Finite Group Databasebeta

← C31×#18820 #21475579

Subgroups — C31×#18820 #21475579

Order 7936.

940 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3511C211yesmaximal, characteristic, complemented
2C31×#3214C211yesmaximal, characteristic, complemented
3C31×#3206C211yesmaximal, characteristic, complemented
4C31×#3115C211yesmaximal, characteristic, complemented
5C31×#1646C212yesmaximal, complemented
6C31×#1643C212yesmaximal, complemented
7C31×#1624C212yesmaximal, complemented
8C31×#1622C212yesmaximal, complemented
9C31×#1558C211yesmaximal, characteristic, complemented
10C31×#1496C211yesmaximal, characteristic, complemented
11C31×#1488C211yesmaximal, characteristic, complemented
12C31×#556—21no—
13C31×#555C2212yes—
14C31×#554C2212yes—
15C31×#553—22no—
16C31×#548—22no—
17C31×#547—21no—
18C31×#542—22no—
19C31×#537—21no—
20C31×#533—21no—
21C31×#532—21no—
22C31×#532C2211yescharacteristic
23C31×#532C2211yescharacteristic
24C2×C31×C4⋊Q8C2211yescharacteristic
25C31×#520—21no—
26C4×#41×C31C2211yescharacteristic
27C31×#496—21no—
28C31×#492C2212yes—
29C31×#489—21no—
30C31×#487C2212yes—
31C31×#482—21no—
32C31×#475C2211yescharacteristic
33C31×#474C2211yescharacteristic
34C31×#461—22no—
35C31×#461C2212yes—
36C31×#460C2211yescharacteristic
37C31×#459C2211yescharacteristic
38C31×#459C2212yes—
39C31×#449—21no—
40C31×#449—22no—
41C31×#448—22no—
42C31×#442—21no—
43C31×#440—21no—
44C31×#432C2211yescharacteristic
45C31×#432C2211yescharacteristic
46C31×#431C2211yescharacteristic
47C31×#333C2212yes—
48C31×#333C2212yes—
49C31×Q8⋊C8C2212yes—
50C31×D4⋊C8C2212yes—
51C31×D4⋊C8C2212yes—
52C31×D4⋊C8C2212yes—
53C2×#41×C31—22no—
54C2×#41×C31—21no—
55C2×#41×C31—21no—
56C2×#41×C31D411yescharacteristic
57C22×Q8×C31D412yes—
58C2×SD16×C31—21no—
59C2×SD16×C31—24no—
60C2×SD16×C31—22no—
61C2×D8×C31—21no—
62C2×D8×C31—22no—
63C31×C4⋊Q8—21no—
64C31×C4⋊Q8—21no—
65C31×C4⋊Q8D412yes—
66C31×C4⋊Q8D411yescharacteristic
67C31×C4⋊Q8C2311yescharacteristic
68C31×C4⋊Q8C2312yes—
69C31×#127D411yescharacteristic
70C31×#127C2311yescharacteristic
71C31×#126—24no—
72C31×#125—22no—
73C31×#124—22no—
74C31×#124—24no—
75C31×#124—21no—
76C31×#124—21no—
77C31×#123—24no—
78C31×C22⋊Q8—22no—
79C31×C22⋊Q8—22no—
80C31×C22⋊Q8—22no—
81C31×C22⋊Q8—24no—
82C31×#121—22no—
83C31×#121—24no—
84C31×#121—22no—
85C31×#121—22no—
86C31×#121—21no—
87C31×#121—21no—
88C31×#121—21no—
89C31×#121D412yes—
90C4×Q8×C31—22no—
91C4×Q8×C31—22no—
92C4×Q8×C31D411yescharacteristic
93C4×Q8×C31C2311yescharacteristic
94C4×D4×C31—22no—
95C4×D4×C31—24no—
96C4×D4×C31—22no—
97C4×D4×C31—21no—
98C4×D4×C31—21no—
99C4×D4×C31—21no—
