Finite Group Databasebeta

← C31×#18216 #21474975

Subgroups — C31×#18216 #21474975

Order 7936.

988 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3493C211yesmaximal, characteristic, complemented
2C31×#3105C212yesmaximal, complemented
3C31×#2598C211yesmaximal, characteristic, complemented
4C31×#1618C212yesmaximal, complemented
5C31×#1617C211yesmaximal, characteristic, complemented
6C31×#1616C211yesmaximal, characteristic, complemented
7C31×#1600C212yesmaximal, complemented
8C31×#1599C211yesmaximal, characteristic, complemented
9C31×#1598C211yesmaximal, characteristic, complemented
10C31×#1521C212yesmaximal, complemented
11C31×#1458C211yesmaximal, characteristic, complemented
12C31×#556—22no—
13C31×#555—22no—
14C31×#553—22no—
15C31×#552—22no—
16C31×#544C2211yescharacteristic
17C31×#538—21no—
18C31×#538C2212yes—
19C31×#537C2211yescharacteristic
20C31×#529—21no—
21C31×#529C2212yes—
22C31×#525—21no—
23C31×#520—21no—
24C2×C4×D4×C31C2211yescharacteristic
25C2×C31×#117C2211yescharacteristic
26C31×#484C2212yes—
27C31×#481C2212yes—
28C31×#478—22no—
29C31×#463—22no—
30C31×#450C2212yes—
31C31×#447C2212yes—
32C4×SD16×C31—22no—
33C4×D8×C31—22no—
34C31×#431C2212yes—
35C31×#400C2211yescharacteristic
36C31×#399C2211yescharacteristic
37C31×#394C2211yescharacteristic
38C31×#392C2211yescharacteristic
39C31×#388—21no—
40C31×#386—21no—
41C31×#386—21no—
42C31×#385—21no—
43C31×#383C2211yescharacteristic
44C31×#328C2212yes—
45C31×#328C2212yes—
46C31×#327C2212yes—
47C31×#327C2212yes—
48C31×#324C2211yescharacteristic
49C31×#324C2211yescharacteristic
50C31×#323C2212yes—
51C2×#41×C31—21no—
52C22×D4×C31—21no—
53C22×D4×C31C2311yescharacteristic
54C23×C4×C31—21no—
55C23×C4×C31D411yescharacteristic
56C2×SD16×C31—22no—
57C2×D8×C31—22no—
58C31×C4⋊Q8—22no—
59C31×#126—22no—
60C31×#126—22no—
61C31×#126—22no—
62C31×#126—22no—
63C31×#125—21no—
64C31×#125—22no—
65C31×#125D412yescomplemented
66C31×#124—21no—
67C31×#124—22no—
68C31×#124—22no—
69C31×#124D412yescomplemented
70C31×#123—22no—
71C31×#123—22no—
72C31×#123—22no—
73C31×C22⋊Q8—21no—
74C31×C22⋊Q8—21no—
75C31×C22⋊Q8—21no—
76C31×C22⋊Q8—22no—
77C31×C22⋊Q8C2312yes—
78C31×#121—21no—
79C31×#121—21no—
80C31×#121—22no—
81C31×#121—22no—
82C31×#121—22no—
83C31×#121—21no—
84C31×#121C2312yes—
85C31×#120—22no—
86C4×Q8×C31—22no—
87C4×Q8×C31—21no—
88C4×Q8×C31D412yescomplemented
89C4×D4×C31—22no—
90C4×D4×C31—22no—
91C4×D4×C31—22no—
92C4×D4×C31—21no—
93C4×D4×C31—22no—
94C4×D4×C31—22no—
95C4×D4×C31—22no—
96C4×D4×C31—22no—
97C4×D4×C31D412yescomplemented
98C31×#117—22no—
99C31×#117—22no—
100C31×#117—21no—
101C31×#117—22no—
102C31×#117—22no—
103C31×#117—22no—
104C2×C4⋊C4×C31—22no—
105C2×C4⋊C4×C31—21no—
106C2×C4⋊C4×C31D411yescharacteristic
