Finite Group Databasebeta

← #16853968

Subgroups — #16853968

Order 3968.

396 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 133 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#14520991C212yesmaximal, complemented
2#14520985C211yesmaximal, characteristic
3#14520983C212yesmaximal, complemented
4#14520981C212yesmaximal, complemented
5#14520972C212yesmaximal
6#14520970C212yesmaximal, complemented
7#14520961C211yesmaximal, characteristic
8#14520909C212yesmaximal
9C31×#381C211yesmaximal, characteristic
10C23×C4×C31C2211yescharacteristic
11C22×Dic62C2211yescharacteristic
12C2×C22⋊C4×C31C2211yescharacteristic
13C2×C22⋊C4×C31C2211yescharacteristic
14C2×C22⋊Dic31C2212yes—
15C2×C22⋊Dic31C2212yes—
16C2×C4⋊Dic31C2212yes—
17C2×C4⋊Dic31C2212yes—
18C2×C4⋊Dic31C2212yes—
19C2×C4⋊Dic31C2211yescharacteristic
20C2×Dic31⋊C4C2212yes—
21C2×Dic31⋊C4C2212yes—
22C2×Dic31⋊C4C2212yes—
23C31×#95C2212yes—
24C31×#95C2212yes—
25C31⋊#95C2212yes—
26C31⋊#95C2212yes—
27C31⋊#95C2212yes—
28C31⋊#95C2211yescharacteristic
29C31⋊#95C2212yes—
30C31⋊#95C2211yescharacteristic
31C24×C31C2311yescharacteristic
32C22×C4×C31—41no—
33C22×C4×C31—41no—
34C22×C4×C31Q814yescomplemented
35C22×C4×C31C2311yescharacteristic
36C22×C4×C31C2312yes—
37C22×C4×C31C2311yescharacteristic
38C22×C4×C31C2312yes—
39C22×Dic31C2312yes—
40C22×Dic31C2312yes—
41C22×Dic31C2312yes—
42C22×Dic31C2312yes—
43C2×Dic62—41no—
44C2×Dic62—42no—
45C22⋊C4×C31—41no—
46C22⋊C4×C31—41no—
47C22⋊Dic31—42no—
48C22⋊Dic31—42no—
49C4⋊Dic31—42no—
50C4⋊Dic31—41no—
51C4⋊Dic31—42no—
52C4⋊Dic31—42no—
53Dic31⋊C4—42no—
54Dic31⋊C4—42no—
55Dic31⋊C4—42no—
56C23×C31—22no—
57C23×C31—21no—
58C23×C31—21no—
59C23×C31—22no—
60C23×C31C2×Q812yes—
61C23×C31C2411yescharacteristic
62C2×C4×C31—22no—
63C2×C4×C31—22no—
64C2×C4×C31—22no—
65C2×C4×C31—22no—
66C2×C4×C31—22no—
67C2×C4×C31—21no—
68C2×C4×C31—21no—
69C2×C4×C31—22no—
70C2×C4×C31—22no—
71C2×C4×C31—44no—
72C2×C4×C31C2×Q814yes—
73C2×Dic31—22no—
74C2×Dic31—22no—
75C2×Dic31—22no—
76C2×Dic31—22no—
77C2×Dic31—22no—
78C2×Dic31—22no—
79C2×Dic31—22no—
80C2×Dic31—22no—
81C2×Dic31—22no—
82C2×Dic31—22no—
83C2×Dic31—22no—
84C2×Dic31—22no—
85Dic62—44no—
86#2841—311nomaximal, Sylow
87C22×C31—22no—
88C22×C31—24no—
89C22×C31—41no—
90C22×C31—42no—
91C22×C31—41no—
92C22×C31C22×Q811yescharacteristic
93C22×C31H(5,𝔽2)11yescharacteristic
94C22×C31H(5,𝔽2)11yescharacteristic
95C22×C31H(5,𝔽2)12yes—
96C22×C312−1+412yes—
97C4×C31—24no—
98C4×C31—41no—
99C4×C31—42no—
100C4×C31—41no—
101Dic31—42no—
102Dic31—42no—
103Dic31—42no—
104Dic31—42no—
105#400—312no—
106#399—311no—
107#398—312no—
108#398—312no—
109#396—312no—
110#395—312no—
111#393—311no—
112#381D3111yescharacteristic
