Finite Group Databasebeta

← C31×#8416 #21465175

Subgroups — C31×#8416 #21465175

Order 7936.

852 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3519C211yesmaximal, characteristic
2C31×#1980C211yesmaximal, characteristic, complemented
3C31×#1762C211yesmaximal, characteristic, complemented
4C31×#1530C211yesmaximal, characteristic, complemented
5C31×#1524C211yesmaximal, characteristic, complemented
6C31×#1509C211yesmaximal, characteristic, complemented
7C31×#1490C211yesmaximal, characteristic, complemented
8C31×#555C411yescharacteristic, complemented
9C31×#554C411yescharacteristic, complemented
10C31×#553—21no—
11C31×#553—21no—
12C31×#550C411yescharacteristic, complemented
13C31×#546—22no—
14C31×#546—21no—
15D42×C31—21no—
16D42×C31C411yescharacteristic, complemented
17C31×#540—21no—
18C31×#538—22no—
19C31×#532C2211yescharacteristic
20C31×#529C2211yescharacteristic
21C31×#525—21no—
22C2×C31×#121—21no—
23C31×#518—21no—
24C2×C4×D4×C31C2211yescharacteristic
25C31×#432C2211yescharacteristic, complemented
26C2×C31×#104—22no—
27C2×C31×#102—21no—
28C4×M16×C31C2211yescharacteristic, complemented
29C2×C31×#97C2211yescharacteristic, complemented
30C4×C4⋊C4×C31C2211yescharacteristic, complemented
31C31×#340—21no—
32C31×#330—21no—
33C31×#329—21no—
34C31×#329—22no—
35C31×#328—21no—
36C31×#327—21no—
37C31×D4⋊C8—21no—
38C31×D4⋊C8—21no—
39C2×#41×C31—21no—
40C2×#41×C31D412yes—
41C22×D4×C31—22no—
42C22×D4×C31—21no—
43C22×D4×C31—21no—
44C22×D4×C31D411yescharacteristic
45C23×C4×C31—21no—
46C2×M16×C31—21no—
47C22×C8×C31—21no—
48C31×C4⋊Q8C2×C411yescharacteristic
49C31×#127—21no—
50C31×#127C2×C411yescharacteristic
51C31×#126—22no—
52C31×#126—21no—
53C31×#125—21no—
54C31×#125D411yescharacteristic
55C31×#124—22no—
56C31×#124D411yescharacteristic
57C31×#124D412yes—
58C31×#123—22no—
59C31×#123—22no—
60C31×#123—22no—
61C31×#123—21no—
62C31×C22⋊Q8—21no—
63C31×C22⋊Q8—22no—
64C31×C22⋊Q8D411yescharacteristic
65C31×#121—22no—
66C31×#121—22no—
67C31×#121—22no—
68C31×#121—22no—
69C31×#121—21no—
70C31×#121—22no—
71C31×#121—21no—
72C31×#121—21no—
73C31×#121—21no—
74C31×#121—21no—
75C31×#121—21no—
76C31×#121—21no—
77C31×#121D411yescharacteristic
78C31×#120—22no—
79C31×#120—22no—
80C31×#120—22no—
81C4×Q8×C31C2×C411yescharacteristic
82C4×D4×C31—22no—
83C4×D4×C31—22no—
84C4×D4×C31—22no—
85C4×D4×C31—21no—
86C4×D4×C31—21no—
87C4×D4×C31—21no—
88C4×D4×C31—21no—
89C4×D4×C31—21no—
90C4×D4×C31—21no—
91C4×D4×C31—21no—
92C4×D4×C31C2×C411yescharacteristic
93C4×D4×C31C2×C411yescharacteristic
94C4×D4×C31C2×C411yescharacteristic
95C4×D4×C31D411yescharacteristic
96C4×D4×C31D411yescharacteristic
97C31×#117—21no—
98C31×#117—21no—
99C31×#117—21no—
100C2×C4⋊C4×C31—21no—
101C2×C4⋊C4×C31—21no—
102C2×C4⋊C4×C31D411yescharacteristic
103C2×C22⋊C4×C31—22no—
104C2×C22⋊C4×C31—22no—
105C2×C22⋊C4×C31—21no—
106C2×C42×C31—21no—
