Finite Group Databasebeta

← D31×#469 #16855303

Subgroups — D31×#469 #16855303

Order 3968.

596 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 135 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×D8×D31C211yesmaximal, characteristic, complemented
2C2×M16×D31C211yesmaximal, characteristic, complemented
3#121×D31C212yesmaximal, complemented
4#14521303C211yesmaximal, characteristic, complemented
5#14521302C211yesmaximal, characteristic, complemented
6#14521301C212yesmaximal, complemented
7#14521300C212yesmaximal, complemented
8#14521299C211yesmaximal, characteristic, complemented
9#106×D31C211yesmaximal, characteristic, complemented
10#102×D31C212yesmaximal, complemented
11C31×#469C211yesmaximal, characteristic, complemented
12C2×D4×D31—22no—
13C2×D4×D31C2212yes—
14C22×C4×D31C2211yescharacteristic
15C2×D8×C31C2211yescharacteristic, complemented
16C2×M16×C31C2211yescharacteristic, complemented
17C31×#121C2212yescomplemented
18#11758516—22no—
19#11758515C2212yescomplemented
20C2×#92479C2212yescomplemented
21#11758507C2212yescomplemented
22C2×#92474C2211yescharacteristic, complemented
23D8×D31—24no—
24M16×D31—24no—
25C2×D248C2211yescharacteristic, complemented
26C2×#92469C2211yescharacteristic, complemented
27C2×#92469C2211yescharacteristic, complemented
28C2×C8×D31C2211yescharacteristic
29C2×C8×D31C2211yescharacteristic
30#11758472C2212yescomplemented
31C4⋊C4×D31C2212yes—
32#11758461—22no—
33C22⋊C4×D31—22no—
34C31×#106C2211yescharacteristic, complemented
35C31×#102C2212yescomplemented
36#11758420C2212yescomplemented
37#11758409C2212yescomplemented
38#11758405C2211yescharacteristic, complemented
39#11758397C2212yescomplemented
40#11758396C2212yescomplemented
41C23×D31—22no—
42C23×D31D411yescharacteristic
43C2×D4×C31—22no—
44C2×D4×C31C2312yes—
45C22×C4×C31C2311yescharacteristic
46C2×C22⋊D31—22no—
47C2×C22⋊D31—22no—
48C2×C22⋊D31—22no—
49C22×Dic31D411yescharacteristic
50D4×D31—24no—
51D4×D31—24no—
52D4×D31—48no—
53C2×D124—22no—
54C2×D124C2312yes—
55C2×C4×D31—22no—
56C2×C4×D31—22no—
57C2×C4×D31—22no—
58C2×C4×D31D411yescharacteristic
59C2×C4×D31D411yescharacteristic
60C2×C4×D31C2311yescharacteristic
61C2×C4×D31C2311yescharacteristic
62D8×C31—22no—
63M16×C31—22no—
64C2×C8×C31C2311yescharacteristic
65C2×C8×C31C2311yescharacteristic
66C4⋊C4×C31C2312yes—
67C22⋊C4×C31—22no—
68C22⋊Dic31—22no—
69#92479—24no—
70(C2×D31)⋊C4—22no—
71(C2×D31)⋊C4—22no—
72C4⋊Dic31C2312yes—
73Dic31⋊C4—22no—
74#92474—22no—
75C2×C31⋊C8C2311yescharacteristic
76C2×C31⋊C8C2311yescharacteristic
77D248—22no—
78#92469—22no—
79#92469—22no—
80C8×D31—22no—
81C8×D31D414yescomplemented
82C23×C31—22no—
83C23×C31C2×D411yescharacteristic
84C22×D31—22no—
85C22×D31—24no—
86C22×D31—22no—
87C22×D31—44no—
88C22×D31—44no—
