Finite Group Databasebeta

← C31×#32881 #21489640

Subgroups — C31×#32881 #21489640

Order 7936.

5942 conjugacy classes of subgroups; only the 438 normal subgroups are shown.

normal subgroups only

This group has 5942 conjugacy classes of subgroups — too many to draw. Only its 438 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
1C24×D4×C31C211yesmaximal, characteristic
2C2×C31×#450C216yesmaximal, complemented
3C31×#2848C211yesmaximal, characteristic, complemented
4C31×#1786C216yesmaximal, complemented
5C31×#1779C211yesmaximal, characteristic, complemented
6C26×C31—21no—
7C23×D4×C31—224no—
8C23×D4×C31—23no—
9C23×D4×C31C2216yes—
10C24×C4×C31C2211yescharacteristic
11C22×SD16×C31C2214yescomplemented
12C2×C31×C22⋊Q8C2213yescomplemented
13C31×#450—224no—
14C2×C31×#102C22112yescomplemented
15C2×C31×C22⋊C8C2216yescomplemented
16C31×#385—23no—
17C31×#380C2213yescomplemented
18C31×#379—21no—
19C25×C31—224no—
20C25×C31—26no—
21C25×C31—412no—
22C25×C31—44no—
23C25×C31D411yescharacteristic
24C22×Q8×C31C2311yescharacteristic, complemented
25C22×D4×C31—2128no—
26C22×D4×C31—224no—
27C22×D4×C31—248no—
28C22×D4×C31—248no—
29C22×D4×C31—26no—
30C22×D4×C31—212no—
31C22×D4×C31D4124yes—
32C22×D4×C31C2314yes—
33C23×C4×C31—212no—
34C23×C4×C31D413yes—
35C23×C4×C31C2313yes—
36C2×SD16×C31—224no—
37C22×C8×C31C2314yescomplemented
38C31×C22⋊Q8—212no—
39C2×C4⋊C4×C31—23no—
40C2×C4⋊C4×C31C2313yescomplemented
41C2×C22⋊C4×C31—23no—
42C2×C22⋊C4×C31—23no—
43C31×#102—224no—
44C31×C22⋊C8—212no—
45C31×#95—23no—
46C24×C31—26no—
47C24×C31—26no—
48C24×C31—264no—
49C24×C31—248no—
50C24×C31—224no—
51C24×C31—24no—
52C24×C31—496no—
53C24×C31—424no—
54C24×C31—496no—
55C24×C31—412no—
56C24×C31—412no—
57C24×C31SD1614yescomplemented
58C24×C31C2×D413yes—
59C2×Q8×C31—26no—
60C2×D4×C31—2192no—
61C2×D4×C31—248no—
62C2×D4×C31—296no—
63C2×D4×C31—296no—
64C2×D4×C31—224no—
65C2×D4×C31—212no—
66C2×D4×C31C2×D4124yes—
67C22×C4×C31—232no—
68C22×C4×C31—26no—
69C22×C4×C31—212no—
70C22×C4×C31—212no—
71C22×C4×C31—23no—
72C22×C4×C31—21no—
73C22×C4×C31C2×D413yes—
74C22×C4×C31C2×D4112yes—
75C22×C4×C31C2411yescharacteristic
76SD16×C31—416no—
77C2×C8×C31—212no—
78C4⋊C4×C31—26no—
79C4⋊C4×C31—46no—
80C22⋊C4×C31—46no—
81C22⋊C4×C31—46no—
82#32881C3111yesmaximal, characteristic, direct product factor, Sylow
83C23×C31—28no—
84C23×C31—224no—
85C23×C31—26no—
86C23×C31—26no—
87C23×C31—224no—
88C23×C31—296no—
89C23×C31—224no—
90C23×C31—4256no—
91C23×C31—496no—
92C23×C31—448no—
93C23×C31—496no—
94C23×C31—44no—
95C23×C31—412no—
96C23×C31—448no—
97C23×C31#12016yes—
98C23×C31C2×SD1616yes—
99C23×C31AGL(1,ℤ/8)16yes—
100C23×C31C22×D411yescharacteristic
101Q8×C31—44no—
102D4×C31—296no—
103D4×C31—224no—
104D4×C31#120116yes—
105C2×C4×C31—224no—
106C2×C4×C31—26no—
107C2×C4×C31—224no—
108C2×C4×C31—43no—
109C2×C4×C31—43no—
110C2×C4×C31—46no—
111C2×C4×C31#12016yes—
112C2×C4×C31C22×D416yes—
113C8×C31—44no—
114C24×D4C2×C3111yescharacteristic
115C2×#450C2×C3116yescomplemented
116#2848C2×C3111yescharacteristic, complemented
117#1786C2×C3116yescomplemented
118#1779C2×C3111yescharacteristic, complemented
119C22×C31—232no—
120C22×C31—26no—
121C22×C31—212no—
122C22×C31—26no—
123C22×C31—212no—
124C22×C31—216no—
125C22×C31—4192no—
126C22×C31—424no—
127C22×C31—424no—
128C22×C31#450112yescomplemented
129C22×C31C2×#12013yes—
130C22×C31C22×SD1611yescharacteristic
131C22×C31C2×AGL(1,ℤ/8)13yes—
132C4×C31—26no—
133C4×C31—81no—
134C4×C31—83no—
135C4×C31C2×#12014yes—
136C26—21no—
137C23×D4—224no—
138C23×D4—23no—
139C23×D4C22×C3116yes—
140C24×C4C22×C3111yescharacteristic
141C22×SD16C22×C3114yescomplemented
142C2×C22⋊Q8C22×C3113yescomplemented
