Finite Group Databasebeta

← C31×#18314 #21475073

Subgroups — C31×#18314 #21475073

Order 7936.

958 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3459C211yesmaximal, characteristic
2C31×#3232C212yesmaximal, complemented
3C31×#2633C211yesmaximal, characteristic, complemented
4C31×#1618C212yesmaximal, complemented
5C31×#1614C211yesmaximal, characteristic, complemented
6C31×#1613C211yesmaximal, characteristic, complemented
7C31×#1612C212yesmaximal, complemented
8C31×#1608C211yesmaximal, characteristic, complemented
9C31×#1607C211yesmaximal, characteristic, complemented
10C31×#1521C212yesmaximal, complemented
11C31×#1460C211yesmaximal, characteristic, complemented
12C31×#543C2211yescharacteristic
13C31×#539—22no—
14C31×#539—21no—
15C31×#538—22no—
16C31×#538C2212yes—
17C31×#534—21no—
18C31×#534C2211yescharacteristic
19C31×#529—22no—
20C31×#529C2212yes—
21C2×C31×#125C2211yescharacteristic, complemented
22C31×#525—21no—
23C31×#525—22no—
24C2×C4×D4×C31—21no—
25C2×C4×D4×C31C2211yescharacteristic
26C31×#499—22no—
27C31×#488—22no—
28C31×#484C2212yescomplemented
29C31×#483C2212yescomplemented
30C31×#481C2212yes—
31C31×#480C2212yes—
32C31×#479—22no—
33C31×#478—22no—
34C31×#431C2212yescomplemented
35C31×#399C2211yescharacteristic
36C31×#399C2211yescharacteristic, complemented
37C31×#398—21no—
38C31×#397C2211yescharacteristic
39C31×#397C2211yescharacteristic, complemented
40C31×#394—21no—
41C31×#385—21no—
42C31×#385—21no—
43C31×#383C2211yescharacteristic, complemented
44C31×#328C2212yescomplemented
45C31×#328C2212yescomplemented
46C31×#327C2212yes—
47C31×#327C2212yes—
48C31×#323C2211yescharacteristic
49C31×#323C2211yescharacteristic
50C31×#323C2212yescomplemented
51C2×#41×C31—21no—
52C22×D4×C31—21no—
53C22×D4×C31D411yescharacteristic
54C23×C4×C31—21no—
55C23×C4×C31D411yescharacteristic
56C23×C4×C31D411yescharacteristic
57C31×#126—22no—
58C31×#126—22no—
59C31×#126—22no—
60C31×#125—22no—
61C31×#125—22no—
62C31×#125—22no—
63C31×#125D412yes—
64C31×#124—22no—
65C31×#124D412yes—
66C31×#123—22no—
67C31×#123—22no—
68C31×#123—22no—
69C31×#123—22no—
70C31×#123—22no—
71C31×#123—22no—
72C31×C22⋊Q8—21no—
73C31×C22⋊Q8—22no—
74C31×C22⋊Q8—22no—
75C31×C22⋊Q8—21no—
76C31×C22⋊Q8—22no—
77C31×C22⋊Q8C2312yes—
78C31×#121—22no—
79C31×#121—21no—
80C31×#121—22no—
81C31×#121—22no—
82C31×#121—21no—
83C31×#121C2312yes—
84C31×#120—22no—
85C31×#120—22no—
86C31×#120—22no—
87C4×Q8×C31D412yes—
88C4×D4×C31—22no—
89C4×D4×C31—22no—
90C4×D4×C31—22no—
91C4×D4×C31—22no—
92C4×D4×C31—22no—
93C4×D4×C31—22no—
94C4×D4×C31—22no—
95C4×D4×C31D412yes—
96C31×#117—21no—
97C31×#117—22no—
98C2×C4⋊C4×C31—21no—
99C2×C4⋊C4×C31—21no—
100C2×C4⋊C4×C31—21no—
101C2×C4⋊C4×C31—21no—
102C2×C4⋊C4×C31—21no—
103C2×C4⋊C4×C31D411yescharacteristic
104C2×C4⋊C4×C31C2311yescharacteristic
105C2×C4⋊C4×C31C2311yescharacteristic
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×C31—21no—
112C2×C22⋊C4×C31C2311yescharacteristic
113C2×C22⋊C4×C31C2311yescharacteristic
114C2×C42×C31—21no—
115C2×C42×C31C2311yescharacteristic
116C31×#107—22no—
117C31×#107—22no—
118C31×#106—22no—
119C31×#106—22no—
120C31×C4⋊C8—22no—
121C31×#103—22no—
122C31×#103—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
