Finite Group Databasebeta

← #16848692

Subgroups — #16848692

Order 3968.

928 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 928 conjugacy classes of subgroups — too many to draw. Only its 209 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
2C2×(C22⋊D31)⋊C4C212yesmaximal, complemented
3C2×C22⋊C4×D31C212yesmaximal, complemented
4#14520937C212yesmaximal, complemented
5#14520929C211yesmaximal, characteristic, complemented
6#14520925C212yesmaximal, complemented
7#14520920C212yesmaximal, complemented
8C4×(C2×D31)⋊C4C212yesmaximal, complemented
9C31×#390C211yesmaximal, characteristic, complemented
10C24×D31C2211yescharacteristic
11C22×D4×C31C2211yescharacteristic
12C22×C22⋊D31C2212yes—
13C23×Dic31C2211yescharacteristic, complemented
14C22×C4×D31C2212yes—
15C22×C4×D31C2212yes—
16C2×C4⋊C4×C31C2211yescharacteristic, complemented
17C2×C22⋊C4×C31C2212yes—
18C2×C22⋊C4×C31C2212yescomplemented
19C2×C42×C31C2211yescharacteristic, complemented
20C2×C22⋊Dic31C2212yes—
21C2×C22⋊Dic31C2211yescharacteristic
22C2×C22⋊Dic31C2211yescharacteristic, complemented
23#11758514C418yescomplemented
24C2×(C2×D31)⋊C4C2212yes—
25C2×(C2×D31)⋊C4C2212yes—
26C2×(C2×D31)⋊C4C2212yes—
27C2×Dic31⋊C4C2212yescomplemented
28C2×C4×Dic31C2212yescomplemented
29(C22⋊D31)⋊C4—28no—
30C22⋊C4×D31—28no—
31C31⋊#95C2212yescomplemented
32C31⋊#95C2212yescomplemented
33C31⋊#95C2212yescomplemented
34C24×C31C2311yescharacteristic
35C24×C31C2311yescharacteristic
36C23×D31—24no—
37C23×D31—24no—
38C23×D31—41no—
39C23×D31—41no—
40C23×D31C2×C414yescomplemented
41C23×D31C2312yes—
42C2×D4×C31—41no—
43C2×D4×C31—41no—
44C2×D4×C31C2×C414yescomplemented
45C22×C4×C31C2312yes—
46C22×C4×C31C2311yescharacteristic
47C22×C4×C31C2312yes—
48C2×C22⋊D31—24no—
49C2×C22⋊D31—24no—
50C2×C22⋊D31C2×C418yescomplemented
51C22×Dic31—24no—
52C22×Dic31—41no—
53C22×Dic31C2312yes—
54C22×Dic31C2312yes—
55C22×Dic31C2312yes—
56C2×C4×D31—24no—
57C2×C4×D31—24no—
58C2×C4×D31—24no—
59C2×C4×D31—24no—
60C2×C4×D31—42no—
61C2×C4×D31—42no—
62C4⋊C4×C31—41no—
63C22⋊C4×C31—24no—
64C22⋊C4×C31—24no—
65C42×C31—41no—
66C22⋊Dic31—24no—
67C22⋊Dic31—41no—
68C22⋊Dic31C2×C414yescomplemented
69(C2×D31)⋊C4—24no—
70(C2×D31)⋊C4C2×C418yescomplemented
71(C2×D31)⋊C4D418yescomplemented
72Dic31⋊C4—24no—
73C4×Dic31—24no—
74C23×C31—22no—
75C23×C31—22no—
76C23×C31—22no—
77C23×C31—22no—
78C23×C31—21no—
79C23×C31—21no—
80C23×C31—21no—
81C23×C31—21no—
82C23×C31C22×C412yes—
83C23×C31C22×C412yes—
84C23×C31C2411yescharacteristic
85C22×D31—22no—
86C22×D31—22no—
87C22×D31—22no—
88C22×D31—22no—
89C22×D31—216no—
90C22×D31—44no—
91C22×D31—44no—
92C22×D31—48no—
93C22×D31—44no—
94C22×D31C22×C414yes—
95C22×D31C2×D414yes—
96C22×D31#4114yes—
97D4×C31—44no—