107C2×C4⋊C4×C31C2311yescharacteristic
108C2×C4⋊C4×C31C2311yescharacteristic
109C2×C22⋊C4×C31—21no—
110C2×C22⋊C4×C31—21no—
111C2×C22⋊C4×C31—21no—
112C2×C22⋊C4×C31—21no—
113C2×C22⋊C4×C31—21no—
114C2×C22⋊C4×C31D411yescharacteristic
115C2×C22⋊C4×C31C2311yescharacteristic
116C2×C42×C31D411yescharacteristic
117C2×C42×C31C2311yescharacteristic
118C31×#107—22no—
119C31×#106—22no—
120C31×C4⋊C8—22no—
121C31×#103—22no—
122C31×#102—22no—
123C31×#102—22no—
124C31×#102—22no—
125C31×C22⋊C8C2312yes—
126C31×C22⋊C8C2312yes—
127C4×C8×C31—22no—
128C31×#95—21no—
129C31×#95—21no—
130C31×#95—21no—
131C31×#95—21no—
132C31×#95C2311yescharacteristic
133C31×#95C2311yescharacteristic
134C24×C31—21no—
135C24×C31C2×D411yescharacteristic
136#41×C31—42no—
137#41×C31—42no—
138C2×Q8×C31—21no—
139C2×Q8×C31—21no—
140C2×Q8×C31C2×D412yes—
141C2×D4×C31—21no—
142C2×D4×C31—22no—
143C2×D4×C31—22no—
144C2×D4×C31—22no—
145C2×D4×C31—21no—
146C2×D4×C31—22no—
147C2×D4×C31—42no—
148C2×D4×C31—42no—
149C2×D4×C31—42no—
150C2×D4×C31—81no—
151C2×D4×C31C2×D412yes—
152C22×C4×C31—22no—
153C22×C4×C31—22no—
154C22×C4×C31—21no—
155C22×C4×C31—21no—
156C22×C4×C31—22no—
157C22×C4×C31—22no—
158C22×C4×C31—21no—
159C22×C4×C31—21no—
160C22×C4×C31—21no—
161C22×C4×C31—41no—
162C22×C4×C31—42no—
163C22×C4×C31—81no—
164C22×C4×C31—81no—
165C22×C4×C31C2×D411yescharacteristic
166C22×C4×C31C2×D411yescharacteristic
167C22×C4×C31C2×D411yescharacteristic
168C22×C4×C31C2×D411yescharacteristic
169C22×C4×C31C2×D411yescharacteristic
170C22×C4×C31C2411yescharacteristic
171SD16×C31—44no—
172D8×C31—44no—
173C2×C8×C31—22no—
174C2×C8×C31—22no—
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—22no—
182C4⋊C4×C31—22no—
183C4⋊C4×C31—22no—
184C4⋊C4×C31—22no—
185C4⋊C4×C31—22no—
186C4⋊C4×C31—22no—
187C4⋊C4×C31—22no—
188C4⋊C4×C31—81no—
189C4⋊C4×C31C2×D412yes—
190C4⋊C4×C31C2×D412yes—
191C22⋊C4×C31—22no—
192C22⋊C4×C31—22no—
193C22⋊C4×C31—22no—
194C22⋊C4×C31—22no—
195C22⋊C4×C31—22no—
196C22⋊C4×C31—22no—
197C22⋊C4×C31—22no—
198C22⋊C4×C31—22no—
199C22⋊C4×C31—22no—
200C22⋊C4×C31—22no—
201C22⋊C4×C31—22no—
202C22⋊C4×C31—42no—
203C22⋊C4×C31—41no—
204C22⋊C4×C31—41no—
205C22⋊C4×C31—42no—
206C22⋊C4×C31—81no—
207C22⋊C4×C31—81no—
208C42×C31—22no—
209C42×C31—22no—
210C42×C31—21no—
211C42×C31—21no—
212C42×C31—22no—
213C42×C31C2×D412yes—
214C42×C31C2×D412yes—
215#18216C3111yesmaximal, characteristic, direct product factor, Sylow
216C23×C31—22no—
217C23×C31—21no—
218C23×C31—22no—
219C23×C31—21no—
220C23×C31—21no—
221C23×C31—41no—
222C23×C31—41no—
223C23×C31—41no—
224C23×C31—41no—
225C23×C31—41no—
226C23×C31—41no—
227C23×C31—41no—
228C23×C31—41no—
229C23×C31C22×D411yescharacteristic
230Q8×C31—42no—
231Q8×C31—42no—
232Q8×C31—81no—
233D4×C31—24no—
234D4×C31—42no—
235D4×C31—42no—
