Finite Group Databasebeta

← D31×#496 #16856141

Subgroups — D31×#496 #16856141

Order 3968.

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

normal subgroups only

This group has 668 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
1C2×SD16×D31C212yesmaximal, complemented
2C2×D8×D31C212yesmaximal, complemented
3#127×D31C211yesmaximal, characteristic, complemented
4#124×D31C211yesmaximal, characteristic, complemented
5#14521439C211yesmaximal, characteristic, complemented
6#14521438C211yesmaximal, characteristic, complemented
7#14521437C212yesmaximal, complemented
8#14521436C211yesmaximal, characteristic, complemented
9#14521435C212yesmaximal, complemented
10#97×D31C211yesmaximal, characteristic, complemented
11C31×#496C211yesmaximal, characteristic, complemented
12C2×Q8×D31C2211yescharacteristic
13C2×D4×D31—22no—
14C2×D4×D31C2211yescharacteristic
15C2×D4×D31C2212yes—
16C2×SD16×C31C2212yescomplemented
17C2×D8×C31C2212yescomplemented
18C31×#127C2211yescharacteristic, complemented
19C31×#124C2211yescharacteristic, complemented
20#11758524C2211yescharacteristic, complemented
21C2×#92481C2212yescomplemented
22#11758517—22no—
23#11758517C2212yescomplemented
24#11758513C2211yescharacteristic, complemented
25C2×#92480C2212yescomplemented
26C2×#92479C2212yescomplemented
27C2×#92479C2212yescomplemented
28SD16×D31—28no—
29D8×D31—28no—
30C2×D248C2212yescomplemented
31C2×#92470C2212yescomplemented
32C2×C8×D31C2212yes—
33#11758462—22no—
34C22⋊C4×D31—22no—
35#11758452C2211yescharacteristic, complemented
36#11758451C2211yescharacteristic, complemented
37C42×D31C2211yescharacteristic
38C31×#97C2211yescharacteristic, complemented
39#11758403C2212yescomplemented
40#11758391C2211yescharacteristic, complemented
41C23×D31—21no—
42C23×D31—22no—
43C2×Q8×C31C2311yescharacteristic
44C2×D4×C31—22no—
45C2×D4×C31C2311yescharacteristic
46C2×D4×C31C2312yes—
47C2×C22⋊D31—21no—
48C2×C22⋊D31—22no—
49C2×C22⋊D31—24no—
50Q8×D31—24no—
51D4×D31—24no—
52D4×D31—24no—
53D4×D31—24no—
54D4×D31—48no—
55C2×D124—22no—
56C2×D124C2312yes—
57C2×D124C2311yescharacteristic
58C2×C4×D31—21no—
59C2×C4×D31D412yes—
60C2×C4×D31C2311yescharacteristic
61C2×Dic62—21no—
62C2×Dic62C2311yescharacteristic
63SD16×C31—24no—
64D8×C31—24no—
65C2×C8×C31C2312yes—
66C22⋊C4×C31—22no—
67C42×C31C2311yescharacteristic
68C22⋊Dic31—22no—
69#92481—24no—
70#92480—24no—
71#92479—24no—
72#92479—24no—
73(C2×D31)⋊C4—22no—
74(C2×D31)⋊C4—22no—
75C4×Dic31D412yes—
76C4×Dic31C2311yescharacteristic
77C2×C31⋊C8C2312yes—
78D248—24no—
79#92470—24no—
80C8×D31D418yescomplemented
81C23×C31—21no—
82C23×C31—22no—
83C22×D31—21no—
84C22×D31—22no—
85C22×D31—42no—
86C22×D31—44no—
87C22×D31—48no—
88C22×D31—44no—
89C22×D31C2×D411yescharacteristic
90Q8×C31—22no—
91D4×C31—22no—
92D4×C31—22no—
93D4×C31—22no—
94D4×C31—44no—
95C2×C4×C31—21no—
96C2×C4×C31C2×D412yes—
97C2×C4×C31C2411yescharacteristic
98C22⋊D31—48no—
99C22⋊D31—42no—
100C22⋊D31—44no—
101C2×Dic31—21no—
102C2×Dic31C2×D412yes—
103C2×Dic31C2×D411yescharacteristic
104D124—22no—
105D124—22no—
106D124—22no—
107D124—44no—
108C4×D31—24no—
109C4×D31—42no—
110C4×D31C2×D414yes—
111Dic62—22no—
112Dic62—42no—
113C8×C31C2×D414yescomplemented
114C31⋊C8C2×D414yescomplemented
115C2×#496—311nomaximal, Sylow
116C22×C31—41no—
117C22×C31—42no—
118C22×C31—44no—
119C22×C31—42no—
120C22×C31C22×D411yescharacteristic
121C2×D31—42no—
122C2×D31—41no—
123C2×D31—44no—
124C2×D31—42no—
125C2×D31—88no—
126C2×D31—84no—
127C2×D31#12712yes—
128C2×D31AGL(1,ℤ/8)14yes—
129C4×C31—22no—
130C4×C31—41no—
131C4×C31C22×D412yes—
132Dic31—22no—
133Dic31—41no—
134Dic31#12712yes—
135C22×SD16—312no—
136C22×D8—312no—
137C2×#127—311no—
138C2×#124—311no—
139#496—312no—
140#496—312no—
141#496—311no—
142#496—311no—
143#496—311no—
144#496D3111yescharacteristic, direct product factor
145C2×#97—311no—
146C2×C31—82no—
147C2×C31—81no—
