Finite Group Databasebeta

← C31×#12216 #21468975

Subgroups — C31×#12216 #21468975

Order 7936.

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

normal subgroups only

This group has 388 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×#2607C211yesmaximal, characteristic
2C31×#2030C211yesmaximal, characteristic
3C31×#1614C212yesmaximal
4C31×#1352C212yesmaximal
5C31×#1317C211yesmaximal, characteristic
6C31×#543C2211yescharacteristic
7C31×#484—22no—
8C31×#399C2212yes—
9C31×#393—21no—
10C31×#391—22no—
11C31×#387—22no—
12C31×#381—21no—
13C31×#381C2211yescharacteristic
14C4×C22⋊C4×C31—21no—
15C31×#342—21no—
16C31×#342C2211yescharacteristic
17C31×#328—22no—
18C31×#323C2212yes—
19C22×Q8×C31—21no—
20C23×C4×C31D411yescharacteristic
21C31×C22⋊Q8—22no—
22C31×C22⋊Q8—42no—
23C2×C4⋊C4×C31—22no—
24C2×C4⋊C4×C31D412yes—
25C2×C22⋊C4×C31—22no—
26C2×C22⋊C4×C31—21no—
27C2×C22⋊C4×C31—21no—
28C2×C22⋊C4×C31D411yescharacteristic
29C2×C22⋊C4×C31C2311yescharacteristic
30C2×C42×C31—21no—
31C2×C42×C31—21no—
32C31×#107—42no—
33C31×#103—42no—
34C31×C22⋊C8—22no—
35C31×#95—22no—
36C31×#95—21no—
37C31×#95—22no—
38C31×#95—22no—
39C31×#95—42no—
40C31×#95D412yes—
41C24×C31C2×D411yescharacteristic
42C2×Q8×C31—22no—
43C2×Q8×C31—44no—
44C2×Q8×C31—81no—
45C22×C4×C31—21no—
46C22×C4×C31—22no—
47C22×C4×C31—21no—
48C22×C4×C31—21no—
49C22×C4×C31—22no—
50C22×C4×C31—22no—
51C22×C4×C31—41no—
52C22×C4×C31—41no—
53C22×C4×C31—41no—
54C22×C4×C31Q1612yes—
55C22×C4×C31C2×D412yes—
56C2×C8×C31—42no—
57C4⋊C4×C31—22no—
58C4⋊C4×C31—44no—
59C4⋊C4×C31—42no—
60C4⋊C4×C31—42no—
61C22⋊C4×C31—41no—
62C22⋊C4×C31—44no—
63C22⋊C4×C31—41no—
64C22⋊C4×C31—41no—
65C22⋊C4×C31—41no—
66C22⋊C4×C31—41no—
67C22⋊C4×C31—41no—
68C42×C31—81no—
69C42×C31—81no—
70#12216C3111yesmaximal, characteristic, direct product factor, Sylow
71C23×C31—21no—
72C23×C31—21no—
73C23×C31—42no—
74C23×C31#12011yescharacteristic
75C23×C31C2×Q1611yescharacteristic
76C23×C31#13711yescharacteristic
77Q8×C31—84no—
78Q8×C31—82no—
79C2×C4×C31—24no—
80C2×C4×C31—22no—
81C2×C4×C31—22no—
82C2×C4×C31—42no—
83C2×C4×C31—41no—
84C2×C4×C31—42no—
85C2×C4×C31—41no—
86C2×C4×C31—42no—
87C2×C4×C31—42no—
88C2×C4×C31—42no—
89C2×C4×C31—42no—
90C2×C4×C31—42no—
91C2×C4×C31—41no—
92C2×C4×C31—42no—
93C2×C4×C31—42no—
94C2×C4×C31—81no—
95C2×C4×C31—82no—
96C2×C4×C31—81no—
97C2×C4×C31—82no—
98C8×C31—82no—
99#2607C2×C3111yescharacteristic
100#2030C2×C3111yescharacteristic
101#1614C2×C3112yes—
102#1352C2×C3112yes—
103#1317C2×C3111yescharacteristic
104C22×C31—22no—
105C22×C31—21no—
106C22×C31—21no—
107C22×C31—41no—
108C22×C31—41no—
109C22×C31—42no—
110C22×C31—81no—
111C22×C31#45111yescharacteristic
112C22×C31#45511yescharacteristic
