Finite Group Databasebeta

← #16849039

Subgroups — #16849039

Order 3968.

1116 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 1116 conjugacy classes of subgroups — too many to draw. Only its 181 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×C22⋊D31C211yesmaximal, characteristic
2C2×#11758463C212yesmaximal, complemented
3C22×C22⋊Dic31C211yesmaximal, characteristic, complemented
4#14520965C214yesmaximal, complemented
5#14520926C212yesmaximal, complemented
6#14520924C212yesmaximal, complemented
7#14520904C212yesmaximal, complemented
8C31×#379C211yesmaximal, characteristic, complemented
9C25×C31C2211yescharacteristic
10C24×D31C2211yescharacteristic
11C22×C22⋊D31—24no—
12C22×C22⋊D31—28no—
13C22×C22⋊D31C2212yes—
14C22×C22⋊D31C2212yes—
15C23×Dic31C2211yescharacteristic
16C23×Dic31C2211yescharacteristic, complemented
17C23×Dic31C2211yescharacteristic, complemented
18C2×C22⋊C4×C31C2212yes—
19C2×C22⋊C4×C31C2212yescomplemented
20C2×C22⋊C4×C31C2212yescomplemented
21C2×C22⋊Dic31—24no—
22C2×C22⋊Dic31C2212yes—
23C2×C22⋊Dic31C2212yescomplemented
24C2×(C2×D31)⋊C4C2214yescomplemented
25C2×(C2×D31)⋊C4C2214yescomplemented
26C2×C4⋊Dic31C2214yescomplemented
27#11758463—28no—
28C31⋊#95C2214yescomplemented
29C24×C31—24no—
30C24×C31—24no—
31C24×C31—42no—
32C24×C31D414yes—
33C24×C31C2311yescharacteristic
34C24×C31C2311yescharacteristic
35C24×C31C2311yescharacteristic
36C23×D31—22no—
37C23×D31—24no—
38C23×D31—24no—
39C23×D31—42no—
40C23×D31C2312yes—
41C22×C4×C31C2314yes—
42C2×C22⋊D31—232no—
43C2×C22⋊D31—24no—
44C2×C22⋊D31—24no—
45C2×C22⋊D31—24no—
46C2×C22⋊D31—24no—
47C2×C22⋊D31—24no—
48C2×C22⋊D31D418yes—
49C22×Dic31—22no—
50C22×Dic31—24no—
51C22×Dic31—24no—
52C22×Dic31—22no—
53C22×Dic31—22no—
54C22×Dic31—42no—
55C22×Dic31C2312yes—
56C22×Dic31C2312yes—
57C22×Dic31C2312yes—
58C22⋊C4×C31—24no—
59C22⋊C4×C31—24no—
60C22⋊C4×C31—42no—
61C22⋊Dic31—24no—
62C22⋊Dic31—24no—
63(C2×D31)⋊C4—28no—
64(C2×D31)⋊C4—44no—
65C4⋊Dic31—28no—
66C23×C31—22no—
67C23×C31—22no—
68C23×C31—22no—
69C23×C31—22no—
70C23×C31—28no—
71C23×C31—28no—
72C23×C31—22no—
73C23×C31—22no—
74C23×C31—22no—
75C23×C31—44no—
76C23×C31—416no—
77C23×C31C2×D412yes—
78C23×C31C2×D412yes—
79C23×C31C2×D412yes—
80C23×C31C2×D414yes—
81C23×C31#4114yes—
82C23×C31C2411yescharacteristic
83C22×D31—28no—
84C22×D31—22no—
85C22×D31—22no—
86C22×D31—24no—
87C22×D31—24no—
88C22×D31—416no—
89C22×D31—44no—
90C22×D31—44no—
91C22×D31C2×D414yes—
92C2×C4×C31—24no—
93C2×C4×C31—24no—
94C2×C4×C31—24no—
95C22⋊D31—216no—
96C22⋊D31—216no—
97C2×Dic31—28no—
98C2×Dic31—22no—
99C2×Dic31—22no—
100C2×Dic31—24no—
101C2×Dic31—22no—
102C2×Dic31—22no—
103C2×Dic31—24no—
104C2×Dic31—44no—
105C2×Dic31—44no—
106C2×Dic31C2×D414yes—
107#2405—311nomaximal, Sylow
108C22×C31—22no—
109C22×C31—22no—
110C22×C31—22no—
111C22×C31—22no—
112C22×C31—28no—
113C22×C31—28no—
114C22×C31—22no—
115C22×C31—22no—
116C22×C31—22no—
117C22×C31—416no—
118C22×C31—44no—
119C22×C31#12014yes—
120C22×C31#12314yescomplemented
121C22×C31C22×D411yescharacteristic
122C22×C31C22×D412yes—
123C22×C31C2×#4112yes—
124C22×C31H(5,𝔽2)12yes—
125C2×D31—24no—
126C2×D31—28no—
127C2×D31—416no—
128C2×D31—42no—
129C2×D31—42no—
130C2×D31—44no—
131C4×C31—44no—
132Dic31—24no—
133Dic31—42no—
134Dic31—42no—
135C23×D4—311no—
136C2×#123—312no—
137C22×C22⋊C4—311no—
138#394—314no—
139#386—312no—
140#385—312no—
141#379—312no—
142#379D3111yescharacteristic, complemented
143C2×C31—24no—
144C2×C31—24no—
145C2×C31—42no—
146C2×C31C2×#12011yescharacteristic