100C4×D4×C31—21no—
101C4×D4×C31—21no—
102C4×D4×C31D411yescharacteristic
103C4×D4×C31C2312yes—
104C4×D4×C31C2311yescharacteristic
105C31×#117—22no—
106C31×#117—22no—
107C31×#117—21no—
108C31×#117D411yescharacteristic
109C2×C4⋊C4×C31—21no—
110C2×C4⋊C4×C31—21no—
111C2×C4⋊C4×C31—22no—
112C2×C42×C31—21no—
113C2×C42×C31C2311yescharacteristic
114C31×#107—21no—
115C31×#106—21no—
116C31×C4⋊C8—21no—
117C31×C4⋊C8C2312yes—
118C31×C4⋊C8C2311yescharacteristic
119C31×C4⋊C8C2311yescharacteristic
120C31×#103—21no—
121C31×#103—24no—
122C31×#103—22no—
123C31×#102—21no—
124C31×#102—21no—
125C31×#102—21no—
126C31×#102—22no—
127C31×C22⋊C8—21no—
128C31×C22⋊C8—21no—
129C31×C22⋊C8—21no—
130C31×#97—21no—
131C31×#97—21no—
132C4×C8×C31C2312yes—
133#41×C31—41no—
134#41×C31—41no—
135#41×C31—42no—
136#41×C31—42no—
137#41×C31—82no—
138#41×C31—84no—
139C2×Q8×C31—21no—
140C2×Q8×C31—22no—
141C2×Q8×C31—21no—
142C2×Q8×C31—82no—
143C2×Q8×C31C2×D411yescharacteristic
144C2×Q8×C31C2×D414yes—
145C2×D4×C31—21no—
146C2×D4×C31—22no—
147C2×D4×C31—24no—
148C2×D4×C31—21no—
149C2×D4×C31—21no—
150C2×D4×C31—21no—
151C2×D4×C31—21no—
152C2×D4×C31—21no—
153C2×D4×C31—41no—
154C2×D4×C31C2×D411yescharacteristic
155C2×D4×C31C2×D412yes—
156C22×C4×C31—24no—
157C22×C4×C31—22no—
158C22×C4×C31—21no—
159C22×C4×C31—21no—
160C22×C4×C31—21no—
161C22×C4×C31—22no—
162C22×C4×C31—41no—
163C22×C4×C31C2×D411yescharacteristic
164C22×C4×C31C2×D411yescharacteristic
165C22×C4×C31C2×D411yescharacteristic
166SD16×C31—42no—
167SD16×C31—82no—
168SD16×C31—84no—
169D8×C31—42no—
170D8×C31—82no—
171C2×C8×C31—22no—
172C2×C8×C31—21no—
173C2×C8×C31—21no—
174C2×C8×C31—22no—
175C4⋊C4×C31—24no—
176C4⋊C4×C31—24no—
177C4⋊C4×C31—22no—
178C4⋊C4×C31—22no—
179C4⋊C4×C31—21no—
180C4⋊C4×C31—21no—
181C4⋊C4×C31—22no—
182C4⋊C4×C31—22no—
183C4⋊C4×C31—21no—
184C4⋊C4×C31—21no—
185C4⋊C4×C31—21no—
186C4⋊C4×C31C2×D412yes—
187C4⋊C4×C31C2×D411yescharacteristic
188C4⋊C4×C31C2×D411yescharacteristic
189C22⋊C4×C31—24no—
190C22⋊C4×C31—24no—
191C22⋊C4×C31—24no—
192C22⋊C4×C31—21no—
193C22⋊C4×C31—21no—
194C22⋊C4×C31—22no—
195C22⋊C4×C31—21no—
196C22⋊C4×C31—22no—
197C22⋊C4×C31—41no—
198C22⋊C4×C31—41no—
199C42×C31—24no—
200C42×C31—21no—
201C42×C31—21no—
202C42×C31—21no—
203C42×C31C2×D411yescharacteristic
204C42×C31C2×D411yescharacteristic
205C42×C31C2×D411yescharacteristic
206C42×C31C2411yescharacteristic
207#18820C3111yesmaximal, characteristic, direct product factor, Sylow
208C23×C31—21no—
209C23×C31—22no—
210C23×C31—41no—
211C23×C31#12011yescharacteristic
212Q8×C31—21no—
213Q8×C31—21no—
214Q8×C31—44no—
215Q8×C31—44no—
216Q8×C31—42no—
217Q8×C31—42no—
