Finite Group Databasebeta

← C31×#18680 #21475439

Subgroups — C31×#18680 #21475439

Order 7936.

1000 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3506C211yesmaximal, characteristic, complemented
2C31×#3245C211yesmaximal, characteristic, complemented
3C31×#3167C211yesmaximal, characteristic, complemented
4C31×#3036C211yesmaximal, characteristic, complemented
5C31×#1654C212yesmaximal, complemented
6C31×#1646C212yesmaximal, complemented
7C31×#1625C212yesmaximal, complemented
8C31×#1624C212yesmaximal, complemented
9C31×#1550C211yesmaximal, characteristic, complemented
10C31×#1492C211yesmaximal, characteristic, complemented
11C2×C31×D4⋊C8C211yesmaximal, characteristic, complemented
12C2×C31×AGL(1,ℤ/8)—21no—
13C31×#559—22no—
14C31×#556—21no—
15C31×#555C2212yes—
16C31×#553—22no—
17C31×#550—21no—
18C31×#547C2212yes—
19C31×#538—22no—
20C31×#535—21no—
21C31×#533C2211yescharacteristic
22C31×#532C2211yescharacteristic
23C31×#532C2211yescharacteristic
24C31×#525—22no—
25C31×#520—21no—
26C2×C4×D4×C31C2211yescharacteristic
27C31×#498C2211yescharacteristic, complemented
28C31×#492C2211yescharacteristic, complemented
29C31×#487C2211yescharacteristic, complemented
30C31×#486C2211yescharacteristic, complemented
31C31×#474—21no—
32C31×#474C2212yescomplemented
33C31×#466—22no—
34C31×#465—22no—
35C31×#463—22no—
36C31×#461C2212yes—
37C31×#459—21no—
38C31×#459C2212yes—
39C31×#459C2212yescomplemented
40C4×SD16×C31—22no—
41C31×#423C2211yescharacteristic
42C2×C31×C4⋊C8C2211yescharacteristic
43C31×#417—21no—
44C2×C4×C8×C31C2211yescharacteristic
45C31×#333C2212yescomplemented
46C31×#332C2212yescomplemented
47C31×D4⋊C8C2212yes—
48C31×D4⋊C8C2212yes—
49C31×D4⋊C8C2212yescomplemented
50C31×D4⋊C8C2212yescomplemented
51C2×#41×C31—22no—
52C2×#41×C31—21no—
53C2×#41×C31—21no—
54C22×D4×C31D411yescharacteristic
55C23×C4×C31—21no—
56C31×AGL(1,ℤ/8)—44no—
57C2×SD16×C31—22no—
58C2×SD16×C31—22no—
59C2×D8×C31—22no—
60C2×D8×C31—22no—
61C2×M16×C31—21no—
62C22×C8×C31—21no—
63C22×C8×C31—21no—
64C31×C4⋊Q8D411yescharacteristic
65C31×C4⋊Q8C2311yescharacteristic
66C31×C4⋊Q8C2311yescharacteristic
67C31×C4⋊Q8C2311yescharacteristic
68C31×#127—22no—
69C31×#127D411yescharacteristic
70C31×#127C2311yescharacteristic
71C31×#126—24no—
72C31×#125—21no—
73C31×#124—22no—
74C31×#124—24no—
75C31×#124—21no—
76C31×#124—21no—
77C31×#123—24no—
78C31×#123—24no—
79C31×C22⋊Q8—21no—
80C31×C22⋊Q8—22no—
81C31×C22⋊Q8D412yes—
82C31×#121—24no—
83C31×#121—22no—
84C31×#121—22no—
85C31×#121—22no—
86C31×#121—21no—
87C31×#121—21no—
88C31×#121—21no—
89C31×#121D412yes—
90C31×#120—24no—
91C4×Q8×C31—22no—
92C4×Q8×C31—21no—
93C4×Q8×C31D412yes—
94C4×D4×C31—24no—
95C4×D4×C31—22no—
96C4×D4×C31—24no—
97C4×D4×C31—21no—
98C4×D4×C31—21no—
99C4×D4×C31—21no—
100C4×D4×C31—21no—
101C4×D4×C31—21no—