107C2×C42×C31C2311yescharacteristic
108C31×C4⋊C8—21no—
109C31×#104—44no—
110C31×#102—24no—
111C31×C22⋊C8—21no—
112C31×#97—21no—
113C31×#97—21no—
114C31×#97—21no—
115C31×#97—21no—
116C4×C8×C31—21no—
117C31×#95—21no—
118C24×C31—22no—
119C24×C31—21no—
120#41×C31—42no—
121#41×C31—42no—
122#41×C31—82no—
123C2×Q8×C31C22⋊C411yescharacteristic
124C2×Q8×C31C22⋊C412yes—
125C2×D4×C31—22no—
126C2×D4×C31—22no—
127C2×D4×C31—22no—
128C2×D4×C31—21no—
129C2×D4×C31—22no—
130C2×D4×C31—22no—
131C2×D4×C31—22no—
132C2×D4×C31—22no—
133C2×D4×C31—21no—
134C2×D4×C31—21no—
135C2×D4×C31—21no—
136C2×D4×C31—41no—
137C2×D4×C31—42no—
138C2×D4×C31—42no—
139C2×D4×C31—82no—
140C2×D4×C31—82no—
141C2×D4×C31—82no—
142C2×D4×C31C22⋊C411yescharacteristic
143C2×D4×C31C22⋊C411yescharacteristic
144C2×D4×C31C22⋊C411yescharacteristic
145C2×D4×C31C22⋊C412yes—
146C22×C4×C31—22no—
147C22×C4×C31—22no—
148C22×C4×C31—21no—
149C22×C4×C31—21no—
150C22×C4×C31—21no—
151C22×C4×C31—21no—
152C22×C4×C31—21no—
153C22×C4×C31—22no—
154C22×C4×C31—21no—
155C22×C4×C31—41no—
156C22×C4×C31—82no—
157C22×C4×C31C2×D411yescharacteristic
158C22×C4×C31C2×D411yescharacteristic
159C22×C4×C31C2×D411yescharacteristic
160M16×C31—22no—
161M16×C31—41no—
162C2×C8×C31—22no—
163C2×C8×C31—41no—
164C2×C8×C31—41no—
165C2×C8×C31—41no—
166C2×C8×C31—41no—
167C4⋊C4×C31—22no—
168C4⋊C4×C31—22no—
169C4⋊C4×C31—21no—
170C4⋊C4×C31—21no—
171C4⋊C4×C31—21no—
172C4⋊C4×C31—21no—
173C4⋊C4×C31—21no—
174C4⋊C4×C31—21no—
175C4⋊C4×C31—21no—
176C4⋊C4×C31—22no—
177C4⋊C4×C31—41no—
178C4⋊C4×C31C22⋊C411yescharacteristic
179C4⋊C4×C31C22⋊C411yescharacteristic
180C4⋊C4×C31C22⋊C411yescharacteristic
181C4⋊C4×C31C22⋊C411yescharacteristic
182C22⋊C4×C31—22no—
183C22⋊C4×C31—22no—
184C22⋊C4×C31—22no—
185C22⋊C4×C31—22no—
186C22⋊C4×C31—22no—
187C22⋊C4×C31—21no—
188C22⋊C4×C31—21no—
189C22⋊C4×C31—21no—
190C22⋊C4×C31—21no—
191C22⋊C4×C31—21no—
192C22⋊C4×C31—21no—
193C22⋊C4×C31—21no—
194C22⋊C4×C31—22no—
195C22⋊C4×C31—21no—
196C22⋊C4×C31—21no—
197C22⋊C4×C31—22no—
198C42×C31—22no—
199C42×C31—21no—
200C42×C31—42no—
201C42×C31C22×C411yescharacteristic
202C42×C31C2×D411yescharacteristic
203C42×C31C2×D411yescharacteristic
204C42×C31C2×D411yescharacteristic
205#8416C3111yesmaximal, characteristic, direct product factor, Sylow
206C23×C31—21no—
207C23×C31—21no—
208C23×C31—22no—
209C23×C31—21no—
210C23×C31—42no—
211C23×C31—44no—
212C23×C31—41no—
213C23×C31—41no—
214C23×C31—81no—
215C23×C31—82no—
216C23×C31—82no—
217C23×C31#12011yescharacteristic
218Q8×C31—42no—
219Q8×C31—41no—
220D4×C31—21no—
221D4×C31—21no—
222D4×C31—21no—
223D4×C31—42no—
224D4×C31—41no—
225D4×C31—42no—
226D4×C31—42no—
227D4×C31—44no—
228D4×C31—41no—
229D4×C31—42no—
230D4×C31—41no—
231D4×C31—42no—
232D4×C31—41no—
233D4×C31—41no—
234D4×C31—82no—
235D4×C31—82no—
236D4×C31—82no—
237D4×C31—82no—
238D4×C31#10412yescomplemented
239C2×C4×C31—21no—
240C2×C4×C31—21no—
