Finite Group Databasebeta

← #16854941

Subgroups — #16854941

Order 3968.

994 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 157 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
1C23×D124C211yesmaximal, characteristic, complemented
2C2×#11758485C212yesmaximal, complemented
3#14521560C211yesmaximal, characteristic, complemented
4#14521202C214yesmaximal, complemented
5#14521185C214yesmaximal, complemented
6#14521055C212yesmaximal, complemented
7C31×#407C211yesmaximal, characteristic, complemented
8C24×D31—21no—
9C23×C4×C31C2211yescharacteristic
10C2×#92495C2211yescharacteristic, complemented
11C22×D124—22no—
12C22×D124—21no—
13C22×D124—28no—
14C22×D124C2214yes—
15C22×D124C2212yes—
16C2×M16×C31C2212yescomplemented
17#11758530—21no—
18#11758507C2212yescomplemented
19#11758506—21no—
20C4×C22⋊D31—22no—
21#11758503C2211yescharacteristic, complemented
22#11758501C2212yescomplemented
23#11758485—28no—
24C2×D248C2214yescomplemented
25C2×#92470C2214yescomplemented
26C31×C22⋊C8C2214yescomplemented
27#11758409C2218yescomplemented
28C24×C31D411yescharacteristic
29C23×D31—28no—
30C23×D31—24no—
31C23×D31—22no—
32C23×D31—48no—
33C22×C4×C31—24no—
34C22×C4×C31D412yes—
35C22×C4×C31D411yescharacteristic
36C22×C4×C31C2312yes—
37C22×C4×C31C2311yescharacteristic
38C2×C22⋊D31—22no—
39C2×C22⋊D31—21no—
40#92495—24no—
41C2×D124—28no—
42C2×D124—216no—
43C2×D124—24no—
44C2×D124—22no—
45C2×D124—24no—
46C2×D124—216no—
47C2×D124D418yescomplemented
48C2×D124C2314yes—
49C2×D124C2311yescharacteristic, complemented
50C2×C4×D31—21no—
51C2×Dic62C2311yescharacteristic, complemented
52M16×C31—24no—
53C2×C8×C31C2314yescomplemented
54C22⋊Dic31—22no—
55C22⋊Dic31—21no—
56(C2×D31)⋊C4—22no—
57C4⋊Dic31C2312yescomplemented
58Dic31⋊C4—22no—
59C4×Dic31—21no—
60D248—28no—
61#92470—28no—
62C23×C31—22no—
63C23×C31—24no—
64C23×C31C2×D411yescharacteristic
65C23×C31C2×D412yes—
66C22×D31—216no—
67C22×D31—28no—
68C22×D31—24no—
69C22×D31—21no—
70C22×D31—432no—
71C22×D31—48no—
72C22×D31—48no—
73C22×D31—48no—
74C2×C4×C31—22no—
75C2×C4×C31—24no—
76C2×C4×C31—24no—
77C2×C4×C31C2×D412yes—
78C2×C4×C31C2×D411yescharacteristic
79C2×C4×C31C2×D414yes—
80C2×C4×C31C2411yescharacteristic
81C22⋊D31—42no—
82C22⋊D31—44no—
83C2×Dic31—22no—
84C2×Dic31—21no—
85D124—216no—
86D124—28no—
87D124—24no—
88D124—22no—
89D124C2×D418yes—
90C4×D31—42no—
91Dic62—22no—
92C8×C31—24no—
93#3003—311nomaximal, Sylow
94C22×C31—28no—
95C22×C31—22no—
96C22×C31—22no—
97C22×C31—22no—
98C22×C31#12012yes—
99C22×C31AGL(1,ℤ/8)14yescomplemented
100C22×C31C22×D411yescharacteristic
101C2×D31—28no—
102C2×D31—416no—
103C2×D31—432no—
104C2×D31—48no—
105C2×D31—41no—
106C2×D31—42no—
107C4×C31—22no—
108C4×C31#12012yes—
109C4×C31C22×D412yes—
110Dic31—41no—
111Dic31—42no—
112C23×D4—311no—
113C2×AGL(1,ℤ/8)—312no—
114#525—311no—
115#450—314no—
116#447—314no—
117#418—312no—
118#407D3111yescharacteristic, complemented
119C2×C31—22no—
120C2×C31—24no—
121C2×C31C2×#12011yescharacteristic
122C2×C31C2×AGL(1,ℤ/8)12yes—
123D31—48no—
124D31—81no—
125C25—621no—
126C2×#41—311no—
127C22×D4—312no—
128C22×D4—314no—
