Finite Group Databasebeta

← D31×#508 #16858825

Subgroups — D31×#508 #16858825

Order 3968.

368 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 115 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#135×D31C212yesmaximal, complemented
2#14521482C211yesmaximal, characteristic, complemented
3#14521481C211yesmaximal, characteristic, complemented
4#14521480C212yesmaximal, complemented
5#14521479C212yesmaximal, complemented
6#14521478C211yesmaximal, characteristic, complemented
7Q32×D31C211yesmaximal, characteristic, complemented
8SD32×D31C212yesmaximal, complemented
9D16×D31C211yesmaximal, characteristic, complemented
10C2×C16×D31C211yesmaximal, characteristic, complemented
11C31×#508C211yesmaximal, characteristic, complemented
12#41×D31—22no—
13C31×#135C2212yescomplemented
14#11758527—22no—
15#11758496C2212yescomplemented
16Q16×D31C2212yes—
17#11758493—22no—
18SD16×D31—22no—
19#11758489C2212yescomplemented
20D8×D31C2212yes—
21#11758481C2212yescomplemented
22C2×C8×D31C2211yescharacteristic
23C31×Q32C2211yescharacteristic, complemented
24C31×SD32C2212yescomplemented
25C31×D16C2211yescharacteristic, complemented
26C2×C16×C31C2211yescharacteristic, complemented
27#11758417C2212yescomplemented
28#11758416C2212yescomplemented
29#11758415C2212yescomplemented
30#11758414C2212yescomplemented
31C2×C31⋊C16C2211yescharacteristic, complemented
32Dic248C2211yescharacteristic, complemented
33#11758388C2212yescomplemented
34D496C2211yescharacteristic, complemented
35C16×D31C2211yescharacteristic
36C16×D31C2211yescharacteristic
37C16×D31C2211yescharacteristic, complemented
38C16×D31C2211yescharacteristic, complemented
39#41×C31—22no—
40#92499—22no—
41Q8×D31—22no—
42#92497—22no—
43#92497—42no—
44D4×D31—22no—
45D4×D31—42no—
46#92495—22no—
47#92495—42no—
48C2×C4×D31—42no—
49C2×C4×D31D411yescharacteristic
50Q16×C31C2312yes—
51SD16×C31—22no—
52D8×C31C2312yes—
53C2×C8×C31C2311yescharacteristic
54#92482—22no—
55#92481—22no—
56#92480—22no—
57#92479—22no—
58C2×C31⋊C8D411yescharacteristic
59Dic124C2312yes—
60D248C2312yes—
61#92470—22no—
62C8×D31D411yescharacteristic
63C8×D31D411yescharacteristic
64C8×D31C2311yescharacteristic
65C8×D31C2311yescharacteristic
66C16×C31C2311yescharacteristic
67C16×C31C2311yescharacteristic
68C31⋊C16C2311yescharacteristic
69C31⋊C16C2311yescharacteristic
70C22×D31—42no—
71C22×D31D811yescharacteristic
72Q8×C31—22no—
73D4×C31—22no—
74D4×C31—42no—
75C2×C4×C31—42no—
76C2×C4×C31C2×D411yescharacteristic
77C22⋊D31—42no—
78C22⋊D31—42no—
79C22⋊D31—42no—
80C2×Dic31—42no—
81C2×Dic31D811yescharacteristic
82D124—22no—
83D124—42no—
84C4×D31—42no—
85C4×D31—42no—
86C4×D31—42no—
87C4×D31D811yescharacteristic
88C4×D31D811yescharacteristic
89C4×D31C2×D411yescharacteristic
90C4×D31C2×D411yescharacteristic
91Dic62—22no—
92Dic62—42no—
93C8×C31C2×D411yescharacteristic
94C8×C31C2411yescharacteristic
95C31⋊C8C2×D411yescharacteristic
96C31⋊C8C2×D411yescharacteristic
97C2×#508—311nomaximal, Sylow
98C22×C31—42no—
99C22×C31C2×D811yescharacteristic
100C2×D31—22no—
101C2×D31—42no—
102C2×D31—84no—
103C2×D31C2×D811yescharacteristic
104C2×D31C2×D811yescharacteristic
105C4×C31—42no—
106C4×C31C2×D811yescharacteristic
107C4×C31C22×D411yescharacteristic
108Dic31—42no—
109Dic31C2×D811yescharacteristic
110Dic31C2×D811yescharacteristic
111C2×#135—312no—
112#508—312no—
113#508—311no—
114#508—312no—
115#508—311no—
116#508—311no—
117#508D3111yescharacteristic, direct product factor
118C2×Q32—311no—
119C2×SD32—312no—
120C2×D16—311no—
121C22×C16—311no—
122C2×C31—21no—
123C2×C31—82no—
124C2×C31C22×D811yescharacteristic
125D31—21no—
126D31—82no—
127D31#50812yesdirect product factor
128C2×#41—622no—
129#135—312no—
130#135—312no—
131#135—312no—
132#135—622no—
133#135—622no—
134#135C2×D3112yescomplemented
135C2×Q16—312no—
136C2×SD16—622no—
137C2×D8—312no—
138C22×C8—311no—
139Q32—312no—
140Q32—311no—
141Q32C2×D3111yescharacteristic, complemented
142SD32—312no—
143SD32—312no—
144SD32—312no—
145SD32C2×D3112yescomplemented
146D16—312no—
147D16—311no—
148D16C2×D3111yescharacteristic, complemented
149C2×C16—311no—
150C2×C16—311no—
151C2×C16—311no—
152C2×C16—311no—
153C2×C16—311no—
154C2×C16C2×D3111yescharacteristic, complemented
155C31C2×#50811yescharacteristic, complemented, Sylow
156#41—22no—
157#41—622no—
158#41—622no—
159#41—622no—
160#41—1242no—
161#41—1242no—
162C2×Q8—622no—
163C2×D4—622no—
164C2×D4—1242no—
165C22×C4—311no—
166C22×C4—1242no—
167Q16—312no—
168Q16—622no—
169Q16C22×D3112yes—
170SD16—22no—
171SD16—622no—
172SD16—622no—
173SD16—622no—
174D8—312no—
175D8—622no—
176D8C22×D3112yes—
177C2×C8—311no—
178C2×C8—311no—
179C2×C8—311no—
180C2×C8—311no—
181C2×C8—311no—
182C2×C8C22×D3111yescharacteristic
183C16—311no—
184C16—311no—
185C16C22×D3111yescharacteristic
186C16C22×D3111yescharacteristic
187C23—311no—
188C23—1242no—
189Q8—22no—
190Q8—622no—
191Q8—1242no—
192D4—22no—
193D4—42no—
194D4—622no—
195D4—1242no—
196D4—1242no—
197D4—1242no—
198D4—1242no—
199C2×C4—42no—
200C2×C4—311no—
201C2×C4—311no—
202C2×C4—311no—
203C2×C4—311no—
204C2×C4—311no—
205C2×C4—1242no—
206C2×C4—1242no—
207C2×C4—1242no—
208C2×C4—1242no—
209C2×C4D4×D3111yescharacteristic
210C8—311no—
211C8—311no—
212C8D4×D3111yescharacteristic
213C8C23×D3111yescharacteristic
214C22—42no—
215C22—311no—
216C22—311no—
217C22—622no—
218C22—1242no—
219C22—2484no—
220C22D8×D3111yescharacteristic
221C4—42no—
222C4—311no—
223C4—311no—
224C4—1242no—
225C4D8×D3111yescharacteristic
226C4C2×D4×D3111yescharacteristic
227C2—21no—
228C2—82no—
229C2—312no—
230C2—621no—
231C2—2482no—
232C2C2×D8×D3111yescharacteristic
subgroupG/H
centerC4D8×D31
commutator, cosocleC8×C31C24
FrattiniC8C23×D31
FittingC31×#508C2
socleC2×C31C22×D8
hypercenter#508D31
hypocenterC31C2×#508
LCSG ▷ C8×C31 ▷ C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×C8 ◁ #508

