Finite Group Databasebeta

← #16848849

Subgroups — #16848849

Order 3968.

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

normal subgroups only

This group has 688 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
1C2×#11758514C211yesmaximal, characteristic
2#14520952C211yesmaximal, characteristic, complemented
3#14520949C212yesmaximal, complemented
4#14520943C212yesmaximal, complemented
5#14520941C212yesmaximal, complemented
6#14520938C212yesmaximal, complemented
7#14520928C212yesmaximal, complemented
8#14520904C212yesmaximal, complemented
9C31×#386C211yesmaximal, characteristic, complemented
10C24×D31C2211yescharacteristic
11C22×D4×C31C2211yescharacteristic
12C23×C4×C31C2211yescharacteristic, complemented
13C22×C22⋊D31C2212yes—
14C2×C22⋊C4×C31C2212yes—
15C2×C22⋊C4×C31C2211yescharacteristic, complemented
16C2×C22⋊Dic31C2212yes—
17C2×C22⋊Dic31C2211yescharacteristic
18C2×C22⋊Dic31C2212yescomplemented
19#11758514C418yescomplemented
20C2×(C2×D31)⋊C4C2212yes—
21C2×(C2×D31)⋊C4C2212yes—
22C2×(C2×D31)⋊C4C2212yes—
23C2×(C2×D31)⋊C4C2212yes—
24C2×(C2×D31)⋊C4C2212yes—
25C2×C4⋊Dic31C2211yescharacteristic, complemented
26C2×Dic31⋊C4C2212yescomplemented
27C2×C4×Dic31C2211yescharacteristic, complemented
28C2×C4×Dic31C2212yescomplemented
29C31×#95C2212yescomplemented
30C31⋊#95C2212yescomplemented
31C31⋊#95C2212yescomplemented
32C24×C31C2311yescharacteristic
33C24×C31C2311yescharacteristic
34C23×D31—41no—
35C23×D31—41no—
36C23×D31—42no—
37C23×D31—42no—
38C23×D31C2×C414yescomplemented
39C23×D31C2312yes—
40C2×D4×C31—41no—
41C2×D4×C31—41no—
42C2×D4×C31C2×C414yes—
43C22×C4×C31—41no—
44C22×C4×C31—42no—
45C22×C4×C31C2312yes—
46C22×C4×C31C2312yes—
47C22×C4×C31C2311yescharacteristic
48C2×C22⋊D31—42no—
49C2×C22⋊D31—42no—
50C2×C22⋊D31C2×C418yes—
51C22×Dic31C2312yes—
52C22×Dic31C2312yes—
53C22×Dic31C2312yes—
54C22⋊C4×C31—42no—
55C22⋊C4×C31—41no—
56C22⋊Dic31—42no—
57C22⋊Dic31—42no—
58C22⋊Dic31C2×C414yescomplemented
59(C2×D31)⋊C4—42no—
60(C2×D31)⋊C4—42no—
61(C2×D31)⋊C4—42no—
62(C2×D31)⋊C4—42no—
63(C2×D31)⋊C4C2×C418yescomplemented
64C4⋊Dic31—41no—
65Dic31⋊C4—42no—
66C4×Dic31—42no—
67C4×Dic31—41no—
68C23×C31—22no—
69C23×C31—22no—
70C23×C31—21no—
71C23×C31—22no—
72C23×C31—22no—
73C23×C31—21no—
74C23×C31—21no—
75C23×C31—21no—
76C23×C31C22×C412yes—
77C23×C31C22×C412yes—
78C23×C31C2411yescharacteristic
79C22×D31—22no—
80C22×D31—22no—
81C22×D31—22no—
82C22×D31—22no—
83C22×D31—22no—
84C22×D31—22no—
85C22×D31—48no—
86C22×D31—44no—
87C22×D31—48no—
88C22×D31—44no—
89C22×D31—44no—
90C22×D31C22×C414yes—
91D4×C31—44no—
92C2×C4×C31—21no—
93C2×C4×C31—21no—
94C2×C4×C31—22no—
95C2×C4×C31—22no—
96C2×C4×C31—22no—
97C2×C4×C31—22no—
98C2×C4×C31—22no—
99C2×C4×C31—22no—
100C2×C4×C31—44no—
101C2×C4×C31C22×C412yes—
102C22⋊D31—48no—
103C2×Dic31—22no—
104C2×Dic31—22no—
105C2×Dic31—22no—
106C2×Dic31—22no—
107C2×Dic31—22no—
108C2×Dic31—22no—
109C2×Dic31—22no—
110C2×Dic31—22no—
111C2×Dic31C22×C414yes—
112#2375—311nomaximal, Sylow
113C22×C31—24no—
114C22×C31—22no—
115C22×C31—22no—
116C22×C31—24no—
117C22×C31—42no—
118C22×C31—41no—
119C22×C31—42no—
120C22×C31—41no—
121C22×C31—41no—
122C22×C31—41no—
123C22×C31C23×C411yescharacteristic
124C22×C31H(5,𝔽2)11yescharacteristic
125C22×C31H(5,𝔽2)12yes—
126C22×C31H(5,𝔽2)11yescharacteristic
127C22×C31H(5,𝔽2)12yes—
128C2×D31—24no—
129C2×D31—24no—
130C2×D31—24no—
131C2×D31—416no—
132C2×D31—42no—
133C2×D31—42no—
134C2×D31—42no—
135C2×D31—42no—
136C4×C31—22no—
137C4×C31—42no—
138C4×C31—42no—
139Dic31—24no—
140Dic31—42no—
141Dic31—42no—
142C2×#120—311no—
143#390—311no—
144#390—312no—
145#388—312no—
146#388—312no—
147#388—312no—
148#386—312no—
149#386D3111yescharacteristic, complemented
150#379—312no—
151C2×C31—42no—
152C2×C31—42no—
153C2×C31#51812yes—
154C2×C31#51811yescharacteristic
