Finite Group Databasebeta

← #16855318

Subgroups — #16855318

Order 3968.

500 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 127 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×#11758489C211yesmaximal, characteristic, complemented
2C2×M16×D31C211yesmaximal, characteristic, complemented
3#14521559C212yesmaximal, complemented
4#14521308C211yesmaximal, characteristic, complemented
5#14521307C211yesmaximal, characteristic, complemented
6#14521299C211yesmaximal, characteristic, complemented
7#14521294C212yesmaximal, complemented
8#14521293C212yesmaximal, complemented
9#14521084C211yesmaximal, characteristic, complemented
10#14521042C212yesmaximal, complemented
11C31×#469C211yesmaximal, characteristic, complemented
12C2×#92497C2212yes—
13C22×C4×D31C2211yescharacteristic
14C2×D8×C31C2211yescharacteristic, complemented
15C2×M16×C31C2211yescharacteristic
16C31×#121C2212yes—
17#11758515C2212yes—
18#11758514—22no—
19D4×Dic31—22no—
20C2×#92480C2212yescomplemented
21#11758501C2212yes—
22C2×#92474C2211yescharacteristic
23#11758489—24no—
24M16×D31—24no—
25C2×Dic124C2211yescharacteristic, complemented
26C2×#92469C2211yescharacteristic
27C2×#92469C2211yescharacteristic
28C2×C8×D31C2211yescharacteristic
29C2×C8×D31C2211yescharacteristic
30#11758474C2212yes—
31#11758469C2212yes—
32#11758460—22no—
33(C22⋊D31)⋊C4—22no—
34C31×#106C2211yescharacteristic, complemented
35C31×#102C2212yescomplemented
36#11758420C2212yescomplemented
37#11758405C2211yescharacteristic, complemented
38#11758404C2212yescomplemented
39#11758398C2212yescomplemented
40#11758395C2212yescomplemented
41C23×D31D411yescharacteristic
42C2×D4×C31—22no—
43C2×D4×C31C2312yes—
44C22×C4×C31C2311yescharacteristic
45C2×C22⋊D31—22no—
46C2×C22⋊D31—22no—
47C22×Dic31—22no—
48C22×Dic31D411yescharacteristic
49#92497—24no—
50#92497—24no—
51C2×C4×D31—22no—
52C2×C4×D31—22no—
53C2×C4×D31D411yescharacteristic
54C2×C4×D31D411yescharacteristic
55C2×C4×D31C2311yescharacteristic
56C2×C4×D31C2311yescharacteristic
57C2×Dic62C2312yes—
58D8×C31—22no—
59M16×C31—22no—
60C2×C8×C31C2311yescharacteristic
61C2×C8×C31C2311yescharacteristic
62C4⋊C4×C31C2312yes—
63C22⋊C4×C31—22no—
64C22⋊Dic31—22no—
65C22⋊Dic31—22no—
66#92480—24no—
67(C2×D31)⋊C4—22no—
68(C2×D31)⋊C4—22no—
69C4⋊Dic31—22no—
70C4⋊Dic31C2312yes—
71Dic31⋊C4—22no—
72C4×Dic31—22no—
73#92474—22no—
74C2×C31⋊C8C2311yescharacteristic
75C2×C31⋊C8C2311yescharacteristic
76Dic124—22no—
77#92469—22no—
78#92469—22no—
79C8×D31—22no—
80C8×D31D414yescomplemented
81C23×C31—22no—
82C23×C31C2×D411yescharacteristic
83C22×D31—22no—
84C22×D31—42no—
85C22×D31C2×D411yescharacteristic
86C22×D31C2×D411yescharacteristic
87D4×C31—22no—
88D4×C31—22no—
89D4×C31—44no—
90C2×C4×C31—21no—
91C2×C4×C31—21no—
92C2×C4×C31—22no—
93C2×C4×C31C2×D411yescharacteristic
94C2×C4×C31C2411yescharacteristic
95C22⋊D31—44no—
96C22⋊D31—44no—
97C2×Dic31—22no—
98C2×Dic31—22no—
99C2×Dic31—22no—
100C2×Dic31—44no—
101C2×Dic31C2×D411yescharacteristic
102C2×Dic31C2×D411yescharacteristic
103C4×D31—22no—
104C4×D31—21no—
105C4×D31—21no—
106C4×D31—22no—
107C4×D31C2×D412yes—
108C4×D31#4112yes—
109Dic62—22no—
110Dic62—22no—
111C8×C31—21no—
112C8×C31C2×D412yes—
113C31⋊C8—21no—
114C31⋊C8C2×D412yes—
115#3054—311nomaximal, Sylow
116C22×C31—21no—
117C22×C31—22no—
118C22×C31—42no—
119C22×C31—42no—
120C22×C31—42no—
121C22×C31C22×D411yescharacteristic
122C2×D31—22no—
123C2×D31—21no—
124C2×D31—22no—
125C2×D31—44no—
126C2×D31#12112yes—
127C4×C31—21no—
128C4×C31—42no—
129C4×C31C22×D411yescharacteristic
130C4×C31C2×#4111yescharacteristic
131Dic31—21no—
132Dic31—42no—
133Dic31—42no—
134Dic31#12112yes—
135C2×#135—311no—
136C22×M16—311no—
137#525—312no—
