Finite Group Databasebeta

← C31×#25642 #21482401

Subgroups — C31×#25642 #21482401

Order 7936.

1276 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#559C211yesmaximal, characteristic, complemented
2C2×C31×#496C211yesmaximal, characteristic, complemented
3C2×C31×#459C211yesmaximal, characteristic, complemented
4C31×#2548C211yesmaximal, characteristic, complemented
5C31×#2012C212yesmaximal, complemented
6C31×#2011C212yesmaximal, complemented
7C31×#1985C211yesmaximal, characteristic, complemented
8C31×#1983C211yesmaximal, characteristic, complemented
9C31×#1880C212yesmaximal, complemented
10C31×#1879C212yesmaximal, complemented
11C31×#1762C211yesmaximal, characteristic, complemented
12C22×SD16×C31C2212yes—
13C22×D8×C31C2212yes—
14C22×D8×C31C2212yes—
15C31×#559—24no—
16C31×#559—24no—
17C2×C31×#127C2211yescharacteristic
18C2×C31×#127C2211yescharacteristic
19C2×C31×#124—21no—
20C2×C31×#124—21no—
21C2×C31×#124C2211yescharacteristic
22C2×C31×#124C2211yescharacteristic
23C2×C31×#123—21no—
24C2×C31×#123—21no—
25C2×C4×Q8×C31C2211yescharacteristic
26C2×C4×Q8×C31C2211yescharacteristic
27C2×C4×D4×C31—21no—
28C2×C4×D4×C31C2211yescharacteristic
29C2×C4×D4×C31C2211yescharacteristic
30C31×#496—24no—
31C31×#459—24no—
32C2×C31×C4⋊C8C2211yescharacteristic
33C2×C31×#103C2212yes—
34C2×C31×#102C2212yes—
35C2×C31×#102C2212yes—
36C2×C31×#102C2212yes—
37C2×C31×#97C2211yescharacteristic
38C31×#394—21no—
39C31×#394—21no—
40C31×#390—21no—
41C31×#390C2212yes—
42C31×#390C2212yes—
43C4×C4⋊C4×C31C2211yescharacteristic
44C31×#340C2212yes—
45C31×#336C2212yes—
46C31×#336C2212yes—
47C22×Q8×C31D411yescharacteristic
48C22×Q8×C31C2311yescharacteristic
49C22×D4×C31—21no—
50C22×D4×C31—21no—
51C22×D4×C31—21no—
52C22×D4×C31—21no—
53C22×D4×C31C2312yes—
54C22×D4×C31C2311yescharacteristic
55C22×D4×C31C2311yescharacteristic
56C23×C4×C31—21no—
57C23×C4×C31—21no—
58C2×SD16×C31—24no—
59C2×SD16×C31—24no—
60C2×SD16×C31—42no—
61C2×D8×C31—24no—
62C2×D8×C31—24no—
63C2×D8×C31—24no—
64C2×D8×C31—42no—
65C2×D8×C31—42no—
66C2×D8×C31—42no—
67C22×C8×C31C2312yes—
68C22×C8×C31C2312yes—
69C31×#127—22no—
70C31×#127—22no—
71C31×#127—22no—
72C31×#127—22no—
73C31×#124—28no—
74C31×#124—22no—
75C31×#124—22no—
76C31×#124—22no—
77C31×#124—44no—
78C31×#124D414yes—
79C31×#123—28no—
80C31×#123—28no—
81C4×Q8×C31—22no—
82C4×Q8×C31—22no—
83C4×Q8×C31—22no—
84C4×Q8×C31—22no—
85C4×D4×C31—28no—
86C4×D4×C31—22no—
87C4×D4×C31—22no—
88C4×D4×C31—22no—
89C4×D4×C31D414yes—
90C2×C4⋊C4×C31—21no—
91C2×C4⋊C4×C31D411yescharacteristic
92C2×C4⋊C4×C31D411yescharacteristic
93C2×C4⋊C4×C31D411yescharacteristic
94C2×C4⋊C4×C31C2312yes—
95C2×C4⋊C4×C31C2311yescharacteristic
96C2×C22⋊C4×C31—21no—
97C2×C22⋊C4×C31—21no—
98C2×C22⋊C4×C31—21no—