113C22×C31B(4,𝔽2)11yescharacteristic
114C4×C31—44no—
115C4×C31—42no—
116C4×C31—42no—
117C4×C31—81no—
118C4×C31—82no—
119C4×C31—82no—
120C4×C31—161no—
121#543C22×C3111yescharacteristic
122#484—22no—
123#399C22×C3112yes—
124#393—21no—
125#391—22no—
126#387—22no—
127#381—21no—
128#381C22×C3111yescharacteristic
129C4×C22⋊C4—21no—
130#342—21no—
131#342C22×C3111yescharacteristic
132#328—22no—
133#323C22×C3112yes—
134C2×C31—21no—
135C2×C31—21no—
136C2×C31—41no—
137C2×C31—41no—
138C2×C31#160211yescharacteristic
139C2×C31#219812yes—
140C22×Q8—21no—
141C23×C4D4×C3111yescharacteristic
142C22⋊Q8—22no—
143C22⋊Q8—42no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4D4×C3112yes—
146C2×C22⋊C4—21no—
147C2×C22⋊C4—22no—
148C2×C22⋊C4—21no—
149C2×C22⋊C4D4×C3111yescharacteristic
150C2×C22⋊C4C23×C3111yescharacteristic
151C2×C42—21no—
152C2×C42—21no—
153#107—42no—
154#103—42no—
155C22⋊C8—22no—
156#95—22no—
157#95—21no—
158#95—22no—
159#95—22no—
160#95—42no—
161#95D4×C3112yes—
162C31#1221611yescharacteristic, direct product factor, Sylow
163C24C2×D4×C3111yescharacteristic
164C2×Q8—22no—
165C2×Q8—44no—
166C2×Q8—81no—
167C22×C4—21no—
168C22×C4—22no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4Q16×C3112yes—
177C22×C4C2×D4×C3112yes—
178C2×C8—42no—
179C4⋊C4—22no—
180C4⋊C4—42no—
181C4⋊C4—44no—
182C4⋊C4—42no—
183C22⋊C4—41no—
184C22⋊C4—44no—
185C22⋊C4—41no—
186C22⋊C4—41no—
187C22⋊C4—41no—
188C22⋊C4—41no—
189C22⋊C4—41no—
190C42—81no—
191C42—81no—
192C23—21no—
193C23—21no—
194C23—42no—
195C23C31×#12011yescharacteristic
196C23C2×Q16×C3111yescharacteristic
197C23C31×#13711yescharacteristic
198Q8—84no—
199Q8—82no—
200C2×C4—24no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—42no—
204C2×C4—41no—
205C2×C4—42no—
206C2×C4—41no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—41no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—81no—
216C2×C4—82no—
217C2×C4—81no—
218C2×C4—82no—
219C8—82no—
220C22—22no—
221C22—21no—
222C22—21no—
223C22—41no—
224C22—41no—
225C22—42no—
226C22—81no—
227C22C31×#45111yescharacteristic
228C22C31×#45511yescharacteristic
229C22C31×B(4,𝔽2)11yescharacteristic
230C4—44no—
231C4—42no—
232C4—42no—
233C4—81no—
234C4—82no—
235C4—82no—
236C4—161no—
237C2—21no—
238C2—21no—
239C2—41no—
240C2—41no—
241C2C31×#160211yescharacteristic
242C2C31×#219812yes—
subgroupG/H
center, socleC22×C31B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C31
prime radicalC24×C31C2×D4
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C31 ◁ C2×C22⋊C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
70. #12216C31direct (N × H)
162. C31#12216direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#26072
C31×#20302
C31×#16142
C31×#13522
C31×#13172
#1221631