Finite Group Databasebeta

← C31×#18107 #21474866

Subgroups — C31×#18107 #21474866

Order 7936.

1000 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3459C211yesmaximal, characteristic
2C31×#3091C212yesmaximal, complemented
3C31×#2395C211yesmaximal, characteristic, complemented
4C31×#1615C212yesmaximal, complemented
5C31×#1614C211yesmaximal, characteristic, complemented
6C31×#1613C211yesmaximal, characteristic, complemented
7C31×#1609C212yesmaximal, complemented
8C31×#1608C211yesmaximal, characteristic, complemented
9C31×#1607C211yesmaximal, characteristic, complemented
10C31×#1501C212yesmaximal, complemented
11C31×#1459C211yesmaximal, characteristic, complemented
12C31×#543C2211yescharacteristic
13C31×#539—22no—
14C31×#539C2212yes—
15C31×#538—22no—
16C31×#538—21no—
17C31×#534—21no—
18C31×#534C2211yescharacteristic
19C31×#529—22no—
20C31×#529—21no—
21C31×#525—22no—
22C31×#525C2212yes—
23C2×C4×D4×C31—21no—
24C2×C4×D4×C31C2211yescharacteristic
25C2×C31×#117C2211yescharacteristic, complemented
26C31×#485—22no—
27C31×#484C2212yescomplemented
28C31×#483C2212yescomplemented
29C31×#482—22no—
30C31×#481C2212yes—
31C31×#480C2212yes—
32C31×#427—22no—
33C31×#416—22no—
34C2×C31×C22⋊C8C2212yescomplemented
35C31×#400C2211yescharacteristic, complemented
36C31×#399C2211yescharacteristic
37C31×#398C2211yescharacteristic, complemented
38C31×#397—21no—
39C31×#397C2211yescharacteristic
40C31×#394—21no—
41C4×C4⋊C4×C31—21no—
42C4×C22⋊C4×C31—21no—
43C4×C22⋊C4×C31C2211yescharacteristic, complemented
44C31×#328C2212yescomplemented
45C31×#328C2212yescomplemented
46C31×#327C2212yes—
47C31×#327C2212yes—
48C31×#324C2212yescomplemented
49C31×#323C2211yescharacteristic
50C31×#323C2211yescharacteristic
51C2×#41×C31D412yes—
52C22×D4×C31—21no—
53C22×D4×C31D411yescharacteristic
54C23×C4×C31—21no—
55C23×C4×C31D411yescharacteristic
56C23×C4×C31C2311yescharacteristic
57C22×C8×C31—22no—
58C31×#126—22no—
59C31×#126—22no—
60C31×#126—22no—
61C31×#125—21no—
62C31×#125—22no—
63C31×#124—22no—
64C31×#124—21no—
65C31×#123—22no—
66C31×#123—22no—
67C31×#123—22no—
68C31×#123—22no—
69C31×#123—22no—
70C31×#123—22no—
71C31×C22⋊Q8—22no—
72C31×C22⋊Q8—22no—
73C31×C22⋊Q8—21no—
74C31×C22⋊Q8—22no—
75C31×C22⋊Q8D412yes—
76C31×C22⋊Q8C2312yes—
77C31×#121—22no—
78C31×#121—22no—
79C31×#121—22no—
80C31×#121—21no—
81C31×#121D412yes—
82C31×#121C2312yes—
83C31×#120—22no—
84C31×#120—22no—
85C31×#120—22no—
86C4×Q8×C31—21no—
87C4×D4×C31—22no—
88C4×D4×C31—22no—
89C4×D4×C31—21no—
90C4×D4×C31—22no—
91C4×D4×C31—22no—
92C4×D4×C31—22no—
93C4×D4×C31—22no—
94C4×D4×C31—22no—
95C31×#117—22no—
96C31×#117—22no—
97C31×#117—22no—
98C31×#117D412yes—
99C2×C4⋊C4×C31—21no—
100C2×C4⋊C4×C31—21no—
101C2×C4⋊C4×C31—21no—
102C2×C4⋊C4×C31C2311yescharacteristic
