Finite Group Databasebeta

← C31×#8572 #21465331

Subgroups — C31×#8572 #21465331

Order 7936.

658 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×C31×C22⋊Q8C211yesmaximal, characteristic
2C31×#1797C211yesmaximal, characteristic
3C2×C31×#328C212yesmaximal
4C31×C4⋊#95C211yesmaximal, characteristic
5C31×#1294C212yesmaximal
6C2×C31×C22⋊Q8C2211yescharacteristic
7C2×C4×Q8×C31C411yescharacteristic, complemented
8C22×C4⋊C4×C31—21no—
9C22×C42×C31C2211yescharacteristic
10C2×C31×C4⋊C8—21no—
11C2×C31×C22⋊C8C2212yes—
12C31×#391C411yescharacteristic, complemented
13C31×#389C411yescharacteristic, complemented
14C31×#387—21no—
15C31×#385—21no—
16C31×#380C2211yescharacteristic
17C4×C4⋊C4×C31—21no—
18C4×C4⋊C4×C31C411yescharacteristic, complemented
19C4×C22⋊C4×C31—21no—
20C2×C31×#95—21no—
21C2×C31×#95C2212yes—
22C31×#336—21no—
23C31×#328—24no—
24C22×Q8×C31C2×C411yescharacteristic
25C23×C4×C31—22no—
26C23×C4×C31D411yescharacteristic
27C23×C4×C31D411yescharacteristic
28C23×C4×C31C2311yescharacteristic
29C22×C8×C31—22no—
30C31×C22⋊Q8—21no—
31C31×C22⋊Q8—21no—
32C31×C22⋊Q8D412yes—
33C31×C22⋊Q8Q812yes—
34C4×Q8×C31—42no—
35C2×C4⋊C4×C31—21no—
36C2×C4⋊C4×C31—24no—
37C2×C4⋊C4×C31—21no—
38C2×C4⋊C4×C31—21no—
39C2×C4⋊C4×C31—42no—
40C2×C4⋊C4×C31—42no—
41C2×C4⋊C4×C31C2×C411yescharacteristic
42C2×C4⋊C4×C31C2×C411yescharacteristic
43C2×C4⋊C4×C31C2×C411yescharacteristic
44C2×C22⋊C4×C31—21no—
45C2×C22⋊C4×C31—21no—
46C2×C42×C31—21no—
47C2×C42×C31—21no—
48C2×C42×C31—42no—
49C2×C42×C31C2×C411yescharacteristic
50C2×C42×C31C2×C411yescharacteristic
51C2×C42×C31D411yescharacteristic
52C2×C42×C31D411yescharacteristic
53C31×C4⋊C8—42no—
54C31×C22⋊C8—22no—
55C31×C22⋊C8—22no—
56C31×#95—21no—
57C31×#95—21no—
58C31×#95—21no—
59C31×#95—21no—
60C31×#95—21no—
61C31×#95—22no—
62C31×#95—22no—
63C31×#95—42no—
64C24×C31C2×D411yescharacteristic
65C2×Q8×C31—21no—
66C2×Q8×C31—41no—
67C2×Q8×C31—41no—
68C2×Q8×C31C4⋊C412yescomplemented
69C22×C4×C31—22no—
70C22×C4×C31—22no—
71C22×C4×C31—21no—
72C22×C4×C31—21no—
73C22×C4×C31—21no—
74C22×C4×C31—21no—
75C22×C4×C31—24no—
76C22×C4×C31—21no—
77C22×C4×C31—21no—
78C22×C4×C31—41no—
79C22×C4×C31—41no—
80C22×C4×C31—41no—
81C22×C4×C31—41no—
82C22×C4×C31—42no—
83C22×C4×C31—42no—
84C22×C4×C31—42no—
85C22×C4×C31—42no—
86C22×C4×C31—42no—
87C22×C4×C31—42no—
88C22×C4×C31C22⋊C411yescharacteristic
89C22×C4×C31C22⋊C411yescharacteristic
90C22×C4×C31C22⋊C411yescharacteristic
91C22×C4×C31C22⋊C411yescharacteristic
92C22×C4×C31SD1612yes—
93C22×C4×C31Q1612yes—
94C22×C4×C31C22×C411yescharacteristic
95C22×C4×C31C2×D411yescharacteristic
96C22×C4×C31C2×D411yescharacteristic
97C22×C4×C31C2×Q811yescharacteristic