147C2×C31C2×#12311yescharacteristic
148C2×C31#53411yescharacteristic
149C2×C31#54614yes—
150D31—44no—
151C25—311no—
152C25C2×D3111yescharacteristic
153C22×D4—312no—
154C22×D4—312no—
155C22×D4—628no—
156C22×D4—624no—
157C23×C4—311no—
158C23×C4—311no—
159C23×C4—311no—
160#123—628no—
161C2×C4⋊C4—314no—
162C2×C22⋊C4—312no—
163C2×C22⋊C4—312no—
164C2×C22⋊C4—314no—
165C2×C22⋊C4—314no—
166C2×C22⋊C4—624no—
167C2×C22⋊C4C2×D3112yes—
168C2×C22⋊C4C2×D3112yescomplemented
169C2×C22⋊C4C2×D3112yescomplemented
170#95—314no—
171C31#240511yescharacteristic, complemented, Sylow
172C24—24no—
173C24—24no—
174C24—42no—
175C24—312no—
176C24—624no—
177C24—622no—
178C24—624no—
179C24—1242no—
180C24D12414yes—
181C24C22×D3111yescharacteristic
182C24C22×D3111yescharacteristic
183C24C22×D3111yescharacteristic
184C2×D4—318no—
185C2×D4—624no—
186C2×D4—624no—
187C2×D4—6232no—
188C2×D4—624no—
189C2×D4—624no—
190C2×D4—624no—
191C22×C4—312no—
192C22×C4—312no—
193C22×C4—312no—
194C22×C4—624no—
195C22×C4—622no—
196C22×C4—622no—
197C22×C4—622no—
198C22×C4—624no—
199C22×C4—1242no—
200C22×C4C22×D3114yes—
201C4⋊C4—628no—
202C22⋊C4—24no—
203C22⋊C4—24no—
204C22⋊C4—42no—
205C22⋊C4—628no—
206C22⋊C4—624no—
207C22⋊C4—624no—
208C22⋊C4—1244no—
209C23—22no—
210C23—22no—
211C23—22no—
212C23—22no—
213C23—28no—
214C23—28no—
215C23—22no—
216C23—22no—
217C23—22no—
218C23—44no—
219C23—416no—
220C23—314no—
221C23—628no—
222C23—622no—
223C23—622no—
224C23—624no—
225C23—624no—
226C23—12416no—
227C23—1244no—
228C23—1244no—
229C23C2×D12412yes—
230C23C2×D12412yes—
231C23C2×D12412yes—
232C23D4×D3114yes—
233C23#9249714yes—
234C23C23×D3111yescharacteristic
235D4—6216no—
236D4—6216no—
237C2×C4—24no—
238C2×C4—24no—
239C2×C4—24no—
240C2×C4—314no—
241C2×C4—628no—
242C2×C4—622no—
243C2×C4—622no—
244C2×C4—622no—
245C2×C4—622no—
246C2×C4—624no—
247C2×C4—624no—
248C2×C4—1244no—
249C2×C4—1244no—
250C22—22no—
251C22—22no—
252C22—28no—
253C22—28no—
254C22—22no—
255C22—22no—
256C22—22no—
257C22—22no—
258C22—22no—
259C22—416no—
260C22—44no—
261C22—628no—
262C22—624no—
263C22—12416no—
264C22—1242no—
265C22—1244no—
266C22—1242no—
267C22#1175845914yes—
268C22#1175846314yescomplemented
269C22C22×D12411yescharacteristic
270C22C2×D4×D3112yes—
271C22C2×#9249712yes—
272C22#1175856112yes—
273C4—44no—
274C4—624no—
275C4—1242no—
276C4—1242no—
277C2—24no—
278C2—24no—
279C2—42no—
280C2—1244no—
281C2C2×#1175845911yescharacteristic
282C2C2×#1175846311yescharacteristic
283C2#1452161111yescharacteristic
284C2#1452168614yes—
subgroupG/H
center, FrattiniC23C23×D31
commutator, socle, cosocleC23×C31C24
FittingC31×#379C2
prime radicalC23×C22⋊D31C2
hypercenter#379D31
hypocenterC31#2405
LCSG ▷ C23×C31 ▷ C31
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
2. C2×#11758463C2semidirect
3. C22×C22⋊Dic31C2semidirect
4. #14520965C2semidirect
5. #14520926C2semidirect
6. #14520924C2semidirect
7. #14520904C2semidirect
8. C31×#379C2semidirect
16. C23×Dic31C22semidirect
17. C23×Dic31C22semidirect
19. C2×C22⋊C4×C31C22semidirect
20. C2×C22⋊C4×C31C22semidirect
23. C2×C22⋊Dic31C22semidirect
24. C2×(C2×D31)⋊C4C22semidirect
25. C2×(C2×D31)⋊C4C22semidirect
26. C2×C4⋊Dic31C22semidirect
28. C31⋊#95C22semidirect
120. C22×C31#123semidirect
142. #379D31semidirect
168. C2×C22⋊C4C2×D31semidirect
169. C2×C22⋊C4C2×D31semidirect
171. C31#2405semidirect
268. C22#11758463semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊D312
C2×#117584632
C22×C22⋊Dic312
#145209652
#145209262
#145209242
#145209042
C31×#3792
#240531