Finite Group Databasebeta

← C31×#18585 #21475344

Subgroups — C31×#18585 #21475344

Order 7936.

938 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3511C211yesmaximal, characteristic, complemented
2C31×#3253C211yesmaximal, characteristic, complemented
3C31×#3141C211yesmaximal, characteristic, complemented
4C31×#3106C211yesmaximal, characteristic, complemented
5C31×#1638C212yesmaximal, complemented
6C31×#1635C212yesmaximal, complemented
7C31×#1628C212yesmaximal, complemented
8C31×#1625C212yesmaximal, complemented
9C31×#1546C211yesmaximal, characteristic, complemented
10C31×#1489C211yesmaximal, characteristic, complemented
11C31×#1484C211yesmaximal, characteristic, complemented
12C31×#556C2212yes—
13C31×#555—21no—
14C31×#554—21no—
15C31×#553—22no—
16C31×#548—22no—
17C31×#547C2212yes—
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×#489—22no—
28C31×#487C2212yes—
29C31×#486C2212yes—
30C31×#482—21no—
31C31×#477C2211yescharacteristic
32C31×#476—21no—
33C31×#476C2211yescharacteristic
34C31×#476C2212yes—
35C31×#469—22no—
36C31×#460C2211yescharacteristic
37C31×#459—21no—
38C31×#459C2211yescharacteristic
39C31×#459C2212yes—
40C31×#449—21no—
41C4×SD16×C31—21no—
42C4×D8×C31—21no—
43C31×#431C2211yescharacteristic
44C31×#423C2211yescharacteristic
45C31×#417—22no—
46C4×M16×C31C2211yescharacteristic
47C31×#332C2212yes—
48C31×#332C2212yes—
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×C31—21no—
58C2×SD16×C31—21no—
59C2×D8×C31—21no—
60C2×D8×C31—22no—
61C2×M16×C31—21no—
62C2×M16×C31—21no—
63C31×C4⋊Q8—21no—
64C31×C4⋊Q8—21no—
65C31×C4⋊Q8D411yescharacteristic
66C31×C4⋊Q8D412yes—
67C31×C4⋊Q8C2311yescharacteristic
68C31×C4⋊Q8C2312yes—
69C31×#127D411yescharacteristic
70C31×#127C2311yescharacteristic
71C31×#126—24no—
72C31×#125—22no—
73C31×#124—24no—
74C31×#124—22no—
75C31×#124—21no—
76C31×#124—21no—
77C31×#123—24no—
78C31×C22⋊Q8—22no—
79C31×C22⋊Q8—24no—
80C31×C22⋊Q8—22no—
81C31×C22⋊Q8—22no—
82C31×#121—22no—
83C31×#121—22no—
84C31×#121—24no—
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×C31C2311yescharacteristic
104C4×D4×C31C2312yes—
105C31×#117—22no—
106C31×#117—22no—
107C31×#117—21no—
108C31×#117D411yescharacteristic
109C2×C4⋊C4×C31—21no—
110C2×C4⋊C4×C31—22no—
111C2×C4⋊C4×C31D412yes—
112C2×C42×C31—21no—
113C2×C42×C31C2311yescharacteristic
114C31×#107—21no—
115C31×#106—21no—
116C31×#106—22no—
117C31×C4⋊C8—21no—
118C31×C4⋊C8C2312yes—
119C31×C4⋊C8C2311yescharacteristic
120C31×C4⋊C8C2311yescharacteristic
121C31×#103—21no—
122C31×#103—24no—
123C31×#102—21no—
124C31×#102—22no—
125C31×#102—24no—
126C31×#102—21no—
127C31×#102—21no—
128C31×#102—22no—
129C31×C22⋊C8—21no—
130C31×#97—21no—
131C4×C8×C31—21no—
132C4×C8×C31C2312yes—
133#41×C31—41no—
134#41×C31—41no—
135#41×C31—42no—
136#41×C31—42no—
137#41×C31—44no—
138#41×C31—82no—
139#41×C31—82no—
140C2×Q8×C31—21no—
141C2×Q8×C31—22no—
142C2×Q8×C31—22no—
143C2×Q8×C31—21no—
144C2×Q8×C31—81no—
145C2×Q8×C31—81no—
146C2×Q8×C31C2×D411yescharacteristic
147C2×D4×C31—21no—
148C2×D4×C31—22no—
149C2×D4×C31—24no—
150C2×D4×C31—21no—
151C2×D4×C31—21no—
152C2×D4×C31—21no—
153C2×D4×C31—21no—
154C2×D4×C31—21no—
155C2×D4×C31—41no—
156C2×D4×C31C2×D411yescharacteristic