103C2×C4⋊C4×C31C2311yescharacteristic
104C2×C4⋊C4×C31C2311yescharacteristic
105C2×C22⋊C4×C31—21no—
106C2×C22⋊C4×C31—21no—
107C2×C22⋊C4×C31—21no—
108C2×C22⋊C4×C31—21no—
109C2×C22⋊C4×C31—21no—
110C2×C22⋊C4×C31—21no—
111C2×C22⋊C4×C31D411yescharacteristic
112C2×C22⋊C4×C31C2311yescharacteristic
113C2×C42×C31—21no—
114C2×C42×C31—21no—
115C2×C42×C31—21no—
116C2×C42×C31D411yescharacteristic
117C31×#107—22no—
118C31×#107—22no—
119C31×#106—22no—
120C31×#106—22no—
121C31×#103—22no—
122C31×#103—22no—
123C31×#102—22no—
124C31×#102—22no—
125C31×C22⋊C8—22no—
126C31×C22⋊C8C2312yes—
127C31×C22⋊C8C2312yes—
128C31×#95—21no—
129C31×#95—21no—
130C31×#95—21no—
131C31×#95—21no—
132C31×#95C2311yescharacteristic
133C31×#95C2311yescharacteristic
134C24×C31—21no—
135C24×C31C2×D411yescharacteristic
136C24×C31C2×D411yescharacteristic
137#41×C31—44no—
138C2×Q8×C31—21no—
139C2×Q8×C31C2×D412yes—
140C2×D4×C31—22no—
141C2×D4×C31—22no—
142C2×D4×C31—22no—
143C2×D4×C31—22no—
144C2×D4×C31—21no—
145C2×D4×C31—22no—
146C2×D4×C31—22no—
147C2×D4×C31—41no—
148C2×D4×C31—42no—
149C2×D4×C31—42no—
150C2×D4×C31C2×D412yes—
151C22×C4×C31—21no—
152C22×C4×C31—21no—
153C22×C4×C31—21no—
154C22×C4×C31—21no—
155C22×C4×C31—22no—
156C22×C4×C31—21no—
157C22×C4×C31—22no—
158C22×C4×C31—21no—
159C22×C4×C31—21no—
160C22×C4×C31—21no—
161C22×C4×C31—21no—
162C22×C4×C31—42no—
163C22×C4×C31—42no—
164C22×C4×C31—82no—
165C22×C4×C31C2×D411yescharacteristic
166C22×C4×C31C2×D411yescharacteristic
167C22×C4×C31C2×D412yes—
168C22×C4×C31C2×D411yescharacteristic
169C22×C4×C31C2×D411yescharacteristic
170C22×C4×C31C2×D412yes—
171C22×C4×C31C2411yescharacteristic
172C2×C8×C31—22no—
173C2×C8×C31—22no—
174C2×C8×C31—44no—
175C4⋊C4×C31—22no—
176C4⋊C4×C31—22no—
177C4⋊C4×C31—22no—
178C4⋊C4×C31—21no—
179C4⋊C4×C31—21no—
180C4⋊C4×C31—22no—
181C4⋊C4×C31—22no—
182C4⋊C4×C31—22no—
183C4⋊C4×C31—22no—
184C4⋊C4×C31—22no—
185C4⋊C4×C31—22no—
186C4⋊C4×C31—42no—
187C4⋊C4×C31—42no—
188C4⋊C4×C31C2×D412yes—
189C4⋊C4×C31C2×D412yes—
190C22⋊C4×C31—22no—
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—42no—
204C22⋊C4×C31—42no—
205C22⋊C4×C31—42no—
206C42×C31—22no—
207C42×C31—21no—
208C42×C31—22no—
209C42×C31—21no—
210C42×C31—42no—
211C42×C31—41no—
212C42×C31—41no—
213#18107C3111yesmaximal, characteristic, direct product factor, Sylow
214C23×C31—21no—
215C23×C31—21no—
216C23×C31—21no—
217C23×C31—21no—
218C23×C31—22no—
219C23×C31—41no—
220C23×C31—41no—
221C23×C31—41no—
222C23×C31—41no—
223C23×C31—42no—
224C23×C31—81no—
225C23×C31—81no—
226C23×C31#12012yes—
227C23×C31#13512yes—
228C23×C31C22×D411yescharacteristic
229Q8×C31—42no—
230Q8×C31—81no—
231D4×C31—42no—
232D4×C31—44no—
233D4×C31—44no—
234D4×C31—42no—
235D4×C31—42no—
236D4×C31—82no—
237D4×C31—82no—
238D4×C31—81no—
239C2×C4×C31—22no—
240C2×C4×C31—24no—
