Finite Group Databasebeta

← C31×#8398 #21465157

Subgroups — C31×#8398 #21465157

Order 7936.

782 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#554C211yesmaximal, characteristic
2C31×#1801C211yesmaximal, characteristic, complemented
3C31×#1549C212yesmaximal, complemented
4C31×#1486C212yesmaximal
5C31×#1461C211yesmaximal, characteristic, complemented
6C31×#554—24no—
7C31×#554—24no—
8C2×C31×C4⋊Q8C2211yescharacteristic
9C2×C31×#125C412yes—
10C2×C31×C22⋊Q8—21no—
11C2×C31×C22⋊Q8—22no—
12C2×C4×Q8×C31C2211yescharacteristic
13C2×C4×D4×C31C412yes—
14C2×C4×D4×C31C2211yescharacteristic
15C22×C4⋊C4×C31—21no—
16C31×#431C2212yes—
17C31×#423C2212yes—
18C2×C31×#101—21no—
19C31×#343—21no—
20C31×#332—24no—
21C31×D4⋊C8—24no—
22C31×#324—22no—
23C22×Q8×C31—21no—
24C22×Q8×C31D411yescharacteristic
25C22×D4×C31C2×C411yescharacteristic
26C23×C4×C31—21no—
27C23×C4×C31—22no—
28C2×M16×C31—22no—
29C31×C4⋊Q8—22no—
30C31×C4⋊Q8D414yes—
31C31×#125—24no—
32C31×#125—24no—
33C31×C22⋊Q8—28no—
34C31×C22⋊Q8—216no—
35C4×Q8×C31—22no—
36C4×Q8×C31—22no—
37C4×D4×C31—24no—
38C4×D4×C31—24no—
39C4×D4×C31—22no—
40C4×D4×C31D414yes—
41C2×C4⋊C4×C31—28no—
42C2×C4⋊C4×C31—22no—
43C2×C4⋊C4×C31—22no—
44C2×C4⋊C4×C31—21no—
45C2×C4⋊C4×C31C2×C412yes—
46C2×C4⋊C4×C31C2×C411yescharacteristic
47C2×C4⋊C4×C31D411yescharacteristic
48C2×C4⋊C4×C31D412yes—
49C2×C22⋊C4×C31—22no—
50C2×C22⋊C4×C31—21no—
51C2×C42×C31C2×C412yes—
52C2×C42×C31C2311yescharacteristic
53C31×C4⋊C8—22no—
54C31×C4⋊C8—22no—
55C31×C4⋊C8—22no—
56C31×#101—42no—
57C31×C22⋊C8—22no—
58C4×C8×C31—22no—
59C24×C31—21no—
60C2×Q8×C31—22no—
61C2×Q8×C31—24no—
62C2×Q8×C31—22no—
63C2×Q8×C31—21no—
64C2×Q8×C31—21no—
65C2×Q8×C31—82no—
66C2×D4×C31—24no—
67C2×D4×C31—21no—
68C2×D4×C31C22⋊C412yes—
69C22×C4×C31—21no—
70C22×C4×C31—22no—
71C22×C4×C31—28no—
72C22×C4×C31—24no—
73C22×C4×C31—42no—
74C22×C4×C31—41no—
75C22×C4×C31—41no—
76C22×C4×C31—84no—
77C22×C4×C31C22⋊C412yes—
78C22×C4×C31C22⋊C412yes—
79C22×C4×C31C22×C411yescharacteristic
80C22×C4×C31C2×D411yescharacteristic
81C22×C4×C31C2×D411yescharacteristic
82M16×C31—42no—
83M16×C31—42no—
84C2×C8×C31—42no—
85C2×C8×C31—42no—
86C4⋊C4×C31—28no—
87C4⋊C4×C31—28no—
88C4⋊C4×C31—21no—
89C4⋊C4×C31—22no—
90C4⋊C4×C31—22no—
91C4⋊C4×C31—24no—
92C4⋊C4×C31—24no—
93C4⋊C4×C31C22⋊C414yes—
94C4⋊C4×C31C22⋊C412yes—
95C22⋊C4×C31—28no—
96C22⋊C4×C31—24no—
97C42×C31—22no—
98C42×C31—22no—
99C42×C31C2×D412yes—
100C42×C31C2×D412yes—
101#8398C3111yesmaximal, characteristic, direct product factor, Sylow
102C23×C31—24no—
103C23×C31—22no—
104C23×C31—42no—
105C23×C31—41no—
