Finite Group Databasebeta

← C31×#8221 #21464980

Subgroups — C31×#8221 #21464980

Order 7936.

640 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×C31×C22⋊Q8C211yesmaximal, characteristic
2C4×C31×C22⋊C8C211yesmaximal, characteristic
3C2×C31×#328C212yesmaximal
4C31×#1434C211yesmaximal, characteristic
5C31×#1294C212yesmaximal
6C2×C31×C22⋊Q8C2211yescharacteristic
7C2×C4×Q8×C31C411yescharacteristic, complemented
8C22×C42×C31C2211yescharacteristic
9C2×C31×C22⋊C8C2212yes—
10C2×C4×C8×C31—21no—
11C31×#391C411yescharacteristic, complemented
12C31×#389C411yescharacteristic, complemented
13C31×#387—21no—
14C31×#385—21no—
15C31×#380C2211yescharacteristic
16C4×C4⋊C4×C31—21no—
17C4×C4⋊C4×C31C411yescharacteristic, complemented
18C4×C22⋊C4×C31—21no—
19C2×C31×#95—21no—
20C2×C31×#95—21no—
21C2×C31×#95C2212yes—
22C31×#336—21no—
23C31×#328C418yescomplemented
24C22×Q8×C31C2×C411yescharacteristic
25C23×C4×C31—22no—
26C23×C4×C31D411yescharacteristic
27C23×C4×C31D411yescharacteristic
28C23×C4×C31C2311yescharacteristic
29C22×C8×C31—22no—
30C31×C22⋊Q8—22no—
31C31×C22⋊Q8C2×C414yes—
32C4×Q8×C31—42no—
33C2×C4⋊C4×C31—21no—
34C2×C4⋊C4×C31—21no—
35C2×C4⋊C4×C31—21no—
36C2×C4⋊C4×C31C2×C411yescharacteristic
37C2×C4⋊C4×C31C2×C411yescharacteristic
38C2×C4⋊C4×C31C2×C411yescharacteristic
39C2×C22⋊C4×C31—21no—
40C2×C22⋊C4×C31—21no—
41C2×C42×C31—21no—
42C2×C42×C31—21no—
43C2×C42×C31—42no—
44C2×C42×C31C2×C411yescharacteristic
45C2×C42×C31C2×C411yescharacteristic
46C2×C42×C31D411yescharacteristic
47C2×C42×C31D411yescharacteristic
48C31×C22⋊C8—22no—
49C31×C22⋊C8C2×C414yes—
50C4×C8×C31—42no—
51C31×#95—21no—
52C31×#95—21no—
53C31×#95—21no—
54C31×#95—21no—
55C31×#95—21no—
56C31×#95—22no—
57C31×#95—42no—
58C31×#95—42no—
59C31×#95C2×C414yes—
60C24×C31C2×D411yescharacteristic
61C2×Q8×C31—21no—
62C2×Q8×C31—41no—
63C2×Q8×C31—41no—
64C2×Q8×C31C4212yes—
65C22×C4×C31—21no—
66C22×C4×C31—21no—
67C22×C4×C31—22no—
68C22×C4×C31—22no—
69C22×C4×C31—21no—
70C22×C4×C31—21no—
71C22×C4×C31—24no—
72C22×C4×C31—41no—
73C22×C4×C31—41no—
74C22×C4×C31—41no—
75C22×C4×C31—41no—
76C22×C4×C31—41no—
77C22×C4×C31—42no—
78C22×C4×C31—42no—
79C22×C4×C31—42no—
80C22×C4×C31—42no—
81C22×C4×C31—42no—
82C22×C4×C31—42no—
83C22×C4×C31C22⋊C411yescharacteristic
84C22×C4×C31C22⋊C411yescharacteristic
85C22×C4×C31C22⋊C411yescharacteristic
86C22×C4×C31C22⋊C411yescharacteristic
87C22×C4×C31SD1612yes—
88C22×C4×C31Q1612yes—
89C22×C4×C31C22×C411yescharacteristic
90C22×C4×C31C22×C412yes—
91C22×C4×C31C2×D411yescharacteristic
92C22×C4×C31#4112yes—
93C2×C8×C31—24no—
94C2×C8×C31—42no—
95C2×C8×C31—42no—
96C4⋊C4×C31—21no—
97C4⋊C4×C31—24no—
98C4⋊C4×C31—42no—
99C4⋊C4×C31—41no—
100C4⋊C4×C31—41no—
101C4⋊C4×C31—42no—
102C4⋊C4×C31C4212yes—
103C22⋊C4×C31—24no—
104C22⋊C4×C31—42no—
105C42×C31—22no—
106C42×C31—22no—
107C42×C31—41no—
108C42×C31—41no—
109C42×C31—42no—
110C42×C31—42no—
111#8221C3111yesmaximal, characteristic, direct product factor, Sylow
112C23×C31—21no—
113C23×C31—21no—
114C23×C31—21no—
115C23×C31—21no—
116C23×C31C2×C22⋊C411yescharacteristic
117C23×C31C4×D412yes—
118C23×C31C2×SD1611yescharacteristic
119C23×C31C2×Q1611yescharacteristic
120C23×C31#13512yes—