241C2×C4×C31—22no—
242C2×C4×C31—21no—
243C2×C4×C31—22no—
244C2×C4×C31—21no—
245C2×C4×C31—22no—
246C2×C4×C31—21no—
247C2×C4×C31—22no—
248C2×C4×C31—21no—
249C2×C4×C31—21no—
250C2×C4×C31—42no—
251C2×C4×C31—41no—
252C2×C4×C31—41no—
253C2×C4×C31—42no—
254C2×C4×C31—41no—
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—81no—
269C2×C4×C31—81no—
270C2×C4×C31—82no—
271C2×C4×C31—81no—
272C2×C4×C31#12011yescharacteristic
273C2×C4×C31#12011yescharacteristic
274C2×C4×C31C22×D412yes—
275C8×C31—42no—
276C8×C31—42no—
277#3459C2×C3111yescharacteristic
278#3091C2×C3112yescomplemented
279#2395C2×C3111yescharacteristic, complemented
280#1615C2×C3112yescomplemented
281#1614C2×C3111yescharacteristic, complemented
282#1613C2×C3111yescharacteristic, complemented
283#1609C2×C3112yescomplemented
284#1608C2×C3111yescharacteristic, complemented
285#1607C2×C3111yescharacteristic, complemented
286#1501C2×C3112yescomplemented
287#1459C2×C3111yescharacteristic, complemented
288C22×C31—21no—
289C22×C31—21no—
290C22×C31—22no—
291C22×C31—22no—
292C22×C31—22no—
293C22×C31—22no—
294C22×C31—42no—
295C22×C31—41no—
296C22×C31—41no—
297C22×C31—42no—
298C22×C31—42no—
299C22×C31—42no—
300C22×C31—81no—
301C22×C31—81no—
302C22×C31—81no—
303C22×C31—84no—
304C22×C31C2×#12011yescharacteristic
305C22×C31C2×#13511yescharacteristic
306C22×C31#57511yescharacteristic
307C4×C31—22no—
308C4×C31—21no—
309C4×C31—42no—
310C4×C31—41no—
311C4×C31—42no—
312C4×C31—41no—
313C4×C31—42no—
314C4×C31—41no—
315C4×C31—42no—
316C4×C31—81no—
317C4×C31—81no—
318C4×C31—81no—
319C4×C31#45812yes—
320#543C22×C3111yescharacteristic
321#539—22no—
322#539C22×C3112yes—
323#538—22no—
324#538—21no—
325#534—21no—
326#534C22×C3111yescharacteristic
327#529—22no—
328#529—21no—
329#525—22no—
330#525C22×C3112yes—
331C2×C4×D4—21no—
332C2×C4×D4C22×C3111yescharacteristic
333C2×#117C22×C3111yescharacteristic, complemented
334#485—22no—
335#484C22×C3112yescomplemented
336#483C22×C3112yescomplemented
337#482—22no—
338#481C22×C3112yes—
339#480C22×C3112yes—
340#427—22no—
341#416—22no—
342C2×C22⋊C8C22×C3112yescomplemented
343#400C22×C3111yescharacteristic, complemented
344#399C22×C3111yescharacteristic
345#398C22×C3111yescharacteristic, complemented
346#397—21no—
347#397C22×C3111yescharacteristic
348#394—21no—
349C4×C4⋊C4—21no—
350C4×C22⋊C4—21no—
351C4×C22⋊C4C22×C3111yescharacteristic, complemented
352#328C22×C3112yescomplemented
353#328C22×C3112yescomplemented
354#327C22×C3112yes—
355#327C22×C3112yes—
356#324C22×C3112yescomplemented
357#323C22×C3111yescharacteristic
358#323C22×C3111yescharacteristic
359C2×C31—21no—
360C2×C31—21no—
361C2×C31—42no—
362C2×C31—42no—
363C2×C31—82no—
364C2×C31#300211yescharacteristic
365C2×C31#301811yescharacteristic
366C2×C31C2×#45811yescharacteristic