236D4×C31—42no—
237D4×C31—42no—
238D4×C31—42no—
239D4×C31—82no—
240D4×C31—81no—
241D4×C31—82no—
242D4×C31—84no—
243C2×C4×C31—21no—
244C2×C4×C31—21no—
245C2×C4×C31—21no—
246C2×C4×C31—21no—
247C2×C4×C31—21no—
248C2×C4×C31—22no—
249C2×C4×C31—21no—
250C2×C4×C31—22no—
251C2×C4×C31—22no—
252C2×C4×C31—41no—
253C2×C4×C31—41no—
254C2×C4×C31—42no—
255C2×C4×C31—42no—
256C2×C4×C31—44no—
257C2×C4×C31—41no—
258C2×C4×C31—41no—
259C2×C4×C31—41no—
260C2×C4×C31—41no—
261C2×C4×C31—41no—
262C2×C4×C31—41no—
263C2×C4×C31—41no—
264C2×C4×C31—41no—
265C2×C4×C31—41no—
266C2×C4×C31—41no—
267C2×C4×C31—41no—
268C2×C4×C31—82no—
269C2×C4×C31—84no—
270C2×C4×C31—82no—
271C2×C4×C31#12011yescharacteristic
272C2×C4×C31#12011yescharacteristic
273C2×C4×C31#12012yes—
274C2×C4×C31#13512yes—
275C2×C4×C31C22×D412yes—
276C8×C31—42no—
277C8×C31—42no—
278#3493C2×C3111yescharacteristic, complemented
279#3105C2×C3112yescomplemented
280#2598C2×C3111yescharacteristic, complemented
281#1618C2×C3112yescomplemented
282#1617C2×C3111yescharacteristic, complemented
283#1616C2×C3111yescharacteristic, complemented
284#1600C2×C3112yescomplemented
285#1599C2×C3111yescharacteristic, complemented
286#1598C2×C3111yescharacteristic, complemented
287#1521C2×C3112yescomplemented
288#1458C2×C3111yescharacteristic, complemented
289C22×C31—21no—
290C22×C31—21no—
291C22×C31—42no—
292C22×C31—42no—
293C22×C31—42no—
294C22×C31—41no—
295C22×C31—42no—
296C22×C31—42no—
297C22×C31—41no—
298C22×C31—42no—
299C22×C31—42no—
300C22×C31—41no—
301C22×C31—81no—
302C22×C31—81no—
303C22×C31—81no—
304C22×C31—81no—
305C22×C31—81no—
306C22×C31C2×#12011yescharacteristic
307C22×C31C2×#13511yescharacteristic
308C22×C31#57511yescharacteristic
309C4×C31—22no—
310C4×C31—21no—
311C4×C31—21no—
312C4×C31—22no—
313C4×C31—22no—
314C4×C31—42no—
315C4×C31—41no—
316C4×C31—42no—
317C4×C31—41no—
318C4×C31—42no—
319C4×C31—81no—
320C4×C31—81no—
321C4×C31—81no—
322#556—22no—
323#555—22no—
324#553—22no—
325#552—22no—
326#544C22×C3111yescharacteristic
327#538—21no—
328#538C22×C3112yes—
329#537C22×C3111yescharacteristic
330#529—21no—
331#529C22×C3112yes—
332#525—21no—
333#520—21no—
334C2×C4×D4C22×C3111yescharacteristic
335C2×#117C22×C3111yescharacteristic
336#484C22×C3112yes—
337#481C22×C3112yes—
338#478—22no—
339#463—22no—
340#450C22×C3112yes—
341#447C22×C3112yes—
342C4×SD16—22no—
343C4×D8—22no—
344#431C22×C3112yes—
345#400C22×C3111yescharacteristic
346#399C22×C3111yescharacteristic
347#394C22×C3111yescharacteristic
348#392C22×C3111yescharacteristic
349#388—21no—
350#386—21no—
351#386—21no—
352#385—21no—
353#383C22×C3111yescharacteristic
354#328C22×C3112yes—
355#328C22×C3112yes—
356#327C22×C3112yes—
357#327C22×C3112yes—
358#324C22×C3111yescharacteristic
