Finite Group Databasebeta

← C31×#12161 #21468920

Subgroups — C31×#12161 #21468920

Order 7936.

398 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 82 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×#3245C211yesmaximal, characteristic
2C31×#2185C212yesmaximal, complemented
3C31×#2056C211yesmaximal, characteristic, complemented
4C31×#1347C212yesmaximal, complemented
5C31×#1322C211yesmaximal, characteristic, complemented
6C22×Q16×C31C2211yescharacteristic
7C31×#532—21no—
8C2×C31×SD32—22no—
9C31×#474—22no—
10C31×#466—21no—
11C31×#466C2212yes—
12C31×#459—22no—
13C2×C31×C4⋊C8C2211yescharacteristic
14C2×C31×#103—21no—
15C31×#391—21no—
16C31×#358—22no—
17C31×C22⋊C16C2212yes—
18C31×#340C2211yescharacteristic
19C2×#41×C31—41no—
20C22×Q8×C31—21no—
21C2×Q16×C31—22no—
22C2×Q16×C31—42no—
23C2×D8×C31—21no—
24C2×D8×C31D412yes—
25C22×C8×C31D411yescharacteristic
26C22×C8×C31C2311yescharacteristic
27C31×C4⋊Q8—21no—
28C31×#127—21no—
29C31×#124—41no—
30C31×#121—22no—
31C31×#121—41no—
32C4×D4×C31—22no—
33C4×D4×C31—41no—
34C2×C4⋊C4×C31—21no—
35C2×C42×C31D411yescharacteristic
36C31×SD32—44no—
37C2×C16×C31—22no—
38C31×#107—21no—
39C31×#107D412yes—
40C31×C4⋊C8—22no—
41C31×#103—42no—
42C31×#102—22no—
43C31×#102—22no—
44C31×#95—41no—
45C31×#95—41no—
46#41×C31—82no—
47#41×C31—82no—
48C2×Q8×C31—22no—
49C2×Q8×C31—41no—
50C2×Q8×C31—41no—
51C2×Q8×C31—41no—
52C2×Q8×C31—41no—
53C2×Q8×C31—41no—
54C2×Q8×C31—41no—
55C2×D4×C31—22no—
56C2×D4×C31—41no—
57C2×D4×C31—41no—
58C2×D4×C31—42no—
59C22×C4×C31—42no—
60C22×C4×C31—41no—
61C22×C4×C31—41no—
62C22×C4×C31D811yescharacteristic
63C22×C4×C31D811yescharacteristic
64C22×C4×C31C2×D411yescharacteristic
65Q16×C31—24no—
66Q16×C31—42no—
67D8×C31—42no—
68D8×C31—81no—
69C2×C8×C31—21no—
70C2×C8×C31—41no—
71C2×C8×C31—41no—
72C2×C8×C31C2×D412yes—
73C4⋊C4×C31—22no—
74C4⋊C4×C31—41no—
75C4⋊C4×C31—41no—
76C4⋊C4×C31—41no—
77C22⋊C4×C31—42no—
78C22⋊C4×C31—42no—
79C42×C31—21no—
80C42×C31—21no—
81C16×C31—42no—
82#12161C3111yesmaximal, characteristic, direct product factor, Sylow
83C23×C31—42no—
84C23×C31C2×D811yescharacteristic
85Q8×C31—42no—
86Q8×C31—42no—
87Q8×C31—81no—
88Q8×C31—82no—
89Q8×C31—81no—
90D4×C31—42no—
91D4×C31—42no—
92D4×C31—82no—
93D4×C31—82no—
94D4×C31—82no—
95D4×C31—82no—
96C2×C4×C31—21no—
97C2×C4×C31—21no—
98C2×C4×C31—21no—
99C2×C4×C31—21no—
100C2×C4×C31—21no—
101C2×C4×C31—41no—
102C2×C4×C31—42no—
103C2×C4×C31—42no—
104C2×C4×C31—84no—
105C2×C4×C31—81no—
106C2×C4×C31—81no—
107C2×C4×C31—81no—
108C2×C4×C31—81no—
109C2×C4×C31—81no—
110C2×C4×C31SD3212yescomplemented
111C2×C4×C31#12011yescharacteristic
112C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
113C8×C31—22no—
114C8×C31—41no—
115#3245C2×C3111yescharacteristic
116#2185C2×C3112yescomplemented
117#2056C2×C3111yescharacteristic, complemented
118#1347C2×C3112yescomplemented
119#1322C2×C3111yescharacteristic, complemented
120C22×C31—21no—
121C22×C31—21no—
122C22×C31—82no—
123C22×C31—82no—
124C22×C31—82no—
125C22×C31#44711yescharacteristic
126C22×C31C2×SD3211yescharacteristic
127C22×C31#50911yescharacteristic
128C4×C31—21no—
129C4×C31—22no—
130C4×C31—41no—
131C4×C31—82no—
132C4×C31—82no—
133C4×C31—161no—
134C4×C31#45411yescharacteristic
135C4×C31B(4,𝔽2)11yescharacteristic
136C22×Q16C22×C3111yescharacteristic