98C22×C4×C31#4111yescharacteristic
99C22×C4×C31#4111yescharacteristic
100C2×C8×C31—42no—
101C2×C8×C31—42no—
102C2×C8×C31—42no—
103C4⋊C4×C31—21no—
104C4⋊C4×C31—22no—
105C4⋊C4×C31—22no—
106C4⋊C4×C31—42no—
107C4⋊C4×C31—48no—
108C4⋊C4×C31—42no—
109C4⋊C4×C31—41no—
110C4⋊C4×C31—41no—
111C4⋊C4×C31C4⋊C412yescomplemented
112C22⋊C4×C31—22no—
113C22⋊C4×C31—22no—
114C22⋊C4×C31—41no—
115C22⋊C4×C31—41no—
116C42×C31—22no—
117C42×C31—22no—
118C42×C31—41no—
119C42×C31—41no—
120C42×C31—42no—
121C42×C31—42no—
122#8572C3111yesmaximal, characteristic, direct product factor, Sylow
123C23×C31—21no—
124C23×C31—21no—
125C23×C31—21no—
126C23×C31—21no—
127C23×C31C2×C22⋊C411yescharacteristic
128C23×C31#12111yescharacteristic
129C23×C31C22⋊Q811yescharacteristic
130C23×C31C2×SD1611yescharacteristic
131C23×C31C2×Q1611yescharacteristic
132C23×C31AGL(1,ℤ/8)11yescharacteristic
133C23×C31#13711yescharacteristic
134Q8×C31—42no—
135Q8×C31—42no—
136C2×C4×C31—21no—
137C2×C4×C31—21no—
138C2×C4×C31—21no—
139C2×C4×C31—21no—
140C2×C4×C31—21no—
141C2×C4×C31—21no—
142C2×C4×C31—24no—
143C2×C4×C31—24no—
144C2×C4×C31—22no—
145C2×C4×C31—22no—
146C2×C4×C31—21no—
147C2×C4×C31—21no—
148C2×C4×C31—21no—
149C2×C4×C31—21no—
150C2×C4×C31—21no—
151C2×C4×C31—21no—
152C2×C4×C31—41no—
153C2×C4×C31—41no—
154C2×C4×C31—41no—
155C2×C4×C31—41no—
156C2×C4×C31—42no—
157C2×C4×C31—42no—
158C2×C4×C31—41no—
159C2×C4×C31—41no—
160C2×C4×C31—41no—
161C2×C4×C31—41no—
162C2×C4×C31—44no—
163C2×C4×C31—44no—
164C2×C4×C31—44no—
165C2×C4×C31—41no—
166C2×C4×C31—41no—
167C2×C4×C31—41no—
168C2×C4×C31—41no—
169C2×C4×C31—41no—
170C2×C4×C31—41no—
171C2×C4×C31—82no—
172C2×C4×C31—82no—
173C2×C4×C31—82no—
174C2×C4×C31—82no—
175C2×C4×C31#9912yes—
176C2×C4×C31#10312yes—
177C2×C4×C31#10312yes—
178C2×C4×C31#10412yes—
179C2×C4×C31C2×C4⋊C411yescharacteristic
180C2×C4×C31#11711yescharacteristic
181C2×C4×C31#12111yescharacteristic
182C2×C4×C31C22⋊Q811yescharacteristic
183C8×C31—82no—
184C4×C22⋊Q8C2×C3111yescharacteristic
185#1797C2×C3111yescharacteristic
186C2×#328C2×C3112yes—
187C4⋊#95C2×C3111yescharacteristic
188#1294C2×C3112yes—
189C22×C31—21no—
190C22×C31—21no—
191C22×C31—21no—
192C22×C31—21no—
193C22×C31—21no—
194C22×C31—21no—
195C22×C31—21no—
196C22×C31—21no—
197C22×C31—21no—
198C22×C31—21no—
199C22×C31—21no—
200C22×C31—21no—
201C22×C31#38011yescharacteristic
202C22×C31C2×#9911yescharacteristic
203C22×C31#41011yescharacteristic
204C22×C31C2×#10311yescharacteristic
205C22×C31#41711yescharacteristic
206C22×C31C2×#10411yescharacteristic
207C22×C31#42111yescharacteristic
208C22×C31#46011yescharacteristic
209C22×C31#46211yescharacteristic
