Finite Group Databasebeta

← C31×#12209 #21468968

Subgroups — C31×#12209 #21468968

Order 7936.

404 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 74 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×#2724C211yesmaximal, characteristic
2C31×#2027C211yesmaximal, characteristic
3C31×#1611C212yesmaximal, complemented
4C31×#1346C212yesmaximal, complemented
5C31×#1318C211yesmaximal, characteristic
6C31×#544C2211yescharacteristic
7C2×C31×#123—21no—
8C31×#483—22no—
9C31×#418—21no—
10C31×#398C2212yes—
11C31×#397—22no—
12C31×#390C2211yescharacteristic
13C31×#389—21no—
14C31×#385—22no—
15C31×#343C2211yescharacteristic
16C31×#339—21no—
17C31×#328—22no—
18C31×#324C2212yes—
19C22×D4×C31D411yescharacteristic
20C23×C4×C31—21no—
21C2×M16×C31—21no—
22C31×C4⋊Q8—41no—
23C31×#125—41no—
24C31×#123—42no—
25C31×#123—42no—
26C31×C22⋊Q8—22no—
27C31×#117—21no—
28C2×C4⋊C4×C31—21no—
29C2×C4⋊C4×C31—22no—
30C2×C4⋊C4×C31—21no—
31C2×C4⋊C4×C31D412yes—
32C2×C4⋊C4×C31C2311yescharacteristic
33C2×C22⋊C4×C31—21no—
34C2×C22⋊C4×C31—22no—
35C2×C22⋊C4×C31—22no—
36C2×C42×C31D411yescharacteristic
37C31×#106—42no—
38C31×#103—42no—
39C31×#102—41no—
40C31×#102—41no—
41C31×C22⋊C8—22no—
42C31×#95—22no—
43C31×#95—22no—
44C31×#95D412yes—
45C24×C31—21no—
46C2×Q8×C31—22no—
47C2×D4×C31—21no—
48C2×D4×C31—41no—
49C2×D4×C31—41no—
50C2×D4×C31SD1612yes—
51C22×C4×C31—21no—
52C22×C4×C31—22no—
53C22×C4×C31—22no—
54C22×C4×C31—42no—
55C22×C4×C31—81no—
56C22×C4×C31—81no—
57C22×C4×C31C2×D411yescharacteristic
58C22×C4×C31C2×D412yes—
59M16×C31—41no—
60M16×C31—41no—
61C2×C8×C31—42no—
62C2×C8×C31—41no—
63C4⋊C4×C31—21no—
64C4⋊C4×C31—21no—
65C4⋊C4×C31—22no—
66C4⋊C4×C31—41no—
67C4⋊C4×C31—42no—
68C4⋊C4×C31—42no—
69C4⋊C4×C31—42no—
70C4⋊C4×C31—42no—
71C4⋊C4×C31—42no—
72C4⋊C4×C31—81no—
73C22⋊C4×C31—44no—
74C22⋊C4×C31—42no—
75C22⋊C4×C31—44no—
76C22⋊C4×C31—41no—
77C42×C31—41no—
78C42×C31—41no—
79#12209C3111yesmaximal, characteristic, direct product factor, Sylow
80C23×C31—22no—
81C23×C31—42no—
82C23×C31—42no—
83C23×C31—41no—
84C23×C31—41no—
85C23×C31#12011yescharacteristic
86Q8×C31—82no—
87D4×C31—42no—
88D4×C31—81no—
89C2×C4×C31—21no—
90C2×C4×C31—21no—
91C2×C4×C31—22no—
92C2×C4×C31—22no—
93C2×C4×C31—41no—
94C2×C4×C31—41no—
95C2×C4×C31—42no—
96C2×C4×C31—42no—
97C2×C4×C31—42no—
98C2×C4×C31—42no—
99C2×C4×C31—42no—
100C2×C4×C31—42no—
101C2×C4×C31—41no—
102C2×C4×C31—41no—
103C2×C4×C31—42no—
104C2×C4×C31—84no—
105C2×C4×C31C2×SD1611yescharacteristic
106C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
107C8×C31—81no—
108C8×C31—82no—
109#2724C2×C3111yescharacteristic
110#2027C2×C3111yescharacteristic
111#1611C2×C3112yescomplemented
112#1346C2×C3112yescomplemented
113#1318C2×C3111yescharacteristic
114C22×C31—21no—
115C22×C31—21no—
116C22×C31—44no—
117C22×C31—42no—
118C22×C31—82no—
119C22×C31—81no—
120C22×C31—81no—
121C22×C31#45011yescharacteristic
122C22×C31#45511yescharacteristic