99C2×C22⋊C4×C31—21no—
100C2×C22⋊C4×C31—21no—
101C2×C22⋊C4×C31—21no—
102C2×C22⋊C4×C31—22no—
103C2×C22⋊C4×C31—22no—
104C2×C22⋊C4×C31—22no—
105C2×C22⋊C4×C31—22no—
106C2×C42×C31—21no—
107C2×C42×C31—21no—
108C2×C42×C31C2311yescharacteristic
109C2×C42×C31C2311yescharacteristic
110C2×C42×C31C2311yescharacteristic
111C31×C4⋊C8—22no—
112C31×#103—42no—
113C31×#102—24no—
114C31×#102—24no—
115C31×#102—42no—
116C31×#97—22no—
117C31×#95—21no—
118C24×C31—21no—
119C24×C31—21no—
120C24×C31—22no—
121C2×Q8×C31—22no—
122C2×Q8×C31—22no—
123C2×Q8×C31—21no—
124C2×Q8×C31—22no—
125C2×Q8×C31—22no—
126C2×Q8×C31—41no—
127C2×Q8×C31C2×D412yes—
128C2×D4×C31—24no—
129C2×D4×C31—24no—
130C2×D4×C31—21no—
131C2×D4×C31—21no—
132C2×D4×C31—22no—
133C2×D4×C31—22no—
134C2×D4×C31—22no—
135C2×D4×C31—22no—
136C2×D4×C31—22no—
137C2×D4×C31—22no—
138C2×D4×C31—22no—
139C2×D4×C31—22no—
140C2×D4×C31—22no—
141C2×D4×C31—22no—
142C2×D4×C31—42no—
143C2×D4×C31—42no—
144C2×D4×C31—42no—
145C2×D4×C31—42no—
146C2×D4×C31—42no—
147C2×D4×C31—42no—
148C2×D4×C31—82no—
149C2×D4×C31—82no—
150C2×D4×C31C2×D412yes—
151C2×D4×C31C2×D412yes—
152C22×C4×C31—21no—
153C22×C4×C31—21no—
154C22×C4×C31—24no—
155C22×C4×C31—24no—
156C22×C4×C31—22no—
157C22×C4×C31—82no—
158C22×C4×C31—82no—
159C22×C4×C31C2×D411yescharacteristic
160C22×C4×C31C2×D411yescharacteristic
161C22×C4×C31C2×D411yescharacteristic
162C22×C4×C31C2×D411yescharacteristic
163C22×C4×C31C2×D411yescharacteristic
164C22×C4×C31C2×D411yescharacteristic
165C22×C4×C31C2411yescharacteristic
166SD16×C31—48no—
167D8×C31—48no—
168D8×C31—48no—
169C2×C8×C31—22no—
170C2×C8×C31—22no—
171C2×C8×C31—22no—
172C2×C8×C31—22no—
173C2×C8×C31—22no—
174C2×C8×C31#4114yes—
175C4⋊C4×C31—24no—
176C4⋊C4×C31—22no—
177C4⋊C4×C31—22no—
178C4⋊C4×C31—22no—
179C4⋊C4×C31—21no—
180C4⋊C4×C31—22no—
181C4⋊C4×C31—22no—
182C4⋊C4×C31C2×D412yes—
183C22⋊C4×C31—24no—
184C22⋊C4×C31—24no—
185C22⋊C4×C31—24no—
186C22⋊C4×C31—24no—
187C22⋊C4×C31—24no—
188C22⋊C4×C31—24no—
189C22⋊C4×C31—44no—
190C22⋊C4×C31—44no—
191C22⋊C4×C31—82no—
192C22⋊C4×C31—82no—
193C42×C31—24no—
194C42×C31—21no—
195C42×C31—21no—
196C42×C31—21no—
197C42×C31—21no—
198C42×C31—42no—
199C42×C31C2×D412yes—
200C42×C31C2×D412yes—
201#25642C3111yesmaximal, characteristic, direct product factor, Sylow
202C23×C31—22no—
203C23×C31—22no—
204C23×C31—42no—
205C23×C31—42no—
206C23×C31—41no—
207C23×C31—41no—
208C23×C31—41no—
209C23×C31—41no—
210C23×C31—42no—
211C23×C31—42no—
212C23×C31—42no—
213C23×C31—42no—
214C23×C31—42no—
215C23×C31—42no—
216C23×C31—42no—
217C23×C31—42no—
218C23×C31—42no—
219C23×C31C22×D411yescharacteristic
220Q8×C31—24no—
221Q8×C31—44no—
222Q8×C31—41no—
223Q8×C31—41no—
224D4×C31—22no—
225D4×C31—22no—
226D4×C31—24no—
227D4×C31—48no—