89C22×D31—44no—
90C22×D31C2×D411yescharacteristic
91C22×D31C2×D411yescharacteristic
92D4×C31—22no—
93D4×C31—22no—
94D4×C31—44no—
95C2×C4×C31—21no—
96C2×C4×C31—21no—
97C2×C4×C31—22no—
98C2×C4×C31C2×D411yescharacteristic
99C2×C4×C31C2411yescharacteristic
100C22⋊D31—44no—
101C22⋊D31—44no—
102C22⋊D31—44no—
103C2×Dic31—22no—
104C2×Dic31—22no—
105C2×Dic31C2×D411yescharacteristic
106C2×Dic31C2×D411yescharacteristic
107D124—22no—
108D124—22no—
109D124—44no—
110C4×D31—22no—
111C4×D31—21no—
112C4×D31—21no—
113C4×D31—22no—
114C4×D31—44no—
115C4×D31C2×D412yes—
116C4×D31#4112yes—
117C8×C31—21no—
118C8×C31C2×D412yescomplemented
119C31⋊C8—21no—
120C31⋊C8C2×D412yescomplemented
121C2×#469—311nomaximal, Sylow
122C22×C31—21no—
123C22×C31—22no—
124C22×C31—42no—
125C22×C31—42no—
126C22×C31—42no—
127C22×C31C22×D411yescharacteristic
128C2×D31—21no—
129C2×D31—22no—
130C2×D31—44no—
131C2×D31—42no—
132C2×D31—42no—
133C2×D31—42no—
134C2×D31—88no—
135C2×D31#12112yes—
136C2×D31AGL(1,ℤ/8)14yes—
137C4×C31—21no—
138C4×C31—42no—
139C4×C31C22×D411yescharacteristic
140C4×C31C2×#4111yescharacteristic
141Dic31—21no—
142Dic31—42no—
143Dic31#12112yes—
144C22×D8—311no—
145C22×M16—311no—
146C2×#121—312no—
147#469—311no—
148#469—311no—
149#469—312no—
150#469—311no—
151#469—312no—
152#469D3111yescharacteristic, direct product factor
153C2×#106—311no—
154C2×#102—312no—
155C2×C31—41no—
156C2×C31—82no—
157C2×C31C2×#12111yescharacteristic
158C2×C31C2×AGL(1,ℤ/8)12yes—
159D31—41no—
160D31—82no—
161D31#46914yesdirect product factor
162C22×D4—312no—
163C22×D4—622no—
164C23×C4—311no—
165C2×D8—312no—
166C2×D8—311no—
167C2×D8—624no—
168C2×D8C2×D3111yescharacteristic, complemented
169C2×M16—311no—
170C2×M16—311no—
171C2×M16—311no—
172C2×M16—624no—
173C2×M16C2×D3111yescharacteristic, complemented
174C22×C8—311no—
175C22×C8—311no—
176#121—312no—
177#121—312no—
178#121—312no—
179#121—622no—
180#121—622no—
181#121C2×D3112yescomplemented
182C2×C4⋊C4—312no—
183C2×C22⋊C4—622no—
184#106—311no—
185#106—312no—
186#106C2×D3111yescharacteristic, complemented
187#102—312no—
188#102—312no—
189#102—312no—
190#102C2×D3112yescomplemented
191C31C2×#46911yescharacteristic, complemented, Sylow
192C24—311no—
193C24—622no—
194C2×D4—22no—
195C2×D4—312no—
196C2×D4—622no—
197C2×D4—624no—
198C2×D4—624no—
199C2×D4—622no—
200C2×D4—622no—
201C2×D4—622no—
202C2×D4—1248no—
203C2×D4C22×D3112yes—
204C22×C4—311no—
205C22×C4—311no—
206C22×C4—311no—
207C22×C4—311no—
208C22×C4—311no—
209C22×C4—622no—
210C22×C4—622no—
211C22×C4—622no—
212C22×C4C22×D3111yescharacteristic
213D8—22no—
214D8—624no—
215D8—622no—
216M16—22no—
217M16—622no—
218M16—622no—
219M16—622no—
220C2×C8—311no—
221C2×C8—314no—
222C2×C8—311no—
223C2×C8—622no—
224C2×C8C22×D3111yescharacteristic