102C4×D4×C31D412yes—
103C4×D4×C31C2312yes—
104C4×D4×C31C2312yes—
105C31×#117—21no—
106C31×#117—22no—
107C31×#117—22no—
108C2×C4⋊C4×C31—22no—
109C2×C4⋊C4×C31D411yescharacteristic
110C2×C22⋊C4×C31—21no—
111C2×C42×C31C2311yescharacteristic
112C31×#107—22no—
113C31×#106—22no—
114C31×C4⋊C8—21no—
115C31×C4⋊C8—21no—
116C31×C4⋊C8C2312yes—
117C31×C4⋊C8C2312yes—
118C31×#103—22no—
119C31×#103—22no—
120C31×#102—22no—
121C31×#102—22no—
122C31×#102—22no—
123C31×#102—22no—
124C4×C8×C31—21no—
125C4×C8×C31C2311yescharacteristic
126C4×C8×C31C2311yescharacteristic
127C24×C31—21no—
128#41×C31—42no—
129#41×C31—42no—
130#41×C31—42no—
131#41×C31—42no—
132#41×C31—82no—
133#41×C31—82no—
134C2×Q8×C31—21no—
135C2×Q8×C31—21no—
136C2×Q8×C31—21no—
137C2×Q8×C31C2×D412yes—
138C2×D4×C31—22no—
139C2×D4×C31—24no—
140C2×D4×C31—24no—
141C2×D4×C31—21no—
142C2×D4×C31—21no—
143C2×D4×C31—21no—
144C2×D4×C31—22no—
145C2×D4×C31—22no—
146C2×D4×C31—21no—
147C2×D4×C31—21no—
148C2×D4×C31—21no—
149C2×D4×C31—41no—
150C2×D4×C31C2×D412yes—
151C2×D4×C31C2×D412yes—
152C22×C4×C31—24no—
153C22×C4×C31—21no—
154C22×C4×C31—22no—
155C22×C4×C31—22no—
156C22×C4×C31—22no—
157C22×C4×C31—41no—
158C22×C4×C31—81no—
159C22×C4×C31—81no—
160C22×C4×C31C2×D411yescharacteristic
161C22×C4×C31C2×D411yescharacteristic
162C22×C4×C31C2×D411yescharacteristic
163SD16×C31—44no—
164SD16×C31—44no—
165D8×C31—44no—
166D8×C31—82no—
167M16×C31—42no—
168C2×C8×C31—21no—
169C2×C8×C31—21no—
170C2×C8×C31—22no—
171C2×C8×C31—22no—
172C2×C8×C31—42no—
173C2×C8×C31—81no—
174C4⋊C4×C31—24no—
175C4⋊C4×C31—21no—
176C4⋊C4×C31—22no—
177C4⋊C4×C31—21no—
178C4⋊C4×C31—22no—
179C4⋊C4×C31—21no—
180C4⋊C4×C31—21no—
181C4⋊C4×C31—21no—
182C4⋊C4×C31—21no—
183C4⋊C4×C31—21no—
184C4⋊C4×C31C2×D412yes—
185C4⋊C4×C31C2×D412yes—
186C4⋊C4×C31C2×D412yes—
187C22⋊C4×C31—24no—
188C22⋊C4×C31—24no—
189C22⋊C4×C31—24no—
190C22⋊C4×C31—22no—
191C22⋊C4×C31—22no—
192C22⋊C4×C31—22no—
193C22⋊C4×C31—22no—
194C22⋊C4×C31—22no—
195C22⋊C4×C31—41no—
196C22⋊C4×C31—41no—
197C42×C31—24no—
198C42×C31—22no—
199C42×C31C2×D411yescharacteristic
200C42×C31C2×D411yescharacteristic
201C42×C31C2×D411yescharacteristic
202C42×C31C2411yescharacteristic
203#18680C3111yesmaximal, characteristic, direct product factor, Sylow
204C23×C31—22no—
205C23×C31—22no—
206C23×C31—42no—
207C23×C31—41no—
208C23×C31—81no—
209C23×C31—81no—
210C23×C31#12011yescharacteristic
211Q8×C31—42no—
212Q8×C31—41no—
213Q8×C31—41no—
214Q8×C31—42no—
215Q8×C31—41no—
216Q8×C31—41no—
217D4×C31—22no—
218D4×C31—22no—
219D4×C31—22no—
220D4×C31—22no—
221D4×C31—42no—
222D4×C31—44no—
223D4×C31—41no—
224D4×C31—41no—
225D4×C31—41no—
226D4×C31—41no—
227D4×C31—42no—
228D4×C31—42no—
229D4×C31—81no—
230D4×C31—84no—
231D4×C31—84no—
232D4×C31—82no—
233C2×C4×C31—21no—
234C2×C4×C31—21no—
235C2×C4×C31—22no—
