Finite Group Databasebeta

← #16848861

Subgroups — #16848861

Order 3968.

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

normal subgroups only

This group has 512 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×#11758512C211yesmaximal, characteristic
2#14520944C212yesmaximal, complemented
3#14520943C212yesmaximal, complemented
4#14520937C212yesmaximal, complemented
5#14520933C212yesmaximal, complemented
6#14520929C211yesmaximal, characteristic
7#14520921C212yesmaximal, complemented
8#14520907C212yesmaximal, complemented
9C31×#390C211yesmaximal, characteristic
10C22×D4×C31C2211yescharacteristic
11C23×Dic31C2211yescharacteristic
12C23×Dic31C2211yescharacteristic
13C2×C4⋊C4×C31C2211yescharacteristic
14C2×C22⋊C4×C31C2212yes—
15C2×C22⋊C4×C31C2212yes—
16C2×C42×C31C2211yescharacteristic
17C2×C22⋊Dic31C2212yes—
18C2×C22⋊Dic31C2211yescharacteristic
19C2×C22⋊Dic31C2212yes—
20C2×C22⋊Dic31C2211yescharacteristic
21#11758512C418yescomplemented
22C2×Dic31⋊C4C2212yes—
23C2×Dic31⋊C4C2212yes—
24C2×Dic31⋊C4C2212yes—
25C2×Dic31⋊C4C2212yes—
26C2×C4×Dic31C2212yes—
27C31⋊#95C2212yes—
28C31⋊#95C2212yes—
29C31⋊#95C2212yes—
30C31⋊#95C2212yes—
31C31⋊#95C2212yes—
32C24×C31C2311yescharacteristic
33C24×C31C2311yescharacteristic
34C2×D4×C31—41no—
35C2×D4×C31—41no—
36C2×D4×C31C2×C414yes—
37C22×C4×C31C2312yes—
38C22×C4×C31C2312yes—
39C22×C4×C31C2311yescharacteristic
40C22×Dic31—41no—
41C22×Dic31—41no—
42C22×Dic31—41no—
43C22×Dic31—42no—
44C22×Dic31C2×C414yescomplemented
45C22×Dic31C2312yes—
46C22×Dic31C2312yes—
47C22×Dic31C2312yes—
48C22×Dic31C2312yes—
49C4⋊C4×C31—41no—
50C22⋊C4×C31—42no—
51C22⋊C4×C31—42no—
52C42×C31—41no—
53C22⋊Dic31—42no—
54C22⋊Dic31—41no—
55C22⋊Dic31C2×C418yes—
56C22⋊Dic31C2×C414yescomplemented
57Dic31⋊C4—42no—
58Dic31⋊C4—42no—
59Dic31⋊C4—42no—
60Dic31⋊C4C2×C418yescomplemented
61C4×Dic31—42no—
62C23×C31—22no—
63C23×C31—21no—
64C23×C31—22no—
65C23×C31—21no—
66C23×C31—21no—
67C23×C31—21no—
68C23×C31—22no—
69C23×C31—22no—
70C23×C31C22×C412yes—
71C23×C31C22×C412yes—
72C23×C31C2411yescharacteristic
73D4×C31—44no—
74C2×C4×C31—21no—
75C2×C4×C31—21no—
76C2×C4×C31—22no—
77C2×C4×C31—22no—
78C2×C4×C31—22no—
79C2×C4×C31—22no—
80C2×C4×C31—22no—
81C2×C4×C31—22no—
82C2×C4×C31C22×C412yes—
83C2×Dic31—22no—
84C2×Dic31—22no—
85C2×Dic31—22no—
86C2×Dic31—22no—
87C2×Dic31—22no—
88C2×Dic31—22no—
89C2×Dic31—22no—
90C2×Dic31—22no—
91C2×Dic31—22no—
92C2×Dic31—22no—
93C2×Dic31—44no—
94C2×Dic31—44no—
95C2×Dic31C22×C414yes—
96C2×Dic31C22×C414yes—
97#2376—311nomaximal, Sylow
98C22×C31—22no—
99C22×C31—22no—
100C22×C31—24no—
101C22×C31—24no—
102C22×C31—41no—
103C22×C31—41no—
104C22×C31—42no—
105C22×C31—42no—
106C22×C31—41no—
107C22×C31—41no—
108C22×C31C23×C411yescharacteristic
109C22×C31H(5,𝔽2)11yescharacteristic
110C22×C31H(5,𝔽2)11yescharacteristic
111C22×C31H(5,𝔽2)12yes—
112C22×C312−1+412yes—
113C4×C31—22no—
114C4×C31—42no—
115C4×C31—42no—
116Dic31—24no—
117Dic31—24no—
118Dic31—42no—
119Dic31—42no—
120C2×#123—311no—
121#390D3111yescharacteristic
122#389—312no—
123#388—312no—
124#388—312no—
125#387—312no—