121Q8×C31—42no—
122Q8×C31—42no—
123C2×C4×C31—21no—
124C2×C4×C31—21no—
125C2×C4×C31—21no—
126C2×C4×C31—21no—
127C2×C4×C31—21no—
128C2×C4×C31—21no—
129C2×C4×C31—21no—
130C2×C4×C31—21no—
131C2×C4×C31—21no—
132C2×C4×C31—21no—
133C2×C4×C31—22no—
134C2×C4×C31—22no—
135C2×C4×C31—21no—
136C2×C4×C31—21no—
137C2×C4×C31—21no—
138C2×C4×C31—42no—
139C2×C4×C31—42no—
140C2×C4×C31—42no—
141C2×C4×C31—42no—
142C2×C4×C31—44no—
143C2×C4×C31—41no—
144C2×C4×C31—41no—
145C2×C4×C31—41no—
146C2×C4×C31—41no—
147C2×C4×C31—44no—
148C2×C4×C31—44no—
149C2×C4×C31—41no—
150C2×C4×C31—41no—
151C2×C4×C31—41no—
152C2×C4×C31—41no—
153C2×C4×C31—41no—
154C2×C4×C31—41no—
155C2×C4×C31—44no—
156C2×C4×C31—82no—
157C2×C4×C31—82no—
158C2×C4×C31—82no—
159C2×C4×C31—82no—
160C2×C4×C31#9912yes—
161C2×C4×C31#10312yes—
162C2×C4×C31#10312yes—
163C2×C4×C31C2×C4211yescharacteristic
164C2×C4×C31#11711yescharacteristic
165C2×C4×C31C4×D412yes—
166C8×C31—44no—
167C4×C22⋊Q8C2×C3111yescharacteristic
168C4×C22⋊C8C2×C3111yescharacteristic
169C2×#328C2×C3112yes—
170#1434C2×C3111yescharacteristic
171#1294C2×C3112yes—
172C22×C31—22no—
173C22×C31—22no—
174C22×C31—22no—
175C22×C31—21no—
176C22×C31—21no—
177C22×C31—21no—
178C22×C31—21no—
179C22×C31—22no—
180C22×C31C4×C22⋊C411yescharacteristic
181C22×C31C2×#9911yescharacteristic
182C22×C31#41011yescharacteristic
183C22×C31C2×#10311yescharacteristic
184C22×C31#41611yescharacteristic
185C22×C31#42111yescharacteristic
186C22×C31#42111yescharacteristic
187C22×C31C4×SD1612yes—
188C22×C31C4×Q1612yes—
189C4×C31—22no—
190C4×C31—21no—
191C4×C31—21no—
192C4×C31—22no—
193C4×C31—22no—
194C4×C31—22no—
195C4×C31—42no—
196C4×C31—42no—
197C4×C31—41no—
198C4×C31—84no—
199C4×C31—81no—
200C4×C31—81no—
201C2×C22⋊Q8C22×C3111yescharacteristic
202C2×C4×Q8C4×C3111yescharacteristic, complemented
203C22×C42C22×C3111yescharacteristic
204C2×C22⋊C8C22×C3112yes—
205C2×C4×C8—21no—
206#391C4×C3111yescharacteristic, complemented
207#389C4×C3111yescharacteristic, complemented
208#387—21no—
209#385—21no—
210#380C22×C3111yescharacteristic
211C4×C4⋊C4—21no—
212C4×C4⋊C4C4×C3111yescharacteristic, complemented
213C4×C22⋊C4—21no—
214C2×#95—21no—
215C2×#95—21no—
216C2×#95C22×C3112yes—
217#336—21no—
218#328C4×C3118yescomplemented
219C2×C31—22no—
220C2×C31—22no—
221C2×C31#151611yescharacteristic
222C2×C31#151811yescharacteristic
223C2×C31#152011yescharacteristic
224C2×C31#152111yescharacteristic
225C2×C31C4×#9911yescharacteristic
226C2×C31#175911yescharacteristic
227C2×C31C4×#10311yescharacteristic
228C22×Q8C2×C4×C3111yescharacteristic
229C23×C4—22no—
230C23×C4D4×C3111yescharacteristic
231C23×C4D4×C3111yescharacteristic
232C23×C4C23×C3111yescharacteristic
233C22×C8—22no—
234C22⋊Q8—22no—
235C22⋊Q8C2×C4×C3114yes—
236C4×Q8—42no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4C2×C4×C3111yescharacteristic
241C2×C4⋊C4C2×C4×C3111yescharacteristic
242C2×C4⋊C4C2×C4×C3111yescharacteristic
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—42no—
248C2×C42C2×C4×C3111yescharacteristic
249C2×C42C2×C4×C3111yescharacteristic
250C2×C42D4×C3111yescharacteristic
251C2×C42D4×C3111yescharacteristic
252C22⋊C8—22no—