241C2×C4×C31—21no—
242C2×C4×C31—21no—
243C2×C4×C31—21no—
244C2×C4×C31—21no—
245C2×C4×C31—22no—
246C2×C4×C31—21no—
247C2×C4×C31—21no—
248C2×C4×C31—21no—
249C2×C4×C31—41no—
250C2×C4×C31—42no—
251C2×C4×C31—42no—
252C2×C4×C31—41no—
253C2×C4×C31—41no—
254C2×C4×C31—42no—
255C2×C4×C31—81no—
256C2×C4×C31—81no—
257C2×C4×C31—82no—
258C2×C4×C31—81no—
259C2×C4×C31—82no—
260C2×C4×C31—82no—
261C2×C4×C31—81no—
262C2×C4×C31—81no—
263C2×C4×C31C2×C22⋊C411yescharacteristic
264C2×C4×C31C2×C22⋊C411yescharacteristic
265C2×C4×C31C2×C22⋊C411yescharacteristic
266C2×C4×C31#12011yescharacteristic
267C2×C4×C31#12011yescharacteristic
268C2×C4×C31#12011yescharacteristic
269C8×C31—42no—
270C8×C31—42no—
271C8×C31—81no—
272#3519C2×C3111yescharacteristic
273#1980C2×C3111yescharacteristic, complemented
274#1762C2×C3111yescharacteristic, complemented
275#1530C2×C3111yescharacteristic, complemented
276#1524C2×C3111yescharacteristic, complemented
277#1509C2×C3111yescharacteristic, complemented
278#1490C2×C3111yescharacteristic, complemented
279C22×C31—21no—
280C22×C31—21no—
281C22×C31—22no—
282C22×C31—41no—
283C22×C31—41no—
284C22×C31—42no—
285C22×C31—41no—
286C22×C31—41no—
287C22×C31—41no—
288C22×C31—42no—
289C22×C31—41no—
290C22×C31—41no—
291C22×C31—41no—
292C22×C31—42no—
293C22×C31—82no—
294C22×C31—82no—
295C22×C31—164no—
296C22×C31#37911yescharacteristic
297C22×C31#45311yescharacteristic
298C22×C31#45511yescharacteristic
299C4×C31—21no—
300C4×C31—21no—
301C4×C31—21no—
302C4×C31—21no—
303C4×C31—21no—
304C4×C31—41no—
305C4×C31—42no—
306C4×C31—41no—
307C4×C31—41no—
308C4×C31—41no—
309C4×C31—41no—
310C4×C31—82no—
311C4×C31C2×#10411yescharacteristic
312C4×C31#42111yescharacteristic
313#555C4×C3111yescharacteristic, complemented
314#554C4×C3111yescharacteristic, complemented
315#553—21no—
316#553—21no—
317#550C4×C3111yescharacteristic, complemented
318#546—22no—
319#546—21no—
320D42—21no—
321D42C4×C3111yescharacteristic, complemented
322#540—21no—
323#538—22no—
324#532C22×C3111yescharacteristic
325#529C22×C3111yescharacteristic
326#525—21no—
327C2×#121—21no—
328#518—21no—
329C2×C4×D4C22×C3111yescharacteristic
330#432C22×C3111yescharacteristic, complemented
331C2×#104—22no—
332C2×#102—21no—
333C4×M16C22×C3111yescharacteristic, complemented
334C2×#97C22×C3111yescharacteristic, complemented
335C4×C4⋊C4C22×C3111yescharacteristic, complemented
336#340—21no—
337#330—21no—
338#329—21no—
339#329—22no—
340#328—21no—
341#327—21no—
342D4⋊C8—21no—
343D4⋊C8—21no—
344C2×C31—21no—
345C2×C31—21no—
346C2×C31—42no—
347C2×C31—81no—
348C2×C31—82no—
349C2×C31—81no—
350C2×C31#178911yescharacteristic
351C2×C31#179211yescharacteristic
352C2×C31#179511yescharacteristic
353C2×#41—21no—
354C2×#41D4×C3112yes—
355C22×D4—22no—
356C22×D4—21no—
357C22×D4—21no—
358C22×D4D4×C3111yescharacteristic
359C23×C4—21no—
360C2×M16—21no—
361C22×C8—21no—
