Finite Group Databasebeta

← #16848307

Subgroups — #16848307

Order 3968.

672 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 672 conjugacy classes of subgroups — too many to draw. Only its 189 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
1C22×C22⋊Dic31C211yesmaximal, characteristic
2#14520943C214yesmaximal, complemented
3#14520937C214yesmaximal, complemented
4#14520907C212yesmaximal, complemented
5#14520905C211yesmaximal, characteristic
6C22⋊C4×Dic31C212yesmaximal, complemented
7C31×#379C211yesmaximal, characteristic
8C25×C31C2211yescharacteristic
9C23×Dic31C2211yescharacteristic
10C23×Dic31C2211yescharacteristic
11C2×C22⋊C4×C31C2212yes—
12C2×C22⋊C4×C31C2212yes—
13C2×C22⋊C4×C31C2212yes—
14C2×C22⋊Dic31C418yescomplemented
15C2×C22⋊Dic31C2212yes—
16C2×C22⋊Dic31C2212yes—
17C2×C22⋊Dic31C2212yes—
18C2×C22⋊Dic31C2212yes—
19C2×C22⋊Dic31C2212yes—
20C2×Dic31⋊C4C2214yescomplemented
21C2×C4×Dic31C2214yescomplemented
22C31⋊#95C2214yescomplemented
23C31⋊#95C2214yescomplemented
24C24×C31—24no—
25C24×C31—42no—
26C24×C31—42no—
27C24×C31C2×C414yes—
28C24×C31C2311yescharacteristic
29C24×C31C2311yescharacteristic
30C24×C31C2311yescharacteristic
31C22×C4×C31C2314yes—
32C22×Dic31—42no—
33C22×Dic31—42no—
34C22×Dic31C2×C414yes—
35C22×Dic31C2×C414yes—
36C22×Dic31C2314yes—
37C22×Dic31C2312yes—
38C22×Dic31C2312yes—
39C22⋊C4×C31—24no—
40C22⋊C4×C31—42no—
41C22⋊C4×C31—42no—
42C22⋊Dic31—24no—
43C22⋊Dic31—42no—
44C22⋊Dic31—42no—
45C22⋊Dic31C2×C418yes—
46C22⋊Dic31C2×C418yes—
47Dic31⋊C4—44no—
48C4×Dic31—44no—
49C23×C31—22no—
50C23×C31—22no—
51C23×C31—28no—
52C23×C31—22no—
53C23×C31—22no—
54C23×C31—22no—
55C23×C31—22no—
56C23×C31—22no—
57C23×C31—22no—
58C23×C31—44no—
59C23×C31—44no—
60C23×C31—416no—
61C23×C31C22×C412yes—
62C23×C31C22×C412yes—
63C23×C31C22×C412yes—
64C23×C31#4114yes—
65C23×C31C2411yescharacteristic
66C2×C4×C31—24no—
67C2×C4×C31—24no—
68C2×C4×C31—24no—
69C2×Dic31—24no—
70C2×Dic31—24no—
71C2×Dic31—24no—
72C2×Dic31—24no—
73C2×Dic31—24no—
74C2×Dic31—44no—
75C2×Dic31—44no—
76C2×Dic31C22×C414yes—
77C2×Dic31C22×C414yes—
78#2312—311nomaximal, Sylow
79C22×C31—22no—
80C22×C31—22no—
81C22×C31—28no—
82C22×C31—22no—
83C22×C31—22no—
84C22×C31—22no—
85C22×C31—22no—
86C22×C31—22no—
87C22×C31—22no—
88C22×C31—416no—
89C22×C31—44no—
90C22×C31—44no—
91C22×C31#11714yes—
92C22×C31C23×C411yescharacteristic
93C22×C31C2×#4112yes—
94C22×C31H(5,𝔽2)12yes—
95C22×C31H(5,𝔽2)12yes—
96C4×C31—44no—
97Dic31—24no—
98Dic31—24no—
99Dic31—44no—
100C22×C22⋊C4—311no—
101#388—314no—
102#388—314no—
103#380—312no—
104#379—311no—
105#379D3111yescharacteristic
106C4×C22⋊C4—312no—
107C2×C31—24no—
