Finite Group Databasebeta

← #16848883

Subgroups — #16848883

Order 3968.

520 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 173 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×#11758524C211yesmaximal, characteristic
2#14520956C211yesmaximal, characteristic, complemented
3#14520940C212yesmaximal, complemented
4#14520938C212yesmaximal, complemented
5#14520927C212yesmaximal
6#14520923C212yesmaximal, complemented
7#14520920C212yesmaximal, complemented
8#14520913C212yesmaximal, complemented
9C31×#391C211yesmaximal, characteristic, complemented
10C22×Q8×C31C2211yescharacteristic
11C22×D124C2211yescharacteristic
12C22×C4×D31C2212yes—
13C2×C4⋊C4×C31C2211yescharacteristic
14C2×C42×C31C2211yescharacteristic
15#11758524C418yescomplemented
16C2×(C2×D31)⋊C4C2212yes—
17C2×(C2×D31)⋊C4C2212yes—
18C2×(C2×D31)⋊C4C2212yes—
19C2×(C2×D31)⋊C4C2212yes—
20C2×(C2×D31)⋊C4C2212yes—
21C2×C4⋊Dic31C2211yescharacteristic
22C2×Dic31⋊C4C2212yes—
23C2×Dic31⋊C4C2212yes—
24C2×C4×Dic31C2211yescharacteristic
25C2×C4×Dic31C2211yescharacteristic
26C31×#95C2212yes—
27C31×#95C2212yes—
28C31⋊#95C2212yes—
29C31⋊#95C2212yes—
30C31⋊#95C2212yes—
31C31⋊#95C2212yes—
32C23×D31C2312yes—
33C2×Q8×C31—41no—
34C2×Q8×C31—41no—
35C2×Q8×C31C2×C414yescomplemented
36C22×C4×C31C2312yes—
37C22×C4×C31C2312yes—
38C22×C4×C31C2311yescharacteristic
39C22×C4×C31C2311yescharacteristic
40C22×C4×C31C2311yescharacteristic
41C22×Dic31C2312yes—
42C22×Dic31C2312yes—
43C22×Dic31C2312yes—
44C2×D124—41no—
45C2×D124—41no—
46C2×D124C2×C414yes—
47C2×C4×D31—42no—
48C2×C4×D31—42no—
49C2×C4×D31—42no—
50C4⋊C4×C31—41no—
51C42×C31—41no—
52(C2×D31)⋊C4—42no—
53(C2×D31)⋊C4—42no—
54(C2×D31)⋊C4—42no—
55(C2×D31)⋊C4C2×C418yescomplemented
56(C2×D31)⋊C4C2×C418yescomplemented
57C4⋊Dic31—41no—
58Dic31⋊C4—42no—
59Dic31⋊C4—42no—
60C4×Dic31—41no—
61C4×Dic31C2×C414yescomplemented
62C23×C31C2411yescharacteristic
63C22×D31—22no—
64C22×D31—22no—
65C22×D31—22no—
66C22×D31—22no—
67C22×D31—22no—
68C22×D31—22no—
69C22×D31C22×C414yes—
70Q8×C31—44no—
71C2×C4×C31—21no—
72C2×C4×C31—21no—
73C2×C4×C31—21no—
74C2×C4×C31—21no—
75C2×C4×C31—21no—
76C2×C4×C31—21no—
77C2×C4×C31—22no—
78C2×C4×C31—22no—
79C2×C4×C31—22no—
80C2×C4×C31—22no—
81C2×C4×C31—22no—
82C2×C4×C31—22no—
83C2×C4×C31C22×C412yes—
84C2×C4×C31C22×C412yes—
85C2×C4×C31C22×C412yes—
86C2×Dic31—22no—
87C2×Dic31—22no—
88C2×Dic31—22no—
89C2×Dic31—22no—
90C2×Dic31—22no—
91C2×Dic31—22no—
92C2×Dic31—22no—
93C2×Dic31—22no—
94C2×Dic31C22×C414yes—
95D124—44no—
96C4×D31—48no—
97#2379—311nomaximal, Sylow
98C22×C31C23×C411yescharacteristic
99C22×C31H(5,𝔽2)11yescharacteristic
100C22×C31H(5,𝔽2)11yescharacteristic
101C22×C31H(5,𝔽2)12yes—
102C22×C312−1+412yes—
103C2×D31—24no—
104C2×D31—24no—
105C2×D31—24no—
106C2×D31—42no—
107C2×D31—42no—
108C2×D31—42no—
109C2×D31—42no—
110C4×C31—22no—
111C4×C31—22no—
112C4×C31—22no—
113C4×C31—42no—
114C4×C31—42no—
115Dic31—24no—
116Dic31—42no—
117Dic31—42no—
118C2×#124—311no—
119#391—311no—
120#391D3111yescharacteristic, complemented
121#388—312no—
122#388—312no—
123#386—312no—
124#385—312no—
