Finite Group Databasebeta

← D31×#509 #16858831

Subgroups — D31×#509 #16858831

Order 3968.

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

normal subgroups only

This group has 400 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×D31C211yesmaximal, characteristic, complemented
2C2×D8×D31C211yesmaximal, characteristic, complemented
3#14521487C211yesmaximal, characteristic, complemented
4#14521486C212yesmaximal, complemented
5#14521485C211yesmaximal, characteristic, complemented
6#14521484C211yesmaximal, characteristic, complemented
7#14521483C212yesmaximal, complemented
8SD32×D31C212yesmaximal, complemented
9D16×D31C212yesmaximal, complemented
10M32×D31C211yesmaximal, characteristic, complemented
11C31×#509C211yesmaximal, characteristic, complemented
12#41×D31—21no—
13C2×D4×D31—21no—
14C31×#135C2211yescharacteristic, complemented
15C2×D8×C31C2211yescharacteristic, complemented
16#11758527—21no—
17C2×#92479—21no—
18#11758496C2211yescharacteristic, complemented
19Q16×D31C2211yescharacteristic, complemented
20#11758493—21no—
21SD16×D31—21no—
22#11758489C2211yescharacteristic, complemented
23D8×D31—21no—
24D8×D31—21no—
25D8×D31C2212yes—
26D8×D31C2212yescomplemented
27D8×D31C2211yescharacteristic, complemented
28#11758481C2211yescharacteristic, complemented
29C2×D248C2211yescharacteristic, complemented
30C2×C8×D31C2211yescharacteristic
31C31×SD32C2212yescomplemented
32C31×D16C2212yescomplemented
33C31×M32C2211yescharacteristic, complemented
34#11758416C2212yescomplemented
35#11758415C2212yescomplemented
36#11758414C2212yescomplemented
37#11758414C2212yescomplemented
38#11758400C2211yescharacteristic, complemented
39#11758388C2212yescomplemented
40D496C2212yescomplemented
41#11758386C2212yescomplemented
42C16×D31C2212yescomplemented
43C23×D31—41no—
44#41×C31—21no—
45C2×D4×C31—21no—
46C2×C22⋊D31—41no—
47#92499—21no—
48Q8×D31—21no—
49#92497—21no—
50#92497—41no—
51D4×D31—22no—
52D4×D31—22no—
53D4×D31—21no—
54D4×D31—41no—
55D4×D31—41no—
56D4×D31—41no—
57#92495—21no—
58#92495—41no—
59C2×D124—21no—
60C2×C4×D31—41no—
61C2×C4×D31D411yescharacteristic
62Q16×C31C2311yescharacteristic, complemented
63SD16×C31—21no—
64D8×C31—21no—
65D8×C31C2312yes—
66D8×C31C2311yescharacteristic, complemented
67C2×C8×C31C2311yescharacteristic
68#92482—21no—
69#92481—21no—
70#92480—21no—
71#92479—22no—
72#92479—22no—
73#92479—21no—
74C2×C31⋊C8D411yescharacteristic
75Dic124C2311yescharacteristic, complemented
76D248—21no—
77D248C2312yes—
78D248C2311yescharacteristic, complemented
79#92470—21no—
80C8×D31D411yescharacteristic
81C8×D31D411yescharacteristic
82C8×D31C2311yescharacteristic
83C8×D31C2311yescharacteristic
84C16×C31C2312yescomplemented
85C31⋊C16C2312yescomplemented
86C23×C31—41no—
87C22×D31—41no—
88C22×D31—41no—
89C22×D31—42no—
90C22×D31—42no—
91C22×D31—84no—
92C22×D31D811yescharacteristic
