Finite Group Databasebeta

← C31×#10012 #21466771

Subgroups — C31×#10012 #21466771

Order 7936.

734 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 734 conjugacy classes of subgroups — too many to draw. Only its 122 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×#3100C212yesmaximal, complemented
2C31×#2397C211yesmaximal, characteristic, complemented
3C31×#1963C211yesmaximal, characteristic, complemented
4C31×#1486C212yesmaximal, complemented
5C31×#1274C211yesmaximal, characteristic, complemented
6C31×#532C2211yescharacteristic
7C2×C4×D4×C31—22no—
8C2×C4×D4×C31C2212yes—
9C31×#480—22no—
10C31×#474—22no—
11C31×#459—22no—
12C31×#447—22no—
13C4×D8×C31—24no—
14C31×#431C2212yes—
15C31×#423C2211yescharacteristic
16C2×C31×#104—22no—
17C31×#399—22no—
18C31×#392—22no—
19C31×#376C2211yescharacteristic
20C31×D4⋊C8—24no—
21C2×#41×C31—21no—
22C22×D4×C31—22no—
23C22×D4×C31D412yes—
24C23×C4×C31—22no—
25C2×D8×C31—24no—
26C2×M16×C31—21no—
27C31×C4⋊Q8D411yescharacteristic, complemented
28C31×#127D411yescharacteristic, complemented
29C31×#124—21no—
30C31×#121—22no—
31C4×D4×C31—22no—
32C4×D4×C31—22no—
33C4×D4×C31—22no—
34C4×D4×C31—22no—
35C4×D4×C31—22no—
36C4×D4×C31—44no—
37C4×D4×C31—44no—
38C2×C4⋊C4×C31—22no—
39C2×C4⋊C4×C31D412yes—
40C2×C22⋊C4×C31—24no—
41C2×C22⋊C4×C31—22no—
42C2×C42×C31D412yes—
43C2×C42×C31C2311yescharacteristic
44C31×#107—24no—
45C31×C4⋊C8—22no—
46C31×C4⋊C8—22no—
47C31×#104—44no—
48C31×#102—24no—
49C31×#102—24no—
50C31×C22⋊C8—22no—
51C4×C8×C31D414yescomplemented
52C31×#95—22no—
53C24×C31—22no—
54#41×C31—42no—
55#41×C31—42no—
56C2×Q8×C31—21no—
57C2×D4×C31—22no—
58C2×D4×C31—22no—
59C2×D4×C31—22no—
60C2×D4×C31—21no—
61C2×D4×C31—44no—
62C2×D4×C31—44no—
63C2×D4×C31—48no—
64C2×D4×C31—41no—
65C22×C4×C31—22no—
66C22×C4×C31—22no—
67C22×C4×C31—42no—
68C22×C4×C31—42no—
69C22×C4×C31—41no—
70C22×C4×C31—44no—
71C22×C4×C31—44no—
72C22×C4×C31C2×D412yes—
73C22×C4×C31C2×D411yescharacteristic
74D8×C31—48no—
75M16×C31—42no—
76C2×C8×C31—24no—
77C2×C8×C31—41no—
78C4⋊C4×C31—22no—
79C4⋊C4×C31—22no—
80C4⋊C4×C31—22no—
81C4⋊C4×C31—44no—
82C22⋊C4×C31—48no—
83C22⋊C4×C31—42no—
84C22⋊C4×C31—42no—
85C22⋊C4×C31—42no—
86C42×C31—22no—
87C42×C31D814yescomplemented
88C42×C31C2×D411yescharacteristic
89C42×C31C2×D412yes—
90C42×C31#4111yescharacteristic
91#10012C3111yesmaximal, characteristic, direct product factor, Sylow
92C23×C31—24no—
93C23×C31—44no—
94C23×C31—42no—
95C23×C31—42no—
96C23×C31—44no—
97C23×C31—41no—
98C23×C31#12011yescharacteristic
99Q8×C31—42no—
100D4×C31—24no—
101D4×C31—24no—
102D4×C31—42no—
103D4×C31—42no—
104D4×C31—44no—
105D4×C31—416no—
106D4×C31—42no—
107D4×C31—82no—
108C2×C4×C31—24no—
109C2×C4×C31—42no—
110C2×C4×C31—44no—
111C2×C4×C31—42no—
112C2×C4×C31—41no—
113C2×C4×C31—44no—
114C2×C4×C31—84no—
115C2×C4×C31—82no—
116C2×C4×C31—81no—
117C2×C4×C31—82no—
118C2×C4×C31—81no—
119C2×C4×C31#12012yes—
120C2×C4×C31#12112yes—
121C2×C4×C31#12111yescharacteristic
122C2×C4×C31#12711yescharacteristic
123C2×C4×C31C2×D812yes—
124C2×C4×C31AGL(1,ℤ/8)12yes—
125C8×C31—44no—
126C8×C31—81no—
127#3100C2×C3112yescomplemented
128#2397C2×C3111yescharacteristic, complemented
129#1963C2×C3111yescharacteristic, complemented
130#1486C2×C3112yescomplemented
131#1274C2×C3111yescharacteristic, complemented
132C22×C31—22no—
133C22×C31—48no—
134C22×C31—44no—
135C22×C31—84no—
136C22×C31—82no—
137C22×C31—82no—
138C22×C31—82no—
139C22×C31—81no—
140C22×C31#39211yescharacteristic
141C22×C31#44712yes—
142C4×C31—22no—
143C4×C31—22no—
144C4×C31—44no—
145C4×C31—82no—
146C4×C31—81no—
147C4×C31#45912yes—