113#380—312no—
114C2×C31—42no—
115C2×C31#54311yescharacteristic
116C2×C31#54412yes—
117C2×C31#56011yescharacteristic
118C2×C31#56212yes—
119C2×C31#56311yescharacteristic
120C22×Q8—311no—
121C23×C4C2×D3111yescharacteristic
122C2×C4⋊C4—312no—
123C2×C4⋊C4—312no—
124C2×C4⋊C4—312no—
125C2×C4⋊C4—312no—
126C2×C4⋊C4—312no—
127C2×C4⋊C4—312no—
128C2×C4⋊C4—311no—
129C2×C22⋊C4—312no—
130C2×C22⋊C4—312no—
131C2×C22⋊C4C2×D3111yescharacteristic
132C2×C22⋊C4C2×D3111yescharacteristic
133#95—312no—
134#95—312no—
135#95—311no—
136#95—312no—
137#95—312no—
138#95—311no—
139#95C2×D3112yes—
140#95C2×D3112yes—
141C31#284111yescharacteristic, complemented, Sylow
142C24C22×D3111yescharacteristic
143C2×Q8—1241no—
144C2×Q8—1242no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4—312no—
148C22×C4—312no—
149C22×C4—312no—
150C22×C4—312no—
151C22×C4Dic6214yescomplemented
152C22×C4C22×D3111yescharacteristic
153C22×C4C22×D3112yes—
154C22×C4C22×D3111yescharacteristic
155C22×C4C22×D3112yes—
156C4⋊C4—1242no—
157C4⋊C4—1242no—
158C4⋊C4—1242no—
159C4⋊C4—1242no—
160C4⋊C4—1242no—
161C4⋊C4—1241no—
162C4⋊C4—1242no—
163C22⋊C4—41no—
164C22⋊C4—41no—
165C22⋊C4—1242no—
166C22⋊C4—1242no—
167C23—21no—
168C23—21no—
169C23—22no—
170C23—22no—
171C23C2×Dic6212yes—
172C23C23×D3111yescharacteristic
173Q8—1244no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—44no—
184C2×C4—622no—
185C2×C4—622no—
186C2×C4—622no—
187C2×C4—622no—
188C2×C4—622no—
189C2×C4—622no—
190C2×C4—622no—
191C2×C4—622no—
192C2×C4—622no—
193C2×C4—622no—
194C2×C4—622no—
195C2×C4—622no—
196C2×C4C2×Dic6214yes—
197C22—22no—
198C22—24no—
199C22—41no—
200C22—42no—
201C22—41no—
202C22C22×Dic6211yescharacteristic
203C22#1175856111yescharacteristic
204C22#1175856111yescharacteristic
205C22#1175856612yes—
206C22#1175856712yes—
207C4—24no—
208C4—41no—
209C4—42no—
210C4—41no—
211C4—1242no—
212C4—1242no—
213C4—1242no—
214C4—1242no—
215C2—42no—
216C2#1452167511yescharacteristic
217C2#1452167712yes—
218C2#1452178911yescharacteristic
219C2#1452179612yes—
220C2#1452180211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D31
commutator, socle, cosocleC23×C31C24
FittingC31×#381C2
prime radicalC24×C31C23
hypercenter#381D31
hypocenterC31#2841
LCSG ▷ C23×C31 ▷ C31
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
1. #14520991C2semidirect
3. #14520983C2semidirect
4. #14520981C2semidirect
6. #14520970C2semidirect
34. C22×C4×C31Q8semidirect
141. C31#2841semidirect
151. C22×C4Dic62semidirect

Maximal subgroups

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

subgroupindex
#145209912
#145209852
#145209832
#145209812
#145209722
#145209702
#145209612
#145209092
C31×#3812
#284131