Finite Group Databasebeta

← C31×#8322 #21465081

Subgroups — C31×#8322 #21465081

Order 7936.

1146 conjugacy classes of subgroups; only the 214 normal subgroups are shown.

normal subgroups only

This group has 1146 conjugacy classes of subgroups — too many to draw. Only its 214 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
1C2×C31×#518C211yesmaximal, characteristic
2C31×#1478C214yesmaximal, complemented
3C31×#1458C212yesmaximal, complemented
4C23×D4×C31C411yescharacteristic
5C31×#518—28no—
6C2×C4×D4×C31—22no—
7C2×C4×D4×C31C412yes—
8C2×C4×D4×C31C2212yes—
9C2×C31×#117C411yescharacteristic
10C2×C31×#117C2211yescharacteristic
11C22×C22⋊C4×C31—22no—
12C31×#431C2214yescomplemented
13C31×D4⋊C8—28no—
14C31×#324—22no—
15C31×#323—22no—
16C25×C31—21no—
17C22×D4×C31—24no—
18C22×D4×C31—24no—
19C22×D4×C31—21no—
20C22×D4×C31—22no—
21C22×D4×C31C814yescomplemented
22C22×D4×C31C2×C412yes—
23C23×C4×C31—22no—
24C23×C4×C31—22no—
25C23×C4×C31C2×C411yescharacteristic
26C4×D4×C31—216no—
27C4×D4×C31—24no—
28C4×D4×C31—24no—
29C4×D4×C31—24no—
30C4×D4×C31D418yes—
31C31×#117—24no—
32C31×#117—24no—
33C2×C4⋊C4×C31C2×C412yes—
34C2×C4⋊C4×C31D412yes—
35C2×C22⋊C4×C31—216no—
36C2×C22⋊C4×C31—21no—
37C2×C22⋊C4×C31—22no—
38C2×C22⋊C4×C31—22no—
39C2×C22⋊C4×C31D412yes—
40C2×C42×C31—21no—
41C2×C42×C31C2×C411yescharacteristic
42C2×C42×C31C2311yescharacteristic
43C31×C4⋊C8—24no—
44C31×C22⋊C8—24no—
45C4×C8×C31—24no—
46C24×C31—24no—
47C24×C31—24no—
48C24×C31—22no—
49C24×C31—24no—
50C24×C31—84no—
51C24×C31C22⋊C411yescharacteristic
52C2×D4×C31—24no—
53C2×D4×C31—24no—
54C2×D4×C31—216no—
55C2×D4×C31—216no—
56C2×D4×C31—24no—
57C2×D4×C31—22no—
58C2×D4×C31—22no—
59C2×D4×C31—22no—
60C2×D4×C31C22⋊C414yes—
61C2×D4×C31C2×C814yes—
62C2×D4×C31M1614yes—
63C22×C4×C31—22no—
64C22×C4×C31—22no—
65C22×C4×C31—22no—
66C22×C4×C31—28no—
67C22×C4×C31—28no—
68C22×C4×C31—84no—
69C22×C4×C31—84no—
70C22×C4×C31C22⋊C411yescharacteristic
71C22×C4×C31C22⋊C412yes—
72C22×C4×C31C2×C812yes—
73C22×C4×C31M1612yes—
74C22×C4×C31C22×C411yescharacteristic
75C22×C4×C31C2×D412yes—
76C2×C8×C31—44no—
77C4⋊C4×C31—22no—
78C4⋊C4×C31—24no—
79C4⋊C4×C31C22⋊C414yes—
80C22⋊C4×C31—24no—
81C22⋊C4×C31—28no—
82C22⋊C4×C31—24no—
83C22⋊C4×C31—28no—
84C42×C31—24no—
85C42×C31—22no—
86C42×C31C2×D414yes—
87#8322C3111yesmaximal, characteristic, direct product factor, Sylow
88C23×C31—216no—
89C23×C31—21no—
90C23×C31—24no—
91C23×C31—24no—
92C23×C31—21no—
93C23×C31—24no—
94C23×C31—42no—
95C23×C31—44no—
96C23×C31—44no—
97C23×C31—88no—
98C23×C31—88no—
99C23×C31—88no—
100C23×C31C22⋊C812yes—
101C23×C31C2×C22⋊C411yescharacteristic
102D4×C31—216no—
103D4×C31—24no—
104D4×C31—24no—
105D4×C31—24no—
106D4×C31C22⋊C818yes—
107C2×C4×C31—21no—
108C2×C4×C31—21no—
109C2×C4×C31—24no—
110C2×C4×C31—24no—
111C2×C4×C31—22no—
112C2×C4×C31—24no—
113C2×C4×C31—42no—
114C2×C4×C31—44no—
115C2×C4×C31—42no—
116C2×C4×C31—88no—
117C2×C4×C31—88no—
118C2×C4×C31C22⋊C812yes—
119C2×C4×C31C2×C22⋊C412yes—
120C2×C4×C31#12014yes—
121C2×C4×C31C22×C811yescharacteristic
122C2×C4×C31C2×M1612yes—
123C2×C4×C31C2×M1611yescharacteristic
124C8×C31—84no—
125C2×#518C2×C3111yescharacteristic
126#1478C2×C3114yescomplemented
127#1458C2×C3112yescomplemented
128C22×C31—22no—
129C22×C31—22no—
130C22×C31—28no—
131C22×C31—42no—
132C22×C31—48no—
133C22×C31—48no—
134C22×C31—42no—
135C22×C31—44no—
136C22×C31—44no—
137C22×C31—42no—
138C22×C31—816no—
139C22×C31#37911yescharacteristic
140C22×C31C2×C22⋊C811yescharacteristic
141C22×C31#40711yescharacteristic
142C4×C31—24no—
143C4×C31—22no—
144C4×C31—44no—