236C2×C4×C31—22no—
237C2×C4×C31—22no—
238C2×C4×C31—22no—
239C2×C4×C31—41no—
240C2×C4×C31—42no—
241C2×C4×C31—42no—
242C2×C4×C31—42no—
243C2×C4×C31—42no—
244C2×C4×C31—41no—
245C2×C4×C31—82no—
246C2×C4×C31—81no—
247C2×C4×C31—84no—
248C2×C4×C31—82no—
249C2×C4×C31#12011yescharacteristic
250C2×C4×C31#12011yescharacteristic
251C2×C4×C31#12011yescharacteristic
252C2×C4×C31#13512yes—
253C2×C4×C31AGL(1,ℤ/8)12yes—
254C2×C4×C31C22×D411yescharacteristic
255C2×C4×C31C22×D411yescharacteristic
256C2×C4×C31C22×D411yescharacteristic
257C8×C31—22no—
258C8×C31—41no—
259C8×C31—42no—
260C8×C31—42no—
261#3506C2×C3111yescharacteristic, complemented
262#3245C2×C3111yescharacteristic, complemented
263#3167C2×C3111yescharacteristic, complemented
264#3036C2×C3111yescharacteristic, complemented
265#1654C2×C3112yescomplemented
266#1646C2×C3112yescomplemented
267#1625C2×C3112yescomplemented
268#1624C2×C3112yescomplemented
269#1550C2×C3111yescharacteristic, complemented
270#1492C2×C3111yescharacteristic, complemented
271C2×D4⋊C8C2×C3111yescharacteristic, complemented
272C22×C31—21no—
273C22×C31—21no—
274C22×C31—42no—
275C22×C31—42no—
276C22×C31—42no—
277C22×C31—42no—
278C22×C31—42no—
279C22×C31—42no—
280C22×C31—84no—
281C22×C31—81no—
282C22×C31—81no—
283C22×C31—81no—
284C22×C31#44912yes—
285C22×C31C2×#12011yescharacteristic
286C4×C31—21no—
287C4×C31—21no—
288C4×C31—21no—
289C4×C31—42no—
290C4×C31—42no—
291C4×C31—42no—
292C4×C31—42no—
293C4×C31—81no—
294C4×C31#44912yes—
295C4×C31C2×#13511yescharacteristic
296C4×C31C2×AGL(1,ℤ/8)11yescharacteristic
297C4×C31#57611yescharacteristic
298C4×C31#57711yescharacteristic
299C2×AGL(1,ℤ/8)—21no—
300#559—22no—
301#556—21no—
302#555C22×C3112yes—
303#553—22no—
304#550—21no—
305#547C22×C3112yes—
306#538—22no—
307#535—21no—
308#533C22×C3111yescharacteristic
309#532C22×C3111yescharacteristic
310#532C22×C3111yescharacteristic
311#525—22no—
312#520—21no—
313C2×C4×D4C22×C3111yescharacteristic
314#498C22×C3111yescharacteristic, complemented
315#492C22×C3111yescharacteristic, complemented
316#487C22×C3111yescharacteristic, complemented
317#486C22×C3111yescharacteristic, complemented
318#474—21no—
319#474C22×C3112yescomplemented
320#466—22no—
321#465—22no—
322#463—22no—
323#461C22×C3112yes—
324#459—21no—
325#459C22×C3112yes—
326#459C22×C3112yescomplemented
327C4×SD16—22no—
328#423C22×C3111yescharacteristic
329C2×C4⋊C8C22×C3111yescharacteristic
330#417—21no—
331C2×C4×C8C22×C3111yescharacteristic
332#333C22×C3112yescomplemented
333#332C22×C3112yescomplemented
334D4⋊C8C22×C3112yes—
335D4⋊C8C22×C3112yes—
336D4⋊C8C22×C3112yescomplemented
337D4⋊C8C22×C3112yescomplemented
338C2×C31—22no—
339C2×C31—82no—
340C2×C31—82no—
341C2×C31—161no—
342C2×C31C2×#44911yescharacteristic
343C2×C31#301211yescharacteristic