136C24×C31C2×D411yescharacteristic
137#41×C31—82no—
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—22no—
155C22×C4×C31—21no—
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—21no—
163C22×C4×C31—21no—
164C22×C4×C31—81no—
165C22×C4×C31—81no—
166C22×C4×C31—82no—
167C22×C4×C31C2×D411yescharacteristic
168C22×C4×C31C2×D411yescharacteristic
169C22×C4×C31C2×D411yescharacteristic
170C22×C4×C31C2×D411yescharacteristic
171C22×C4×C31C2411yescharacteristic
172C2×C8×C31—22no—
173C2×C8×C31—22no—
174C4⋊C4×C31—22no—
175C4⋊C4×C31—21no—
176C4⋊C4×C31—21no—
177C4⋊C4×C31—22no—
178C4⋊C4×C31—22no—
179C4⋊C4×C31—22no—
180C4⋊C4×C31—22no—
181C4⋊C4×C31—22no—
182C4⋊C4×C31—22no—
183C4⋊C4×C31—22no—
184C4⋊C4×C31—22no—
185C4⋊C4×C31—41no—
186C4⋊C4×C31—42no—
187C4⋊C4×C31—41no—
188C4⋊C4×C31—81no—
189C4⋊C4×C31—81no—
190C4⋊C4×C31C2×D412yes—
191C4⋊C4×C31C2×D412yes—
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—22no—
203C22⋊C4×C31—22no—
204C22⋊C4×C31—42no—
205C22⋊C4×C31—42no—
206C22⋊C4×C31—81no—
207C22⋊C4×C31—81no—
208C42×C31—22no—
209C42×C31—22no—
210C42×C31C2×D412yes—
211C42×C31C2×D412yes—
212#18314C3111yesmaximal, characteristic, direct product factor, Sylow
213C23×C31—21no—
214C23×C31—21no—
215C23×C31—21no—
216C23×C31—22no—
217C23×C31—21no—
218C23×C31—21no—
219C23×C31—41no—
220C23×C31—41no—
221C23×C31—41no—
222C23×C31—41no—
223C23×C31—42no—
224C23×C31—81no—
225C23×C31—81no—
226C23×C31#13512yes—
227C23×C31C22×D411yescharacteristic
228Q8×C31—42no—
229Q8×C31—81no—
230D4×C31—44no—
231D4×C31—42no—
232D4×C31—44no—
233D4×C31—42no—
234D4×C31—42no—
235D4×C31—82no—
236D4×C31—81no—
237D4×C31—82no—
238C2×C4×C31—21no—
239C2×C4×C31—21no—
240C2×C4×C31—22no—
241C2×C4×C31—22no—
242C2×C4×C31—21no—
243C2×C4×C31—21no—
244C2×C4×C31—22no—
245C2×C4×C31—41no—
246C2×C4×C31—41no—
247C2×C4×C31—42no—
248C2×C4×C31—42no—
249C2×C4×C31—44no—
250C2×C4×C31—42no—
251C2×C4×C31—41no—
252C2×C4×C31—41no—
253C2×C4×C31—41no—
254C2×C4×C31—41no—
255C2×C4×C31—41no—
256C2×C4×C31—41no—
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—81no—
266C2×C4×C31—81no—
267C2×C4×C31—81no—
268C2×C4×C31—82no—
269C2×C4×C31—81no—
270C2×C4×C31#12011yescharacteristic
271C2×C4×C31#12012yes—
272C2×C4×C31#12011yescharacteristic
273C2×C4×C31C22×D412yes—
274C8×C31—42no—
275C8×C31—42no—
276#3459C2×C3111yescharacteristic
277#3232C2×C3112yescomplemented
278#2633C2×C3111yescharacteristic, complemented
279#1618C2×C3112yescomplemented
280#1614C2×C3111yescharacteristic, complemented
281#1613C2×C3111yescharacteristic, complemented
282#1612C2×C3112yescomplemented
283#1608C2×C3111yescharacteristic, complemented
284#1607C2×C3111yescharacteristic, complemented
285#1521C2×C3112yescomplemented
286#1460C2×C3111yescharacteristic, complemented
287C22×C31—21no—
288C22×C31—21no—
289C22×C31—22no—
290C22×C31—42no—
291C22×C31—42no—
292C22×C31—41no—
293C22×C31—41no—
294C22×C31—41no—
295C22×C31—42no—
296C22×C31—41no—
297C22×C31—42no—
298C22×C31—42no—
299C22×C31—41no—
300C22×C31—81no—
301C22×C31—81no—
302C22×C31—84no—
303C22×C31C2×#12011yescharacteristic
304C22×C31C2×#13511yescharacteristic
305C22×C31#57511yescharacteristic
306C4×C31—22no—
307C4×C31—21no—
308C4×C31—21no—
309C4×C31—22no—
310C4×C31—42no—
311C4×C31—41no—
312C4×C31—41no—
313C4×C31—42no—
