Finite Group Databasebeta

← C31×#3021 #16842954

Subgroups — C31×#3021 #16842954

Order 3968.

736 conjugacy classes of subgroups; only the 212 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 212 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
1C31×#585C211yesmaximal, characteristic, complemented
2C2×C31×AGL(1,ℤ/8)C212yesmaximal, complemented
3C31×#535C211yesmaximal, characteristic, complemented
4C31×#454C214yesmaximal, complemented
5C31×#453C214yesmaximal, complemented
6C31×#421C212yesmaximal, complemented
7C31×#413C211yesmaximal, characteristic, complemented
8C31×2−1+4—22no—
9C31×2−1+4C2212yescomplemented
10C31×H(5,𝔽2)—22no—
11C31×H(5,𝔽2)—22no—
12C31×H(5,𝔽2)C2212yescomplemented
13C2×#41×C31—24no—
14C2×#41×C31—22no—
15C2×#41×C31C2212yes—
16C2×#41×C31C2211yescharacteristic
17C22×D4×C31C2211yescharacteristic, complemented
18C31×AGL(1,ℤ/8)—24no—
19C31×AGL(1,ℤ/8)C2218yescomplemented
20C2×SD16×C31—22no—
21C2×D8×C31—22no—
22C2×M16×C31C2212yescomplemented
23C31×#127C2212yescomplemented
24C31×#124C2212yescomplemented
25C31×#121—22no—
26C31×#120—22no—
27C31×#117C2211yescharacteristic, complemented
28C31×#104C2218yescomplemented
29C31×#101C2212yescomplemented
30C31×#100C2212yescomplemented
31C24×C31—21no—
32#41×C31—24no—
33#41×C31—28no—
34#41×C31—24no—
35#41×C31—24no—
36#41×C31—24no—
37#41×C31—22no—
38#41×C31—22no—
39#41×C31—24no—
40#41×C31—22no—
41#41×C31—22no—
42#41×C31D414yescomplemented
43#41×C31C2314yes—
44C2×Q8×C31—22no—
45C2×Q8×C31—24no—
46C2×Q8×C31D412yescomplemented
47C2×Q8×C31C2311yescharacteristic
48C2×D4×C31—28no—
49C2×D4×C31—22no—
50C2×D4×C31—24no—
51C2×D4×C31—22no—
52C2×D4×C31—24no—
53C2×D4×C31—24no—
54C2×D4×C31—22no—
55C2×D4×C31—22no—
56C2×D4×C31—21no—
57C2×D4×C31D412yes—
58C2×D4×C31D412yescomplemented
59C2×D4×C31C2312yes—
60C2×D4×C31C2311yescharacteristic
61C22×C4×C31—24no—
62C22×C4×C31—22no—
63C22×C4×C31D412yes—
64C22×C4×C31C2311yescharacteristic
65SD16×C31—28no—
66D8×C31—28no—
67M16×C31—22no—
68M16×C31C2314yes—
69C2×C8×C31—22no—
70C4⋊C4×C31—21no—
71C22⋊C4×C31—21no—
72C22⋊C4×C31—24no—
73C42×C31C2312yes—
74C23×C31—24no—
75C23×C31—22no—
76C23×C31—22no—
77C23×C31—42no—
78C23×C31—82no—
79C23×C31C2×D412yes—
80C23×C31C2×D411yescharacteristic
81Q8×C31—24no—
82Q8×C31—22no—
83Q8×C31—22no—
84Q8×C31C2×D414yes—
85D4×C31—28no—
86D4×C31—24no—
87D4×C31—24no—
88D4×C31—24no—
89D4×C31—24no—
90D4×C31—22no—
91D4×C31—22no—
92D4×C31—24no—
93D4×C31—41no—
94D4×C31—41no—
95D4×C31—84no—
96D4×C31C2×D414yes—
97C2×C4×C31—24no—
98C2×C4×C31—28no—
99C2×C4×C31—22no—
100C2×C4×C31—24no—
101C2×C4×C31—22no—
102C2×C4×C31—22no—
103C2×C4×C31—24no—
104C2×C4×C31—22no—
105C2×C4×C31C2×D412yes—
106C2×C4×C31C2×D411yescharacteristic
107C2×C4×C31C2×D412yes—
108C2×C4×C31C2×D412yes—
109C2×C4×C31C2411yescharacteristic
110C8×C31—24no—
111#3021C3111yesmaximal, characteristic, direct product factor, Sylow
112C22×C31—22no—
113C22×C31—24no—
114C22×C31—41no—
115C22×C31—42no—
116C22×C31—44no—
117C22×C31—42no—
118C22×C31—82no—
119C22×C31—84no—
120C22×C31—84no—
121C22×C31#12012yes—
122C22×C31C22×D411yescharacteristic
123C4×C31—24no—
124C4×C31—22no—
125C4×C31—42no—
126C4×C31#12011yescharacteristic
127C4×C31#12011yescharacteristic
128C4×C31C22×D412yes—
129#585C2×C3111yescharacteristic, complemented
130C2×AGL(1,ℤ/8)C2×C3112yescomplemented
131#535C2×C3111yescharacteristic, complemented
132#454C2×C3114yescomplemented
133#453C2×C3114yescomplemented
134#421C2×C3112yescomplemented
135#413C2×C3111yescharacteristic, complemented
136C2×C31—21no—
137C2×C31—22no—
138C2×C31—42no—
139C2×C31—44no—
140C2×C31—82no—
141C2×C31C2×#12011yescharacteristic
1422−1+4—22no—
1432−1+4C22×C3112yescomplemented
144H(5,𝔽2)—22no—
145H(5,𝔽2)—22no—
146H(5,𝔽2)C22×C3112yescomplemented
147C2×#41—24no—
148C2×#41—22no—
149C2×#41C22×C3112yes—
150C2×#41C22×C3111yescharacteristic
