Finite Group Databasebeta

← #16855102

Subgroups — #16855102

Order 3968.

766 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 766 conjugacy classes of subgroups — too many to draw. Only its 145 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
1#14521885C211yesmaximal, characteristic, complemented
2#14521560C212yesmaximal, complemented
3#14521238C214yesmaximal, complemented
4#14521235C214yesmaximal, complemented
5#14521023C211yesmaximal, characteristic, complemented
6#14521022C212yesmaximal, complemented
7C31×#525C211yesmaximal, characteristic, complemented
8C2×#41×C31C2211yescharacteristic
9C23×C4×C31C2211yescharacteristic, complemented
10#11758567—22no—
11#11758566—22no—
12#41×D31—24no—
13#11758564—21no—
14#11758561—21no—
15#11758561C2214yes—
16C2×#92495—22no—
17C2×#92495C2212yes—
18C31×#123C2212yescomplemented
19C31×C22⋊Q8—21no—
20C31×#121—21no—
21C31×#120C2212yescomplemented
22C4×D4×C31—22no—
23C31×#117C2211yescharacteristic, complemented
24#11758530C2214yescomplemented
25#11758507—22no—
26#11758506C2214yescomplemented
27C4×C22⋊D31—24no—
28#11758503C2212yescomplemented
29#11758501—22no—
30#11758422C2214yescomplemented
31#11758394C2218yescomplemented
32C24×C31C2311yescharacteristic
33#41×C31—22no—
34#41×C31—22no—
35C2×Q8×C31D411yescharacteristic, complemented
36C2×D4×C31—22no—
37C2×D4×C31D411yescharacteristic, complemented
38C2×D4×C31C2312yes—
39C22×C4×C31—22no—
40C22×C4×C31—22no—
41C22×C4×C31—21no—
42C22×C4×C31—21no—
43C22×C4×C31D412yescomplemented
44C22×C4×C31C2311yescharacteristic
45C22×C4×C31C2311yescharacteristic
46C2×C22⋊D31—24no—
47C2×C22⋊D31—24no—
48C2×C22⋊D31C2314yes—
49#92499—24no—
50Q8×D31—24no—
51#92497—24no—
52#92497—28no—
53D4×D31—24no—
54D4×D31—28no—
55#92495—24no—
56#92495—24no—
57#92495—22no—
58#92495—24no—
59#92495—22no—
60#92495—22no—
61#92495—22no—
62C2×D124—22no—
63C2×D124D412yescomplemented
64C2×C4×D31—24no—
65C2×C4×D31D414yescomplemented
66C2×Dic62—22no—
67C2×Dic62D412yescomplemented
68C4⋊C4×C31—21no—
69C4⋊C4×C31—22no—
70C22⋊C4×C31—22no—
71C22⋊C4×C31—22no—
72C22⋊C4×C31C2312yes—
73C42×C31—21no—
74C22⋊Dic31—24no—
75C22⋊Dic31C2314yes—
76(C2×D31)⋊C4—24no—
77C4⋊Dic31—22no—
78Dic31⋊C4—24no—
79C4×Dic31—22no—
80C23×C31—22no—
81C23×C31—21no—
82C23×C31—42no—
83C23×C31C2×D412yes—
84C23×C31C2411yescharacteristic
85C22×D31—24no—
86C22×D31C2×D414yes—
87Q8×C31—22no—
88D4×C31—24no—
89D4×C31—22no—
90D4×C31—44no—
91C2×C4×C31—22no—
92C2×C4×C31—22no—
93C2×C4×C31—22no—
94C2×C4×C31—22no—
95C2×C4×C31—21no—
96C2×C4×C31—41no—
97C2×C4×C31—42no—
98C2×C4×C31—41no—
99C2×C4×C31C2×D412yes—
100C2×C4×C31C2×D412yes—
101C2×C4×C31C2×D412yes—
102C2×C4×C31C2×D411yescharacteristic
103C2×C4×C31C2×D411yescharacteristic
104C22⋊D31—28no—
105C22⋊D31—24no—
106C22⋊D31—24no—
107C22⋊D31—48no—
108C2×Dic31—24no—
109C2×Dic31—24no—
110C2×Dic31C2×D414yes—
111D124—24no—
112D124—22no—
113D124—22no—
114C4×D31—28no—
115C4×D31—24no—
116C4×D31—24no—
117Dic62—24no—
118Dic62—22no—
119Dic62—22no—
120#3025—311nomaximal, Sylow
121C22×C31—24no—
122C22×C31—22no—
123C22×C31—22no—
124C22×C31—41no—
125C22×C31—42no—
126C22×C31—42no—
127C22×C31—41no—
128C22×C31—41no—
129C22×C31C22×D412yes—
130C22×C31C22×D411yescharacteristic
131C2×D31—24no—
132C2×D31—44no—
133C2×D31—84no—
134C4×C31—22no—
135C4×C31—42no—
136C4×C31—41no—
137C4×C31#12011yescharacteristic
138C4×C31#12012yes—
139C4×C31#12011yescharacteristic
140Dic31—24no—
141Dic31—44no—
142#585—311no—
143#525—312no—
144#525D3111yescharacteristic, complemented
145#458—314no—
146B(4,𝔽2)—314no—
147#410—311no—
148#410—312no—
149C2×C31—22no—
150C2×C31—21no—
151C2×C31—42no—
152C2×C31—42no—
153C2×C31C2×#12011yescharacteristic
154D31—44no—
1552−1+4—622no—
1562−1+4—621no—
157H(5,𝔽2)—314no—
158H(5,𝔽2)—622no—