228D4×C31—41no—
229D4×C31—42no—
230D4×C31—41no—
231D4×C31—42no—
232D4×C31—42no—
233D4×C31—42no—
234D4×C31—42no—
235D4×C31—84no—
236D4×C31—84no—
237D4×C31—84no—
238C2×C4×C31—21no—
239C2×C4×C31—21no—
240C2×C4×C31—21no—
241C2×C4×C31—21no—
242C2×C4×C31—22no—
243C2×C4×C31—22no—
244C2×C4×C31—21no—
245C2×C4×C31—21no—
246C2×C4×C31—22no—
247C2×C4×C31—21no—
248C2×C4×C31—21no—
249C2×C4×C31—21no—
250C2×C4×C31—22no—
251C2×C4×C31—22no—
252C2×C4×C31—42no—
253C2×C4×C31—42no—
254C2×C4×C31—42no—
255C2×C4×C31—42no—
256C2×C4×C31—42no—
257C2×C4×C31—84no—
258C2×C4×C31—84no—
259C2×C4×C31#12012yes—
260C2×C4×C31#12012yes—
261C2×C4×C31AGL(1,ℤ/8)12yes—
262C2×C4×C31C22×D411yescharacteristic
263C2×C4×C31C22×D411yescharacteristic
264C2×C4×C31C2×#4112yes—
265C2×C4×C31H(5,𝔽2)12yes—
266C8×C31—24no—
267C8×C31—42no—
268C2×#559C2×C3111yescharacteristic, complemented
269C2×#496C2×C3111yescharacteristic, complemented
270C2×#459C2×C3111yescharacteristic, complemented
271#2548C2×C3111yescharacteristic, complemented
272#2012C2×C3112yescomplemented
273#2011C2×C3112yescomplemented
274#1985C2×C3111yescharacteristic, complemented
275#1983C2×C3111yescharacteristic, complemented
276#1880C2×C3112yescomplemented
277#1879C2×C3112yescomplemented
278#1762C2×C3111yescharacteristic, complemented
279C22×C31—42no—
280C22×C31—42no—
281C22×C31—42no—
282C22×C31—42no—
283C22×C31—42no—
284C22×C31—42no—
285C22×C31—82no—
286C22×C31—82no—
287C22×C31—82no—
288C22×C31—82no—
289C22×C31—82no—
290C22×C31—82no—
291C22×C31—82no—
292C22×C31—82no—
293C22×C31—82no—
294C22×C31—82no—
295C22×C31—82no—
296C22×C31C2×#12011yescharacteristic
297C22×C31#55012yes—
298C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
299C22×C31#57511yescharacteristic
300C22×C31#57612yes—
301C4×C31—22no—
302C4×C31—22no—
303C4×C31—22no—
304C4×C31—42no—
305C4×C31—41no—
306C4×C31—41no—
307C4×C31—41no—
308C4×C31—42no—
309C4×C31—82no—
310C4×C31#53211yescharacteristic
311C4×C31#53511yescharacteristic
312C4×C31#54612yes—
313C22×SD16C22×C3112yes—
314C22×D8C22×C3112yes—
315C22×D8C22×C3112yes—
316#559—24no—
317#559—24no—
318C2×#127C22×C3111yescharacteristic
319C2×#127C22×C3111yescharacteristic
320C2×#124—21no—
321C2×#124—21no—
322C2×#124C22×C3111yescharacteristic
323C2×#124C22×C3111yescharacteristic
324C2×#123—21no—
325C2×#123—21no—
326C2×C4×Q8C22×C3111yescharacteristic
327C2×C4×Q8C22×C3111yescharacteristic
328C2×C4×D4—21no—
329C2×C4×D4C22×C3111yescharacteristic
330C2×C4×D4C22×C3111yescharacteristic
331#496—24no—
332#459—24no—
333C2×C4⋊C8C22×C3111yescharacteristic
334C2×#103C22×C3112yes—
335C2×#102C22×C3112yes—
336C2×#102C22×C3112yes—
337C2×#102C22×C3112yes—
338C2×#97C22×C3111yescharacteristic
339#394—21no—
340#394—21no—
341#390—21no—