367C2×#41D4×C3112yes—
368C22×D4—21no—
369C22×D4D4×C3111yescharacteristic
370C23×C4—21no—
371C23×C4D4×C3111yescharacteristic
372C23×C4C23×C3111yescharacteristic
373C22×C8—22no—
374#126—22no—
375#126—22no—
376#126—22no—
377#125—21no—
378#125—22no—
379#124—22no—
380#124—21no—
381#123—22no—
382#123—22no—
383#123—22no—
384#123—22no—
385#123—22no—
386#123—22no—
387C22⋊Q8—22no—
388C22⋊Q8—22no—
389C22⋊Q8—21no—
390C22⋊Q8—22no—
391C22⋊Q8D4×C3112yes—
392C22⋊Q8C23×C3112yes—
393#121—22no—
394#121—22no—
395#121—22no—
396#121—21no—
397#121D4×C3112yes—
398#121C23×C3112yes—
399#120—22no—
400#120—22no—
401#120—22no—
402C4×Q8—21no—
403C4×D4—22no—
404C4×D4—22no—
405C4×D4—21no—
406C4×D4—22no—
407C4×D4—22no—
408C4×D4—22no—
409C4×D4—22no—
410C4×D4—22no—
411#117—22no—
412#117—22no—
413#117—22no—
414#117D4×C3112yes—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—21no—
418C2×C4⋊C4C23×C3111yescharacteristic
419C2×C4⋊C4C23×C3111yescharacteristic
420C2×C4⋊C4C23×C3111yescharacteristic
421C2×C22⋊C4—21no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—21no—
424C2×C22⋊C4—21no—
425C2×C22⋊C4—21no—
426C2×C22⋊C4—21no—
427C2×C22⋊C4D4×C3111yescharacteristic
428C2×C22⋊C4C23×C3111yescharacteristic
429C2×C42—21no—
430C2×C42—21no—
431C2×C42—21no—
432C2×C42D4×C3111yescharacteristic
433#107—22no—
434#107—22no—
435#106—22no—
436#106—22no—
437#103—22no—
438#103—22no—
439#102—22no—
440#102—22no—
441C22⋊C8—22no—
442C22⋊C8C23×C3112yes—
443C22⋊C8C23×C3112yes—
444#95—21no—
445#95—21no—
446#95—21no—
447#95—21no—
448#95C23×C3111yescharacteristic
449#95C23×C3111yescharacteristic
450C31#1810711yescharacteristic, direct product factor, Sylow
451C24—21no—
452C24C2×D4×C3111yescharacteristic
453C24C2×D4×C3111yescharacteristic
454#41—44no—
455C2×Q8—21no—
456C2×Q8C2×D4×C3112yes—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—22no—
460C2×D4—22no—
461C2×D4—21no—
462C2×D4—22no—
463C2×D4—22no—
464C2×D4—41no—
465C2×D4—42no—
466C2×D4—42no—
467C2×D4C2×D4×C3112yes—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—22no—
473C22×C4—21no—
474C22×C4—22no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—82no—
482C22×C4C2×D4×C3111yescharacteristic
483C22×C4C2×D4×C3111yescharacteristic
484C22×C4C2×D4×C3112yes—
485C22×C4C2×D4×C3111yescharacteristic
486C22×C4C2×D4×C3111yescharacteristic
487C22×C4C2×D4×C3112yes—
488C22×C4C24×C3111yescharacteristic
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—44no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4C2×D4×C3112yes—
506C4⋊C4C2×D4×C3112yes—
507C22⋊C4—22no—
508C22⋊C4—22no—
509C22⋊C4—22no—
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—42no—
520C22⋊C4—42no—
521C22⋊C4—42no—
522C22⋊C4—42no—
523C42—22no—
524C42—22no—
525C42—21no—
526C42—21no—
527C42—42no—
528C42—41no—