314C4×C31—42no—
315C4×C31—81no—
316C4×C31—81no—
317C4×C31—81no—
318C4×C31—81no—
319#543C22×C3111yescharacteristic
320#539—22no—
321#539—21no—
322#538—22no—
323#538C22×C3112yes—
324#534—21no—
325#534C22×C3111yescharacteristic
326#529—22no—
327#529C22×C3112yes—
328C2×#125C22×C3111yescharacteristic, complemented
329#525—21no—
330#525—22no—
331C2×C4×D4—21no—
332C2×C4×D4C22×C3111yescharacteristic
333#499—22no—
334#488—22no—
335#484C22×C3112yescomplemented
336#483C22×C3112yescomplemented
337#481C22×C3112yes—
338#480C22×C3112yes—
339#479—22no—
340#478—22no—
341#431C22×C3112yescomplemented
342#399C22×C3111yescharacteristic
343#399C22×C3111yescharacteristic, complemented
344#398—21no—
345#397C22×C3111yescharacteristic
346#397C22×C3111yescharacteristic, complemented
347#394—21no—
348#385—21no—
349#385—21no—
350#383C22×C3111yescharacteristic, complemented
351#328C22×C3112yescomplemented
352#328C22×C3112yescomplemented
353#327C22×C3112yes—
354#327C22×C3112yes—
355#323C22×C3111yescharacteristic
356#323C22×C3111yescharacteristic
357#323C22×C3112yescomplemented
358C2×C31—21no—
359C2×C31—21no—
360C2×C31—42no—
361C2×C31—81no—
362C2×C31—82no—
363C2×C31#300211yescharacteristic
364C2×C31#301911yescharacteristic
365C2×C31#302811yescharacteristic
366C2×#41—21no—
367C22×D4—21no—
368C22×D4D4×C3111yescharacteristic
369C23×C4—21no—
370C23×C4D4×C3111yescharacteristic
371C23×C4D4×C3111yescharacteristic
372#126—22no—
373#126—22no—
374#126—22no—
375#125—22no—
376#125—22no—
377#125—22no—
378#125D4×C3112yes—
379#124—22no—
380#124D4×C3112yes—
381#123—22no—
382#123—22no—
383#123—22no—
384#123—22no—
385#123—22no—
386#123—22no—
387C22⋊Q8—21no—
388C22⋊Q8—22no—
389C22⋊Q8—22no—
390C22⋊Q8—21no—
391C22⋊Q8—22no—
392C22⋊Q8C23×C3112yes—
393#121—22no—
394#121—21no—
395#121—22no—
396#121—22no—
397#121—21no—
398#121C23×C3112yes—
399#120—22no—
400#120—22no—
401#120—22no—
402C4×Q8D4×C3112yes—
403C4×D4—22no—
404C4×D4—22no—
405C4×D4—22no—
406C4×D4—22no—
407C4×D4—22no—
408C4×D4—22no—
409C4×D4—22no—
410C4×D4D4×C3112yes—
411#117—21no—
412#117—22no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—21no—
418C2×C4⋊C4D4×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⋊C4C23×C3111yescharacteristic
428C2×C22⋊C4C23×C3111yescharacteristic
429C2×C42—21no—
430C2×C42C23×C3111yescharacteristic
431#107—22no—
432#107—22no—
433#106—22no—
434#106—22no—
435C4⋊C8—22no—
436#103—22no—
437#103—22no—
438#102—22no—
439#102—22no—
440C22⋊C8C23×C3112yes—
441C22⋊C8C23×C3112yes—
442C4×C8—22no—
443#95—21no—
444#95—21no—
445#95—21no—
446#95—21no—
447#95C23×C3111yescharacteristic
448#95C23×C3111yescharacteristic
449C31#1831411yescharacteristic, direct product factor, Sylow
450C24—21no—
451C24C2×D4×C3111yescharacteristic
452C24C2×D4×C3111yescharacteristic
453#41—82no—
454C2×Q8—21no—
455C2×Q8C2×D4×C3112yes—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—22no—
460C2×D4—21no—
461C2×D4—22no—
462C2×D4—22no—
463C2×D4—41no—
464C2×D4—42no—
465C2×D4—42no—
466C2×D4C2×D4×C3112yes—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—22no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—22no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—81no—
481C22×C4—81no—
482C22×C4—82no—
483C22×C4C2×D4×C3111yescharacteristic