362C4⋊Q8C2×C4×C3111yescharacteristic
363#127—21no—
364#127C2×C4×C3111yescharacteristic
365#126—22no—
366#126—21no—
367#125—21no—
368#125D4×C3111yescharacteristic
369#124—22no—
370#124D4×C3111yescharacteristic
371#124D4×C3112yes—
372#123—22no—
373#123—22no—
374#123—22no—
375#123—21no—
376C22⋊Q8—21no—
377C22⋊Q8—22no—
378C22⋊Q8D4×C3111yescharacteristic
379#121—22no—
380#121—22no—
381#121—22no—
382#121—22no—
383#121—21no—
384#121—22no—
385#121—21no—
386#121—21no—
387#121—21no—
388#121—21no—
389#121—21no—
390#121—21no—
391#121D4×C3111yescharacteristic
392#120—22no—
393#120—22no—
394#120—22no—
395C4×Q8C2×C4×C3111yescharacteristic
396C4×D4—22no—
397C4×D4—22no—
398C4×D4—22no—
399C4×D4—21no—
400C4×D4—21no—
401C4×D4—21no—
402C4×D4—21no—
403C4×D4—21no—
404C4×D4—21no—
405C4×D4—21no—
406C4×D4C2×C4×C3111yescharacteristic
407C4×D4C2×C4×C3111yescharacteristic
408C4×D4C2×C4×C3111yescharacteristic
409C4×D4D4×C3111yescharacteristic
410C4×D4D4×C3111yescharacteristic
411#117—21no—
412#117—21no—
413#117—21no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4D4×C3111yescharacteristic
417C2×C22⋊C4—22no—
418C2×C22⋊C4—22no—
419C2×C22⋊C4—21no—
420C2×C42—21no—
421C2×C42C23×C3111yescharacteristic
422C4⋊C8—21no—
423#104—44no—
424#102—24no—
425C22⋊C8—21no—
426#97—21no—
427#97—21no—
428#97—21no—
429#97—21no—
430C4×C8—21no—
431#95—21no—
432C31#841611yescharacteristic, direct product factor, Sylow
433C24—22no—
434C24—21no—
435#41—42no—
436#41—42no—
437#41—82no—
438C2×Q8C22⋊C4×C3111yescharacteristic
439C2×Q8C22⋊C4×C3112yes—
440C2×D4—22no—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—21no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—41no—
452C2×D4—42no—
453C2×D4—42no—
454C2×D4—82no—
455C2×D4—82no—
456C2×D4—82no—
457C2×D4C22⋊C4×C3111yescharacteristic
458C2×D4C22⋊C4×C3111yescharacteristic
459C2×D4C22⋊C4×C3111yescharacteristic
460C2×D4C22⋊C4×C3112yes—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—21no—
470C22×C4—41no—
471C22×C4—82no—
472C22×C4C2×D4×C3111yescharacteristic
473C22×C4C2×D4×C3111yescharacteristic
474C22×C4C2×D4×C3111yescharacteristic
475M16—22no—
476M16—41no—
477C2×C8—22no—
478C2×C8—41no—
479C2×C8—41no—
480C2×C8—41no—
481C2×C8—41no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—22no—
492C4⋊C4—41no—
493C4⋊C4C22⋊C4×C3111yescharacteristic
494C4⋊C4C22⋊C4×C3111yescharacteristic
495C4⋊C4C22⋊C4×C3111yescharacteristic
496C4⋊C4C22⋊C4×C3111yescharacteristic
497C22⋊C4—22no—
498C22⋊C4—22no—
499C22⋊C4—22no—
500C22⋊C4—22no—
501C22⋊C4—22no—
502C22⋊C4—21no—
503C22⋊C4—21no—
504C22⋊C4—21no—
505C22⋊C4—21no—
506C22⋊C4—21no—
507C22⋊C4—21no—
508C22⋊C4—21no—
509C22⋊C4—22no—
510C22⋊C4—21no—
511C22⋊C4—21no—
512C22⋊C4—22no—
513C42—22no—
514C42—21no—
515C42—42no—
516C42C22×C4×C3111yescharacteristic
517C42C2×D4×C3111yescharacteristic
518C42C2×D4×C3111yescharacteristic
519C42C2×D4×C3111yescharacteristic