218D4×C31—21no—
219D4×C31—21no—
220D4×C31—22no—
221D4×C31—22no—
222D4×C31—42no—
223D4×C31—42no—
224D4×C31—42no—
225D4×C31—42no—
226D4×C31—44no—
227D4×C31—41no—
228D4×C31—41no—
229D4×C31—42no—
230D4×C31—84no—
231D4×C31—82no—
232C2×C4×C31—21no—
233C2×C4×C31—21no—
234C2×C4×C31—22no—
235C2×C4×C31—21no—
236C2×C4×C31—21no—
237C2×C4×C31—22no—
238C2×C4×C31—21no—
239C2×C4×C31—24no—
240C2×C4×C31—41no—
241C2×C4×C31—42no—
242C2×C4×C31—42no—
243C2×C4×C31—42no—
244C2×C4×C31—42no—
245C2×C4×C31—42no—
246C2×C4×C31—41no—
247C2×C4×C31—82no—
248C2×C4×C31—82no—
249C2×C4×C31—84no—
250C2×C4×C31#12011yescharacteristic
251C2×C4×C31#12011yescharacteristic
252C2×C4×C31#12011yescharacteristic
253C2×C4×C31C22×D411yescharacteristic
254C2×C4×C31C22×D411yescharacteristic
255C2×C4×C31C22×D411yescharacteristic
256C8×C31—42no—
257C8×C31—41no—
258C8×C31—41no—
259C8×C31—42no—
260#3511C2×C3111yescharacteristic, complemented
261#3214C2×C3111yescharacteristic, complemented
262#3206C2×C3111yescharacteristic, complemented
263#3115C2×C3111yescharacteristic, complemented
264#1646C2×C3112yescomplemented
265#1643C2×C3112yescomplemented
266#1624C2×C3112yescomplemented
267#1622C2×C3112yescomplemented
268#1558C2×C3111yescharacteristic, complemented
269#1496C2×C3111yescharacteristic, complemented
270#1488C2×C3111yescharacteristic, complemented
271C22×C31—21no—
272C22×C31—21no—
273C22×C31—21no—
274C22×C31—41no—
275C22×C31—42no—
276C22×C31—41no—
277C22×C31—42no—
278C22×C31—41no—
279C22×C31—42no—
280C22×C31—81no—
281C22×C31—81no—
282C22×C31—81no—
283C22×C31C2×#12011yescharacteristic
284C4×C31—21no—
285C4×C31—21no—
286C4×C31—21no—
287C4×C31—21no—
288C4×C31—41no—
289C4×C31—42no—
290C4×C31—41no—
291C4×C31—41no—
292C4×C31—44no—
293C4×C31—81no—
294C4×C31#57511yescharacteristic
295C4×C31#57511yescharacteristic
296C4×C31#57712yes—
297#556—21no—
298#555C22×C3112yes—
299#554C22×C3112yes—
300#553—22no—
301#548—22no—
302#547—21no—
303#542—22no—
304#537—21no—
305#533—21no—
306#532—21no—
307#532C22×C3111yescharacteristic
308#532C22×C3111yescharacteristic
309C2×C4⋊Q8C22×C3111yescharacteristic
310#520—21no—
311C4×#41C22×C3111yescharacteristic
312#496—21no—
313#492C22×C3112yes—
314#489—21no—
315#487C22×C3112yes—
316#482—21no—
317#475C22×C3111yescharacteristic
318#474C22×C3111yescharacteristic
319#461—22no—
320#461C22×C3112yes—
321#460C22×C3111yescharacteristic
322#459C22×C3111yescharacteristic
323#459C22×C3112yes—
324#449—21no—
325#449—22no—
326#448—22no—
327#442—21no—
328#440—21no—
329#432C22×C3111yescharacteristic
330#432C22×C3111yescharacteristic
331#431C22×C3111yescharacteristic
332#333C22×C3112yes—