98C2×C4×C31—22no—
99C2×C4×C31—22no—
100C2×C4×C31—21no—
101C2×C4×C31—21no—
102C2×C4×C31—22no—
103C2×C4×C31—22no—
104C2×C4×C31—22no—
105C2×C4×C31C22×C412yes—
106C2×C4×C31C2×D414yes—
107C22⋊D31—216no—
108C2×Dic31—22no—
109C2×Dic31—22no—
110C2×Dic31—22no—
111C2×Dic31—22no—
112C2×Dic31—22no—
113C2×Dic31—22no—
114C2×Dic31—22no—
115C2×Dic31—44no—
116C2×Dic31C22×C414yes—
117C2×Dic31C2×D414yes—
118C4×D31—48no—
119C4×D31—48no—
120#2358—311nomaximal, Sylow
121C22×C31—24no—
122C22×C31—24no—
123C22×C31—22no—
124C22×C31—22no—
125C22×C31—41no—
126C22×C31—41no—
127C22×C31—42no—
128C22×C31—42no—
129C22×C31—41no—
130C22×C31—41no—
131C22×C31C23×C411yescharacteristic
132C22×C31C22×D412yes—
133C22×C31C2×#4112yes—
134C22×C31H(5,𝔽2)11yescharacteristic
135C22×C31H(5,𝔽2)11yescharacteristic
136C2×D31—24no—
137C2×D31—24no—
138C2×D31—24no—
139C2×D31—24no—
140C2×D31—24no—
141C2×D31—24no—
142C2×D31—416no—
143C2×D31C4×D418yescomplemented
144C4×C31—22no—
145C4×C31—24no—
146C4×C31—42no—
147Dic31—24no—
148Dic31—24no—
149Dic31—42no—
150C2×#120—311no—
151C2×C4×D4—312no—
152C22×C22⋊C4—312no—
153#390D3111yescharacteristic, complemented
154#388—312no—
155#386—311no—
156#386—312no—
157#385—312no—
158C4×C22⋊C4—312no—
159C2×C31—42no—
160C2×C31—42no—
161C2×C31C2×C4×D412yes—
162C2×C31#51811yescharacteristic
163C2×C31D4211yescharacteristic
164C2×C31#54612yes—
165C2×C31#55111yescharacteristic
166D31—28no—
167C25—311no—
168C22×D4—312no—
169C22×D4C2×D3111yescharacteristic
170C23×C4—311no—
171C23×C4—312no—
172C23×C4—312no—
173#120—318no—
174C4×D4—628no—
175C2×C4⋊C4—312no—
176C2×C4⋊C4C2×D3111yescharacteristic, complemented
177C2×C22⋊C4—311no—
178C2×C22⋊C4—312no—
179C2×C22⋊C4—312no—
180C2×C22⋊C4—312no—
181C2×C22⋊C4—312no—
182C2×C22⋊C4—311no—
183C2×C22⋊C4—628no—
184C2×C22⋊C4C2×D3112yes—
185C2×C22⋊C4C2×D3112yescomplemented
186C2×C42—312no—
187C2×C42C2×D3111yescharacteristic, complemented
188#95—312no—
189#95—312no—
190#95—312no—
191C31#235811yescharacteristic, complemented, Sylow
192C24—312no—
193C24—314no—
194C24—624no—
195C24—624no—
196C24—1241no—
197C24—1241no—
198C24C22×D3111yescharacteristic
199C24C22×D3111yescharacteristic
200C2×D4—41no—
201C2×D4—41no—
202C2×D4—318no—
203C2×D4—624no—
204C2×D4—624no—
205C2×D4C4×D3114yescomplemented
206C22×C4—312no—
207C22×C4—312no—
208C22×C4—312no—
209C22×C4—624no—
210C22×C4—624no—
211C22×C4—624no—
212C22×C4—624no—
213C22×C4—624no—
214C22×C4—1242no—
215C22×C4—1242no—
216C22×C4—1241no—
217C22×C4C22×D3112yes—
218C22×C4C22×D3111yescharacteristic
219C22×C4C22×D3112yes—
220C4⋊C4—41no—
221C4⋊C4—624no—
222C22⋊C4—24no—
223C22⋊C4—24no—
224C22⋊C4—314no—
225C22⋊C4—318no—
226C22⋊C4—318no—
227C22⋊C4—624no—
228C22⋊C4—624no—
229C22⋊C4—1241no—
230C42—41no—