129C22×D4—628no—
130C22×D4—622no—
131C22×D4—621no—
132C23×C4C2×D3111yescharacteristic
133AGL(1,ℤ/8)—628no—
134C2×SD16—314no—
135C2×D8—314no—
136C2×M16C2×D3112yescomplemented
137#123—621no—
138C22⋊Q8—312no—
139#121—312no—
140#120—621no—
141C4×D4—622no—
142#117—311no—
143#102—318no—
144C22⋊C8C2×D3114yescomplemented
145C31#300311yescharacteristic, complemented, Sylow
146C24—628no—
147C24—624no—
148C24—622no—
149C24—1248no—
150C24D12411yescharacteristic
151#41—624no—
152C2×Q8—311no—
153C2×D4—318no—
154C2×D4—314no—
155C2×D4—311no—
156C2×D4—6216no—
157C2×D4—628no—
158C2×D4—6216no—
159C2×D4—622no—
160C2×D4—621no—
161C2×D4—624no—
162C2×D4—622no—
163C2×D4—624no—
164C22×C4—24no—
165C22×C4—621no—
166C22×C4D12412yes—
167C22×C4D12411yescharacteristic
168C22×C4C22×D3112yes—
169C22×C4C22×D3111yescharacteristic
170SD16—628no—
171D8—628no—
172M16—24no—
173C2×C8C22×D3114yescomplemented
174C4⋊C4—312no—
175C4⋊C4—622no—
176C22⋊C4—621no—
177C22⋊C4—622no—
178C22⋊C4—622no—
179C42—621no—
180C23—22no—
181C23—24no—
182C23—6216no—
183C23—628no—
184C23—624no—
185C23—621no—
186C23—12432no—
187C23—1248no—
188C23—1248no—
189C23—1248no—
190C23C2×D12411yescharacteristic
191C23C2×D12412yes—
192Q8—622no—
193D4—318no—
194D4—6216no—
195D4—628no—
196D4—624no—
197D4—622no—
198D4—1242no—
199D4—1244no—
200C2×C4—22no—
201C2×C4—24no—
202C2×C4—24no—
203C2×C4—622no—
204C2×C4—621no—
205C2×C4—1242no—
206C2×C4C2×D12412yes—
207C2×C4C2×D12411yescharacteristic
208C2×C4D4×D3114yes—
209C2×C4C23×D3111yescharacteristic
210C8—24no—
211C22—28no—
212C22—22no—
213C22—22no—
214C22—22no—
215C22—628no—
216C22—12416no—
217C22—12432no—
218C22—1248no—
219C22—1241no—
220C22—1242no—
221C22#1175845912yes—
222C22#1175848514yescomplemented
223C22C22×D12411yescharacteristic
224C4—22no—
225C4—1241no—
226C4—1242no—
227C4#1175845912yes—
228C4C2×D4×D3112yes—
229C2—22no—
230C2—24no—
231C2—1248no—
232C2—2481no—
233C2C2×#1175845911yescharacteristic
234C2C2×#1175848512yes—
subgroupG/H
centerC22C22×D124
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#407C2
socleC22×C31C22×D4
hypercenter#407D31
hypocenterC31#3003
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
1. C23×D124C2semidirect
2. C2×#11758485C2semidirect
3. #14521560C2semidirect
4. #14521202C2semidirect
5. #14521185C2semidirect
6. #14521055C2semidirect
7. C31×#407C2semidirect
10. C2×#92495C22semidirect
16. C2×M16×C31C22semidirect
18. #11758507C22semidirect
21. #11758503C22semidirect
22. #11758501C22semidirect
24. C2×D248C22semidirect
25. C2×#92470C22semidirect
26. C31×C22⋊C8C22semidirect
27. #11758409C22semidirect
47. C2×D124D4semidirect
49. C2×D124C23semidirect
51. C2×Dic62C23semidirect
53. C2×C8×C31C23semidirect
57. C4⋊Dic31C23semidirect
99. C22×C31AGL(1,ℤ/8)semidirect
118. #407D31semidirect
136. C2×M16C2×D31semidirect
144. C22⋊C8C2×D31semidirect
145. C31#3003semidirect
173. C2×C8C22×D31semidirect
222. C22#11758485semidirect

Maximal subgroups

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

subgroupindex
C23×D1242
C2×#117584852
#145215602
#145212022
#145211852
#145210552
C31×#4072
#300331