Finite Group Databasebeta

← #16849030

Subgroups — #16849030

Order 3968.

1056 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 1056 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
1C22×#92495C211yesmaximal, characteristic, complemented
2C2×#11758530C212yesmaximal, complemented
3#14520961C212yesmaximal, complemented
4#14520958C212yesmaximal, complemented
5#14520928C212yesmaximal, complemented
6#14520925C214yesmaximal, complemented
7#14520912C211yesmaximal, characteristic, complemented
8C22×C22⋊C4×C31C211yesmaximal, characteristic, complemented
9C25×C31C2211yescharacteristic
10C23×C4×C31C2211yescharacteristic
11C23×C4×C31C2211yescharacteristic
12C22×C22⋊D31C2214yes—
13C22×C22⋊D31C2212yes—
14C2×#92495—24no—
15C2×#92495—28no—
16C22×D124C2211yescharacteristic, complemented
17C22×C4×D31C2212yescomplemented
18C22×Dic62C2211yescharacteristic, complemented
19C2×C22⋊C4×C31—24no—
20C2×C22⋊C4×C31C2212yes—
21C2×C22⋊C4×C31C2212yes—
22#11758530—28no—
23C2×C22⋊Dic31C2212yes—
24C2×C22⋊Dic31C2214yes—
25C2×(C2×D31)⋊C4C2214yescomplemented
26C31⋊#95C2212yescomplemented
27C31⋊#95C2212yescomplemented
28C31⋊#95C2214yescomplemented
29C24×C31—22no—
30C24×C31—24no—
31C24×C31—24no—
32C24×C31—42no—
33C24×C31C2312yes—
34C24×C31C2311yescharacteristic
35C23×D31C2312yes—
36C22×C4×C31—24no—
37C22×C4×C31—22no—
38C22×C4×C31—42no—
39C22×C4×C31D414yescomplemented
40C22×C4×C31C2312yes—
41C22×C4×C31C2311yescharacteristic
42C22×C4×C31C2311yescharacteristic
43C2×C22⋊D31—24no—
44C2×C22⋊D31—24no—
45C2×C22⋊D31—28no—
46C2×C22⋊D31—28no—
47C2×C22⋊D31—44no—
48C2×C22⋊D31D418yescomplemented
49C22×Dic31C2314yes—
50C22×Dic31C2312yes—
51#92495—232no—
52C2×D124—24no—
53C2×D124—22no—
54C2×C4×D31—24no—
55C2×C4×D31—24no—
56C2×C4×D31—24no—
57C2×Dic62—22no—
58C2×Dic62—24no—
59C22⋊C4×C31—24no—
60C22⋊C4×C31—24no—
61C22⋊Dic31—24no—
62C22⋊Dic31—28no—
63(C2×D31)⋊C4—44no—
64C23×C31—22no—
65C23×C31—22no—
66C23×C31—28no—
67C23×C31—24no—
68C23×C31—22no—
69C23×C31—22no—
70C23×C31—24no—
71C23×C31—416no—
72C23×C31—44no—
73C23×C31—44no—
74C23×C31C2×D414yes—
75C23×C31C2×D412yes—
76C23×C31#4114yescomplemented
77C23×C31C2411yescharacteristic
78C22×D31—22no—
79C22×D31—22no—
80C22×D31—24no—
81C22×D31—24no—
82C22×D31C2×D414yes—
83C2×C4×C31—28no—
84C2×C4×C31—22no—
85C2×C4×C31—22no—
86C2×C4×C31—22no—
87C2×C4×C31—24no—
88C2×C4×C31—44no—
89C2×C4×C31C2×D412yes—
90C2×C4×C31C2×D412yes—
91C22⋊D31—216no—
92C22⋊D31—416no—
93C2×Dic31—22no—
94C2×Dic31—22no—
95C2×Dic31—24no—
96C2×Dic31—24no—
97C2×Dic31—24no—
98C2×Dic31C2×D414yes—
99D124—28no—
100C4×D31—216no—
101Dic62—28no—
102#2404—311nomaximal, Sylow
103C22×C31—22no—
104C22×C31—22no—
105C22×C31—28no—
106C22×C31—24no—
107C22×C31—28no—
108C22×C31—416no—
109C22×C31—42no—
110C22×C31—42no—
111C22×C31—44no—
112C22×C31#12014yes—
113C22×C31C22×D412yes—
114C22×C31C22×D411yescharacteristic
115C22×C31C2×#4112yes—
116C22×C31H(5,𝔽2)12yes—
117C2×D31—24no—
118C2×D31—28no—
119C2×D31—42no—
120C2×D31—42no—
121C2×D31—44no—
122C4×C31—22no—
123C4×C31—22no—
124C4×C31—42no—
125Dic31—24no—
126Dic31—44no—
127C22×#41—311no—
128C2×#120—312no—
129C22×C22⋊C4D3111yescharacteristic, complemented
130#393—312no—
131#392—312no—
132#386—312no—
133#386—314no—
134#381—311no—
135C2×C31—24no—
136C2×C31—44no—
137C2×C31C2×#12011yescharacteristic
138C2×C31#52511yescharacteristic
139C2×C31#53511yescharacteristic