484C22×C4C2×D4×C3111yescharacteristic
485C22×C4C2×D4×C3111yescharacteristic
486C22×C4C2×D4×C3111yescharacteristic
487C22×C4C24×C3111yescharacteristic
488C2×C8—22no—
489C2×C8—22no—
490C4⋊C4—22no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—41no—
502C4⋊C4—42no—
503C4⋊C4—41no—
504C4⋊C4—81no—
505C4⋊C4—81no—
506C4⋊C4C2×D4×C3112yes—
507C4⋊C4C2×D4×C3112yes—
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—22no—
520C22⋊C4—42no—
521C22⋊C4—42no—
522C22⋊C4—81no—
523C22⋊C4—81no—
524C42—22no—
525C42—22no—
526C42C2×D4×C3112yes—
527C42C2×D4×C3112yes—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—22no—
532C23—21no—
533C23—21no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—42no—
539C23—81no—
540C23—81no—
541C23C31×#13512yes—
542C23C22×D4×C3111yescharacteristic
543Q8—42no—
544Q8—81no—
545D4—44no—
546D4—42no—
547D4—44no—
548D4—42no—
549D4—42no—
550D4—82no—
551D4—81no—
552D4—82no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—81no—
581C2×C4—81no—
582C2×C4—81no—
583C2×C4—82no—
584C2×C4—81no—
585C2×C4C31×#12011yescharacteristic
586C2×C4C31×#12012yes—
587C2×C4C31×#12011yescharacteristic
588C2×C4C22×D4×C3112yes—
589C8—42no—
590C8—42no—
591C22—21no—
592C22—21no—
593C22—22no—
594C22—42no—
595C22—42no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—42no—
600C22—41no—
601C22—42no—
602C22—42no—
603C22—41no—
604C22—81no—
605C22—81no—
606C22—84no—
607C22C2×C31×#12011yescharacteristic
608C22C2×C31×#13511yescharacteristic
609C22C31×#57511yescharacteristic
610C4—22no—
611C4—21no—
612C4—21no—
613C4—22no—
614C4—42no—
615C4—41no—
616C4—41no—
617C4—42no—
618C4—42no—
619C4—81no—
620C4—81no—
621C4—81no—
622C4—81no—
623C2—21no—
624C2—21no—
625C2—42no—
626C2—81no—
627C2—82no—
628C2C31×#300211yescharacteristic
629C2C31×#301911yescharacteristic
630C2C31×#302811yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#3459C2
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
2. C31×#3232C2semidirect
3. C31×#2633C2semidirect
4. C31×#1618C2semidirect
5. C31×#1614C2semidirect
6. C31×#1613C2semidirect
7. C31×#1612C2semidirect
8. C31×#1608C2semidirect
9. C31×#1607C2semidirect
10. C31×#1521C2semidirect
11. C31×#1460C2semidirect
21. C2×C31×#125C22semidirect
28. C31×#484C22semidirect
29. C31×#483C22semidirect
34. C31×#431C22semidirect
36. C31×#399C22semidirect
39. C31×#397C22semidirect
43. C31×#383C22semidirect
44. C31×#328C22semidirect
45. C31×#328C22semidirect
50. C31×#323C22semidirect
212. #18314C31direct (N × H)
277. #3232C2×C31semidirect
278. #2633C2×C31semidirect
279. #1618C2×C31semidirect
280. #1614C2×C31semidirect
281. #1613C2×C31semidirect
282. #1612C2×C31semidirect
283. #1608C2×C31semidirect
284. #1607C2×C31semidirect
285. #1521C2×C31semidirect
286. #1460C2×C31semidirect
328. C2×#125C22×C31semidirect
335. #484C22×C31semidirect
336. #483C22×C31semidirect
341. #431C22×C31semidirect
343. #399C22×C31semidirect
346. #397C22×C31semidirect
350. #383C22×C31semidirect
351. #328C22×C31semidirect
352. #328C22×C31semidirect
357. #323C22×C31semidirect
449. C31#18314direct (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×#32322
C31×#26332
C31×#16182
C31×#16142
C31×#16132
C31×#16122
C31×#16082
C31×#16072
C31×#15212
C31×#14602
#1831431