Finite Group Databasebeta

← #16856095

Subgroups — #16856095

Order 3968.

400 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 125 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#14521602C211yesmaximal, characteristic, complemented
2C2×#11758513C211yesmaximal, characteristic
3#14521407C214yesmaximal, complemented
4#14521402C214yesmaximal, complemented
5#14521059C211yesmaximal, characteristic, complemented
6#14521057C211yesmaximal, characteristic, complemented
7#14521054C211yesmaximal, characteristic
8M16×Dic31C211yesmaximal, characteristic, complemented
9C31×#418C211yesmaximal, characteristic
10C22×D4×C31C2211yescharacteristic
11C22×Dic62C2211yescharacteristic
12C2×M16×C31C2211yescharacteristic
13C31×#117C2211yescharacteristic
14C2×C22⋊Dic31—22no—
15#11758513—22no—
16#11758513C2212yes—
17#11758513C2212yes—
18#11758503C2211yescharacteristic
19C2×C4×Dic31C2211yescharacteristic
20C2×#92474C2211yescharacteristic
21#11758467C2212yescomplemented
22#11758465C2212yescomplemented
23#11758455—22no—
24C4×Dic62—22no—
25C31×#102C2214yes—
26#11758420C2214yes—
27#11758404C2214yescomplemented
28#11758403C2212yescomplemented
29C8×Dic31C2212yescomplemented
30#11758398C2214yescomplemented
31C24×C31—21no—
32C2×D4×C31—21no—
33C2×D4×C31—22no—
34C2×D4×C31—22no—
35C2×D4×C31C2312yes—
36C22×C4×C31C2311yescharacteristic
37C22×Dic31—21no—
38C22×Dic31D412yes—
39C2×Dic62—22no—
40C2×Dic62—22no—
41C2×Dic62—21no—
42C2×Dic62—22no—
43C2×Dic62C2312yes—
44M16×C31—22no—
45C2×C8×C31C2312yes—
46C4⋊C4×C31C2312yes—
47C22⋊C4×C31—21no—
48C42×C31—21no—
49C22⋊Dic31—21no—
50C22⋊Dic31—24no—
51C22⋊Dic31—24no—
52C4⋊Dic31—22no—
53C4⋊Dic31C2312yes—
54Dic31⋊C4—22no—
55Dic31⋊C4—22no—
56C4×Dic31—21no—
57C4×Dic31D412yes—
58C4×Dic31C2311yescharacteristic
59C4×Dic31C2311yescharacteristic
60#92474—22no—
61C2×C31⋊C8C2312yes—
62C23×C31—22no—
63C23×C31—44no—
64C23×C31—42no—
65C23×C31C2×D411yescharacteristic
66D4×C31—22no—
67D4×C31—22no—
68D4×C31—42no—
69C2×C4×C31—22no—
70C2×C4×C31C2×D411yescharacteristic
71C2×C4×C31#4112yes—
72C2×C4×C31#4112yes—
73C2×C4×C31C2411yescharacteristic
74C2×Dic31—22no—
75C2×Dic31—22no—
76C2×Dic31—42no—
77C2×Dic31—42no—
78C2×Dic31C2×D412yes—
79C2×Dic31C2×D412yes—
80Dic62—22no—
81Dic62—22no—
82Dic62—44no—
83Dic62—42no—
84C8×C31—22no—
85C31⋊C8—22no—
86#3140—311nomaximal, Sylow
87C22×C31—22no—
88C22×C31—42no—
89C22×C31—44no—
90C22×C31—84no—
91C22×C31#12412yes—
92C22×C31C22×D411yescharacteristic
93C4×C31—42no—
94C4×C31#12412yes—
95C4×C31C2×#4111yescharacteristic
96C4×C31C2×#4111yescharacteristic
97Dic31—22no—
98Dic31—22no—
99Dic31—42no—
100Dic31—42no—
101#533—311no—
102C2×#124—311no—
103#489—314no—
104#488—314no—