137#532—21no—
138C2×SD32—22no—
139#474—22no—
140#466—21no—
141#466C22×C3112yes—
142#459—22no—
143C2×C4⋊C8C22×C3111yescharacteristic
144C2×#103—21no—
145#391—21no—
146#358—22no—
147C22⋊C16C22×C3112yes—
148#340C22×C3111yescharacteristic
149C2×C31—21no—
150C2×C31—21no—
151C2×C31—162no—
152C2×C31#159811yescharacteristic
153C2×C31#218711yescharacteristic
154C2×C31#219111yescharacteristic
155C2×#41—41no—
156C22×Q8—21no—
157C2×Q16—22no—
158C2×Q16—42no—
159C2×D8—21no—
160C2×D8D4×C3112yes—
161C22×C8D4×C3111yescharacteristic
162C22×C8C23×C3111yescharacteristic
163C4⋊Q8—21no—
164#127—21no—
165#124—41no—
166#121—22no—
167#121—41no—
168C4×D4—22no—
169C4×D4—41no—
170C2×C4⋊C4—21no—
171C2×C42D4×C3111yescharacteristic
172SD32—44no—
173C2×C16—22no—
174#107—21no—
175#107D4×C3112yes—
176C4⋊C8—22no—
177#103—42no—
178#102—22no—
179#102—22no—
180#95—41no—
181#95—41no—
182C31#1216111yescharacteristic, direct product factor, Sylow
183#41—82no—
184#41—82no—
185C2×Q8—22no—
186C2×Q8—41no—
187C2×Q8—41no—
188C2×Q8—41no—
189C2×Q8—41no—
190C2×Q8—41no—
191C2×Q8—41no—
192C2×D4—22no—
193C2×D4—41no—
194C2×D4—41no—
195C2×D4—42no—
196C22×C4—42no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4D8×C3111yescharacteristic
200C22×C4D8×C3111yescharacteristic
201C22×C4C2×D4×C3111yescharacteristic
202Q16—24no—
203Q16—42no—
204D8—42no—
205D8—81no—
206C2×C8—21no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8C2×D4×C3112yes—
210C4⋊C4—22no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C22⋊C4—42no—
215C22⋊C4—42no—
216C42—21no—
217C42—21no—
218C16—42no—
219C23—42no—
220C23C2×D8×C3111yescharacteristic
221Q8—42no—
222Q8—42no—
223Q8—81no—
224Q8—82no—
225Q8—81no—
226D4—42no—
227D4—42no—
228D4—82no—
229D4—82no—
230D4—82no—
231D4—82no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—41no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—84no—
241C2×C4—81no—
242C2×C4—81no—
243C2×C4—81no—
244C2×C4—81no—
245C2×C4—81no—
246C2×C4C31×SD3212yescomplemented
247C2×C4C31×#12011yescharacteristic
248C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
249C8—22no—
250C8—41no—
251C22—21no—
252C22—21no—
253C22—82no—
254C22—82no—
255C22—82no—
256C22C31×#44711yescharacteristic
257C22C2×C31×SD3211yescharacteristic
258C22C31×#50911yescharacteristic
259C4—21no—
260C4—22no—
261C4—41no—
262C4—82no—
263C4—82no—
264C4—161no—
265C4C31×#45411yescharacteristic
266C4C31×B(4,𝔽2)11yescharacteristic
267C2—21no—
268C2—21no—
269C2—162no—
270C2C31×#159811yescharacteristic
271C2C31×#218711yescharacteristic
272C2C31×#219111yescharacteristic
subgroupG/H
center, socleC22×C31#447
commutator, Frattini, cosocleC22×C8C23×C31
prime radicalC31×#3245C2
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C22×C4×C31 ◁ C2×C31×C4⋊C8 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#2185C2semidirect
3. C31×#2056C2semidirect
4. C31×#1347C2semidirect
5. C31×#1322C2semidirect
82. #12161C31direct (N × H)
110. C2×C4×C31SD32semidirect
116. #2185C2×C31semidirect
117. #2056C2×C31semidirect
118. #1347C2×C31semidirect
119. #1322C2×C31semidirect
182. C31#12161direct (N × H)
246. C2×C4C31×SD32semidirect

Maximal subgroups

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

subgroupindex
C31×#32452
C31×#21852
C31×#20562
C31×#13472
C31×#13222
#1216131