106C23×C31—41no—
107C23×C31C2×C22⋊C411yescharacteristic
108Q8×C31—42no—
109Q8×C31—42no—
110Q8×C31—84no—
111D4×C31—24no—
112D4×C31—22no—
113D4×C31#10114yescomplemented
114C2×C4×C31—22no—
115C2×C4×C31—22no—
116C2×C4×C31—22no—
117C2×C4×C31—22no—
118C2×C4×C31—24no—
119C2×C4×C31—22no—
120C2×C4×C31—22no—
121C2×C4×C31—41no—
122C2×C4×C31—44no—
123C2×C4×C31—41no—
124C2×C4×C31—41no—
125C2×C4×C31—44no—
126C2×C4×C31—44no—
127C2×C4×C31—42no—
128C2×C4×C31—88no—
129C2×C4×C31C2×C22⋊C411yescharacteristic
130C2×C4×C31C2×C22⋊C411yescharacteristic
131C2×C4×C31#12012yes—
132C2×C4×C31#12012yes—
133C2×C4×C31#13512yes—
134C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
135C2×C4×C31#13711yescharacteristic
136C8×C31—82no—
137C8×C31—82no—
138C2×#554C2×C3111yescharacteristic
139#1801C2×C3111yescharacteristic, complemented
140#1549C2×C3112yescomplemented
141#1486C2×C3112yes—
142#1461C2×C3111yescharacteristic, complemented
143C22×C31—21no—
144C22×C31—21no—
145C22×C31—24no—
146C22×C31—44no—
147C22×C31—44no—
148C22×C31—42no—
149C22×C31—42no—
150C22×C31#37911yescharacteristic
151C22×C31#41611yescharacteristic
152C22×C31#41711yescharacteristic
153C4×C31—22no—
154C4×C31—21no—
155C4×C31—44no—
156C4×C31—42no—
157C4×C31—42no—
158C4×C31—42no—
159C4×C31C2×#10112yes—
160C4×C31#44912yes—
161C4×C31#45212yes—
162#554—24no—
163#554—24no—
164C2×C4⋊Q8C22×C3111yescharacteristic
165C2×#125C4×C3112yes—
166C2×C22⋊Q8—21no—
167C2×C22⋊Q8—22no—
168C2×C4×Q8C22×C3111yescharacteristic
169C2×C4×D4C4×C3112yes—
170C2×C4×D4C22×C3111yescharacteristic
171C22×C4⋊C4—21no—
172#431C22×C3112yes—
173#423C22×C3112yes—
174C2×#101—21no—
175#343—21no—
176#332—24no—
177D4⋊C8—24no—
178#324—22no—
179C2×C31—21no—
180C2×C31—21no—
181C2×C31—44no—
182C2×C31#178411yescharacteristic
183C2×C31#178811yescharacteristic
184C2×C31#179511yescharacteristic
185C22×Q8—21no—
186C22×Q8D4×C3111yescharacteristic
187C22×D4C2×C4×C3111yescharacteristic
188C23×C4—21no—
189C23×C4—22no—
190C2×M16—22no—
191C4⋊Q8—22no—
192C4⋊Q8D4×C3114yes—
193#125—24no—
194#125—24no—
195C22⋊Q8—28no—
196C22⋊Q8—216no—
197C4×Q8—22no—
198C4×Q8—22no—
199C4×D4—24no—
200C4×D4—24no—
201C4×D4—22no—
202C4×D4D4×C3114yes—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4C2×C4×C3112yes—
208C2×C4⋊C4C2×C4×C3111yescharacteristic
209C2×C4⋊C4D4×C3111yescharacteristic
210C2×C4⋊C4D4×C3112yes—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—21no—
213C2×C42C2×C4×C3112yes—
214C2×C42C23×C3111yescharacteristic
215C4⋊C8—22no—
216C4⋊C8—22no—
217C4⋊C8—22no—
218#101—42no—
219C22⋊C8—22no—
220C4×C8—22no—