138#470—311no—
139#470—311no—
140#469—311no—
141#469D3111yescharacteristic, complemented
142#468—312no—
143#468—312no—
144#427—311no—
145#416—312no—
146C2×C31—41no—
147C2×C31—82no—
148C2×C31C2×#12111yescharacteristic
149C2×C31#57512yes—
150D31—22no—
151D31—41no—
152C2×#41—312no—
153C23×C4—311no—
154#135—624no—
155C2×Q16—311no—
156C2×SD16—312no—
157C2×D8C2×D3111yescharacteristic, complemented
158C2×M16—311no—
159C2×M16—311no—
160C2×M16—311no—
161C2×M16—624no—
162C2×M16C2×D3111yescharacteristic
163C22×C8—311no—
164C22×C8—311no—
165#123—622no—
166C22⋊Q8—312no—
167C22⋊Q8—312no—
168#121—312no—
169#121C2×D3112yes—
170#120—622no—
171C4×D4—622no—
172C4×D4—622no—
173#117—312no—
174#107—312no—
175#106—311no—
176#106C2×D3111yescharacteristic, complemented
177#103—312no—
178#103—312no—
179#102—312no—
180#102C2×D3112yescomplemented
181C31#305411yescharacteristic, complemented, Sylow
182C24—311no—
183#41—624no—
184#41—624no—
185C2×Q8—312no—
186C2×D4—22no—
187C2×D4—622no—
188C2×D4—622no—
189C2×D4C22×D3112yes—
190C22×C4—311no—
191C22×C4—311no—
192C22×C4—311no—
193C22×C4—311no—
194C22×C4—311no—
195C22×C4—622no—
196C22×C4—622no—
197C22×C4—622no—
198C22×C4C22×D3111yescharacteristic
199Q16—622no—
200SD16—624no—
201D8—22no—
202M16—22no—
203M16—622no—
204M16—622no—
205M16—622no—
206C2×C8—311no—
207C2×C8—314no—
208C2×C8—311no—
209C2×C8—622no—
210C2×C8C22×D3111yescharacteristic
211C2×C8C22×D3111yescharacteristic
212C4⋊C4—312no—
213C4⋊C4—622no—
214C4⋊C4—622no—
215C4⋊C4C22×D3112yes—
216C22⋊C4—22no—
217C22⋊C4—622no—
218C22⋊C4—622no—
219C22⋊C4—622no—
220C22⋊C4—622no—
221C42—622no—
222C23—22no—
223C23—311no—
224C23—311no—
225C23—622no—
226C23—1242no—
227C23D4×D3111yescharacteristic
228Q8—622no—
229Q8—622no—
230D4—22no—
231D4—22no—
232D4—44no—
233D4—1244no—
234D4—1244no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—311no—
239C2×C4—312no—
240C2×C4—312no—
241C2×C4—311no—
242C2×C4—622no—
243C2×C4—622no—
244C2×C4—622no—
245C2×C4—621no—
246C2×C4—621no—
247C2×C4—622no—
248C2×C4—622no—
249C2×C4—1244no—
250C2×C4D4×D3111yescharacteristic
251C2×C4C23×D3111yescharacteristic
252C8—21no—
253C8—312no—
254C8—621no—
255C8D4×D3112yes—
256C22—21no—
257C22—22no—
258C22—42no—
259C22—42no—
260C22—42no—
261C22—312no—
262C22—622no—
263C22—621no—
264C22—622no—
265C22—1244no—
266C22C2×D4×D3111yescharacteristic
267C4—21no—
268C4—42no—
269C4—312no—
270C4—621no—
271C4—1242no—
272C4—1242no—
273C4C2×D4×D3111yescharacteristic
274C4#41×D3111yescharacteristic
275C2—41no—
276C2—82no—
277C2—622no—
278C2—1241no—
279C2#121×D3111yescharacteristic
280C2#1452184512yes—
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#469C2
socleC22×C31C22×D4
hypercenter#469D31
hypocenterC31#3054
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. C2×#11758489C2semidirect
2. C2×M16×D31C2semidirect
3. #14521559C2semidirect
4. #14521308C2semidirect
5. #14521307C2semidirect
6. #14521299C2semidirect
7. #14521294C2semidirect
8. #14521293C2semidirect
9. #14521084C2semidirect
10. #14521042C2semidirect
11. C31×#469C2semidirect
14. C2×D8×C31C22semidirect
20. C2×#92480C22semidirect
25. C2×Dic124C22semidirect
34. C31×#106C22semidirect
35. C31×#102C22semidirect
36. #11758420C22semidirect
37. #11758405C22semidirect
38. #11758404C22semidirect
39. #11758398C22semidirect
40. #11758395C22semidirect
80. C8×D31D4semidirect
141. #469D31semidirect
157. C2×D8C2×D31semidirect
176. #106C2×D31semidirect
180. #102C2×D31semidirect
181. C31#3054semidirect

Maximal subgroups

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

subgroupindex
C2×#117584892
C2×M16×D312
#145215592
#145213082
#145213072
#145212992
#145212942
#145212932
#145210842
#145210422
C31×#4692
#305431