210C22×C31#47511yescharacteristic
211C22×C31#47711yescharacteristic
212C4×C31—21no—
213C4×C31—21no—
214C4×C31—21no—
215C4×C31—21no—
216C4×C31—22no—
217C4×C31—42no—
218C4×C31—42no—
219C4×C31—41no—
220C4×C31—41no—
221C4×C31—84no—
222C4×C31—81no—
223C4×C31—81no—
224C4×C31#32814yescomplemented
225C2×C22⋊Q8C22×C3111yescharacteristic
226C2×C4×Q8C4×C3111yescharacteristic, complemented
227C22×C4⋊C4—21no—
228C22×C42C22×C3111yescharacteristic
229C2×C4⋊C8—21no—
230C2×C22⋊C8C22×C3112yes—
231#391C4×C3111yescharacteristic, complemented
232#389C4×C3111yescharacteristic, complemented
233#387—21no—
234#385—21no—
235#380C22×C3111yescharacteristic
236C4×C4⋊C4—21no—
237C4×C4⋊C4C4×C3111yescharacteristic, complemented
238C4×C22⋊C4—21no—
239C2×#95—21no—
240C2×#95C22×C3112yes—
241#336—21no—
242#328—24no—
243C2×C31—21no—
244C2×C31—21no—
245C2×C31—21no—
246C2×C31—21no—
247C2×C31C2×#32811yescharacteristic
248C2×C31#150911yescharacteristic
249C2×C31#151311yescharacteristic
250C2×C31#151811yescharacteristic
251C2×C31#179911yescharacteristic
252C2×C31#180311yescharacteristic
253C2×C31#180611yescharacteristic
254C22×Q8C2×C4×C3111yescharacteristic
255C23×C4—22no—
256C23×C4D4×C3111yescharacteristic
257C23×C4D4×C3111yescharacteristic
258C23×C4C23×C3111yescharacteristic
259C22×C8—22no—
260C22⋊Q8—21no—
261C22⋊Q8—21no—
262C22⋊Q8D4×C3112yes—
263C22⋊Q8Q8×C3112yes—
264C4×Q8—42no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4C2×C4×C3111yescharacteristic
272C2×C4⋊C4C2×C4×C3111yescharacteristic
273C2×C4⋊C4C2×C4×C3111yescharacteristic
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—42no—
279C2×C42C2×C4×C3111yescharacteristic
280C2×C42C2×C4×C3111yescharacteristic
281C2×C42D4×C3111yescharacteristic
282C2×C42D4×C3111yescharacteristic
283C4⋊C8—42no—
284C22⋊C8—22no—
285C22⋊C8—22no—
286#95—21no—
287#95—21no—
288#95—21no—
289#95—21no—
290#95—21no—
291#95—22no—
292#95—22no—
293#95—42no—
294C31#857211yescharacteristic, direct product factor, Sylow
295C24C2×D4×C3111yescharacteristic
296C2×Q8—21no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8C4⋊C4×C3112yescomplemented
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—24no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4C22⋊C4×C3111yescharacteristic
320C22×C4C22⋊C4×C3111yescharacteristic
321C22×C4C22⋊C4×C3111yescharacteristic
322C22×C4C22⋊C4×C3111yescharacteristic
323C22×C4SD16×C3112yes—
324C22×C4Q16×C3112yes—
325C22×C4C22×C4×C3111yescharacteristic
326C22×C4C2×D4×C3111yescharacteristic
327C22×C4C2×D4×C3111yescharacteristic
328C22×C4C2×Q8×C3111yescharacteristic
329C22×C4#41×C3111yescharacteristic
330C22×C4#41×C3111yescharacteristic