155C2×C31#56012yes—
156C2×C31#56011yescharacteristic
157C2×C31H(3,𝔽4)11yescharacteristic
158D31—44no—
159C25—311no—
160C22×D4—312no—
161C22×D4C2×D3111yescharacteristic
162C23×C4C2×D3111yescharacteristic, complemented
163#120—318no—
164C2×C4⋊C4—312no—
165C2×C4⋊C4—311no—
166C2×C22⋊C4—311no—
167C2×C22⋊C4—312no—
168C2×C22⋊C4—312no—
169C2×C22⋊C4—312no—
170C2×C22⋊C4—312no—
171C2×C22⋊C4—312no—
172C2×C22⋊C4—312no—
173C2×C22⋊C4—312no—
174C2×C22⋊C4C2×D3112yes—
175C2×C22⋊C4C2×D3111yescharacteristic, complemented
176C2×C42—312no—
177C2×C42—311no—
178#95—312no—
179#95—312no—
180#95C2×D3112yescomplemented
181C31#237511yescharacteristic, complemented, Sylow
182C24—312no—
183C24—314no—
184C24—1242no—
185C24—1241no—
186C24—1241no—
187C24—1242no—
188C24C22×D3111yescharacteristic
189C24C22×D3111yescharacteristic
190C2×D4—41no—
191C2×D4—41no—
192C2×D4—318no—
193C2×D4—1242no—
194C2×D4—1242no—
195C2×D4C4×D3114yes—
196C22×C4—42no—
197C22×C4—41no—
198C22×C4—312no—
199C22×C4—312no—
200C22×C4—312no—
201C22×C4C22×D3112yes—
202C22×C4C22×D3112yes—
203C22×C4C22×D3111yescharacteristic
204C4⋊C4—1242no—
205C4⋊C4—1241no—
206C22⋊C4—42no—
207C22⋊C4—41no—
208C22⋊C4—314no—
209C22⋊C4—318no—
210C22⋊C4—1242no—
211C22⋊C4—1242no—
212C22⋊C4—1242no—
213C22⋊C4—1242no—
214C22⋊C4—1242no—
215C22⋊C4—1242no—
216C42—1241no—
217C42—1242no—
218C23—21no—
219C23—22no—
220C23—21no—
221C23—22no—
222C23—21no—
223C23—21no—
224C23—22no—
225C23—22no—
226C23—314no—
227C23—622no—
228C23—622no—
229C23—622no—
230C23—622no—
231C23—622no—
232C23—622no—
233C23—1248no—
234C23—1244no—
235C23—1248no—
236C23—1244no—
237C23—1244no—
238C23C2×C4×D3112yes—
239C23C2×C4×D3112yes—
240C23C23×D3111yescharacteristic
241D4—44no—
242D4—1248no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—44no—
252C2×C4—314no—
253C2×C4—622no—
254C2×C4—622no—
255C2×C4—622no—
256C2×C4—622no—
257C2×C4—622no—
258C2×C4—622no—
259C2×C4—622no—
260C2×C4—622no—
261C2×C4C2×C4×D3112yes—
262C22—24no—
263C22—24no—
264C22—22no—
265C22—22no—
266C22—42no—
267C22—42no—
268C22—41no—
269C22—41no—
270C22—41no—
271C22—41no—
272C22—624no—
273C22—624no—
274C22—624no—
275C22—12416no—
276C22—1242no—
277C22—1242no—
278C22—1242no—
279C22—1242no—
280C22C22×C4×D3111yescharacteristic
281C22#1175856112yes—
282C22#1175856111yescharacteristic
283C22#1175856111yescharacteristic
284C22#1175856612yes—
285C4—22no—
286C4—42no—
287C4—42no—
288C4—624no—
289C4—1242no—
290C4—1242no—
291C2—42no—
292C2—42no—
293C2—1244no—
294C2#1452152112yes—
295C2#1452152211yescharacteristic
296C2#1452178712yes—
297C2#1452178911yescharacteristic
298C2#1452179211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D31
commutatorC22×C31C23×C4
FittingC31×#386C2
socle, cosocleC23×C31C24
prime radicalC2×#11758514C2
hypercenter#386D31
hypocenterC31#2375
LCSG ▷ C22×C31 ▷ C31
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. #14520952C2semidirect
3. #14520949C2semidirect
4. #14520943C2semidirect
5. #14520941C2semidirect
6. #14520938C2semidirect
7. #14520928C2semidirect
8. #14520904C2semidirect
9. C31×#386C2semidirect
12. C23×C4×C31C22semidirect
15. C2×C22⋊C4×C31C22semidirect
18. C2×C22⋊Dic31C22semidirect
19. #11758514C4semidirect
25. C2×C4⋊Dic31C22semidirect
26. C2×Dic31⋊C4C22semidirect
27. C2×C4×Dic31C22semidirect
28. C2×C4×Dic31C22semidirect
29. C31×#95C22semidirect
30. C31⋊#95C22semidirect
31. C31⋊#95C22semidirect
38. C23×D31C2×C4semidirect
58. C22⋊Dic31C2×C4semidirect
63. (C2×D31)⋊C4C2×C4semidirect
149. #386D31semidirect
162. C23×C4C2×D31semidirect
175. C2×C22⋊C4C2×D31semidirect
180. #95C2×D31semidirect
181. C31#2375semidirect

Maximal subgroups

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

subgroupindex
C2×#117585142
#145209522
#145209492
#145209432
#145209412
#145209382
#145209282
#145209042
C31×#3862
#237531