Finite Group Databasebeta

← #16855850

Subgroups — #16855850

Order 3968.

450 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 121 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#14521585C211yesmaximal, characteristic, complemented
2C2×D4×Dic31C211yesmaximal, characteristic, complemented
3#14521377C211yesmaximal, characteristic, complemented
4#14521363C211yesmaximal, characteristic, complemented
5#14521347C212yesmaximal, complemented
6#14521266C212yesmaximal, complemented
7#14521183C211yesmaximal, characteristic
8#14521158C212yesmaximal, complemented
9#14521150C212yesmaximal, complemented
10#14521103C211yesmaximal, characteristic, complemented
11C31×#450C211yesmaximal, characteristic, complemented
12C22×D4×C31C2211yescharacteristic
13C23×Dic31—21no—
14C2×SD16×C31C2212yes—
15C31×C22⋊Q8C2211yescharacteristic
16C2×C22⋊Dic31—21no—
17#11758524C2211yescharacteristic, complemented
18Q8×Dic31C2211yescharacteristic, complemented
19D4×Dic31—21no—
20D4×Dic31—21no—
21D4×Dic31C2212yes—
22D4×Dic31C2212yes—
23C2×#92479C2212yes—
24#11758507C2211yescharacteristic
25C2×C4⋊Dic31C2211yescharacteristic
26C2×C4×Dic31C2211yescharacteristic
27#11758475—21no—
28#11758470C2211yescharacteristic, complemented
29#11758469—21no—
30#11758467C2211yescharacteristic, complemented
31#11758463—21no—
32(C22⋊D31)⋊C4—21no—
33C31×#102C2212yes—
34C31×C22⋊C8C2211yescharacteristic
35#11758423C2212yescomplemented
36#11758420C2212yes—
37C22⋊(C31⋊C8)C2211yescharacteristic
38#11758409C2212yescomplemented
39#11758406C2212yescomplemented
40#11758403C2212yescomplemented
41Dic31⋊C8C2212yescomplemented
42#11758396C2212yescomplemented
43C24×C31—21no—
44C2×Q8×C31C2311yescharacteristic
45C2×D4×C31—24no—
46C2×D4×C31—21no—
47C2×D4×C31C2312yes—
48C22×C4×C31C2311yescharacteristic
49C2×C22⋊D31—21no—
50C22×Dic31—21no—
51C22×Dic31—22no—
52C22×Dic31—22no—
53C22×Dic31—44no—
54C22×Dic31D411yescharacteristic
55C22×Dic31D411yescharacteristic
56C2×D124C2311yescharacteristic
57C2×C4×D31—21no—
58SD16×C31—24no—
59C2×C8×C31C2312yes—
60C4⋊C4×C31—21no—
61C4⋊C4×C31C2311yescharacteristic
62C22⋊C4×C31—21no—
63C22⋊Dic31—22no—
64C22⋊Dic31—22no—
65#92479—24no—
66(C2×D31)⋊C4—21no—
67(C2×D31)⋊C4—21no—
68(C2×D31)⋊C4—21no—
69(C2×D31)⋊C4—21no—
70C4⋊Dic31—21no—
71C4⋊Dic31—21no—
72C4⋊Dic31—21no—
73C4⋊Dic31C2312yes—
74C4⋊Dic31C2311yescharacteristic
75Dic31⋊C4—21no—
76C4×Dic31—21no—
77C4×Dic31D411yescharacteristic
78C4×Dic31D411yescharacteristic
79C4×Dic31C2311yescharacteristic
80C4×Dic31C2311yescharacteristic
81C2×C31⋊C8C2312yes—
82C23×C31—24no—
83C23×C31—22no—
84C23×C31—42no—
85C23×C31—42no—
86C23×C31C2×D411yescharacteristic
87C22×D31—21no—
88Q8×C31—22no—
89D4×C31—24no—
90D4×C31—22no—
91D4×C31#4114yes—
92C2×C4×C31—22no—
93C2×C4×C31—21no—
94C2×C4×C31—21no—
95C2×C4×C31C2×D411yescharacteristic
96C2×C4×C31C2411yescharacteristic
97C22⋊D31—42no—
98C2×Dic31—22no—
99C2×Dic31—21no—
100C2×Dic31—22no—
101C2×Dic31—41no—
102C2×Dic31—44no—
103C2×Dic31—44no—
104C2×Dic31—42no—
105C2×Dic31C2×D411yescharacteristic
106C2×Dic31C2×D411yescharacteristic
107C2×Dic31C2×D411yescharacteristic
108C2×Dic31C2×D411yescharacteristic
109D124—22no—
110C4×D31—42no—
111C8×C31—22no—
112C31⋊C8—22no—
113#3114—311nomaximal, Sylow
114C22×C31—21no—
115C22×C31—21no—
116C22×C31—24no—
117C22×C31—48no—
118C22×C31—42no—
119C22×C31—42no—
120C22×C31AGL(1,ℤ/8)12yes—
121C22×C31C22×D411yescharacteristic
122C2×D31—41no—
123C2×D31—41no—
124C2×D31—41no—
125C4×C31—21no—
126C4×C31—41no—
127C4×C31—41no—
128C4×C31C2×#4112yes—
129Dic31—21no—
130Dic31—21no—
131Dic31—21no—
132Dic31—21no—
133Dic31—42no—
134Dic31—41no—
135#529—311no—