344C2×C31#301811yescharacteristic
345C2×#41—22no—
346C2×#41—21no—
347C2×#41—21no—
348C22×D4D4×C3111yescharacteristic
349C23×C4—21no—
350AGL(1,ℤ/8)—44no—
351C2×SD16—22no—
352C2×SD16—22no—
353C2×D8—22no—
354C2×D8—22no—
355C2×M16—21no—
356C22×C8—21no—
357C22×C8—21no—
358C4⋊Q8D4×C3111yescharacteristic
359C4⋊Q8C23×C3111yescharacteristic
360C4⋊Q8C23×C3111yescharacteristic
361C4⋊Q8C23×C3111yescharacteristic
362#127—22no—
363#127D4×C3111yescharacteristic
364#127C23×C3111yescharacteristic
365#126—24no—
366#125—21no—
367#124—22no—
368#124—24no—
369#124—21no—
370#124—21no—
371#123—24no—
372#123—24no—
373C22⋊Q8—21no—
374C22⋊Q8—22no—
375C22⋊Q8D4×C3112yes—
376#121—24no—
377#121—22no—
378#121—22no—
379#121—22no—
380#121—21no—
381#121—21no—
382#121—21no—
383#121D4×C3112yes—
384#120—24no—
385C4×Q8—22no—
386C4×Q8—21no—
387C4×Q8D4×C3112yes—
388C4×D4—24no—
389C4×D4—22no—
390C4×D4—24no—
391C4×D4—21no—
392C4×D4—21no—
393C4×D4—21no—
394C4×D4—21no—
395C4×D4—21no—
396C4×D4D4×C3112yes—
397C4×D4C23×C3112yes—
398C4×D4C23×C3112yes—
399#117—21no—
400#117—22no—
401#117—22no—
402C2×C4⋊C4—22no—
403C2×C4⋊C4D4×C3111yescharacteristic
404C2×C22⋊C4—21no—
405C2×C42C23×C3111yescharacteristic
406#107—22no—
407#106—22no—
408C4⋊C8—21no—
409C4⋊C8—21no—
410C4⋊C8C23×C3112yes—
411C4⋊C8C23×C3112yes—
412#103—22no—
413#103—22no—
414#102—22no—
415#102—22no—
416#102—22no—
417#102—22no—
418C4×C8—21no—
419C4×C8C23×C3111yescharacteristic
420C4×C8C23×C3111yescharacteristic
421C31#1868011yescharacteristic, direct product factor, Sylow
422C24—21no—
423#41—42no—
424#41—42no—
425#41—42no—
426#41—42no—
427#41—82no—
428#41—82no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8C2×D4×C3112yes—
433C2×D4—22no—
434C2×D4—24no—
435C2×D4—24no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—41no—
445C2×D4C2×D4×C3112yes—
446C2×D4C2×D4×C3112yes—
447C22×C4—24no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—41no—
453C22×C4—81no—
454C22×C4—81no—
455C22×C4C2×D4×C3111yescharacteristic
456C22×C4C2×D4×C3111yescharacteristic
457C22×C4C2×D4×C3111yescharacteristic
458SD16—44no—
459SD16—44no—
460D8—44no—
461D8—82no—
462M16—42no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—42no—
468C2×C8—81no—
469C4⋊C4—24no—
470C4⋊C4—21no—
471C4⋊C4—22no—
472C4⋊C4—21no—
473C4⋊C4—22no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4C2×D4×C3112yes—
480C4⋊C4C2×D4×C3112yes—
481C4⋊C4C2×D4×C3112yes—
482C22⋊C4—24no—
483C22⋊C4—24no—
484C22⋊C4—24no—
485C22⋊C4—22no—
486C22⋊C4—22no—
487C22⋊C4—22no—
488C22⋊C4—22no—
489C22⋊C4—22no—
490C22⋊C4—41no—
491C22⋊C4—41no—
492C42—24no—
493C42—22no—
494C42C2×D4×C3111yescharacteristic
495C42C2×D4×C3111yescharacteristic
496C42C2×D4×C3111yescharacteristic
497C42C24×C3111yescharacteristic
498C23—22no—
499C23—22no—
500C23—42no—
501C23—41no—
502C23—81no—