123C22×C31B(4,𝔽2)11yescharacteristic
124C4×C31—21no—
125C4×C31—21no—
126C4×C31—42no—
127C4×C31—42no—
128C4×C31—82no—
129C4×C31—82no—
130C4×C31—81no—
131C4×C31—81no—
132#544C22×C3111yescharacteristic
133C2×#123—21no—
134#483—22no—
135#418—21no—
136#398C22×C3112yes—
137#397—22no—
138#390C22×C3111yescharacteristic
139#389—21no—
140#385—22no—
141#343C22×C3111yescharacteristic
142#339—21no—
143#328—22no—
144#324C22×C3112yes—
145C2×C31—21no—
146C2×C31—21no—
147C2×C31—82no—
148C2×C31#160111yescharacteristic
149C2×C31#219811yescharacteristic
150C2×C31#220311yescharacteristic
151C22×D4D4×C3111yescharacteristic
152C23×C4—21no—
153C2×M16—21no—
154C4⋊Q8—41no—
155#125—41no—
156#123—42no—
157#123—42no—
158C22⋊Q8—22no—
159#117—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4D4×C3112yes—
164C2×C4⋊C4C23×C3111yescharacteristic
165C2×C22⋊C4—22no—
166C2×C22⋊C4—21no—
167C2×C22⋊C4—22no—
168C2×C42D4×C3111yescharacteristic
169#106—42no—
170#103—42no—
171#102—41no—
172#102—41no—
173C22⋊C8—22no—
174#95—22no—
175#95—22no—
176#95D4×C3112yes—
177C31#1220911yescharacteristic, direct product factor, Sylow
178C24—21no—
179C2×Q8—22no—
180C2×D4—21no—
181C2×D4—41no—
182C2×D4—41no—
183C2×D4SD16×C3112yes—
184C22×C4—21no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—42no—
188C22×C4—81no—
189C22×C4—81no—
190C22×C4C2×D4×C3111yescharacteristic
191C22×C4C2×D4×C3112yes—
192M16—41no—
193M16—41no—
194C2×C8—42no—
195C2×C8—41no—
196C4⋊C4—21no—
197C4⋊C4—21no—
198C4⋊C4—22no—
199C4⋊C4—41no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—81no—
206C22⋊C4—44no—
207C22⋊C4—42no—
208C22⋊C4—44no—
209C22⋊C4—41no—
210C42—41no—
211C42—41no—
212C23—22no—
213C23—42no—
214C23—42no—
215C23—41no—
216C23—41no—
217C23C31×#12011yescharacteristic
218Q8—82no—
219D4—42no—
220D4—81no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—42no—
236C2×C4—84no—
237C2×C4C2×SD16×C3111yescharacteristic
238C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
239C8—81no—
240C8—82no—
241C22—21no—
242C22—21no—
243C22—44no—
244C22—42no—
245C22—82no—
246C22—81no—
247C22—81no—
248C22C31×#45011yescharacteristic
249C22C31×#45511yescharacteristic
250C22C31×B(4,𝔽2)11yescharacteristic
251C4—21no—
252C4—21no—
253C4—42no—
254C4—42no—
255C4—82no—
256C4—82no—
257C4—81no—
258C4—81no—
259C2—21no—
260C2—21no—
261C2—82no—
262C2C31×#160111yescharacteristic
263C2C31×#219811yescharacteristic
264C2C31×#220311yescharacteristic
subgroupG/H
center, socleC22×C31B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C31
prime radicalC22×D4×C31D4
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C31 ◁ C2×C4⋊C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
3. C31×#1611C2semidirect
4. C31×#1346C2semidirect
79. #12209C31direct (N × H)
111. #1611C2×C31semidirect
112. #1346C2×C31semidirect
177. C31#12209direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#27242
C31×#20272
C31×#16112
C31×#13462
C31×#13182
#1220931