105#419—311no—
106#419—311no—
107#418—311no—
108#418D3111yescharacteristic
109C4×M16—311no—
110C2×C31—22no—
111C2×C31—82no—
112C2×C31C2×#12411yescharacteristic
113C2×C31#57512yes—
114C22×Q8—311no—
115C22×D4C2×D3111yescharacteristic
116C2×M16—311no—
117C2×M16C2×D3111yescharacteristic
118C4⋊Q8—312no—
119#125—312no—
120#124—312no—
121#124—312no—
122#124—622no—
123C22⋊Q8—622no—
124C4×Q8—622no—
125#117—311no—
126#117C2×D3111yescharacteristic
127C2×C22⋊C4—622no—
128C2×C42—311no—
129#103—314no—
130#103—314no—
131#102—314no—
132#102C2×D3114yes—
133#97—312no—
134C4×C8—312no—
135C31#314011yescharacteristic, complemented, Sylow
136C24—21no—
137C2×Q8—312no—
138C2×Q8—622no—
139C2×Q8—621no—
140C2×Q8—622no—
141C2×Q8—622no—
142C2×D4—22no—
143C2×D4—21no—
144C2×D4—22no—
145C2×D4C22×D3112yes—
146C22×C4—312no—
147C22×C4—621no—
148C22×C4C22×D3111yescharacteristic
149M16—22no—
150M16—622no—
151C2×C8—312no—
152C2×C8C22×D3112yes—
153C4⋊C4—312no—
154C4⋊C4—622no—
155C4⋊C4—622no—
156C4⋊C4—622no—
157C4⋊C4C22×D3112yes—
158C22⋊C4—21no—
159C22⋊C4—624no—
160C22⋊C4—624no—
161C22⋊C4—621no—
162C42—21no—
163C42—312no—
164C42—311no—
165C42—311no—
166C42—621no—
167C23—22no—
168C23—44no—
169C23—42no—
170C23D4×D3111yescharacteristic
171Q8—622no—
172Q8—622no—
173Q8—1244no—
174Q8—1242no—
175D4—22no—
176D4—22no—
177D4—42no—
178C2×C4—22no—
179C2×C4—312no—
180C2×C4—312no—
181C2×C4—622no—
182C2×C4—622no—
183C2×C4—1242no—
184C2×C4—1242no—
185C2×C4#9249512yes—
186C2×C4D4×D3111yescharacteristic
187C2×C4#9249712yes—
188C2×C4C23×D3111yescharacteristic
189C8—22no—
190C8—622no—
191C22—22no—
192C22—42no—
193C22—44no—
194C22—84no—
195C22#1175846212yes—
196C22C2×D4×D3111yescharacteristic
197C4—42no—
198C4—622no—
199C4—622no—
200C4—1242no—
201C4—1242no—
202C4#1175846212yes—
203C4C2×#9249511yescharacteristic
204C4C2×#9249711yescharacteristic
205C2—22no—
206C2—82no—
207C2C2×#1175846211yescharacteristic
208C2#1452184512yes—
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#418C2
socleC22×C31C22×D4
prime radicalC22×D4×C31C22
hypercenter#418D31
hypocenterC31#3140
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. #14521602C2semidirect
3. #14521407C2semidirect
4. #14521402C2semidirect
5. #14521059C2semidirect
6. #14521057C2semidirect
8. M16×Dic31C2semidirect
21. #11758467C22semidirect
22. #11758465C22semidirect
27. #11758404C22semidirect
28. #11758403C22semidirect
29. C8×Dic31C22semidirect
30. #11758398C22semidirect
135. C31#3140semidirect

Maximal subgroups

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

subgroupindex
#145216022
C2×#117585132
#145214072
#145214022
#145210592
#145210572
#145210542
M16×Dic312
C31×#4182
#314031