Complements

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

NH ≅ G/Ntype
1. #135×D31C2semidirect
2. #14521482C2semidirect
3. #14521481C2semidirect
4. #14521480C2semidirect
5. #14521479C2semidirect
6. #14521478C2semidirect
7. Q32×D31C2semidirect
8. SD32×D31C2semidirect
9. D16×D31C2semidirect
10. C2×C16×D31C2semidirect
11. C31×#508C2semidirect
13. C31×#135C22semidirect
15. #11758496C22semidirect
19. #11758489C22semidirect
21. #11758481C22semidirect
23. C31×Q32C22semidirect
24. C31×SD32C22semidirect
25. C31×D16C22semidirect
26. C2×C16×C31C22semidirect
27. #11758417C22semidirect
28. #11758416C22semidirect
29. #11758415C22semidirect
30. #11758414C22semidirect
31. C2×C31⋊C16C22semidirect
32. Dic248C22semidirect
33. #11758388C22semidirect
34. D496C22semidirect
37. C16×D31C22semidirect
38. C16×D31C22semidirect
117. #508D31direct (N × H)
127. D31#508direct (N × H)
134. #135C2×D31semidirect
141. Q32C2×D31semidirect
145. SD32C2×D31semidirect
148. D16C2×D31semidirect
154. C2×C16C2×D31semidirect
155. C31C2×#508semidirect

Maximal subgroups

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

subgroupindex
#135×D312
#145214822
#145214812
#145214802
#145214792
#145214782
Q32×D312
SD32×D312
D16×D312
C2×C16×D312
C31×#5082
C2×#50831