93Q8×C31—21no—
94D4×C31—22no—
95D4×C31—21no—
96D4×C31—41no—
97D4×C31—41no—
98C2×C4×C31—41no—
99C2×C4×C31C2×D411yescharacteristic
100C22⋊D31—41no—
101C22⋊D31—41no—
102C22⋊D31—41no—
103C22⋊D31—42no—
104C22⋊D31—42no—
105C2×Dic31—41no—
106C2×Dic31D811yescharacteristic
107D124—21no—
108D124—22no—
109D124—41no—
110D124—41no—
111C4×D31—41no—
112C4×D31—41no—
113C4×D31—41no—
114C4×D31D811yescharacteristic
115C4×D31D811yescharacteristic
116C4×D31C2×D411yescharacteristic
117C4×D31C2×D411yescharacteristic
118Dic62—21no—
119Dic62—41no—
120C8×C31C2×D411yescharacteristic
121C8×C31C2411yescharacteristic
122C31⋊C8C2×D411yescharacteristic
123C31⋊C8C2×D411yescharacteristic
124C2×#509—311nomaximal, Sylow
125C22×C31—41no—
126C22×C31—42no—
127C22×C31—82no—
128C22×C31C2×D811yescharacteristic
129C2×D31—22no—
130C2×D31—41no—
131C2×D31—42no—
132C2×D31—84no—
133C2×D31—82no—
134C2×D31—82no—
135C2×D31—84no—
136C2×D31C2×D811yescharacteristic
137C2×D31C2×D811yescharacteristic
138C4×C31—41no—
139C4×C31C2×D811yescharacteristic
140C4×C31C22×D411yescharacteristic
141Dic31—41no—
142Dic31C2×D811yescharacteristic
143Dic31C2×D811yescharacteristic
144C2×#135—311no—
145C22×D8—311no—
146#509—311no—
147#509—311no—
148#509—312no—
149#509—312no—
150#509—311no—
151#509D3111yescharacteristic, direct product factor
152C2×SD32—312no—
153C2×D16—312no—
154C2×M32—311no—
155C2×C31—21no—
156C2×C31—82no—
157C2×C31—81no—
158C2×C31C22×D811yescharacteristic
159D31—21no—
160D31—81no—
161D31—82no—
162D31#50912yesdirect product factor
163C2×#41—621no—
164C22×D4—621no—
165#135—311no—
166#135—311no—
167#135—311no—
168#135—621no—
169#135—621no—
170#135C2×D3111yescharacteristic, complemented
171C2×Q16—311no—
172C2×SD16—621no—
173C2×D8—312no—
174C2×D8—312no—
175C2×D8—311no—
176C2×D8—311no—
177C2×D8—621no—
178C2×D8—621no—
179C2×D8—621no—
180C2×D8C2×D3111yescharacteristic, complemented
181C22×C8—311no—
182SD32—312no—
183SD32—312no—
184SD32—312no—
185SD32C2×D3112yescomplemented
186D16—312no—
187D16—312no—
188D16—312no—
189D16C2×D3112yescomplemented
190M32—311no—
191M32—312no—
192M32C2×D3111yescharacteristic, complemented
193C2×C16—312no—
194C31C2×#50911yescharacteristic, complemented, Sylow
195C24—1241no—
196#41—21no—
197#41—621no—
198#41—621no—
199#41—621no—
200#41—1241no—
201#41—1241no—
202C2×Q8—621no—
203C2×D4—21no—
204C2×D4—622no—
205C2×D4—622no—
206C2×D4—621no—
207C2×D4—621no—
208C2×D4—1241no—
209C2×D4—1241no—
210C2×D4—1241no—
211C2×D4—1241no—
212C22×C4—311no—
213C22×C4—1241no—
214Q16—311no—
215Q16—621no—
216Q16C22×D3111yescharacteristic, complemented
217SD16—21no—
218SD16—621no—
219SD16—621no—
220SD16—621no—
221D8—21no—
222D8—312no—
223D8—311no—
224D8—622no—
225D8—622no—
226D8—621no—
227D8—621no—
228D8C22×D3112yes—
229D8C22×D3111yescharacteristic, complemented
230C2×C8—311no—
231C2×C8—311no—
232C2×C8—311no—
233C2×C8—311no—