221C31#839811yescharacteristic, direct product factor, Sylow
222C24—21no—
223C2×Q8—22no—
224C2×Q8—24no—
225C2×Q8—22no—
226C2×Q8—21no—
227C2×Q8—21no—
228C2×Q8—82no—
229C2×D4—24no—
230C2×D4—21no—
231C2×D4C22⋊C4×C3112yes—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—28no—
235C22×C4—24no—
236C22×C4—42no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—84no—
240C22×C4C22⋊C4×C3112yes—
241C22×C4C22⋊C4×C3112yes—
242C22×C4C22×C4×C3111yescharacteristic
243C22×C4C2×D4×C3111yescharacteristic
244C22×C4C2×D4×C3111yescharacteristic
245M16—42no—
246M16—42no—
247C2×C8—42no—
248C2×C8—42no—
249C4⋊C4—28no—
250C4⋊C4—28no—
251C4⋊C4—21no—
252C4⋊C4—22no—
253C4⋊C4—22no—
254C4⋊C4—24no—
255C4⋊C4—24no—
256C4⋊C4C22⋊C4×C3114yes—
257C4⋊C4C22⋊C4×C3112yes—
258C22⋊C4—28no—
259C22⋊C4—24no—
260C42—22no—
261C42—22no—
262C42C2×D4×C3112yes—
263C42C2×D4×C3112yes—
264C23—24no—
265C23—22no—
266C23—42no—
267C23—41no—
268C23—41no—
269C23C2×C22⋊C4×C3111yescharacteristic
270Q8—42no—
271Q8—42no—
272Q8—84no—
273D4—24no—
274D4—22no—
275D4C31×#10114yescomplemented
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—24no—
283C2×C4—41no—
284C2×C4—44no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—42no—
290C2×C4—88no—
291C2×C4C2×C22⋊C4×C3111yescharacteristic
292C2×C4C2×C22⋊C4×C3111yescharacteristic
293C2×C4C31×#12012yes—
294C2×C4C31×#12012yes—
295C2×C4C31×#13512yes—
296C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
297C2×C4C31×#13711yescharacteristic
298C8—82no—
299C8—82no—
300C22—21no—
301C22—21no—
302C22—24no—
303C22—44no—
304C22—44no—
305C22—42no—
306C22—42no—
307C22C31×#37911yescharacteristic
308C22C31×#41611yescharacteristic
309C22C31×#41711yescharacteristic
310C4—22no—
311C4—21no—
312C4—44no—
313C4—42no—
314C4—42no—
315C4—42no—
316C4C2×C31×#10112yes—
317C4C31×#44912yes—
318C4C31×#45212yes—
319C2—21no—
320C2—21no—
321C2—44no—
322C2C31×#178411yescharacteristic
323C2C31×#178811yescharacteristic
324C2C31×#179511yescharacteristic
subgroupG/H
center, socleC22×C31#379
commutatorC22×C4C22×C4×C31
Frattini, cosocleC2×C42C23×C31
prime radicalC22×D4×C31C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#1801C2semidirect
3. C31×#1549C2semidirect
5. C31×#1461C2semidirect
101. #8398C31direct (N × H)
113. D4×C31#101semidirect
139. #1801C2×C31semidirect
140. #1549C2×C31semidirect
142. #1461C2×C31semidirect
221. C31#8398direct (N × H)
275. D4C31×#101semidirect

Maximal subgroups

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

subgroupindex
C2×C31×#5542
C31×#18012
C31×#15492
C31×#14862
C31×#14612
#839831