225C2×C8C22×D3111yescharacteristic
226C4⋊C4—312no—
227C4⋊C4—622no—
228C4⋊C4C22×D3112yes—
229C22⋊C4—22no—
230C22⋊C4—622no—
231C22⋊C4—622no—
232C22⋊C4—622no—
233C23—22no—
234C23—311no—
235C23—311no—
236C23—622no—
237C23—624no—
238C23—622no—
239C23—1244no—
240C23—1244no—
241C23—1244no—
242C23D4×D3111yescharacteristic
243D4—22no—
244D4—22no—
245D4—44no—
246D4—622no—
247D4—622no—
248D4—1244no—
249D4—1244no—
250D4—1244no—
251D4—1244no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—311no—
256C2×C4—312no—
257C2×C4—312no—
258C2×C4—311no—
259C2×C4—622no—
260C2×C4—622no—
261C2×C4—622no—
262C2×C4—621no—
263C2×C4—621no—
264C2×C4—622no—
265C2×C4—1244no—
266C2×C4D4×D3111yescharacteristic
267C2×C4C23×D3111yescharacteristic
268C8—21no—
269C8—312no—
270C8—621no—
271C8D4×D3112yescomplemented
272C22—21no—
273C22—22no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—312no—
278C22—314no—
279C22—621no—
280C22—622no—
281C22—1244no—
282C22—1242no—
283C22—1242no—
284C22—1242no—
285C22—2488no—
286C22C2×D4×D3111yescharacteristic
287C4—21no—
288C4—42no—
289C4—312no—
290C4—621no—
291C4—1242no—
292C4C2×D4×D3111yescharacteristic
293C4#41×D3111yescharacteristic
294C2—41no—
295C2—82no—
296C2—314no—
297C2—1241no—
298C2—2482no—
299C2#121×D3111yescharacteristic
300C2AGL(1,ℤ/8)×D3112yes—
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#469C2
socleC22×C31C22×D4
hypercenter#469D31
hypocenterC31C2×#469
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×D8×D31C2semidirect
2. C2×M16×D31C2semidirect
3. #121×D31C2semidirect
4. #14521303C2semidirect
5. #14521302C2semidirect
6. #14521301C2semidirect
7. #14521300C2semidirect
8. #14521299C2semidirect
9. #106×D31C2semidirect
10. #102×D31C2semidirect
11. C31×#469C2semidirect
15. C2×D8×C31C22semidirect
16. C2×M16×C31C22semidirect
17. C31×#121C22semidirect
19. #11758515C22semidirect
20. C2×#92479C22semidirect
21. #11758507C22semidirect
22. C2×#92474C22semidirect
25. C2×D248C22semidirect
26. C2×#92469C22semidirect
27. C2×#92469C22semidirect
30. #11758472C22semidirect
34. C31×#106C22semidirect
35. C31×#102C22semidirect
36. #11758420C22semidirect
37. #11758409C22semidirect
38. #11758405C22semidirect
39. #11758397C22semidirect
40. #11758396C22semidirect
81. C8×D31D4semidirect
118. C8×C31C2×D4semidirect
120. C31⋊C8C2×D4semidirect
152. #469D31direct (N × H)
161. D31#469direct (N × H)
168. C2×D8C2×D31semidirect
173. C2×M16C2×D31semidirect
181. #121C2×D31semidirect
186. #106C2×D31semidirect
190. #102C2×D31semidirect
191. C31C2×#469semidirect
271. C8D4×D31semidirect

Maximal subgroups

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

subgroupindex
C2×D8×D312
C2×M16×D312
#121×D312
#145213032
#145213022
#145213012
#145213002
#145212992
#106×D312
#102×D312
C31×#4692
C2×#46931