Finite Group Databasebeta

← D31×#489 #16856037

Subgroups — D31×#489 #16856037

Order 3968.

436 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 125 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
1C4⋊Q8×D31C211yesmaximal, characteristic, complemented
2#124×D31C211yesmaximal, characteristic
3#14521411C211yesmaximal, characteristic, complemented
4#14521410C212yesmaximal, complemented
5#14521409C211yesmaximal, characteristic, complemented
6#14521408C211yesmaximal, characteristic, complemented
7#14521407C212yesmaximal, complemented
8#103×D31C212yesmaximal, complemented
9#102×D31C212yesmaximal
10#97×D31C211yesmaximal, characteristic, complemented
11C31×#489C211yesmaximal, characteristic, complemented
12C2×Q8×D31—21no—
13C2×Q8×D31C2211yescharacteristic
14C2×D4×D31C2211yescharacteristic
15C31×C4⋊Q8C2211yescharacteristic, complemented
16C31×#124C2211yescharacteristic
17#11758524C2211yescharacteristic
18#11758521—21no—
19#11758513C2211yescharacteristic
20C2×C8×D31C2212yes—
21C4⋊C4×D31—21no—
22C4⋊C4×D31C2212yes—
23#11758465—21no—
24#11758465C2212yescomplemented
25#11758462—21no—
26#11758462—21no—
27C22⋊C4×D31—21no—
28C22⋊C4×D31—21no—
29#11758452C2211yescharacteristic
30C42×D31C2211yescharacteristic
31#11758446C2211yescharacteristic, complemented
32C31×#103C2212yescomplemented
33C31×#102C2212yes—
34C31×#97C2211yescharacteristic, complemented
35#11758423C2212yescomplemented
36#11758420C2212yes—
37#11758409C2212yes—
38#11758404C2212yescomplemented
39#11758403C2212yescomplemented
40#11758398C2212yescomplemented
41#11758397C2212yes—
42#11758391C2211yescharacteristic, complemented
43C23×D31—21no—
44C2×Q8×C31—21no—
45C2×Q8×C31C2311yescharacteristic
46C2×D4×C31C2311yescharacteristic
47C2×C22⋊D31—21no—
48Q8×D31—24no—
49Q8×D31—44no—
50D4×D31—24no—
51C2×D124C2311yescharacteristic
52C2×C4×D31—21no—
53C2×C4×D31—22no—
54C2×C4×D31D411yescharacteristic
55C2×C4×D31D411yescharacteristic
56C2×C4×D31C2311yescharacteristic
57C2×Dic62—21no—
58C2×Dic62—21no—
59C2×Dic62—22no—
60C2×Dic62C2311yescharacteristic
61C2×C8×C31C2312yes—
62C4⋊C4×C31—21no—
63C4⋊C4×C31C2312yes—
64C22⋊C4×C31—21no—
65C22⋊C4×C31—21no—
66C42×C31C2311yescharacteristic
67C22⋊Dic31—21no—
68C22⋊Dic31—21no—
69(C2×D31)⋊C4—21no—
70(C2×D31)⋊C4—21no—
71(C2×D31)⋊C4—21no—
72(C2×D31)⋊C4—21no—
73C4⋊Dic31—21no—
74C4⋊Dic31C2312yes—
75Dic31⋊C4—22no—
76Dic31⋊C4—22no—
77C4×Dic31D411yescharacteristic
78C4×Dic31D411yescharacteristic
79C4×Dic31C2311yescharacteristic
80C2×C31⋊C8C2312yes—
81C8×D31—24no—
82C23×C31—21no—
83C22×D31—21no—
84C22×D31—42no—
85C22×D31—42no—
86C22×D31—42no—
87C22×D31C2×D411yescharacteristic
88Q8×C31—22no—
89Q8×C31—42no—
90D4×C31—22no—
91C2×C4×C31—22no—
92C2×C4×C31—21no—
93C2×C4×C31C2×D411yescharacteristic
94C2×C4×C31C2×D411yescharacteristic
95C2×C4×C31C2411yescharacteristic
96C22⋊D31—42no—
97C2×Dic31—22no—
98C2×Dic31—21no—
99C2×Dic31C2×D411yescharacteristic
100C2×Dic31C2×D411yescharacteristic
101C2×Dic31C2×D411yescharacteristic
102D124—22no—
103C4×D31—22no—
104C4×D31—22no—
105C4×D31—42no—
106C4×D31—44no—
107C4×D31#4114yes—
108Dic62—22no—
109Dic62—44no—
110Dic62—42no—
111Dic62—42no—
112C8×C31—22no—
113C31⋊C8—22no—
114C2×#489—311nomaximal, Sylow
115C22×C31—41no—
116C22×C31—41no—
117C22×C31—41no—
118C22×C31C22×D411yescharacteristic
119C2×D31—41no—
120C2×D31—41no—
121C2×D31—41no—
122C2×D31—84no—
123C2×D31#12412yes—
124C2×D31AGL(1,ℤ/8)12yes—
125C2×D31#13712yes—
126C4×C31—21no—
127C4×C31—21no—
128C4×C31—41no—
129C4×C31—42no—
130C4×C31C2×#4112yes—
131Dic31—21no—
132Dic31—21no—
133Dic31—41no—
134Dic31—42no—
135Dic31#12412yes—
136C2×C4⋊Q8—311no—
137C2×#124—311no—
138#489—311no—
139#489—311no—
140#489—312no—
141#489—312no—
142#489—311no—