342#390C22×C3112yes—
343#390C22×C3112yes—
344C4×C4⋊C4C22×C3111yescharacteristic
345#340C22×C3112yes—
346#336C22×C3112yes—
347#336C22×C3112yes—
348C2×C31—82no—
349C2×C31—82no—
350C2×C31—162no—
351C2×C31#243111yescharacteristic
352C2×C31#300611yescharacteristic
353C2×C31#300811yescharacteristic
354C2×C31#339912yes—
355C2×C31#340312yes—
356C22×Q8D4×C3111yescharacteristic
357C22×Q8C23×C3111yescharacteristic
358C22×D4—21no—
359C22×D4—21no—
360C22×D4—21no—
361C22×D4—21no—
362C22×D4C23×C3112yes—
363C22×D4C23×C3111yescharacteristic
364C22×D4C23×C3111yescharacteristic
365C23×C4—21no—
366C23×C4—21no—
367C2×SD16—24no—
368C2×SD16—24no—
369C2×SD16—42no—
370C2×D8—24no—
371C2×D8—24no—
372C2×D8—24no—
373C2×D8—42no—
374C2×D8—42no—
375C2×D8—42no—
376C22×C8C23×C3112yes—
377C22×C8C23×C3112yes—
378#127—22no—
379#127—22no—
380#127—22no—
381#127—22no—
382#124—28no—
383#124—22no—
384#124—22no—
385#124—22no—
386#124—44no—
387#124D4×C3114yes—
388#123—28no—
389#123—28no—
390C4×Q8—22no—
391C4×Q8—22no—
392C4×Q8—22no—
393C4×Q8—22no—
394C4×D4—28no—
395C4×D4—22no—
396C4×D4—22no—
397C4×D4—22no—
398C4×D4D4×C3114yes—
399C2×C4⋊C4—21no—
400C2×C4⋊C4D4×C3111yescharacteristic
401C2×C4⋊C4D4×C3111yescharacteristic
402C2×C4⋊C4D4×C3111yescharacteristic
403C2×C4⋊C4C23×C3112yes—
404C2×C4⋊C4C23×C3111yescharacteristic
405C2×C22⋊C4—21no—
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C22⋊C4—21no—
409C2×C22⋊C4—21no—
410C2×C22⋊C4—21no—
411C2×C22⋊C4—22no—
412C2×C22⋊C4—22no—
413C2×C22⋊C4—22no—
414C2×C22⋊C4—22no—
415C2×C42—21no—
416C2×C42—21no—
417C2×C42C23×C3111yescharacteristic
418C2×C42C23×C3111yescharacteristic
419C2×C42C23×C3111yescharacteristic
420C4⋊C8—22no—
421#103—42no—
422#102—24no—
423#102—24no—
424#102—42no—
425#97—22no—
426#95—21no—
427C31#2564211yescharacteristic, direct product factor, Sylow
428C24—21no—
429C24—21no—
430C24—22no—
431C2×Q8—22no—
432C2×Q8—22no—
433C2×Q8—21no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—41no—
437C2×Q8C2×D4×C3112yes—
438C2×D4—24no—
439C2×D4—24no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—42no—
453C2×D4—42no—
454C2×D4—42no—
455C2×D4—42no—
456C2×D4—42no—
457C2×D4—42no—
458C2×D4—82no—
459C2×D4—82no—
460C2×D4C2×D4×C3112yes—
461C2×D4C2×D4×C3112yes—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—82no—
468C22×C4—82no—
469C22×C4C2×D4×C3111yescharacteristic
470C22×C4C2×D4×C3111yescharacteristic
471C22×C4C2×D4×C3111yescharacteristic
472C22×C4C2×D4×C3111yescharacteristic
473C22×C4C2×D4×C3111yescharacteristic
474C22×C4C2×D4×C3111yescharacteristic
475C22×C4C24×C3111yescharacteristic
476SD16—48no—
477D8—48no—
478D8—48no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8#41×C3114yes—