145C4×C31—42no—
146C4×C31C2×C22⋊C812yes—
147C4×C31#40712yes—
148C23×D4C4×C3111yescharacteristic
149#518—28no—
150C2×C4×D4—22no—
151C2×C4×D4C4×C3112yes—
152C2×C4×D4C22×C3112yes—
153C2×#117C4×C3111yescharacteristic
154C2×#117C22×C3111yescharacteristic
155C22×C22⋊C4—22no—
156#431C22×C3114yescomplemented
157D4⋊C8—28no—
158#324—22no—
159#323—22no—
160C2×C31—21no—
161C2×C31—21no—
162C2×C31—48no—
163C2×C31—42no—
164C2×C31#177911yescharacteristic
165C2×C31#179512yes—
166C25—21no—
167C22×D4—24no—
168C22×D4—24no—
169C22×D4—21no—
170C22×D4—22no—
171C22×D4C8×C3114yescomplemented
172C22×D4C2×C4×C3112yes—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4C2×C4×C3111yescharacteristic
176C4×D4—216no—
177C4×D4—24no—
178C4×D4—24no—
179C4×D4—24no—
180C4×D4D4×C3118yes—
181#117—24no—
182#117—24no—
183C2×C4⋊C4C2×C4×C3112yes—
184C2×C4⋊C4D4×C3112yes—
185C2×C22⋊C4—216no—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4D4×C3112yes—
190C2×C42—21no—
191C2×C42C2×C4×C3111yescharacteristic
192C2×C42C23×C3111yescharacteristic
193C4⋊C8—24no—
194C22⋊C8—24no—
195C4×C8—24no—
196C31#832211yescharacteristic, direct product factor, Sylow
197C24—24no—
198C24—24no—
199C24—22no—
200C24—24no—
201C24—84no—
202C24C22⋊C4×C3111yescharacteristic
203C2×D4—24no—
204C2×D4—24no—
205C2×D4—216no—
206C2×D4—216no—
207C2×D4—24no—
208C2×D4—22no—
209C2×D4—22no—
210C2×D4—22no—
211C2×D4C22⋊C4×C3114yes—
212C2×D4C2×C8×C3114yes—
213C2×D4M16×C3114yes—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—28no—
218C22×C4—28no—
219C22×C4—84no—
220C22×C4—84no—
221C22×C4C22⋊C4×C3111yescharacteristic
222C22×C4C22⋊C4×C3112yes—
223C22×C4C2×C8×C3112yes—
224C22×C4M16×C3112yes—
225C22×C4C22×C4×C3111yescharacteristic
226C22×C4C2×D4×C3112yes—
227C2×C8—44no—
228C4⋊C4—22no—
229C4⋊C4—24no—
230C4⋊C4C22⋊C4×C3114yes—
231C22⋊C4—24no—
232C22⋊C4—28no—
233C22⋊C4—24no—
234C22⋊C4—28no—
235C42—24no—
236C42—22no—
237C42C2×D4×C3114yes—
238C23—216no—
239C23—24no—
240C23—21no—
241C23—24no—
242C23—21no—
243C23—24no—
244C23—42no—
245C23—44no—
246C23—44no—
247C23—88no—
248C23—88no—
249C23—88no—
250C23C31×C22⋊C812yes—
251C23C2×C22⋊C4×C3111yescharacteristic
252D4—216no—
253D4—24no—
254D4—24no—
255D4—24no—
256D4C31×C22⋊C818yes—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—24no—
260C2×C4—24no—
261C2×C4—22no—
262C2×C4—24no—
263C2×C4—42no—
264C2×C4—44no—
265C2×C4—42no—
266C2×C4—88no—
267C2×C4—88no—
268C2×C4C31×C22⋊C812yes—
269C2×C4C2×C22⋊C4×C3112yes—
270C2×C4C31×#12014yes—
271C2×C4C22×C8×C3111yescharacteristic
272C2×C4C2×M16×C3112yes—
273C2×C4C2×M16×C3111yescharacteristic
274C8—84no—
275C22—22no—
276C22—22no—
277C22—28no—
278C22—42no—
279C22—42no—
280C22—48no—
281C22—48no—
282C22—44no—
283C22—44no—
284C22—42no—
285C22—816no—
286C22C31×#37911yescharacteristic
287C22C2×C31×C22⋊C811yescharacteristic
288C22C31×#40711yescharacteristic
289C4—24no—
290C4—22no—
291C4—44no—
292C4—42no—
293C4C2×C31×C22⋊C812yes—
294C4C31×#40712yes—
295C2—21no—
296C2—21no—
297C2—48no—
298C2—42no—
299C2C31×#177911yescharacteristic
300C2C31×#179512yes—
subgroupG/H
center, socleC22×C31#379
commutatorC2×C4C22×C8×C31
Frattini, cosocleC2×C42C23×C31
prime radicalC23×D4×C31C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#1478C2semidirect
3. C31×#1458C2semidirect
12. C31×#431C22semidirect
21. C22×D4×C31C8semidirect
87. #8322C31direct (N × H)
126. #1478C2×C31semidirect
127. #1458C2×C31semidirect
156. #431C22×C31semidirect
171. C22×D4C8×C31semidirect
196. C31#8322direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5182
C31×#14782
C31×#14582
#832231