148C4×C31#49311yescharacteristic
149C4×C31#49611yescharacteristic
150#532C22×C3111yescharacteristic
151C2×C4×D4—22no—
152C2×C4×D4C22×C3112yes—
153#480—22no—
154#474—22no—
155#459—22no—
156#447—22no—
157C4×D8—24no—
158#431C22×C3112yes—
159#423C22×C3111yescharacteristic
160C2×#104—22no—
161#399—22no—
162#392—22no—
163#376C22×C3111yescharacteristic
164D4⋊C8—24no—
165C2×C31—22no—
166C2×C31—84no—
167C2×C31—161no—
168C2×C31#201012yes—
169C2×C31#201111yescharacteristic
170C2×#41—21no—
171C22×D4—22no—
172C22×D4D4×C3112yes—
173C23×C4—22no—
174C2×D8—24no—
175C2×M16—21no—
176C4⋊Q8D4×C3111yescharacteristic, complemented
177#127D4×C3111yescharacteristic, complemented
178#124—21no—
179#121—22no—
180C4×D4—22no—
181C4×D4—22no—
182C4×D4—22no—
183C4×D4—22no—
184C4×D4—22no—
185C4×D4—44no—
186C4×D4—44no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4D4×C3112yes—
189C2×C22⋊C4—24no—
190C2×C22⋊C4—22no—
191C2×C42D4×C3112yes—
192C2×C42C23×C3111yescharacteristic
193#107—24no—
194C4⋊C8—22no—
195C4⋊C8—22no—
196#104—44no—
197#102—24no—
198#102—24no—
199C22⋊C8—22no—
200C4×C8D4×C3114yescomplemented
201#95—22no—
202C31#1001211yescharacteristic, direct product factor, Sylow
203C24—22no—
204#41—42no—
205#41—42no—
206C2×Q8—21no—
207C2×D4—22no—
208C2×D4—22no—
209C2×D4—22no—
210C2×D4—21no—
211C2×D4—44no—
212C2×D4—44no—
213C2×D4—48no—
214C2×D4—41no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—41no—
220C22×C4—44no—
221C22×C4—44no—
222C22×C4C2×D4×C3112yes—
223C22×C4C2×D4×C3111yescharacteristic
224D8—48no—
225M16—42no—
226C2×C8—24no—
227C2×C8—41no—
228C4⋊C4—22no—
229C4⋊C4—22no—
230C4⋊C4—22no—
231C4⋊C4—44no—
232C22⋊C4—48no—
233C22⋊C4—42no—
234C22⋊C4—42no—
235C22⋊C4—42no—
236C42—22no—
237C42D8×C3114yescomplemented
238C42C2×D4×C3111yescharacteristic
239C42C2×D4×C3112yes—
240C42#41×C3111yescharacteristic
241C23—24no—
242C23—44no—
243C23—42no—
244C23—44no—
245C23—42no—
246C23—41no—
247C23C31×#12011yescharacteristic
248Q8—42no—
249D4—24no—
250D4—24no—
251D4—42no—
252D4—42no—
253D4—44no—
254D4—416no—
255D4—42no—
256D4—82no—
257C2×C4—24no—
258C2×C4—42no—
259C2×C4—44no—
260C2×C4—44no—
261C2×C4—42no—
262C2×C4—41no—
263C2×C4—84no—
264C2×C4—82no—
265C2×C4—81no—
266C2×C4—82no—
267C2×C4—81no—
268C2×C4C31×#12012yes—
269C2×C4C31×#12112yes—
270C2×C4C31×#12111yescharacteristic
271C2×C4C31×#12711yescharacteristic
272C2×C4C2×D8×C3112yes—
273C2×C4C31×AGL(1,ℤ/8)12yes—
274C8—44no—
275C8—81no—
276C22—22no—
277C22—48no—
278C22—44no—
279C22—84no—
280C22—82no—
281C22—82no—
282C22—82no—
283C22—81no—
284C22C31×#39211yescharacteristic
285C22C31×#44712yes—
286C4—22no—
287C4—22no—
288C4—44no—
289C4—82no—
290C4—81no—
291C4C31×#45912yes—
292C4C31×#49311yescharacteristic
293C4C31×#49611yescharacteristic
294C2—22no—
295C2—84no—
296C2—161no—
297C2C31×#201012yes—
298C2C31×#201111yescharacteristic
subgroupG/H
centerC2×C4×C31#127
commutator, Frattini, cosocleC2×C42C23×C31
socleC22×C31#392
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#3100C2semidirect
2. C31×#2397C2semidirect
3. C31×#1963C2semidirect
4. C31×#1486C2semidirect
5. C31×#1274C2semidirect
27. C31×C4⋊Q8D4semidirect
28. C31×#127D4semidirect
51. C4×C8×C31D4semidirect
87. C42×C31D8semidirect
91. #10012C31direct (N × H)
127. #3100C2×C31semidirect
128. #2397C2×C31semidirect
129. #1963C2×C31semidirect
130. #1486C2×C31semidirect
131. #1274C2×C31semidirect
176. C4⋊Q8D4×C31semidirect
177. #127D4×C31semidirect
200. C4×C8D4×C31semidirect
202. C31#10012direct (N × H)
237. C42D8×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#31002
C31×#23972
C31×#19632
C31×#14862
C31×#12742
#1001231