359#324C22×C3111yescharacteristic
360#323C22×C3112yes—
361C2×C31—21no—
362C2×C31—21no—
363C2×C31—82no—
364C2×C31—81no—
365C2×C31—82no—
366C2×C31#300211yescharacteristic
367C2×C31#302111yescharacteristic
368C2×C31#302511yescharacteristic
369C2×#41—21no—
370C22×D4—21no—
371C22×D4C23×C3111yescharacteristic
372C23×C4—21no—
373C23×C4D4×C3111yescharacteristic
374C2×SD16—22no—
375C2×D8—22no—
376C4⋊Q8—22no—
377#126—22no—
378#126—22no—
379#126—22no—
380#126—22no—
381#125—21no—
382#125—22no—
383#125D4×C3112yescomplemented
384#124—21no—
385#124—22no—
386#124—22no—
387#124D4×C3112yescomplemented
388#123—22no—
389#123—22no—
390#123—22no—
391C22⋊Q8—21no—
392C22⋊Q8—21no—
393C22⋊Q8—21no—
394C22⋊Q8—22no—
395C22⋊Q8C23×C3112yes—
396#121—21no—
397#121—21no—
398#121—22no—
399#121—22no—
400#121—22no—
401#121—21no—
402#121C23×C3112yes—
403#120—22no—
404C4×Q8—22no—
405C4×Q8—21no—
406C4×Q8D4×C3112yescomplemented
407C4×D4—22no—
408C4×D4—22no—
409C4×D4—22no—
410C4×D4—21no—
411C4×D4—22no—
412C4×D4—22no—
413C4×D4—22no—
414C4×D4—22no—
415C4×D4D4×C3112yescomplemented
416#117—22no—
417#117—22no—
418#117—21no—
419#117—22no—
420#117—22no—
421#117—22no—
422C2×C4⋊C4—22no—
423C2×C4⋊C4—21no—
424C2×C4⋊C4D4×C3111yescharacteristic
425C2×C4⋊C4C23×C3111yescharacteristic
426C2×C4⋊C4C23×C3111yescharacteristic
427C2×C22⋊C4—21no—
428C2×C22⋊C4—21no—
429C2×C22⋊C4—21no—
430C2×C22⋊C4—21no—
431C2×C22⋊C4—21no—
432C2×C22⋊C4D4×C3111yescharacteristic
433C2×C22⋊C4C23×C3111yescharacteristic
434C2×C42D4×C3111yescharacteristic
435C2×C42C23×C3111yescharacteristic
436#107—22no—
437#106—22no—
438C4⋊C8—22no—
439#103—22no—
440#102—22no—
441#102—22no—
442#102—22no—
443C22⋊C8C23×C3112yes—
444C22⋊C8C23×C3112yes—
445C4×C8—22no—
446#95—21no—
447#95—21no—
448#95—21no—
449#95—21no—
450#95C23×C3111yescharacteristic
451#95C23×C3111yescharacteristic
452C31#1821611yescharacteristic, direct product factor, Sylow
453C24—21no—
454C24C2×D4×C3111yescharacteristic
455#41—42no—
456#41—42no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8C2×D4×C3112yes—
460C2×D4—21no—
461C2×D4—22no—
462C2×D4—22no—
463C2×D4—22no—
464C2×D4—21no—
465C2×D4—22no—
466C2×D4—42no—
467C2×D4—42no—
468C2×D4—42no—
469C2×D4—81no—
470C2×D4C2×D4×C3112yes—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—41no—
481C22×C4—42no—
482C22×C4—81no—
483C22×C4—81no—
484C22×C4C2×D4×C3111yescharacteristic
485C22×C4C2×D4×C3111yescharacteristic
486C22×C4C2×D4×C3111yescharacteristic
487C22×C4C2×D4×C3111yescharacteristic
488C22×C4C2×D4×C3111yescharacteristic
489C22×C4C24×C3111yescharacteristic
490SD16—44no—
491D8—44no—
492C2×C8—22no—
493C2×C8—22no—
494C4⋊C4—21no—
495C4⋊C4—22no—
496C4⋊C4—21no—
497C4⋊C4—22no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—81no—
508C4⋊C4C2×D4×C3112yes—