529C42—41no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—22no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—42no—
540C23—81no—
541C23—81no—
542C23C31×#12012yes—
543C23C31×#13512yes—
544C23C22×D4×C3111yescharacteristic
545Q8—42no—
546Q8—81no—
547D4—42no—
548D4—44no—
549D4—44no—
550D4—42no—
551D4—42no—
552D4—82no—
553D4—82no—
554D4—81no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—41no—
574C2×C4—41no—
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—81no—
585C2×C4—81no—
586C2×C4—82no—
587C2×C4—81no—
588C2×C4C31×#12011yescharacteristic
589C2×C4C31×#12011yescharacteristic
590C2×C4C22×D4×C3112yes—
591C8—42no—
592C8—42no—
593C22—21no—
594C22—21no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—42no—
600C22—41no—
601C22—41no—
602C22—42no—
603C22—42no—
604C22—42no—
605C22—81no—
606C22—81no—
607C22—81no—
608C22—84no—
609C22C2×C31×#12011yescharacteristic
610C22C2×C31×#13511yescharacteristic
611C22C31×#57511yescharacteristic
612C4—22no—
613C4—21no—
614C4—42no—
615C4—41no—
616C4—42no—
617C4—41no—
618C4—42no—
619C4—41no—
620C4—42no—
621C4—81no—
622C4—81no—
623C4—81no—
624C4C31×#45812yes—
625C2—21no—
626C2—21no—
627C2—42no—
628C2—42no—
629C2—82no—
630C2C31×#300211yescharacteristic
631C2C31×#301811yescharacteristic
632C2C2×C31×#45811yescharacteristic
subgroupG/H
centerC2×C4×C31#120
commutator, Frattini, cosocleC22×C4C24×C31
socleC22×C31C2×#120
prime radicalC31×#3459C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#3091C2semidirect
3. C31×#2395C2semidirect
4. C31×#1615C2semidirect
5. C31×#1614C2semidirect
6. C31×#1613C2semidirect
7. C31×#1609C2semidirect
8. C31×#1608C2semidirect
9. C31×#1607C2semidirect
10. C31×#1501C2semidirect
11. C31×#1459C2semidirect
25. C2×C31×#117C22semidirect
27. C31×#484C22semidirect
28. C31×#483C22semidirect
34. C2×C31×C22⋊C8C22semidirect
35. C31×#400C22semidirect
37. C31×#398C22semidirect
43. C4×C22⋊C4×C31C22semidirect
44. C31×#328C22semidirect
45. C31×#328C22semidirect
48. C31×#324C22semidirect
213. #18107C31direct (N × H)
278. #3091C2×C31semidirect
279. #2395C2×C31semidirect
280. #1615C2×C31semidirect
281. #1614C2×C31semidirect
282. #1613C2×C31semidirect
283. #1609C2×C31semidirect
284. #1608C2×C31semidirect
285. #1607C2×C31semidirect
286. #1501C2×C31semidirect
287. #1459C2×C31semidirect
333. C2×#117C22×C31semidirect
335. #484C22×C31semidirect
336. #483C22×C31semidirect
342. C2×C22⋊C8C22×C31semidirect
343. #400C22×C31semidirect
345. #398C22×C31semidirect
351. C4×C22⋊C4C22×C31semidirect
352. #328C22×C31semidirect
353. #328C22×C31semidirect
356. #324C22×C31semidirect
450. C31#18107direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#34592
C31×#30912
C31×#23952
C31×#16152
C31×#16142
C31×#16132
C31×#16092
C31×#16082
C31×#16072
C31×#15012
C31×#14592
#1810731