520C23—21no—
521C23—21no—
522C23—21no—
523C23—22no—
524C23—42no—
525C23—44no—
526C23—41no—
527C23—41no—
528C23—81no—
529C23—82no—
530C23—82no—
531C23C31×#12011yescharacteristic
532Q8—42no—
533Q8—41no—
534D4—21no—
535D4—21no—
536D4—21no—
537D4—42no—
538D4—41no—
539D4—42no—
540D4—42no—
541D4—44no—
542D4—41no—
543D4—42no—
544D4—41no—
545D4—42no—
546D4—41no—
547D4—41no—
548D4—82no—
549D4—82no—
550D4—82no—
551D4—82no—
552D4C31×#10412yescomplemented
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—81no—
570C2×C4—81no—
571C2×C4—82no—
572C2×C4—81no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4—81no—
576C2×C4—81no—
577C2×C4C2×C22⋊C4×C3111yescharacteristic
578C2×C4C2×C22⋊C4×C3111yescharacteristic
579C2×C4C2×C22⋊C4×C3111yescharacteristic
580C2×C4C31×#12011yescharacteristic
581C2×C4C31×#12011yescharacteristic
582C2×C4C31×#12011yescharacteristic
583C8—42no—
584C8—42no—
585C8—81no—
586C22—21no—
587C22—21no—
588C22—22no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—42no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—42no—
597C22—42no—
598C22—41no—
599C22—41no—
600C22—82no—
601C22—82no—
602C22—164no—
603C22C31×#37911yescharacteristic
604C22C31×#45311yescharacteristic
605C22C31×#45511yescharacteristic
606C4—21no—
607C4—21no—
608C4—21no—
609C4—21no—
610C4—21no—
611C4—41no—
612C4—41no—
613C4—41no—
614C4—41no—
615C4—41no—
616C4—42no—
617C4—82no—
618C4C2×C31×#10411yescharacteristic
619C4C31×#42111yescharacteristic
620C2—21no—
621C2—21no—
622C2—42no—
623C2—81no—
624C2—81no—
625C2—82no—
626C2C31×#178911yescharacteristic
627C2C31×#179211yescharacteristic
628C2C31×#179511yescharacteristic
subgroupG/H
center, socleC22×C31#379
commutatorC42C22×C4×C31
Frattini, cosocleC2×C42C23×C31
prime radicalC31×#3519C2
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
2. C31×#1980C2semidirect
3. C31×#1762C2semidirect
4. C31×#1530C2semidirect
5. C31×#1524C2semidirect
6. C31×#1509C2semidirect
7. C31×#1490C2semidirect
8. C31×#555C4semidirect
9. C31×#554C4semidirect
12. C31×#550C4semidirect
16. D42×C31C4semidirect
25. C31×#432C22semidirect
28. C4×M16×C31C22semidirect
29. C2×C31×#97C22semidirect
30. C4×C4⋊C4×C31C22semidirect
205. #8416C31direct (N × H)
238. D4×C31#104semidirect
273. #1980C2×C31semidirect
274. #1762C2×C31semidirect
275. #1530C2×C31semidirect
276. #1524C2×C31semidirect
277. #1509C2×C31semidirect
278. #1490C2×C31semidirect
313. #555C4×C31semidirect
314. #554C4×C31semidirect
317. #550C4×C31semidirect
321. D42C4×C31semidirect
330. #432C22×C31semidirect
333. C4×M16C22×C31semidirect
334. C2×#97C22×C31semidirect
335. C4×C4⋊C4C22×C31semidirect
432. C31#8416direct (N × H)
552. D4C31×#104semidirect

Maximal subgroups

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

subgroupindex
C31×#35192
C31×#19802
C31×#17622
C31×#15302
C31×#15242
C31×#15092
C31×#14902
#841631