503C23—81no—
504C23C31×#12011yescharacteristic
505Q8—42no—
506Q8—42no—
507Q8—41no—
508Q8—41no—
509Q8—41no—
510Q8—41no—
511D4—22no—
512D4—22no—
513D4—22no—
514D4—22no—
515D4—42no—
516D4—44no—
517D4—42no—
518D4—42no—
519D4—41no—
520D4—41no—
521D4—41no—
522D4—41no—
523D4—81no—
524D4—84no—
525D4—84no—
526D4—82no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—82no—
540C2×C4—81no—
541C2×C4—84no—
542C2×C4—82no—
543C2×C4C31×#12011yescharacteristic
544C2×C4C31×#12011yescharacteristic
545C2×C4C31×#12011yescharacteristic
546C2×C4C31×#13512yes—
547C2×C4C31×AGL(1,ℤ/8)12yes—
548C2×C4C22×D4×C3111yescharacteristic
549C2×C4C22×D4×C3111yescharacteristic
550C2×C4C22×D4×C3111yescharacteristic
551C8—22no—
552C8—41no—
553C8—42no—
554C8—42no—
555C22—21no—
556C22—21no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22—42no—
562C22—42no—
563C22—84no—
564C22—81no—
565C22—81no—
566C22—81no—
567C22C31×#44912yes—
568C22C2×C31×#12011yescharacteristic
569C4—21no—
570C4—21no—
571C4—21no—
572C4—42no—
573C4—42no—
574C4—42no—
575C4—42no—
576C4—81no—
577C4C31×#44912yes—
578C4C2×C31×#13511yescharacteristic
579C4C2×C31×AGL(1,ℤ/8)11yescharacteristic
580C4C31×#57611yescharacteristic
581C4C31×#57711yescharacteristic
582C2—22no—
583C2—82no—
584C2—82no—
585C2—161no—
586C2C2×C31×#44911yescharacteristic
587C2C31×#301211yescharacteristic
588C2C31×#301811yescharacteristic
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×#3506C2semidirect
2. C31×#3245C2semidirect
3. C31×#3167C2semidirect
4. C31×#3036C2semidirect
5. C31×#1654C2semidirect
6. C31×#1646C2semidirect
7. C31×#1625C2semidirect
8. C31×#1624C2semidirect
9. C31×#1550C2semidirect
10. C31×#1492C2semidirect
11. C2×C31×D4⋊C8C2semidirect
27. C31×#498C22semidirect
28. C31×#492C22semidirect
29. C31×#487C22semidirect
30. C31×#486C22semidirect
32. C31×#474C22semidirect
39. C31×#459C22semidirect
45. C31×#333C22semidirect
46. C31×#332C22semidirect
49. C31×D4⋊C8C22semidirect
50. C31×D4⋊C8C22semidirect
203. #18680C31direct (N × H)
261. #3506C2×C31semidirect
262. #3245C2×C31semidirect
263. #3167C2×C31semidirect
264. #3036C2×C31semidirect
265. #1654C2×C31semidirect
266. #1646C2×C31semidirect
267. #1625C2×C31semidirect
268. #1624C2×C31semidirect
269. #1550C2×C31semidirect
270. #1492C2×C31semidirect
271. C2×D4⋊C8C2×C31semidirect
314. #498C22×C31semidirect
315. #492C22×C31semidirect
316. #487C22×C31semidirect
317. #486C22×C31semidirect
319. #474C22×C31semidirect
326. #459C22×C31semidirect
332. #333C22×C31semidirect
333. #332C22×C31semidirect
336. D4⋊C8C22×C31semidirect
337. D4⋊C8C22×C31semidirect
421. C31#18680direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35062
C31×#32452
C31×#31672
C31×#30362
C31×#16542
C31×#16462
C31×#16252
C31×#16242
C31×#15502
C31×#14922
C2×C31×D4⋊C82
#1868031