125#385—312no—
126#382—312no—
127C2×C31#51811yescharacteristic
128C2×C31#52012yes—
129C2×C31H(3,𝔽4)11yescharacteristic
130C2×C31#56212yes—
131C2×C31#56311yescharacteristic
132D31—44no—
133C22×Q8C2×D3111yescharacteristic
134C22×D4—311no—
135C23×C4—312no—
136#124—318no—
137C2×C4⋊C4—312no—
138C2×C4⋊C4—312no—
139C2×C4⋊C4—311no—
140C2×C4⋊C4C2×D3111yescharacteristic
141C2×C22⋊C4—312no—
142C2×C22⋊C4—312no—
143C2×C22⋊C4—312no—
144C2×C22⋊C4—312no—
145C2×C22⋊C4—312no—
146C2×C42—311no—
147C2×C42—311no—
148C2×C42C2×D3111yescharacteristic
149#95—312no—
150#95—312no—
151#95—312no—
152#95—312no—
153#95C2×D3112yes—
154#95C2×D3112yes—
155C31#237911yescharacteristic, complemented, Sylow
156C24—312no—
157C2×Q8—41no—
158C2×Q8—41no—
159C2×Q8C4×D3114yescomplemented
160C2×D4—314no—
161C2×D4—1241no—
162C2×D4—1241no—
163C22×C4—312no—
164C22×C4—312no—
165C22×C4—312no—
166C22×C4—1242no—
167C22×C4—1242no—
168C22×C4—1242no—
169C22×C4C22×D3112yes—
170C22×C4C22×D3112yes—
171C22×C4C22×D3111yescharacteristic
172C22×C4C22×D3111yescharacteristic
173C22×C4C22×D3111yescharacteristic
174C4⋊C4—41no—
175C4⋊C4—1242no—
176C4⋊C4—1242no—
177C4⋊C4—1241no—
178C22⋊C4—318no—
179C22⋊C4—318no—
180C22⋊C4—1242no—
181C22⋊C4—1242no—
182C22⋊C4—1242no—
183C42—41no—
184C42—314no—
185C42—1241no—
186C23—314no—
187C23—622no—
188C23—622no—
189C23—622no—
190C23—622no—
191C23—622no—
192C23—622no—
193C23C23×D3111yescharacteristic
194Q8—44no—
195D4—1244no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—314no—
209C2×C4—622no—
210C2×C4—622no—
211C2×C4—622no—
212C2×C4—622no—
213C2×C4—622no—
214C2×C4—622no—
215C2×C4—622no—
216C2×C4—622no—
217C2×C4—1248no—
218C2×C4C2×C4×D3112yes—
219C2×C4C2×C4×D3112yes—
220C2×C4C2×C4×D3112yes—
221C22—624no—
222C22—624no—
223C22—624no—
224C22—1242no—
225C22—1242no—
226C22—1242no—
227C22—1242no—
228C22C22×C4×D3111yescharacteristic
229C22#1175856111yescharacteristic
230C22#1175856111yescharacteristic
231C22#1175856412yes—
232C22#1175856612yes—
233C4—22no—
234C4—22no—
235C4—22no—
236C4—42no—
237C4—42no—
238C4—624no—
239C4—1242no—
240C4—1242no—
241C2—1244no—
242C2#1452152211yescharacteristic
243C2#1452153112yes—
244C2#1452179111yescharacteristic
245C2#1452179512yes—
246C2#1452179811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D31
commutatorC22×C31C23×C4
FittingC31×#391C2
socle, cosocleC23×C31C24
prime radicalC22×D124C22
hypercenter#391D31
hypocenterC31#2379
LCSG ▷ C22×C31 ▷ C31
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
2. #14520956C2semidirect
3. #14520940C2semidirect
4. #14520938C2semidirect
6. #14520923C2semidirect
7. #14520920C2semidirect
8. #14520913C2semidirect
9. C31×#391C2semidirect
15. #11758524C4semidirect
35. C2×Q8×C31C2×C4semidirect
55. (C2×D31)⋊C4C2×C4semidirect
56. (C2×D31)⋊C4C2×C4semidirect
61. C4×Dic31C2×C4semidirect
120. #391D31semidirect
155. C31#2379semidirect
159. C2×Q8C4×D31semidirect

Maximal subgroups

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

subgroupindex
C2×#117585242
#145209562
#145209402
#145209382
#145209272
#145209232
#145209202
#145209132
C31×#3912
#237931