148C2×C31C2×#12711yescharacteristic
149C2×C31C2×AGL(1,ℤ/8)12yes—
150D31—82no—
151D31—81no—
152D31#49614yesdirect product factor
153C22×Q8—311no—
154C22×D4—311no—
155C22×D4—312no—
156C22×D4—622no—
157C2×SD16—312no—
158C2×SD16—312no—
159C2×SD16—312no—
160C2×SD16—628no—
161C2×SD16C2×D3112yescomplemented
162C2×D8—312no—
163C2×D8—312no—
164C2×D8—312no—
165C2×D8—628no—
166C2×D8C2×D3112yescomplemented
167C22×C8—312no—
168#127—312no—
169#127—311no—
170#127—622no—
171#127C2×D3111yescharacteristic, complemented
172#124—311no—
173#124—311no—
174#124—311no—
175#124—622no—
176#124C2×D3111yescharacteristic, complemented
177C2×C22⋊C4—622no—
178C2×C42—311no—
179#97—312no—
180#97—311no—
181#97C2×D3111yescharacteristic, complemented
182C31C2×#49611yescharacteristic, complemented, Sylow
183C24—621no—
184C24—622no—
185C2×Q8—311no—
186C2×Q8—621no—
187C2×Q8—624no—
188C2×Q8C22×D3111yescharacteristic
189C2×D4—22no—
190C2×D4—311no—
191C2×D4—312no—
192C2×D4—624no—
193C2×D4—624no—
194C2×D4—624no—
195C2×D4—621no—
196C2×D4—624no—
197C2×D4—622no—
198C2×D4—622no—
199C2×D4—1248no—
200C2×D4C22×D3111yescharacteristic
201C2×D4C22×D3112yes—
202C22×C4—311no—
203C22×C4—312no—
204C22×C4—621no—
205SD16—24no—
206SD16—624no—
207SD16—624no—
208SD16—624no—
209D8—24no—
210D8—624no—
211D8—624no—
212D8—624no—
213C2×C8—312no—
214C2×C8—318no—
215C2×C8C22×D3112yes—
216C22⋊C4—22no—
217C22⋊C4—622no—
218C22⋊C4—622no—
219C22⋊C4—622no—
220C42—312no—
221C42—311no—
222C42C22×D3111yescharacteristic
223C23—21no—
224C23—22no—
225C23—311no—
226C23—621no—
227C23—622no—
228C23—1242no—
229C23—1244no—
230C23—1248no—
231C23—1244no—
232Q8—22no—
233Q8—622no—
234Q8—1242no—
235D4—22no—
236D4—22no—
237D4—22no—
238D4—44no—
239D4—622no—
240D4—622no—
241D4—622no—
242D4—1244no—
243D4—1242no—
244D4—1244no—
245D4—1248no—
246C2×C4—21no—
247C2×C4—311no—
248C2×C4—314no—
249C2×C4—312no—
250C2×C4—624no—
251C2×C4—621no—
252C2×C4—1242no—
253C2×C4D4×D3112yes—
254C2×C4C23×D3111yescharacteristic
255C8—314no—
256C8D4×D3114yescomplemented
257C22—44no—
258C22—42no—
259C22—41no—
260C22—42no—
261C22—314no—
262C22—312no—
263C22—1244no—
264C22—1242no—
265C22—1241no—
266C22—1242no—
267C22—2488no—
268C22—2484no—
269C22C2×D4×D3111yescharacteristic
270C4—22no—
271C4—41no—
272C4—312no—
273C4—622no—
274C4—1241no—
275C4C2×D4×D3112yes—
276C2—82no—
277C2—81no—
278C2—314no—
279C2—2482no—
280C2—2481no—
281C2#127×D3111yescharacteristic
282C2AGL(1,ℤ/8)×D3112yes—
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#496C2
socleC22×C31C22×D4
hypercenter#496D31
hypocenterC31C2×#496
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×SD16×D31C2semidirect
2. C2×D8×D31C2semidirect
3. #127×D31C2semidirect
4. #124×D31C2semidirect
5. #14521439C2semidirect
6. #14521438C2semidirect
7. #14521437C2semidirect
8. #14521436C2semidirect
9. #14521435C2semidirect
10. #97×D31C2semidirect
11. C31×#496C2semidirect
16. C2×SD16×C31C22semidirect
17. C2×D8×C31C22semidirect
18. C31×#127C22semidirect
19. C31×#124C22semidirect
20. #11758524C22semidirect
21. C2×#92481C22semidirect
23. #11758517C22semidirect
24. #11758513C22semidirect
25. C2×#92480C22semidirect
26. C2×#92479C22semidirect
27. C2×#92479C22semidirect
30. C2×D248C22semidirect
31. C2×#92470C22semidirect
35. #11758452C22semidirect
36. #11758451C22semidirect
38. C31×#97C22semidirect
39. #11758403C22semidirect
40. #11758391C22semidirect
80. C8×D31D4semidirect
113. C8×C31C2×D4semidirect
114. C31⋊C8C2×D4semidirect
144. #496D31direct (N × H)
152. D31#496direct (N × H)
161. C2×SD16C2×D31semidirect
166. C2×D8C2×D31semidirect
171. #127C2×D31semidirect
176. #124C2×D31semidirect
181. #97C2×D31semidirect
182. C31C2×#496semidirect
256. C8D4×D31semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D312
C2×D8×D312
#127×D312
#124×D312
#145214392
#145214382
#145214372
#145214362
#145214352
#97×D312
C31×#4962
C2×#49631