108C2×C31—42no—
109C2×C31—42no—
110C2×C31C2×#11711yescharacteristic
111C2×C31#51811yescharacteristic
112C2×C31#51811yescharacteristic
113C2×C31#53814yes—
114C25C2×D3111yescharacteristic
115C23×C4—311no—
116C23×C4—311no—
117C2×C4⋊C4—314no—
118C2×C22⋊C4—318no—
119C2×C22⋊C4—312no—
120C2×C22⋊C4—312no—
121C2×C22⋊C4—312no—
122C2×C22⋊C4—312no—
123C2×C22⋊C4—312no—
124C2×C22⋊C4C2×D3112yes—
125C2×C22⋊C4C2×D3112yes—
126C2×C22⋊C4C2×D3112yes—
127C2×C42—314no—
128#95—314no—
129#95—314no—
130C31#231211yescharacteristic, complemented, Sylow
131C24—24no—
132C24—42no—
133C24—42no—
134C24C4×D3114yes—
135C24C22×D3111yescharacteristic
136C24C22×D3111yescharacteristic
137C24C22×D3111yescharacteristic
138C22×C4—312no—
139C22×C4—312no—
140C22×C4—314no—
141C22×C4—314no—
142C22×C4—314no—
143C22×C4—1242no—
144C22×C4—1242no—
145C22×C4C22×D3114yes—
146C4⋊C4—1244no—
147C22⋊C4—24no—
148C22⋊C4—42no—
149C22⋊C4—42no—
150C22⋊C4—318no—
151C22⋊C4—318no—
152C22⋊C4—624no—
153C22⋊C4—1242no—
154C22⋊C4—1242no—
155C42—1244no—
156C23—22no—
157C23—22no—
158C23—28no—
159C23—22no—
160C23—22no—
161C23—22no—
162C23—22no—
163C23—22no—
164C23—22no—
165C23—44no—
166C23—44no—
167C23—416no—
168C23C2×C4×D3112yes—
169C23C2×C4×D3112yes—
170C23C2×C4×D3112yes—
171C23#9249714yes—
172C23C23×D3111yescharacteristic
173C2×C4—24no—
174C2×C4—24no—
175C2×C4—24no—
176C2×C4—314no—
177C2×C4—314no—
178C2×C4—624no—
179C2×C4—624no—
180C2×C4—624no—
181C2×C4—624no—
182C2×C4—624no—
183C2×C4—1244no—
184C2×C4—1244no—
185C22—28no—
186C22—22no—
187C22—22no—
188C22—22no—
189C22—22no—
190C22—22no—
191C22—22no—
192C22—22no—
193C22—22no—
194C22—416no—
195C22—44no—
196C22—44no—
197C22#1175845414yes—
198C22C22×C4×D3111yescharacteristic
199C22C2×#9249712yes—
200C22#1175856112yes—
201C22#1175856112yes—
202C4—44no—
203C4—624no—
204C4—624no—
205C4—1244no—
206C2—24no—
207C2—42no—
208C2—42no—
209C2C2×#1175845411yescharacteristic
210C2#1452152211yescharacteristic
211C2#1452152211yescharacteristic
212C2#1452163414yes—
subgroupG/H
center, FrattiniC23C23×D31
commutatorC22×C31C23×C4
FittingC31×#379C2
socle, cosocleC23×C31C24
prime radicalC25×C31C22
hypercenter#379D31
hypocenterC31#2312
LCSG ▷ C22×C31 ▷ C31
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
2. #14520943C2semidirect
3. #14520937C2semidirect
4. #14520907C2semidirect
6. C22⋊C4×Dic31C2semidirect
14. C2×C22⋊Dic31C4semidirect
20. C2×Dic31⋊C4C22semidirect
21. C2×C4×Dic31C22semidirect
22. C31⋊#95C22semidirect
23. C31⋊#95C22semidirect
130. C31#2312semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊Dic312
#145209432
#145209372
#145209072
#145209052
C22⋊C4×Dic312
C31×#3792
#231231