136C2×C4×D4—311no—
137#483—311no—
138#480—311no—
139#478—312no—
140#463—312no—
141#450D3111yescharacteristic, complemented
142#447—311no—
143#442—312no—
144#440—312no—
145#432—311no—
146C2×C31—22no—
147C2×C31—44no—
148C2×C31#52511yescharacteristic
149C2×C31C2×AGL(1,ℤ/8)11yescharacteristic
150C2×C31#57511yescharacteristic
151D31—81no—
152C22×D4C2×D3111yescharacteristic
153C23×C4—621no—
154C2×SD16C2×D3112yes—
155C2×D8—312no—
156#126—621no—
157#125—311no—
158#124—311no—
159#123—621no—
160C22⋊Q8C2×D3111yescharacteristic
161#121—311no—
162C4×Q8—311no—
163C4×D4—312no—
164C4×D4—312no—
165C4×D4—311no—
166C4×D4—621no—
167C4×D4—621no—
168C4×D4—621no—
169#117—621no—
170C2×C4⋊C4—311no—
171C2×C22⋊C4—621no—
172C2×C42—311no—
173#107—312no—
174#106—312no—
175C4⋊C8—312no—
176#103—312no—
177#102—312no—
178#102—312no—
179#102C2×D3112yes—
180C22⋊C8—311no—
181C22⋊C8C2×D3111yescharacteristic
182#97—312no—
183C31#311411yescharacteristic, complemented, Sylow
184C24—21no—
185C2×Q8C22×D3111yescharacteristic
186C2×D4—24no—
187C2×D4—21no—
188C2×D4—311no—
189C2×D4—621no—
190C2×D4C22×D3112yes—
191C22×C4—311no—
192C22×C4—311no—
193C22×C4—621no—
194C22×C4—622no—
195C22×C4—622no—
196C22×C4—621no—
197C22×C4—1244no—
198C22×C4C22×D3111yescharacteristic
199SD16—24no—
200D8—624no—
201C2×C8—312no—
202C2×C8C22×D3112yes—
203C4⋊C4—21no—
204C4⋊C4—311no—
205C4⋊C4—312no—
206C4⋊C4—621no—
207C4⋊C4—621no—
208C4⋊C4—621no—
209C4⋊C4—621no—
210C4⋊C4C22×D3111yescharacteristic
211C22⋊C4—21no—
212C22⋊C4—621no—
213C22⋊C4—621no—
214C22⋊C4—621no—
215C22⋊C4—622no—
216C22⋊C4—622no—
217C22⋊C4—621no—
218C42—311no—
219C42—311no—
220C42—311no—
221C42—311no—
222C42—621no—
223C23—24no—
224C23—22no—
225C23—42no—
226C23—42no—
227C23—621no—
228C23D4×D3111yescharacteristic
229Q8—22no—
230D4—24no—
231D4—22no—
232D4—622no—
233D4—1242no—
234D4#9249714yes—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—311no—
239C2×C4—311no—
240C2×C4—311no—
241C2×C4—311no—
242C2×C4—622no—
243C2×C4—621no—
244C2×C4—622no—
245C2×C4—1242no—
246C2×C4—1241no—
247C2×C4—1244no—
248C2×C4—1244no—
249C2×C4—1242no—
250C2×C4D4×D3111yescharacteristic
251C2×C4C23×D3111yescharacteristic
252C8—22no—
253C8—622no—
254C22—21no—
255C22—21no—
256C22—24no—
257C22—48no—
258C22—42no—
259C22—42no—
260C22—1241no—
261C22—1241no—
262C22—1241no—
263C22#1175849112yes—
264C22C2×D4×D3111yescharacteristic
265C4—21no—
266C4—41no—
267C4—41no—
268C4—621no—
269C4—621no—
270C4—621no—
271C4—621no—
272C4—1242no—
273C4—1241no—
274C4C2×#9249712yes—
275C2—22no—
276C2—44no—
277C2—2481no—
278C2#1452155511yescharacteristic
279C2C2×#1175849111yescharacteristic
280C2#1452184511yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#450C2
socleC22×C31C22×D4
prime radical#14521183C2
hypercenter#450D31
hypocenterC31#3114
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #14521585C2semidirect
2. C2×D4×Dic31C2semidirect
3. #14521377C2semidirect
4. #14521363C2semidirect
5. #14521347C2semidirect
6. #14521266C2semidirect
8. #14521158C2semidirect
9. #14521150C2semidirect
10. #14521103C2semidirect
11. C31×#450C2semidirect
17. #11758524C22semidirect
18. Q8×Dic31C22semidirect
28. #11758470C22semidirect
30. #11758467C22semidirect
35. #11758423C22semidirect
38. #11758409C22semidirect
39. #11758406C22semidirect
40. #11758403C22semidirect
41. Dic31⋊C8C22semidirect
42. #11758396C22semidirect
141. #450D31semidirect
183. C31#3114semidirect

Maximal subgroups

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

subgroupindex
#145215852
C2×D4×Dic312
#145213772
#145213632
#145213472
#145212662
#145211832
#145211582
#145211502
#145211032
C31×#4502
#311431