509C4⋊C4C2×D4×C3112yes—
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—22no—
513C22⋊C4—22no—
514C22⋊C4—22no—
515C22⋊C4—22no—
516C22⋊C4—22no—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—42no—
522C22⋊C4—41no—
523C22⋊C4—41no—
524C22⋊C4—42no—
525C22⋊C4—81no—
526C22⋊C4—81no—
527C42—22no—
528C42—22no—
529C42—21no—
530C42—21no—
531C42—22no—
532C42C2×D4×C3112yes—
533C42C2×D4×C3112yes—
534C23—22no—
535C23—21no—
536C23—22no—
537C23—21no—
538C23—21no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23C22×D4×C3111yescharacteristic
548Q8—42no—
549Q8—42no—
550Q8—81no—
551D4—24no—
552D4—42no—
553D4—42no—
554D4—42no—
555D4—42no—
556D4—42no—
557D4—82no—
558D4—81no—
559D4—82no—
560D4—84no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—82no—
587C2×C4—84no—
588C2×C4—82no—
589C2×C4C31×#12011yescharacteristic
590C2×C4C31×#12011yescharacteristic
591C2×C4C31×#12012yes—
592C2×C4C31×#13512yes—
593C2×C4C22×D4×C3112yes—
594C8—42no—
595C8—42no—
596C22—21no—
597C22—21no—
598C22—42no—
599C22—42no—
600C22—42no—
601C22—41no—
602C22—42no—
603C22—42no—
604C22—41no—
605C22—42no—
606C22—42no—
607C22—41no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22C2×C31×#12011yescharacteristic
614C22C2×C31×#13511yescharacteristic
615C22C31×#57511yescharacteristic
616C4—22no—
617C4—21no—
618C4—21no—
619C4—22no—
620C4—22no—
621C4—42no—
622C4—41no—
623C4—42no—
624C4—41no—
625C4—42no—
626C4—81no—
627C4—81no—
628C4—81no—
629C2—21no—
630C2—21no—
631C2—81no—
632C2—82no—
633C2—82no—
634C2C31×#300211yescharacteristic
635C2C31×#302111yescharacteristic
636C2C31×#302511yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
LCSG ▷ C22×C4 ▷ 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×#3493C2semidirect
2. C31×#3105C2semidirect
3. C31×#2598C2semidirect
4. C31×#1618C2semidirect
5. C31×#1617C2semidirect
6. C31×#1616C2semidirect
7. C31×#1600C2semidirect
8. C31×#1599C2semidirect
9. C31×#1598C2semidirect
10. C31×#1521C2semidirect
11. C31×#1458C2semidirect
65. C31×#125D4semidirect
69. C31×#124D4semidirect
88. C4×Q8×C31D4semidirect
97. C4×D4×C31D4semidirect
215. #18216C31direct (N × H)
278. #3493C2×C31semidirect
279. #3105C2×C31semidirect
280. #2598C2×C31semidirect
281. #1618C2×C31semidirect
282. #1617C2×C31semidirect
283. #1616C2×C31semidirect
284. #1600C2×C31semidirect
285. #1599C2×C31semidirect
286. #1598C2×C31semidirect
287. #1521C2×C31semidirect
288. #1458C2×C31semidirect
383. #125D4×C31semidirect
387. #124D4×C31semidirect
406. C4×Q8D4×C31semidirect
415. C4×D4D4×C31semidirect
452. C31#18216direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#34932
C31×#31052
C31×#25982
C31×#16182
C31×#16172
C31×#16162
C31×#16002
C31×#15992
C31×#15982
C31×#15212
C31×#14582
#1821631