no—
206C23C2×C22⋊D3112yes—
207C23C2×C22⋊D3111yescharacteristic
208Q8—24no—
209Q8—22no—
210Q8—22no—
211Q8—624no—
212Q8—1241no—
213Q8—1241no—
214Q8C2×C22⋊D3114yes—
215D4—28no—
216D4—24no—
217D4—24no—
218D4—24no—
219D4—24no—
220D4—22no—
221D4—22no—
222D4C2×C22⋊D3114yes—
223C2×C4—24no—
224C2×C4—28no—
225C2×C4—22no—
226C2×C4—24no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—24no—
230C2×C4—622no—
231C2×C4—622no—
232C2×C4—1242no—
233C2×C4C2×C22⋊D3112yes—
234C2×C4C2×C22⋊D3111yescharacteristic
235C2×C4C2×C22⋊D3112yes—
236C2×C4C2×C22⋊D3112yes—
237C2×C4C23×D3111yescharacteristic
238C8—624no—
239C22—22no—
240C22—24no—
241C22—41no—
242C22—42no—
243C22—82no—
244C22—84no—
245C22#1175853012yes—
246C22C22×C22⋊D3111yescharacteristic
247C4—24no—
248C4—22no—
249C4—1242no—
250C4—1242no—
251C4#1175853011yescharacteristic
252C4#1175853011yescharacteristic
253C4C22×C22⋊D3112yes—
254C2—21no—
255C2—22no—
256C2—42no—
257C2—44no—
258C2C2×#1175853011yescharacteristic
subgroupG/H
centerC2C2×#11758530
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
Fitting, prime radicalC31×#585C2
socleC2×C31C2×#120
hypercenter#585D31
hypocenterC31#3022
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C22×C4 ◁ #585

Complements

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

NH ≅ G/Ntype
1. C2×#11758528C2semidirect
2. #14521620C2semidirect
3. #14521232C2semidirect
4. #14521227C2semidirect
5. #14521066C2semidirect
6. #14521033C2semidirect
17. C22×Dic62C22semidirect
19. #11758528C22semidirect
20. #11758521C22semidirect
22. #11758513C22semidirect
25. #11758503C22semidirect
27. C2×#92474C22semidirect
28. #11758425C22semidirect
29. #11758424C22semidirect
30. #11758421C22semidirect
160. C31#3022semidirect

Maximal subgroups

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

subgroupindex
C2×#117585282
#145216202
#145212322
#145212272
#145210662
#145210332
C31×#5852
#302231