333#333C22×C3112yes—
334Q8⋊C8C22×C3112yes—
335D4⋊C8C22×C3112yes—
336D4⋊C8C22×C3112yes—
337D4⋊C8C22×C3112yes—
338C2×C31—41no—
339C2×C31—81no—
340C2×C31—82no—
341C2×C31—161no—
342C2×C31#300511yescharacteristic
343C2×C31#301011yescharacteristic
344C2×C31#301911yescharacteristic
345C2×#41—22no—
346C2×#41—21no—
347C2×#41—21no—
348C2×#41D4×C3111yescharacteristic
349C22×Q8D4×C3112yes—
350C2×SD16—21no—
351C2×SD16—24no—
352C2×SD16—22no—
353C2×D8—21no—
354C2×D8—22no—
355C4⋊Q8—21no—
356C4⋊Q8—21no—
357C4⋊Q8D4×C3112yes—
358C4⋊Q8D4×C3111yescharacteristic
359C4⋊Q8C23×C3111yescharacteristic
360C4⋊Q8C23×C3112yes—
361#127D4×C3111yescharacteristic
362#127C23×C3111yescharacteristic
363#126—24no—
364#125—22no—
365#124—22no—
366#124—24no—
367#124—21no—
368#124—21no—
369#123—24no—
370C22⋊Q8—22no—
371C22⋊Q8—22no—
372C22⋊Q8—22no—
373C22⋊Q8—24no—
374#121—22no—
375#121—24no—
376#121—22no—
377#121—22no—
378#121—21no—
379#121—21no—
380#121—21no—
381#121D4×C3112yes—
382C4×Q8—22no—
383C4×Q8—22no—
384C4×Q8D4×C3111yescharacteristic
385C4×Q8C23×C3111yescharacteristic
386C4×D4—22no—
387C4×D4—24no—
388C4×D4—22no—
389C4×D4—21no—
390C4×D4—21no—
391C4×D4—21no—
392C4×D4—21no—
393C4×D4—21no—
394C4×D4D4×C3111yescharacteristic
395C4×D4C23×C3111yescharacteristic
396C4×D4C23×C3112yes—
397#117—22no—
398#117—22no—
399#117—21no—
400#117D4×C3111yescharacteristic
401C2×C4⋊C4—21no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4—22no—
404C2×C42—21no—
405C2×C42C23×C3111yescharacteristic
406#107—21no—
407#106—21no—
408C4⋊C8—21no—
409C4⋊C8C23×C3111yescharacteristic
410C4⋊C8C23×C3111yescharacteristic
411C4⋊C8C23×C3112yes—
412#103—21no—
413#103—24no—
414#103—22no—
415#102—21no—
416#102—21no—
417#102—21no—
418#102—22no—
419C22⋊C8—21no—
420C22⋊C8—21no—
421C22⋊C8—21no—
422#97—21no—
423#97—21no—
424C4×C8C23×C3112yes—
425C31#1882011yescharacteristic, direct product factor, Sylow
426#41—41no—
427#41—41no—
428#41—42no—
429#41—42no—
430#41—82no—
431#41—84no—
432C2×Q8—21no—
433C2×Q8—22no—
434C2×Q8—21no—
435C2×Q8—82no—
436C2×Q8C2×D4×C3111yescharacteristic
437C2×Q8C2×D4×C3114yes—
438C2×D4—21no—
439C2×D4—22no—
440C2×D4—24no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—41no—
447C2×D4C2×D4×C3111yescharacteristic
448C2×D4C2×D4×C3112yes—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—41no—
456C22×C4C2×D4×C3111yescharacteristic
457C22×C4C2×D4×C3111yescharacteristic
458C22×C4C2×D4×C3111yescharacteristic
459SD16—42no—
460SD16—82no—
461SD16—84no—
462D8—42no—
463D8—82no—
464C2×C8—22no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—22no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4C2×D4×C3112yes—