140C2×C31#54614yes—
141D31—44no—
142C25C2×D3111yescharacteristic
143C2×#41—628no—
144C2×#41—624no—
145C22×Q8—311no—
146C22×D4—312no—
147C22×D4—314no—
148C22×D4—311no—
149C23×C4—312no—
150C23×C4C2×D3111yescharacteristic
151C23×C4C2×D3111yescharacteristic
152#120—628no—
153C2×C22⋊C4—24no—
154C2×C22⋊C4—312no—
155C2×C22⋊C4—314no—
156C2×C22⋊C4—314no—
157C2×C22⋊C4C2×D3112yes—
158C2×C22⋊C4C2×D3112yes—
159#95—312no—
160#95—312no—
161#95—314no—
162C31#240411yescharacteristic, complemented, Sylow
163C24—24no—
164C24—22no—
165C24—24no—
166C24—42no—
167C24—312no—
168C24C22×D3112yes—
169C24C22×D3111yescharacteristic
170#41—6232no—
171C2×Q8—624no—
172C2×Q8—622no—
173C2×D4—318no—
174C2×D4—624no—
175C2×D4—624no—
176C2×D4—624no—
177C2×D4—628no—
178C2×D4—628no—
179C2×D4—622no—
180C2×D4—1244no—
181C22×C4—24no—
182C22×C4—22no—
183C22×C4—42no—
184C22×C4—312no—
185C22×C4—314no—
186C22×C4—624no—
187C22×C4—624no—
188C22×C4—624no—
189C22×C4C22⋊D3114yescomplemented
190C22×C4C22×D3112yes—
191C22×C4C22×D3111yescharacteristic
192C22×C4C22×D3111yescharacteristic
193C22⋊C4—24no—
194C22⋊C4—24no—
195C22⋊C4—628no—
196C22⋊C4—624no—
197C22⋊C4—1244no—
198C23—22no—
199C23—22no—
200C23—28no—
201C23—22no—
202C23—24no—
203C23—24no—
204C23—22no—
205C23—416no—
206C23—44no—
207C23—44no—
208C23—314no—
209C23—624no—
210C23—622no—
211C23—624no—
212C23—622no—
213C23#9249514yescomplemented
214C23D4×D3114yes—
215C23C2×C22⋊D3112yes—
216C23C23×D3111yescharacteristic
217Q8—628no—
218D4—628no—
219D4—6216no—
220D4—12416no—
221C2×C4—28no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—24no—
225C2×C4—22no—
226C2×C4—44no—
227C2×C4—314no—
228C2×C4—6216no—
229C2×C4—622no—
230C2×C4—622no—
231C2×C4—624no—
232C2×C4—624no—
233C2×C4—624no—
234C2×C4C2×C22⋊D3112yes—
235C2×C4C2×C22⋊D3112yes—
236C22—22no—
237C22—28no—
238C22—22no—
239C22—28no—
240C22—24no—
241C22—416no—
242C22—42no—
243C22—44no—
244C22—42no—
245C22—628no—
246C22—624no—
247C22—1242no—
248C22—1244no—
249C22—1242no—
250C22#1175851414yes—
251C22C2×#9249512yes—
252C22C2×D4×D3112yes—
253C22#1175856112yes—
254C22C22×C22⋊D3111yescharacteristic
255C4—22no—
256C4—22no—
257C4—42no—
258C4—624no—
259C4—1244no—
260C2—24no—
261C2—44no—
262C2—1244no—
263C2C2×#1175851411yescharacteristic
264C2#1452156011yescharacteristic
265C2#1452161611yescharacteristic
266C2#1452168914yes—
subgroupG/H
center, FrattiniC23C23×D31
commutator, socle, cosocleC23×C31C24
FittingC22×C22⋊C4×C31C2
hypercenterC22×C22⋊C4D31
hypocenterC31#2404
LCSG ▷ C23×C31 ▷ C31
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×#92495C2semidirect
2. C2×#11758530C2semidirect
3. #14520961C2semidirect
4. #14520958C2semidirect
5. #14520928C2semidirect
6. #14520925C2semidirect
7. #14520912C2semidirect
8. C22×C22⋊C4×C31C2semidirect
16. C22×D124C22semidirect
17. C22×C4×D31C22semidirect
18. C22×Dic62C22semidirect
25. C2×(C2×D31)⋊C4C22semidirect
26. C31⋊#95C22semidirect
27. C31⋊#95C22semidirect
28. C31⋊#95C22semidirect
39. C22×C4×C31D4semidirect
48. C2×C22⋊D31D4semidirect
76. C23×C31#41semidirect
129. C22×C22⋊C4D31semidirect
162. C31#2404semidirect
189. C22×C4C22⋊D31semidirect
213. C23#92495semidirect

Maximal subgroups

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

subgroupindex
C22×#924952
C2×#117585302
#145209612
#145209582
#145209282
#145209252
#145209122
C22×C22⋊C4×C312
#240431