157C2×D4×C31C2×D412yes—
158C22×C4×C31—24no—
159C22×C4×C31—22no—
160C22×C4×C31—21no—
161C22×C4×C31—21no—
162C22×C4×C31—21no—
163C22×C4×C31—22no—
164C22×C4×C31—41no—
165C22×C4×C31C2×D411yescharacteristic
166C22×C4×C31C2×D411yescharacteristic
167C22×C4×C31C2×D411yescharacteristic
168SD16×C31—42no—
169D8×C31—42no—
170D8×C31—82no—
171M16×C31—42no—
172M16×C31—81no—
173C2×C8×C31—21no—
174C2×C8×C31—21no—
175C2×C8×C31—22no—
176C2×C8×C31—22no—
177C4⋊C4×C31—24no—
178C4⋊C4×C31—24no—
179C4⋊C4×C31—22no—
180C4⋊C4×C31—22no—
181C4⋊C4×C31—21no—
182C4⋊C4×C31—21no—
183C4⋊C4×C31—22no—
184C4⋊C4×C31—21no—
185C4⋊C4×C31—21no—
186C4⋊C4×C31—21no—
187C4⋊C4×C31C2×D414yes—
188C4⋊C4×C31C2×D411yescharacteristic
189C4⋊C4×C31C2×D412yes—
190C4⋊C4×C31C2×D411yescharacteristic
191C22⋊C4×C31—24no—
192C22⋊C4×C31—24no—
193C22⋊C4×C31—24no—
194C22⋊C4×C31—21no—
195C22⋊C4×C31—21no—
196C22⋊C4×C31—22no—
197C22⋊C4×C31—21no—
198C22⋊C4×C31—22no—
199C22⋊C4×C31—41no—
200C22⋊C4×C31—41no—
201C42×C31—24no—
202C42×C31—21no—
203C42×C31—21no—
204C42×C31—21no—
205C42×C31C2×D411yescharacteristic
206C42×C31C2×D411yescharacteristic
207C42×C31C2×D411yescharacteristic
208C42×C31C2411yescharacteristic
209#18585C3111yesmaximal, characteristic, direct product factor, Sylow
210C23×C31—21no—
211C23×C31—22no—
212C23×C31—41no—
213C23×C31#12011yescharacteristic
214Q8×C31—21no—
215Q8×C31—21no—
216Q8×C31—42no—
217Q8×C31—42no—
218Q8×C31—42no—
219Q8×C31—42no—
220Q8×C31—82no—
221D4×C31—21no—
222D4×C31—21no—
223D4×C31—22no—
224D4×C31—22no—
225D4×C31—42no—
226D4×C31—42no—
227D4×C31—42no—
228D4×C31—44no—
229D4×C31—41no—
230D4×C31—41no—
231D4×C31—41no—
232D4×C31—41no—
233D4×C31—42no—
234D4×C31—84no—
235D4×C31—82no—
236C2×C4×C31—21no—
237C2×C4×C31—21no—
238C2×C4×C31—21no—
239C2×C4×C31—21no—
240C2×C4×C31—22no—
241C2×C4×C31—21no—
242C2×C4×C31—21no—
243C2×C4×C31—22no—
244C2×C4×C31—21no—
245C2×C4×C31—24no—
246C2×C4×C31—42no—
247C2×C4×C31—42no—
248C2×C4×C31—42no—
249C2×C4×C31—41no—
250C2×C4×C31—41no—
251C2×C4×C31—42no—
252C2×C4×C31—41no—
253C2×C4×C31—82no—
254C2×C4×C31—84no—
255C2×C4×C31—82no—
256C2×C4×C31#12011yescharacteristic
257C2×C4×C31#12011yescharacteristic
258C2×C4×C31#12011yescharacteristic
259C2×C4×C31C22×D411yescharacteristic
260C2×C4×C31C22×D411yescharacteristic
261C2×C4×C31C22×D411yescharacteristic
262C8×C31—42no—
263C8×C31—41no—
264C8×C31—41no—
265C8×C31—42no—
266#3511C2×C3111yescharacteristic, complemented
267#3253C2×C3111yescharacteristic, complemented
268#3141C2×C3111yescharacteristic, complemented
269#3106C2×C3111yescharacteristic, complemented
270#1638C2×C3112yescomplemented
271#1635C2×C3112yescomplemented
272#1628C2×C3112yescomplemented
273#1625C2×C3112yescomplemented
274#1546C2×C3111yescharacteristic, complemented
275#1489C2×C3111yescharacteristic, complemented
276#1484C2×C3111yescharacteristic, complemented
277C22×C31—21no—
278C22×C31—21no—
279C22×C31—21no—
280C22×C31—41no—
281C22×C31—42no—
282C22×C31—41no—
283C22×C31—42no—
284C22×C31—41no—
285C22×C31—42no—
286C22×C31—81no—
287C22×C31—81no—
288C22×C31—81no—
289C22×C31C2×#12011yescharacteristic
290C4×C31—21no—
291C4×C31—21no—
292C4×C31—21no—
293C4×C31—41no—
294C4×C31—42no—
295C4×C31—41no—