480C4⋊C4C2×D4×C3111yescharacteristic
481C4⋊C4C2×D4×C3111yescharacteristic
482C22⋊C4—24no—
483C22⋊C4—24no—
484C22⋊C4—24no—
485C22⋊C4—21no—
486C22⋊C4—21no—
487C22⋊C4—22no—
488C22⋊C4—21no—
489C22⋊C4—22no—
490C22⋊C4—41no—
491C22⋊C4—41no—
492C42—24no—
493C42—21no—
494C42—21no—
495C42—21no—
496C42C2×D4×C3111yescharacteristic
497C42C2×D4×C3111yescharacteristic
498C42C2×D4×C3111yescharacteristic
499C42C24×C3111yescharacteristic
500C23—21no—
501C23—22no—
502C23—41no—
503C23C31×#12011yescharacteristic
504Q8—21no—
505Q8—21no—
506Q8—44no—
507Q8—44no—
508Q8—42no—
509Q8—42no—
510D4—21no—
511D4—21no—
512D4—22no—
513D4—22no—
514D4—42no—
515D4—42no—
516D4—42no—
517D4—42no—
518D4—44no—
519D4—41no—
520D4—41no—
521D4—42no—
522D4—84no—
523D4—82no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—24no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—84no—
542C2×C4C31×#12011yescharacteristic
543C2×C4C31×#12011yescharacteristic
544C2×C4C31×#12011yescharacteristic
545C2×C4C22×D4×C3111yescharacteristic
546C2×C4C22×D4×C3111yescharacteristic
547C2×C4C22×D4×C3111yescharacteristic
548C8—42no—
549C8—41no—
550C8—41no—
551C8—42no—
552C22—21no—
553C22—21no—
554C22—21no—
555C22—41no—
556C22—42no—
557C22—41no—
558C22—42no—
559C22—41no—
560C22—42no—
561C22—81no—
562C22—81no—
563C22—81no—
564C22C2×C31×#12011yescharacteristic
565C4—21no—
566C4—21no—
567C4—21no—
568C4—21no—
569C4—41no—
570C4—42no—
571C4—41no—
572C4—41no—
573C4—44no—
574C4—81no—
575C4C31×#57511yescharacteristic
576C4C31×#57511yescharacteristic
577C4C31×#57712yes—
578C2—41no—
579C2—81no—
580C2—82no—
581C2—161no—
582C2C31×#300511yescharacteristic
583C2C31×#301011yescharacteristic
584C2C31×#301911yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC42C24×C31
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#3511C2semidirect
2. C31×#3214C2semidirect
3. C31×#3206C2semidirect
4. C31×#3115C2semidirect
5. C31×#1646C2semidirect
6. C31×#1643C2semidirect
7. C31×#1624C2semidirect
8. C31×#1622C2semidirect
9. C31×#1558C2semidirect
10. C31×#1496C2semidirect
11. C31×#1488C2semidirect
207. #18820C31direct (N × H)
260. #3511C2×C31semidirect
261. #3214C2×C31semidirect
262. #3206C2×C31semidirect
263. #3115C2×C31semidirect
264. #1646C2×C31semidirect
265. #1643C2×C31semidirect
266. #1624C2×C31semidirect
267. #1622C2×C31semidirect
268. #1558C2×C31semidirect
269. #1496C2×C31semidirect
270. #1488C2×C31semidirect
425. C31#18820direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35112
C31×#32142
C31×#32062
C31×#31152
C31×#16462
C31×#16432
C31×#16242
C31×#16222
C31×#15582
C31×#14962
C31×#14882
#1882031