151C22×D4C22×C3111yescharacteristic, complemented
152AGL(1,ℤ/8)—24no—
153AGL(1,ℤ/8)C22×C3118yescomplemented
154C2×SD16—22no—
155C2×D8—22no—
156C2×M16C22×C3112yescomplemented
157#127C22×C3112yescomplemented
158#124C22×C3112yescomplemented
159#121—22no—
160#120—22no—
161#117C22×C3111yescharacteristic, complemented
162#104C22×C3118yescomplemented
163#101C22×C3112yescomplemented
164#100C22×C3112yescomplemented
165C31#302111yescharacteristic, direct product factor, Sylow
166C24—21no—
167#41—24no—
168#41—28no—
169#41—24no—
170#41—24no—
171#41—24no—
172#41—22no—
173#41—22no—
174#41—24no—
175#41—22no—
176#41—22no—
177#41D4×C3114yescomplemented
178#41C23×C3114yes—
179C2×Q8—22no—
180C2×Q8—24no—
181C2×Q8D4×C3112yescomplemented
182C2×Q8C23×C3111yescharacteristic
183C2×D4—28no—
184C2×D4—22no—
185C2×D4—24no—
186C2×D4—22no—
187C2×D4—24no—
188C2×D4—24no—
189C2×D4—22no—
190C2×D4—22no—
191C2×D4—21no—
192C2×D4D4×C3112yes—
193C2×D4D4×C3112yescomplemented
194C2×D4C23×C3112yes—
195C2×D4C23×C3111yescharacteristic
196C22×C4—24no—
197C22×C4—22no—
198C22×C4D4×C3112yes—
199C22×C4C23×C3111yescharacteristic
200SD16—28no—
201D8—28no—
202M16—22no—
203M16C23×C3114yes—
204C2×C8—22no—
205C4⋊C4—21no—
206C22⋊C4—21no—
207C22⋊C4—24no—
208C42C23×C3112yes—
209C23—24no—
210C23—22no—
211C23—22no—
212C23—42no—
213C23—82no—
214C23C2×D4×C3112yes—
215C23C2×D4×C3111yescharacteristic
216Q8—24no—
217Q8—22no—
218Q8—22no—
219Q8C2×D4×C3114yes—
220D4—28no—
221D4—24no—
222D4—24no—
223D4—24no—
224D4—24no—
225D4—22no—
226D4—22no—
227D4—24no—
228D4—41no—
229D4—41no—
230D4—84no—
231D4C2×D4×C3114yes—
232C2×C4—24no—
233C2×C4—28no—
234C2×C4—22no—
235C2×C4—24no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—24no—
239C2×C4—22no—
240C2×C4C2×D4×C3112yes—
241C2×C4C2×D4×C3111yescharacteristic
242C2×C4C2×D4×C3112yes—
243C2×C4C2×D4×C3112yes—
244C2×C4C24×C3111yescharacteristic
245C8—24no—
246C22—22no—
247C22—24no—
248C22—41no—
249C22—42no—
250C22—44no—
251C22—42no—
252C22—82no—
253C22—84no—
254C22—84no—
255C22C31×#12012yes—
256C22C22×D4×C3111yescharacteristic
257C4—24no—
258C4—22no—
259C4—42no—
260C4C31×#12011yescharacteristic
261C4C31×#12011yescharacteristic
262C4C22×D4×C3112yes—
263C2—21no—
264C2—22no—
265C2—42no—
266C2—44no—
267C2—82no—
268C2C2×C31×#12011yescharacteristic
subgroupG/H
center, socleC2×C31C2×#120
commutator, Frattini, cosocleC2×C4C24×C31
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C31 ◁ C22×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#585C2semidirect
2. C2×C31×AGL(1,ℤ/8)C2semidirect
3. C31×#535C2semidirect
4. C31×#454C2semidirect
5. C31×#453C2semidirect
6. C31×#421C2semidirect
7. C31×#413C2semidirect
9. C31×2−1+4C22semidirect
12. C31×H(5,𝔽2)C22semidirect
17. C22×D4×C31C22semidirect
19. C31×AGL(1,ℤ/8)C22semidirect
22. C2×M16×C31C22semidirect
23. C31×#127C22semidirect
24. C31×#124C22semidirect
27. C31×#117C22semidirect
28. C31×#104C22semidirect
29. C31×#101C22semidirect
30. C31×#100C22semidirect
42. #41×C31D4semidirect
46. C2×Q8×C31D4semidirect
58. C2×D4×C31D4semidirect
111. #3021C31direct (N × H)
129. #585C2×C31semidirect
130. C2×AGL(1,ℤ/8)C2×C31semidirect
131. #535C2×C31semidirect
132. #454C2×C31semidirect
133. #453C2×C31semidirect
134. #421C2×C31semidirect
135. #413C2×C31semidirect
143. 2−1+4C22×C31semidirect
146. H(5,𝔽2)C22×C31semidirect
151. C22×D4C22×C31semidirect
153. AGL(1,ℤ/8)C22×C31semidirect
156. C2×M16C22×C31semidirect
157. #127C22×C31semidirect
158. #124C22×C31semidirect
161. #117C22×C31semidirect
162. #104C22×C31semidirect
163. #101C22×C31semidirect
164. #100C22×C31semidirect
165. C31#3021direct (N × H)
177. #41D4×C31semidirect
181. C2×Q8D4×C31semidirect
193. C2×D4D4×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#5852
C2×C31×AGL(1,ℤ/8)2
C31×#5352
C31×#4542
C31×#4532
C31×#4212
C31×#4132
#302131