296C4×C31—44no—
297C4×C31—41no—
298C4×C31—81no—
299C4×C31#45412yes—
300C4×C31#57511yescharacteristic
301C4×C31#57511yescharacteristic
302C4×C31#57611yescharacteristic
303C4×C31#57711yescharacteristic
304#556C22×C3112yes—
305#555—21no—
306#554—21no—
307#553—22no—
308#548—22no—
309#547C22×C3112yes—
310#542—22no—
311#537—21no—
312#533—21no—
313#532—21no—
314#532C22×C3111yescharacteristic
315#532C22×C3111yescharacteristic
316C2×C4⋊Q8C22×C3111yescharacteristic
317#520—21no—
318C4×#41C22×C3111yescharacteristic
319#489—22no—
320#487C22×C3112yes—
321#486C22×C3112yes—
322#482—21no—
323#477C22×C3111yescharacteristic
324#476—21no—
325#476C22×C3111yescharacteristic
326#476C22×C3112yes—
327#469—22no—
328#460C22×C3111yescharacteristic
329#459—21no—
330#459C22×C3111yescharacteristic
331#459C22×C3112yes—
332#449—21no—
333C4×SD16—21no—
334C4×D8—21no—
335#431C22×C3111yescharacteristic
336#423C22×C3111yescharacteristic
337#417—22no—
338C4×M16C22×C3111yescharacteristic
339#332C22×C3112yes—
340#332C22×C3112yes—
341Q8⋊C8C22×C3112yes—
342D4⋊C8C22×C3112yes—
343D4⋊C8C22×C3112yes—
344D4⋊C8C22×C3112yes—
345C2×C31—41no—
346C2×C31—81no—
347C2×C31—82no—
348C2×C31—161no—
349C2×C31#300511yescharacteristic
350C2×C31#301211yescharacteristic
351C2×C31C2×#45411yescharacteristic
352C2×#41—22no—
353C2×#41—21no—
354C2×#41—21no—
355C2×#41D4×C3111yescharacteristic
356C22×Q8—21no—
357C2×SD16—21no—
358C2×D8—21no—
359C2×D8—22no—
360C2×M16—21no—
361C2×M16—21no—
362C4⋊Q8—21no—
363C4⋊Q8—21no—
364C4⋊Q8D4×C3112yes—
365C4⋊Q8D4×C3111yescharacteristic
366C4⋊Q8C23×C3111yescharacteristic
367C4⋊Q8C23×C3112yes—
368#127D4×C3111yescharacteristic
369#127C23×C3111yescharacteristic
370#126—24no—
371#125—22no—
372#124—24no—
373#124—22no—
374#124—21no—
375#124—21no—
376#123—24no—
377C22⋊Q8—22no—
378C22⋊Q8—24no—
379C22⋊Q8—22no—
380C22⋊Q8—22no—
381#121—22no—
382#121—22no—
383#121—24no—
384#121—22no—
385#121—21no—
386#121—21no—
387#121—21no—
388#121D4×C3112yes—
389C4×Q8—22no—
390C4×Q8—22no—
391C4×Q8D4×C3111yescharacteristic
392C4×Q8C23×C3111yescharacteristic
393C4×D4—22no—
394C4×D4—24no—
395C4×D4—22no—
396C4×D4—21no—
397C4×D4—21no—
398C4×D4—21no—
399C4×D4—21no—
400C4×D4—21no—
401C4×D4D4×C3111yescharacteristic
402C4×D4C23×C3111yescharacteristic
403C4×D4C23×C3112yes—
404#117—22no—
405#117—22no—
406#117—21no—
407#117D4×C3111yescharacteristic
408C2×C4⋊C4—21no—
409C2×C4⋊C4—22no—
410C2×C4⋊C4D4×C3112yes—
411C2×C42—21no—
412C2×C42C23×C3111yescharacteristic
413#107—21no—
414#106—21no—
415#106—22no—
416C4⋊C8—21no—
417C4⋊C8C23×C3111yescharacteristic
418C4⋊C8C23×C3111yescharacteristic
419C4⋊C8C23×C3112yes—
420#103—21no—
421#103—24no—
422#102—21no—
423#102—22no—
424#102—24no—
425#102—21no—
426#102—21no—
427#102—22no—
428C22⋊C8—21no—
429#97—21no—
430C4×C8—21no—
431C4×C8C23×C3112yes—
432C31#1858511yescharacteristic, direct product factor, Sylow
433#41—41no—
434#41—41no—
435#41—42no—
436#41—42no—
437#41—44no—
438#41—82no—
439#41—82no—
440C2×Q8—21no—
441C2×Q8—22no—
442C2×Q8—22no—
443C2×Q8—21no—
444C2×Q8—81no—
445C2×Q8—81no—
446C2×Q8C2×D4×C3111yescharacteristic
447C2×D4—21no—
448C2×D4—22no—
449C2×D4—24no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—41no—