485C4⋊C4—24no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—21no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4C2×D4×C3112yes—
493C22⋊C4—24no—
494C22⋊C4—24no—
495C22⋊C4—24no—
496C22⋊C4—24no—
497C22⋊C4—24no—
498C22⋊C4—24no—
499C22⋊C4—44no—
500C22⋊C4—44no—
501C22⋊C4—82no—
502C22⋊C4—82no—
503C42—24no—
504C42—21no—
505C42—21no—
506C42—21no—
507C42—21no—
508C42—42no—
509C42C2×D4×C3112yes—
510C42C2×D4×C3112yes—
511C23—22no—
512C23—22no—
513C23—42no—
514C23—42no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23C22×D4×C3111yescharacteristic
529Q8—24no—
530Q8—44no—
531Q8—41no—
532Q8—41no—
533D4—22no—
534D4—22no—
535D4—24no—
536D4—48no—
537D4—41no—
538D4—42no—
539D4—41no—
540D4—42no—
541D4—42no—
542D4—42no—
543D4—42no—
544D4—84no—
545D4—84no—
546D4—84no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—84no—
567C2×C4—84no—
568C2×C4C31×#12012yes—
569C2×C4C31×#12012yes—
570C2×C4C31×AGL(1,ℤ/8)12yes—
571C2×C4C22×D4×C3111yescharacteristic
572C2×C4C22×D4×C3111yescharacteristic
573C2×C4C2×#41×C3112yes—
574C2×C4C31×H(5,𝔽2)12yes—
575C8—24no—
576C8—42no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—82no—
584C22—82no—
585C22—82no—
586C22—82no—
587C22—82no—
588C22—82no—
589C22—82no—
590C22—82no—
591C22—82no—
592C22—82no—
593C22—82no—
594C22C2×C31×#12011yescharacteristic
595C22C31×#55012yes—
596C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
597C22C31×#57511yescharacteristic
598C22C31×#57612yes—
599C4—22no—
600C4—22no—
601C4—22no—
602C4—42no—
603C4—41no—
604C4—41no—
605C4—41no—
606C4—42no—
607C4—82no—
608C4C31×#53211yescharacteristic
609C4C31×#53511yescharacteristic
610C4C31×#54612yes—
611C2—82no—
612C2—82no—
613C2—162no—
614C2C31×#243111yescharacteristic
615C2C31×#300611yescharacteristic
616C2C31×#300811yescharacteristic
617C2C31×#339912yes—
618C2C31×#340312yes—
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C2×C4×Q8×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C31×#559C2semidirect
2. C2×C31×#496C2semidirect
3. C2×C31×#459C2semidirect
4. C31×#2548C2semidirect
5. C31×#2012C2semidirect
6. C31×#2011C2semidirect
7. C31×#1985C2semidirect
8. C31×#1983C2semidirect
9. C31×#1880C2semidirect
10. C31×#1879C2semidirect
11. C31×#1762C2semidirect
201. #25642C31direct (N × H)
268. C2×#559C2×C31semidirect
269. C2×#496C2×C31semidirect
270. C2×#459C2×C31semidirect
271. #2548C2×C31semidirect
272. #2012C2×C31semidirect
273. #2011C2×C31semidirect
274. #1985C2×C31semidirect
275. #1983C2×C31semidirect
276. #1880C2×C31semidirect
277. #1879C2×C31semidirect
278. #1762C2×C31semidirect
427. C31#25642direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5592
C2×C31×#4962
C2×C31×#4592
C31×#25482
C31×#20122
C31×#20112
C31×#19852
C31×#19832
C31×#18802
C31×#18792
C31×#17622
#2564231