143#489D3111yescharacteristic, direct product factor
144C2×#103—312no—
145C2×#102—312no—
146C2×#97—311no—
147C2×C31—81no—
148C2×C31C2×#12411yescharacteristic
149C2×C31C2×AGL(1,ℤ/8)11yescharacteristic
150C2×C31C2×#13711yescharacteristic
151D31—81no—
152D31#48914yesdirect product factor
153C22×Q8—311no—
154C22×Q8—621no—
155C22×D4—311no—
156C22×C8—312no—
157C4⋊Q8—312no—
158C4⋊Q8—311no—
159C4⋊Q8—621no—
160C4⋊Q8—621no—
161C4⋊Q8C2×D3111yescharacteristic, complemented
162#124—311no—
163#124—311no—
164#124—311no—
165#124—621no—
166#124—621no—
167#124C2×D3111yescharacteristic
168C2×C4⋊C4—312no—
169C2×C4⋊C4—621no—
170C2×C22⋊C4—621no—
171C2×C22⋊C4—621no—
172C2×C42—311no—
173#103—312no—
174#103—312no—
175#103—312no—
176#103C2×D3112yescomplemented
177#102—312no—
178#102—312no—
179#102—312no—
180#102C2×D3112yes—
181#97—312no—
182#97—311no—
183#97C2×D3111yescharacteristic, complemented
184C31C2×#48911yescharacteristic, complemented, Sylow
185C24—621no—
186C2×Q8—21no—
187C2×Q8—311no—
188C2×Q8—621no—
189C2×Q8—622no—
190C2×Q8—624no—
191C2×Q8—621no—
192C2×Q8—1244no—
193C2×Q8C22×D3111yescharacteristic
194C2×D4—311no—
195C2×D4—624no—
196C2×D4—621no—
197C2×D4C22×D3111yescharacteristic
198C22×C4—311no—
199C22×C4—311no—
200C22×C4—311no—
201C22×C4—621no—
202C22×C4—622no—
203C2×C8—312no—
204C2×C8—624no—
205C2×C8C22×D3112yes—
206C4⋊C4—21no—
207C4⋊C4—312no—
208C4⋊C4—622no—
209C4⋊C4—622no—
210C4⋊C4—621no—
211C4⋊C4C22×D3112yes—
212C22⋊C4—21no—
213C22⋊C4—21no—
214C22⋊C4—621no—
215C22⋊C4—621no—
216C22⋊C4—621no—
217C22⋊C4—621no—
218C22⋊C4—621no—
219C22⋊C4—621no—
220C42—311no—
221C42—311no—
222C42—311no—
223C42C22×D3111yescharacteristic
224C23—21no—
225C23—311no—
226C23—621no—
227C23—1242no—
228C23—1242no—
229C23—1242no—
230Q8—22no—
231Q8—42no—
232Q8—622no—
233Q8—1242no—
234Q8—1242no—
235Q8—1244no—
236D4—22no—
237D4—622no—
238D4—1242no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—311no—
242C2×C4—314no—
243C2×C4—311no—
244C2×C4—311no—
245C2×C4—622no—
246C2×C4—622no—
247C2×C4—621no—
248C2×C4—622no—
249C2×C4—1242no—
250C2×C4—1244no—
251C2×C4D4×D3111yescharacteristic
252C2×C4D4×D3111yescharacteristic
253C2×C4C23×D3111yescharacteristic
254C8—22no—
255C8—622no—
256C22—41no—
257C22—41no—
258C22—41no—
259C22—312no—
260C22—312no—
261C22—312no—
262C22—1241no—
263C22—1241no—
264C22—1241no—
265C22—2484no—
266C22C2×D4×D3111yescharacteristic
267C4—21no—
268C4—21no—
269C4—41no—
270C4—42no—
271C4—312no—
272C4—621no—
273C4—621no—
274C4—1241no—
275C4—1242no—
276C4#41×D3112yes—
277C2—81no—
278C2—314no—
279C2—2481no—
280C2#124×D3111yescharacteristic
281C2AGL(1,ℤ/8)×D3111yescharacteristic
282C2#137×D3111yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#489C2
socleC22×C31C22×D4
prime radicalC2×D4×D31C22
hypercenter#489D31
hypocenterC31C2×#489
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. C4⋊Q8×D31C2semidirect
3. #14521411C2semidirect
4. #14521410C2semidirect
5. #14521409C2semidirect
6. #14521408C2semidirect
7. #14521407C2semidirect
8. #103×D31C2semidirect
10. #97×D31C2semidirect
11. C31×#489C2semidirect
15. C31×C4⋊Q8C22semidirect
24. #11758465C22semidirect
31. #11758446C22semidirect
32. C31×#103C22semidirect
34. C31×#97C22semidirect
35. #11758423C22semidirect
38. #11758404C22semidirect
39. #11758403C22semidirect
40. #11758398C22semidirect
42. #11758391C22semidirect
143. #489D31direct (N × H)
152. D31#489direct (N × H)
161. C4⋊Q8C2×D31semidirect
176. #103C2×D31semidirect
183. #97C2×D31semidirect
184. C31C2×#489semidirect

Maximal subgroups

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

subgroupindex
C4⋊Q8×D312
#124×D312
#145214112
#145214102
#145214092
#145214082
#145214072
#103×D312
#102×D312
#97×D312
C31×#4892
C2×#48931