456C2×D4C2×D4×C3111yescharacteristic
457C2×D4C2×D4×C3112yes—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—22no—
464C22×C4—41no—
465C22×C4C2×D4×C3111yescharacteristic
466C22×C4C2×D4×C3111yescharacteristic
467C22×C4C2×D4×C3111yescharacteristic
468SD16—42no—
469D8—42no—
470D8—82no—
471M16—42no—
472M16—81no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—22no—
476C2×C8—22no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—22no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4C2×D4×C3114yes—
488C4⋊C4C2×D4×C3111yescharacteristic
489C4⋊C4C2×D4×C3112yes—
490C4⋊C4C2×D4×C3111yescharacteristic
491C22⋊C4—24no—
492C22⋊C4—24no—
493C22⋊C4—24no—
494C22⋊C4—21no—
495C22⋊C4—21no—
496C22⋊C4—22no—
497C22⋊C4—21no—
498C22⋊C4—22no—
499C22⋊C4—41no—
500C22⋊C4—41no—
501C42—24no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42C2×D4×C3111yescharacteristic
506C42C2×D4×C3111yescharacteristic
507C42C2×D4×C3111yescharacteristic
508C42C24×C3111yescharacteristic
509C23—21no—
510C23—22no—
511C23—41no—
512C23C31×#12011yescharacteristic
513Q8—21no—
514Q8—21no—
515Q8—42no—
516Q8—42no—
517Q8—42no—
518Q8—42no—
519Q8—82no—
520D4—21no—
521D4—21no—
522D4—22no—
523D4—22no—
524D4—42no—
525D4—42no—
526D4—42no—
527D4—44no—
528D4—42no—
529D4—41no—
530D4—41no—
531D4—41no—
532D4—41no—
533D4—84no—
534D4—82no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—24no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—82no—
553C2×C4—84no—
554C2×C4—82no—
555C2×C4C31×#12011yescharacteristic
556C2×C4C31×#12011yescharacteristic
557C2×C4C31×#12011yescharacteristic
558C2×C4C22×D4×C3111yescharacteristic
559C2×C4C22×D4×C3111yescharacteristic
560C2×C4C22×D4×C3111yescharacteristic
561C8—42no—
562C8—41no—
563C8—41no—
564C8—42no—
565C22—21no—
566C22—21no—
567C22—21no—
568C22—41no—
569C22—42no—
570C22—41no—
571C22—42no—
572C22—41no—
573C22—42no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22C2×C31×#12011yescharacteristic
578C4—21no—
579C4—21no—
580C4—21no—
581C4—41no—
582C4—42no—
583C4—41no—
584C4—44no—
585C4—41no—
586C4—81no—
587C4C31×#45412yes—
588C4C31×#57511yescharacteristic
589C4C31×#57511yescharacteristic
590C4C31×#57611yescharacteristic
591C4C31×#57711yescharacteristic
592C2—41no—
593C2—81no—
594C2—82no—
595C2—161no—
596C2C31×#300511yescharacteristic
597C2C31×#301211yescharacteristic
598C2C2×C31×#45411yescharacteristic
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×#3253C2semidirect
3. C31×#3141C2semidirect
4. C31×#3106C2semidirect
5. C31×#1638C2semidirect
6. C31×#1635C2semidirect
7. C31×#1628C2semidirect
8. C31×#1625C2semidirect
9. C31×#1546C2semidirect
10. C31×#1489C2semidirect
11. C31×#1484C2semidirect
209. #18585C31direct (N × H)
266. #3511C2×C31semidirect
267. #3253C2×C31semidirect
268. #3141C2×C31semidirect
269. #3106C2×C31semidirect
270. #1638C2×C31semidirect
271. #1635C2×C31semidirect
272. #1628C2×C31semidirect
273. #1625C2×C31semidirect
274. #1546C2×C31semidirect
275. #1489C2×C31semidirect
276. #1484C2×C31semidirect
432. C31#18585direct (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×#32532
C31×#31412
C31×#31062
C31×#16382
C31×#16352
C31×#16282
C31×#16252
C31×#15462
C31×#14892
C31×#14842
#1858531