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—42no—
334C4⋊C4—21no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—42no—
340C4⋊C4—48no—
341C4⋊C4—42no—
342C4⋊C4C4⋊C4×C3112yescomplemented
343C22⋊C4—22no—
344C22⋊C4—22no—
345C22⋊C4—41no—
346C22⋊C4—41no—
347C42—22no—
348C42—22no—
349C42—41no—
350C42—41no—
351C42—42no—
352C42—42no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23C2×C22⋊C4×C3111yescharacteristic
358C23C31×#12111yescharacteristic
359C23C31×C22⋊Q811yescharacteristic
360C23C2×SD16×C3111yescharacteristic
361C23C2×Q16×C3111yescharacteristic
362C23C31×AGL(1,ℤ/8)11yescharacteristic
363C23C31×#13711yescharacteristic
364Q8—42no—
365Q8—42no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4C31×#9912yes—
406C2×C4C31×#10312yes—
407C2×C4C31×#10312yes—
408C2×C4C31×#10412yes—
409C2×C4C2×C4⋊C4×C3111yescharacteristic
410C2×C4C31×#11711yescharacteristic
411C2×C4C31×#12111yescharacteristic
412C2×C4C31×C22⋊Q811yescharacteristic
413C8—82no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22C31×#38011yescharacteristic
427C22C2×C31×#9911yescharacteristic
428C22C31×#41011yescharacteristic
429C22C2×C31×#10311yescharacteristic
430C22C31×#41711yescharacteristic
431C22C2×C31×#10411yescharacteristic
432C22C31×#42111yescharacteristic
433C22C31×#46011yescharacteristic
434C22C31×#46211yescharacteristic
435C22C31×#47511yescharacteristic
436C22C31×#47711yescharacteristic
437C4—21no—
438C4—21no—
439C4—21no—
440C4—21no—
441C4—22no—
442C4—42no—
443C4—42no—
444C4—41no—
445C4—41no—
446C4—84no—
447C4—81no—
448C4—81no—
449C4C31×#32814yescomplemented
450C2—21no—
451C2—21no—
452C2—21no—
453C2—21no—
454C2C2×C31×#32811yescharacteristic
455C2C31×#150911yescharacteristic
456C2C31×#151311yescharacteristic
457C2C31×#151811yescharacteristic
458C2C31×#179911yescharacteristic
459C2C31×#180311yescharacteristic
460C2C31×#180611yescharacteristic
subgroupG/H
center, socleC23×C31C2×C22⋊C4
commutatorC22×C4C22×C4×C31
Frattini, cosocleC23×C4C23×C31
prime radicalC24×C31C2×D4
LCSG ▷ C22×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
12. C31×#391C4semidirect
13. C31×#389C4semidirect
18. C4×C4⋊C4×C31C4semidirect
68. C2×Q8×C31C4⋊C4semidirect
111. C4⋊C4×C31C4⋊C4semidirect
122. #8572C31direct (N × H)
224. C4×C31#328semidirect
226. C2×C4×Q8C4×C31semidirect
231. #391C4×C31semidirect
232. #389C4×C31semidirect
237. C4×C4⋊C4C4×C31semidirect
294. C31#8572direct (N × H)
299. C2×Q8C4⋊C4×C31semidirect
342. C4⋊C4C4⋊C4×C31semidirect
449. C4C31×#328semidirect

Maximal subgroups

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

subgroupindex
C4×C31×C22⋊Q82
C31×#17972
C2×C31×#3282
C31×C4⋊#952
C31×#12942
#857231