Finite Group Databasebeta

← #16849026

Subgroups — #16849026

Order 3968.

1056 conjugacy classes of subgroups; only the 185 normal subgroups are shown.

normal subgroups only

This group has 1056 conjugacy classes of subgroups — too many to draw. Only its 185 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
1C2×#11758514C212yesmaximal, complemented
2C22×(C2×D31)⋊C4C211yesmaximal, characteristic, complemented
3#14520960C212yesmaximal, complemented
4#14520957C212yesmaximal, complemented
5#14520929C212yesmaximal, complemented
6#14520928C214yesmaximal, complemented
7#14520912C211yesmaximal, characteristic, complemented
8C22×#41×C31C211yesmaximal, characteristic, complemented
9C24×D31C2211yescharacteristic
10C2×#41×C31—28no—
11C2×#41×C31—24no—
12C22×Q8×C31C2211yescharacteristic, complemented
13C22×D4×C31C2212yes—
14C22×D4×C31C2211yescharacteristic, complemented
15C23×C4×C31C2211yescharacteristic
16C23×C4×C31C2212yescomplemented
17C22×C22⋊D31C2214yes—
18C23×Dic31C2211yescharacteristic
19C2×C22⋊Dic31C2212yes—
20C2×C22⋊Dic31C2214yescomplemented
21#11758514—28no—
22C2×(C2×D31)⋊C4—24no—
23C2×(C2×D31)⋊C4C2212yes—
24C2×(C2×D31)⋊C4C2214yes—
25C2×(C2×D31)⋊C4C2212yes—
26C31⋊#95C2212yescomplemented
27C31⋊#95C2212yescomplemented
28C31⋊#95C2214yescomplemented
29C24×C31C2312yes—
30C24×C31C2311yescharacteristic
31C23×D31—22no—
32C23×D31—24no—
33C23×D31—24no—
34C23×D31—42no—
35C23×D31C2312yes—
36#41×C31—232no—
37C2×Q8×C31—24no—
38C2×Q8×C31—22no—
39C2×D4×C31—24no—
40C2×D4×C31—24no—
41C2×D4×C31—24no—
42C2×D4×C31—22no—
43C2×D4×C31D418yescomplemented
44C22×C4×C31—24no—
45C22×C4×C31—24no—
46C22×C4×C31—24no—
47C22×C4×C31—24no—
48C22×C4×C31D414yescomplemented
49C22×C4×C31C2311yescharacteristic
50C22×C4×C31C2311yescharacteristic
51C22×C4×C31C2312yes—
52C2×C22⋊D31—28no—
53C2×C22⋊D31—28no—
54C2×C22⋊D31—44no—
55C22×Dic31—22no—
56C22×Dic31—42no—
57C22×Dic31C2314yes—
58C22×Dic31C2312yes—
59C22⋊Dic31—24no—
60C22⋊Dic31—44no—
61(C2×D31)⋊C4—24no—
62(C2×D31)⋊C4—24no—
63(C2×D31)⋊C4—28no—
64C23×C31—22no—
65C23×C31—22no—
66C23×C31—22no—
67C23×C31—22no—
68C23×C31—24no—
69C23×C31—24no—
70C23×C31C2×D414yes—
71C23×C31C2×D414yes—
72C23×C31C2×D412yes—
73C23×C31C2411yescharacteristic
74C22×D31—28no—
75C22×D31—22no—
76C22×D31—24no—
77C22×D31—22no—
78C22×D31—24no—
79C22×D31—416no—
80C22×D31—44no—
81C22×D31—44no—
82C22×D31#4114yescomplemented
83Q8×C31—28no—
84D4×C31—28no—
85D4×C31—216no—
86C2×C4×C31—216no—
87C2×C4×C31—28no—
88C2×C4×C31—22no—
89C2×C4×C31—22no—
90C2×C4×C31—22no—
91C2×C4×C31—22no—
92C2×C4×C31C2×D412yes—
93C2×C4×C31C2×D414yes—
94C2×C4×C31C2×D412yes—
95C22⋊D31—416no—
96C2×Dic31—22no—
97C2×Dic31—24no—
98C2×Dic31—24no—
99C2×Dic31—24no—
100C2×Dic31—24no—
101C2×Dic31—44no—
102#2404—311nomaximal, Sylow
103C22×C31—22no—
104C22×C31—22no—
105C22×C31—28no—
106C22×C31—24no—
107C22×C31—28no—
108C22×C31—42no—
109C22×C31—42no—
110C22×C31—44no—
111C22×C31#12014yes—
112C22×C31C22×D411yescharacteristic
113C22×C31C22×D412yes—
114C22×C31C2×#4112yes—
115C22×C31H(5,𝔽2)12yes—
116C2×D31—24no—
117C2×D31—28no—
118C2×D31—416no—
119C2×D31—42no—
120C2×D31—42no—
121C2×D31—44no—
122C4×C31—22no—
123C4×C31—24no—
124C4×C31—22no—
125Dic31—42no—
126Dic31—44no—
127C22×#41D3111yescharacteristic, complemented
128C2×#120—312no—
129C22×C22⋊C4—311no—
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—
142C25—311no—
143C2×#41—28no—
144C2×#41—24no—