234C2×C8—311no—
235C2×C8C22×D3111yescharacteristic
236C16—312no—
237C16C22×D3112yescomplemented
238C23—41no—
239C23—311no—
240C23—1241no—
241C23—1242no—
242C23—1242no—
243C23—1241no—
244C23—2484no—
245Q8—21no—
246Q8—621no—
247Q8—1241no—
248D4—22no—
249D4—21no—
250D4—41no—
251D4—41no—
252D4—622no—
253D4—621no—
254D4—1241no—
255D4—1242no—
256D4—1242no—
257D4—1241no—
258D4—1241no—
259D4—1241no—
260D4—1241no—
261C2×C4—41no—
262C2×C4—311no—
263C2×C4—311no—
264C2×C4—311no—
265C2×C4—311no—
266C2×C4—311no—
267C2×C4—1241no—
268C2×C4—1241no—
269C2×C4—1241no—
270C2×C4—1241no—
271C2×C4D4×D3111yescharacteristic
272C8—311no—
273C8—311no—
274C8D4×D3111yescharacteristic
275C8C23×D3111yescharacteristic
276C22—41no—
277C22—42no—
278C22—82no—
279C22—311no—
280C22—311no—
281C22—622no—
282C22—1241no—
283C22—1242no—
284C22—2482no—
285C22—2484no—
286C22—2482no—
287C22—2484no—
288C22D8×D3111yescharacteristic
289C4—41no—
290C4—311no—
291C4—311no—
292C4—1241no—
293C4D8×D3111yescharacteristic
294C4C2×D4×D3111yescharacteristic
295C2—21no—
296C2—82no—
297C2—81no—
298C2—312no—
299C2—621no—
300C2—2482no—
301C2—2481no—
302C2C2×D8×D3111yescharacteristic
subgroupG/H
centerC2C2×D8×D31
commutator, cosocleC8×C31C24
FrattiniC8C23×D31
FittingC31×#509C2
socleC2×C31C22×D8
hypercenter#509D31
hypocenterC31C2×#509
LCSG ▷ C8×C31 ▷ C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. #135×D31C2semidirect
2. C2×D8×D31C2semidirect
3. #14521487C2semidirect
4. #14521486C2semidirect
5. #14521485C2semidirect
6. #14521484C2semidirect
7. #14521483C2semidirect
8. SD32×D31C2semidirect
9. D16×D31C2semidirect
10. M32×D31C2semidirect
11. C31×#509C2semidirect
14. C31×#135C22semidirect
15. C2×D8×C31C22semidirect
18. #11758496C22semidirect
19. Q16×D31C22semidirect
22. #11758489C22semidirect
26. D8×D31C22semidirect
27. D8×D31C22semidirect
28. #11758481C22semidirect
29. C2×D248C22semidirect
31. C31×SD32C22semidirect
32. C31×D16C22semidirect
33. C31×M32C22semidirect
34. #11758416C22semidirect
35. #11758415C22semidirect
36. #11758414C22semidirect
37. #11758414C22semidirect
38. #11758400C22semidirect
39. #11758388C22semidirect
40. D496C22semidirect
41. #11758386C22semidirect
42. C16×D31C22semidirect
62. Q16×C31C23semidirect
66. D8×C31C23semidirect
75. Dic124C23semidirect
78. D248C23semidirect
84. C16×C31C23semidirect
85. C31⋊C16C23semidirect
151. #509D31direct (N × H)
162. D31#509direct (N × H)
170. #135C2×D31semidirect
180. C2×D8C2×D31semidirect
185. SD32C2×D31semidirect
189. D16C2×D31semidirect
192. M32C2×D31semidirect
194. C31C2×#509semidirect
216. Q16C22×D31semidirect
229. D8C22×D31semidirect
237. C16C22×D31semidirect

Maximal subgroups

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

subgroupindex
#135×D312
C2×D8×D312
#145214872
#145214862
#145214852
#145214842
#145214832
SD32×D312
D16×D312
M32×D312
C31×#5092
C2×#50931