231C42—624no—
232C23—22no—
233C23—22no—
234C23—21no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—22no—
239C23—22no—
240C23—314no—
241C23—314no—
242C23—314no—
243C23—6216no—
244C23—622no—
245C23—622no—
246C23—622no—
247C23—622no—
248C23—1244no—
249C23—1244no—
250C23—1248no—
251C23—1244no—
252C23C2×C4×D3112yes—
253C23C2×C4×D3112yes—
254C23C23×D3111yescharacteristic
255D4—44no—
256D4—6216no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—314no—
265C2×C4—314no—
266C2×C4—622no—
267C2×C4—622no—
268C2×C4—622no—
269C2×C4—622no—
270C2×C4—622no—
271C2×C4—622no—
272C2×C4—622no—
273C2×C4—1248no—
274C2×C4—1248no—
275C2×C4—1244no—
276C2×C4C2×C4×D3112yes—
277C2×C4D4×D3114yes—
278C22—24no—
279C22—24no—
280C22—22no—
281C22—22no—
282C22—41no—
283C22—41no—
284C22—42no—
285C22—42no—
286C22—41no—
287C22—41no—
288C22—318no—
289C22—624no—
290C22—624no—
291C22—624no—
292C22—624no—
293C22—624no—
294C22—624no—
295C22—12416no—
296C22C22×C4×D3111yescharacteristic
297C22C2×D4×D3112yes—
298C22#1175856111yescharacteristic
299C22#1175856111yescharacteristic
300C22#41×D3112yes—
301C4—24no—
302C4—22no—
303C4—42no—
304C4—624no—
305C4—624no—
306C4—1242no—
307C2—42no—
308C2—42no—
309C2—628no—
310C2C4×D4×D3112yes—
311C2#1452152211yescharacteristic
312C2#1452168411yescharacteristic
313C2#1452168812yes—
314C2#1452172911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D31
commutatorC22×C31C23×C4
FittingC31×#390C2
socle, cosocleC23×C31C24
prime radicalC2×#11758514C2
hypercenter#390D31
hypocenterC31#2358
LCSG ▷ C22×C31 ▷ C31
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×(C22⋊D31)⋊C4C2semidirect
3. C2×C22⋊C4×D31C2semidirect
4. #14520937C2semidirect
5. #14520929C2semidirect
6. #14520925C2semidirect
7. #14520920C2semidirect
8. C4×(C2×D31)⋊C4C2semidirect
9. C31×#390C2semidirect
13. C23×Dic31C22semidirect
16. C2×C4⋊C4×C31C22semidirect
18. C2×C22⋊C4×C31C22semidirect
19. C2×C42×C31C22semidirect
22. C2×C22⋊Dic31C22semidirect
23. #11758514C4semidirect
27. C2×Dic31⋊C4C22semidirect
28. C2×C4×Dic31C22semidirect
31. C31⋊#95C22semidirect
32. C31⋊#95C22semidirect
33. C31⋊#95C22semidirect
40. C23×D31C2×C4semidirect
44. C2×D4×C31C2×C4semidirect
50. C2×C22⋊D31C2×C4semidirect
68. C22⋊Dic31C2×C4semidirect
70. (C2×D31)⋊C4C2×C4semidirect
71. (C2×D31)⋊C4D4semidirect
143. C2×D31C4×D4semidirect
153. #390D31semidirect
176. C2×C4⋊C4C2×D31semidirect
185. C2×C22⋊C4C2×D31semidirect
187. C2×C42C2×D31semidirect
191. C31#2358semidirect
205. C2×D4C4×D31semidirect

Maximal subgroups

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

subgroupindex
C2×#117585142
C2×(C22⋊D31)⋊C42
C2×C22⋊C4×D312
#145209372
#145209292
#145209252
#145209202
C4×(C2×D31)⋊C42
C31×#3902
#235831