145C22×Q8C2×D3111yescharacteristic, complemented
146C22×D4—314no—
147C22×D4C2×D3112yes—
148C22×D4C2×D3111yescharacteristic, complemented
149C23×C4—311no—
150C23×C4C2×D3111yescharacteristic
151C23×C4C2×D3112yescomplemented
152#120—628no—
153C2×C22⋊C4—312no—
154C2×C22⋊C4—312no—
155C2×C22⋊C4—312no—
156C2×C22⋊C4—314no—
157C2×C22⋊C4—314no—
158C2×C22⋊C4—624no—
159#95—312no—
160#95—312no—
161#95—314no—
162C31#240411yescharacteristic, complemented, Sylow
163C24—312no—
164C24—624no—
165C24—622no—
166C24—624no—
167C24—1242no—
168C24C22×D3112yes—
169C24C22×D3111yescharacteristic
170#41—232no—
171C2×Q8—24no—
172C2×Q8—22no—
173C2×D4—24no—
174C2×D4—24no—
175C2×D4—24no—
176C2×D4—22no—
177C2×D4—628no—
178C2×D4—628no—
179C2×D4—1244no—
180C2×D4C22⋊D3118yescomplemented
181C22×C4—24no—
182C22×C4—24no—
183C22×C4—24no—
184C22×C4—24no—
185C22×C4—314no—
186C22×C4—312no—
187C22×C4—622no—
188C22×C4—1242no—
189C22×C4C22⋊D3114yescomplemented
190C22×C4C22×D3111yescharacteristic
191C22×C4C22×D3111yescharacteristic
192C22×C4C22×D3112yes—
193C22⋊C4—624no—
194C22⋊C4—624no—
195C22⋊C4—624no—
196C22⋊C4—628no—
197C22⋊C4—1244no—
198C23—22no—
199C23—22no—
200C23—24no—
201C23—24no—
202C23—22no—
203C23—22no—
204C23—314no—
205C23—628no—
206C23—622no—
207C23—624no—
208C23—622no—
209C23—624no—
210C23—12416no—
211C23—1244no—
212C23—1244no—
213C23C2×C22⋊D3114yes—
214C23C2×C22⋊D3114yes—
215C23C2×C22⋊D3112yes—
216C23C23×D3111yescharacteristic
217Q8—28no—
218D4—28no—
219D4—216no—
220D4—12416no—
221C2×C4—216no—
222C2×C4—28no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—624no—
228C2×C4—624no—
229C2×C4—624no—
230C2×C4—622no—
231C2×C4—624no—
232C2×C4—1244no—
233C2×C4C2×C22⋊D3112yes—
234C2×C4C2×C22⋊D3114yes—
235C2×C4C2×C22⋊D3112yes—
236C22—22no—
237C22—28no—
238C22—22no—
239C22—28no—
240C22—24no—
241C22—42no—
242C22—44no—
243C22—42no—
244C22—628no—
245C22—624no—
246C22—12416no—
247C22—1242no—
248C22—1244no—
249C22—1242no—
250C22#1175853014yes—
251C22#41×D3112yes—
252C22#1175856612yes—
253C22C22×C22⋊D3111yescharacteristic
254C22C22×C22⋊D3112yes—
255C4—22no—
256C4—24no—
257C4—22no—
258C4—1242no—
259C4—1244no—
260C2—24no—
261C2—44no—
262C2—1244no—
263C2C2×#1175853011yescharacteristic
264C2#1452155611yescharacteristic
265C2#1452161311yescharacteristic
266C2#1452168714yes—
subgroupG/H
center, FrattiniC23C23×D31
commutator, socle, cosocleC23×C31C24
FittingC22×#41×C31C2
hypercenterC22×#41D31
hypocenterC31#2404
LCSG ▷ C23×C31 ▷ C31
UCSC1 ◁ C23 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C2×#11758514C2semidirect
2. C22×(C2×D31)⋊C4C2semidirect
3. #14520960C2semidirect
4. #14520957C2semidirect
5. #14520929C2semidirect
6. #14520928C2semidirect
7. #14520912C2semidirect
8. C22×#41×C31C2semidirect
12. C22×Q8×C31C22semidirect
14. C22×D4×C31C22semidirect
16. C23×C4×C31C22semidirect
20. C2×C22⋊Dic31C22semidirect
26. C31⋊#95C22semidirect
27. C31⋊#95C22semidirect
28. C31⋊#95C22semidirect
43. C2×D4×C31D4semidirect
48. C22×C4×C31D4semidirect
82. C22×D31#41semidirect
127. C22×#41D31semidirect
145. C22×Q8C2×D31semidirect
148. C22×D4C2×D31semidirect
151. C23×C4C2×D31semidirect
162. C31#2404semidirect
180. C2×D4C22⋊D31semidirect
189. C22×C4C22⋊D31semidirect

Maximal subgroups

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

subgroupindex
C2×#117585142
C22×(C2×D31)⋊C42
#145209602
#145209572
#145209292
#145209282
#145209122
C22×#41×C312
#240431