159H(5,𝔽2)—621no—
160C2×#41—312no—
161C2×#41—624no—
162C2×#41—622no—
163C2×#41C2×D3111yescharacteristic
164C23×C4C2×D3111yescharacteristic, complemented
165#123—314no—
166#123C2×D3112yescomplemented
167C22⋊Q8—21no—
168C22⋊Q8—622no—
169#121—21no—
170#121—622no—
171#120—314no—
172#120C2×D3112yescomplemented
173C4×D4—22no—
174C4×D4—624no—
175#117—312no—
176#117C2×D3111yescharacteristic, complemented
177#99—314no—
178#99—318no—
179C31#302511yescharacteristic, complemented, Sylow
180C24C22×D3111yescharacteristic
181#41—22no—
182#41—22no—
183#41—624no—
184#41—624no—
185#41—628no—
186#41—624no—
187#41—624no—
188#41—622no—
189#41—622no—
190#41—622no—
191#41—622no—
192#41—624no—
193C2×Q8—312no—
194C2×Q8—622no—
195C2×Q8—624no—
196C2×Q8C22⋊D3111yescharacteristic, complemented
197C2×D4—22no—
198C2×D4—314no—
199C2×D4—312no—
200C2×D4—622no—
201C2×D4—628no—
202C2×D4—624no—
203C2×D4—624no—
204C2×D4—624no—
205C2×D4C22⋊D3111yescharacteristic, complemented
206C2×D4C22×D3112yes—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—314no—
212C22×C4—624no—
213C22×C4C22⋊D3112yescomplemented
214C22×C4C22×D3111yescharacteristic
215C22×C4C22×D3111yescharacteristic
216C4⋊C4—21no—
217C4⋊C4—22no—
218C4⋊C4—624no—
219C4⋊C4—622no—
220C22⋊C4—22no—
221C22⋊C4—22no—
222C22⋊C4—314no—
223C22⋊C4—624no—
224C22⋊C4—624no—
225C22⋊C4C22×D3112yes—
226C42—21no—
227C42—622no—
228C23—22no—
229C23—21no—
230C23—42no—
231C23—314no—
232C23—624no—
233C23C2×C22⋊D3112yes—
234C23C23×D3111yescharacteristic
235Q8—22no—
236Q8—624no—
237Q8—622no—
238Q8—622no—
239D4—24no—
240D4—22no—
241D4—44no—
242D4—624no—
243D4—628no—
244D4—624no—
245D4—624no—
246D4—622no—
247D4—622no—
248D4—1248no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—41no—
255C2×C4—42no—
256C2×C4—41no—
257C2×C4—314no—
258C2×C4—628no—
259C2×C4—624no—
260C2×C4—624no—
261C2×C4—624no—
262C2×C4—624no—
263C2×C4D4×D3112yes—
264C2×C4D4×D3112yes—
265C2×C4C2×C22⋊D3112yes—
266C2×C4C2×C22⋊D3111yescharacteristic
267C2×C4C2×C22⋊D3111yescharacteristic
268C22—22no—
269C22—24no—
270C22—22no—
271C22—41no—
272C22—42no—
273C22—41no—
274C22—42no—
275C22—41no—
276C22—624no—
277C22—1244no—
278C22—2484no—
279C22C2×D4×D3112yes—
280C22C22×C22⋊D3111yescharacteristic
281C4—22no—
282C4—42no—
283C4—41no—
284C4—624no—
285C4—1244no—
286C4#1175851411yescharacteristic
287C4#1175851412yes—
288C4#1175851411yescharacteristic
289C2—21no—
290C2—22no—
291C2—42no—
292C2—42no—
293C2—1244no—
294C2C2×#1175851411yescharacteristic
subgroupG/H
centerC4#11758514
commutator, cosocleC23×C31C24
FrattiniC23C23×D31
FittingC31×#525C2
socleC2×C31C2×#120
hypercenter#525D31
hypocenterC31#3025
LCSG ▷ C23×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C4 ◁ C22×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. #14521885C2semidirect
2. #14521560C2semidirect
3. #14521238C2semidirect
4. #14521235C2semidirect
5. #14521023C2semidirect
6. #14521022C2semidirect
7. C31×#525C2semidirect
9. C23×C4×C31C22semidirect
18. C31×#123C22semidirect
21. C31×#120C22semidirect
23. C31×#117C22semidirect
24. #11758530C22semidirect
26. #11758506C22semidirect
28. #11758503C22semidirect
30. #11758422C22semidirect
31. #11758394C22semidirect
35. C2×Q8×C31D4semidirect
37. C2×D4×C31D4semidirect
43. C22×C4×C31D4semidirect
63. C2×D124D4semidirect
65. C2×C4×D31D4semidirect
67. C2×Dic62D4semidirect
144. #525D31semidirect
164. C23×C4C2×D31semidirect
166. #123C2×D31semidirect
172. #120C2×D31semidirect
176. #117C2×D31semidirect
179. C31#3025semidirect
196. C2×Q8C22⋊D31semidirect
205. C2×D4C22⋊D31semidirect
213. C22×C4C22⋊D31semidirect

Maximal subgroups

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

subgroupindex
#145218852
#145215602
#145212382
#145212352
#145210232
#145210222
C31×#5252
#302531