126#386—311no—
127#385—312no—
128#380—312no—
129C2×C31—42no—
130C2×C31—42no—
131C2×C31#51811yescharacteristic
132C2×C31#52012yes—
133C2×C31#56011yescharacteristic
134C2×C31#56212yes—
135C2×C31#56311yescharacteristic
136C22×D4C2×D3111yescharacteristic
137C23×C4—311no—
138C23×C4—311no—
139#123—318no—
140C2×C4⋊C4—312no—
141C2×C4⋊C4—312no—
142C2×C4⋊C4—312no—
143C2×C4⋊C4—312no—
144C2×C4⋊C4C2×D3111yescharacteristic
145C2×C22⋊C4—312no—
146C2×C22⋊C4—311no—
147C2×C22⋊C4—312no—
148C2×C22⋊C4—311no—
149C2×C22⋊C4C2×D3112yes—
150C2×C22⋊C4C2×D3112yes—
151C2×C42—312no—
152C2×C42C2×D3111yescharacteristic
153#95—312no—
154#95—312no—
155#95—312no—
156#95—312no—
157#95—312no—
158C31#237611yescharacteristic, complemented, Sylow
159C24C22×D3111yescharacteristic
160C24C22×D3111yescharacteristic
161C2×D4—41no—
162C2×D4—41no—
163C2×D4C4×D3114yes—
164C22×C4—312no—
165C22×C4—312no—
166C22×C4—314no—
167C22×C4—312no—
168C22×C4—312no—
169C22×C4—1241no—
170C22×C4—1241no—
171C22×C4—1242no—
172C22×C4—1241no—
173C22×C4C22×D3112yes—
174C22×C4C22×D3112yes—
175C22×C4C22×D3111yescharacteristic
176C4⋊C4—41no—
177C4⋊C4—318no—
178C4⋊C4—1242no—
179C4⋊C4—1242no—
180C4⋊C4—1242no—
181C22⋊C4—42no—
182C22⋊C4—42no—
183C22⋊C4—318no—
184C22⋊C4—314no—
185C22⋊C4—1242no—
186C22⋊C4—1241no—
187C42—41no—
188C42—1242no—
189C23—22no—
190C23—22no—
191C23—21no—
192C23—21no—
193C23—21no—
194C23—21no—
195C23—22no—
196C23—22no—
197C23C2×C4×D3112yes—
198C23C2×C4×D3112yes—
199C23C23×D3111yescharacteristic
200D4—44no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—314no—
210C2×C4—314no—
211C2×C4—622no—
212C2×C4—622no—
213C2×C4—622no—
214C2×C4—622no—
215C2×C4—622no—
216C2×C4—622no—
217C2×C4—622no—
218C2×C4—622no—
219C2×C4—622no—
220C2×C4—622no—
221C2×C4—1244no—
222C2×C4—1244no—
223C2×C4C2×C4×D3112yes—
224C22—22no—
225C22—22no—
226C22—24no—
227C22—24no—
228C22—41no—
229C22—41no—
230C22—42no—
231C22—42no—
232C22—41no—
233C22—41no—
234C22C22×C4×D3111yescharacteristic
235C22#1175856111yescharacteristic
236C22#1175856111yescharacteristic
237C22#1175856112yes—
238C22#1175856712yes—
239C4—22no—
240C4—42no—
241C4—42no—
242C4—624no—
243C4—624no—
244C4—1242no—
245C4—1242no—
246C2—42no—
247C2—42no—
248C2#1452152211yescharacteristic
249C2#1452152812yes—
250C2#1452178611yescharacteristic
251C2#1452179412yes—
252C2#1452179911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D31
commutatorC22×C31C23×C4
FittingC31×#390C2
socle, cosocleC23×C31C24
prime radicalC22×D4×C31C22
hypercenter#390D31
hypocenterC31#2376
LCSG ▷ C22×C31 ▷ C31
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. #14520944C2semidirect
3. #14520943C2semidirect
4. #14520937C2semidirect
5. #14520933C2semidirect
7. #14520921C2semidirect
8. #14520907C2semidirect
21. #11758512C4semidirect
44. C22×Dic31C2×C4semidirect
56. C22⋊Dic31C2×C4semidirect
60. Dic31⋊C4C2×C4semidirect
158. C31#2376semidirect

Maximal subgroups

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

subgroupindex
C2×#117585122
#145209442
#145209432
#145209372
#145209332
#145209292
#145209212
#145209072
C31×#3902
#237631