253C22⋊C8C2×C4×C3114yes—
254C4×C8—42no—
255#95—21no—
256#95—21no—
257#95—21no—
258#95—21no—
259#95—21no—
260#95—22no—
261#95—42no—
262#95—42no—
263#95C2×C4×C3114yes—
264C31#822111yescharacteristic, direct product factor, Sylow
265C24C2×D4×C3111yescharacteristic
266C2×Q8—21no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×Q8C42×C3112yes—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—24no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4C22⋊C4×C3111yescharacteristic
289C22×C4C22⋊C4×C3111yescharacteristic
290C22×C4C22⋊C4×C3111yescharacteristic
291C22×C4C22⋊C4×C3111yescharacteristic
292C22×C4SD16×C3112yes—
293C22×C4Q16×C3112yes—
294C22×C4C22×C4×C3111yescharacteristic
295C22×C4C22×C4×C3112yes—
296C22×C4C2×D4×C3111yescharacteristic
297C22×C4#41×C3112yes—
298C2×C8—24no—
299C2×C8—42no—
300C2×C8—42no—
301C4⋊C4—21no—
302C4⋊C4—24no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4C42×C3112yes—
308C22⋊C4—24no—
309C22⋊C4—42no—
310C42—22no—
311C42—22no—
312C42—41no—
313C42—41no—
314C42—42no—
315C42—42no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23C2×C22⋊C4×C3111yescharacteristic
321C23C4×D4×C3112yes—
322C23C2×SD16×C3111yescharacteristic
323C23C2×Q16×C3111yescharacteristic
324C23C31×#13512yes—
325Q8—42no—
326Q8—42no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—44no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—44no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4C31×#9912yes—
365C2×C4C31×#10312yes—
366C2×C4C31×#10312yes—
367C2×C4C2×C42×C3111yescharacteristic
368C2×C4C31×#11711yescharacteristic
369C2×C4C4×D4×C3112yes—
370C8—44no—
371C22—22no—
372C22—22no—
373C22—22no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—22no—
379C22C4×C22⋊C4×C3111yescharacteristic
380C22C2×C31×#9911yescharacteristic
381C22C31×#41011yescharacteristic
382C22C2×C31×#10311yescharacteristic
383C22C31×#41611yescharacteristic
384C22C31×#42111yescharacteristic
385C22C31×#42111yescharacteristic
386C22C4×SD16×C3112yes—
387C22C4×Q16×C3112yes—
388C4—22no—
389C4—21no—
390C4—21no—
391C4—22no—
392C4—22no—
393C4—22no—
394C4—42no—
395C4—42no—
396C4—41no—
397C4—84no—
398C4—81no—
399C4—81no—
400C2—22no—
401C2—22no—
402C2C31×#151611yescharacteristic
403C2C31×#151811yescharacteristic
404C2C31×#152011yescharacteristic
405C2C31×#152111yescharacteristic
406C2C4×C31×#9911yescharacteristic
407C2C31×#175911yescharacteristic
408C2C4×C31×#10311yescharacteristic
subgroupG/H
center, socleC23×C31C2×C22⋊C4
commutatorC2×C4C2×C42×C31
Frattini, cosocleC23×C4C23×C31
prime radicalC24×C31C2×D4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C31 ◁ C22×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
7. C2×C4×Q8×C31C4semidirect
11. C31×#391C4semidirect
12. C31×#389C4semidirect
17. C4×C4⋊C4×C31C4semidirect
23. C31×#328C4semidirect
111. #8221C31direct (N × H)
202. C2×C4×Q8C4×C31semidirect
206. #391C4×C31semidirect
207. #389C4×C31semidirect
212. C4×C4⋊C4C4×C31semidirect
218. #328C4×C31semidirect
264. C31#8221direct (N × H)

Maximal subgroups

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

subgroupindex
C4×C31×C22⋊Q82
C4×C31×C22⋊C82
C2×C31×#3282
C31×#14342
C31×#12942
#822131