143#450—224no—
144C2×#102C22×C31112yescomplemented
145C2×C22⋊C8C22×C3116yescomplemented
146#385—23no—
147#380C22×C3113yescomplemented
148#379—21no—
149C2×C31—212no—
150C2×C31—416no—
151C2×C31#284611yescharacteristic
152C2×C31C2×#45013yes—
153C2×C31#300313yes—
154C25—224no—
155C25—26no—
156C25—412no—
157C25—44no—
158C25D4×C3111yescharacteristic
159C22×Q8C23×C3111yescharacteristic, complemented
160C22×D4—2128no—
161C22×D4—224no—
162C22×D4—248no—
163C22×D4—248no—
164C22×D4—26no—
165C22×D4—212no—
166C22×D4D4×C31124yes—
167C22×D4C23×C3114yes—
168C23×C4—212no—
169C23×C4D4×C3113yes—
170C23×C4C23×C3113yes—
171C2×SD16—224no—
172C22×C8C23×C3114yescomplemented
173C22⋊Q8—212no—
174C2×C4⋊C4—23no—
175C2×C4⋊C4C23×C3113yescomplemented
176C2×C22⋊C4—23no—
177C2×C22⋊C4—23no—
178#102—224no—
179C22⋊C8—212no—
180#95—23no—
181C31#3288111yescharacteristic, direct product factor, Sylow
182C24—26no—
183C24—26no—
184C24—264no—
185C24—248no—
186C24—224no—
187C24—24no—
188C24—496no—
189C24—496no—
190C24—424no—
191C24—412no—
192C24—412no—
193C24SD16×C3114yescomplemented
194C24C2×D4×C3113yes—
195C2×Q8—26no—
196C2×D4—2192no—
197C2×D4—248no—
198C2×D4—296no—
199C2×D4—296no—
200C2×D4—224no—
201C2×D4—212no—
202C2×D4C2×D4×C31124yes—
203C22×C4—232no—
204C22×C4—26no—
205C22×C4—212no—
206C22×C4—212no—
207C22×C4—23no—
208C22×C4—21no—
209C22×C4C2×D4×C3113yes—
210C22×C4C2×D4×C31112yes—
211C22×C4C24×C3111yescharacteristic
212SD16—416no—
213C2×C8—212no—
214C4⋊C4—26no—
215C4⋊C4—46no—
216C22⋊C4—46no—
217C22⋊C4—46no—
218C23—28no—
219C23—224no—
220C23—224no—
221C23—26no—
222C23—26no—
223C23—296no—
224C23—224no—
225C23—4256no—
226C23—496no—
227C23—448no—
228C23—496no—
229C23—448no—
230C23—44no—
231C23—412no—
232C23C31×#12016yes—
233C23C2×SD16×C3116yes—
234C23C31×AGL(1,ℤ/8)16yes—
235C23C22×D4×C3111yescharacteristic
236Q8—44no—
237D4—296no—
238D4—224no—
239D4C31×#120116yes—
240C2×C4—224no—
241C2×C4—26no—
242C2×C4—224no—
243C2×C4—43no—
244C2×C4—43no—
245C2×C4—46no—
246C2×C4C31×#12016yes—
247C2×C4C22×D4×C3116yes—
248C8—44no—
249C22—232no—
250C22—26no—
251C22—212no—
252C22—26no—
253C22—212no—
254C22—216no—
255C22—4192no—
256C22—424no—
257C22—424no—
258C22C31×#450112yescomplemented
259C22C2×C31×#12013yes—
260C22C22×SD16×C3111yescharacteristic
261C22C2×C31×AGL(1,ℤ/8)13yes—
262C4—26no—
263C4—81no—
264C4—83no—
265C4C2×C31×#12014yes—
266C2—212no—
267C2—416no—
268C2C31×#284611yescharacteristic
269C2C2×C31×#45013yes—
270C2C31×#300313yes—
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC24×D4×C31C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C24×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×#450C2semidirect
3. C31×#2848C2semidirect
4. C31×#1786C2semidirect
5. C31×#1779C2semidirect
11. C22×SD16×C31C22semidirect
12. C2×C31×C22⋊Q8C22semidirect
14. C2×C31×#102C22semidirect
15. C2×C31×C22⋊C8C22semidirect
17. C31×#380C22semidirect
24. C22×Q8×C31C23semidirect
37. C22×C8×C31C23semidirect
40. C2×C4⋊C4×C31C23semidirect
57. C24×C31SD16semidirect
82. #32881C31direct (N × H)
115. C2×#450C2×C31semidirect
116. #2848C2×C31semidirect
117. #1786C2×C31semidirect
118. #1779C2×C31semidirect
128. C22×C31#450semidirect
141. C22×SD16C22×C31semidirect
142. C2×C22⋊Q8C22×C31semidirect
144. C2×#102C22×C31semidirect
145. C2×C22⋊C8C22×C31semidirect
147. #380C22×C31semidirect
159. C22×Q8C23×C31semidirect
172. C22×C8C23×C31semidirect
175. C2×C4⋊C4C23×C31semidirect
181. C31#32881direct (N × H)
193. C24SD16×C31semidirect
258. C22C31×#450semidirect

Maximal subgroups

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

subgroupindex
C24×D4×C312
C2×C31×#4502
C31×#28482
C31×#17862
C31×#17792
#3288131