Finite Group Databasebeta

← C31×#18245 #21475004

Subgroups — C31×#18245 #21475004

Order 7936.

908 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3559C211yesmaximal, characteristic
2C31×#3117C211yesmaximal, characteristic, complemented
3C31×#3103C211yesmaximal, characteristic, complemented
4C31×#2634C211yesmaximal, characteristic, complemented
5C31×#1616C212yesmaximal, complemented
6C31×#1615C212yesmaximal, complemented
7C31×#1604C212yesmaximal, complemented
8C31×#1603C212yesmaximal, complemented
9C31×#1521C211yesmaximal, characteristic, complemented
10C31×#1514C211yesmaximal, characteristic, complemented
11C31×#1470C211yesmaximal, characteristic, complemented
12C31×#563—21no—
13C31×#562—21no—
14C31×#554—21no—
15C31×#553—21no—
16C31×#553—21no—
17C31×#547—22no—
18C31×#547—21no—
19C31×#544—21no—
20C31×#540—21no—
21C31×#539—22no—
22C31×#539C2212yes—
23C31×#537C2212yes—
24C31×#532C2211yescharacteristic
25C31×#529C2211yescharacteristic
26C2×C31×#125C2211yescharacteristic
27C2×C4×Q8×C31C2211yescharacteristic
28C31×#484C2211yescharacteristic
29C31×#481C2212yes—
30C31×#481C2211yescharacteristic
31C31×#479—21no—
32C31×#475—21no—
33C31×#464—21no—
34C31×#460—21no—
35C31×#451C2211yescharacteristic
36C31×#448C2212yes—
37C31×#448C2211yescharacteristic
38C31×#441—21no—
39C31×#440—21no—
40C4×SD16×C31—22no—
41C31×#432C2211yescharacteristic
42C31×#431C2211yescharacteristic
43C31×#400—21no—
44C31×#400C2212yes—
45C31×#398—21no—
46C31×#395C2212yes—
47C31×#387—22no—
48C31×#383C2211yescharacteristic
49C31×#328C2212yes—
50C31×#327C2212yes—
51C31×#327C2212yes—
52C31×#327C2212yes—
53C31×#324C2212yes—
54C31×#324C2212yes—
55C2×#41×C31—21no—
56C2×#41×C31—22no—
57C22×Q8×C31C2311yescharacteristic
58C2×Q16×C31—21no—
59C2×SD16×C31—22no—
60C2×SD16×C31—21no—
61C31×C4⋊Q8—21no—
62C31×C4⋊Q8—22no—
63C31×C4⋊Q8D411yescharacteristic
64C31×#127D411yescharacteristic
65C31×#126—22no—
66C31×#126—22no—
67C31×#126—21no—
68C31×#125—21no—
69C31×#125—22no—
70C31×#125—21no—
71C31×#125—21no—
72C31×#125—21no—
73C31×#125—21no—
74C31×#125D411yescharacteristic
75C31×#124—21no—
76C31×#124D411yescharacteristic
77C31×#123—22no—
78C31×#123—22no—
79C31×#123—22no—
80C31×#123—22no—
81C31×#123—22no—
82C31×#123—21no—
83C31×#123—22no—
84C31×C22⋊Q8—22no—
85C31×C22⋊Q8—22no—
86C31×C22⋊Q8—22no—
87C31×C22⋊Q8—22no—
88C31×C22⋊Q8—22no—
89C31×C22⋊Q8—22no—
90C31×C22⋊Q8C2311yescharacteristic
91C31×#121—21no—
92C31×#121—22no—
93C31×#121—21no—
94C31×#121—22no—
95C31×#121—22no—
96C31×#121—21no—
97C31×#121—22no—
98C31×#121C2311yescharacteristic
99C31×#121C2312yes—
100C4×Q8×C31—21no—
101C4×Q8×C31—21no—
102C4×Q8×C31—21no—
103C4×Q8×C31—21no—
104C4×Q8×C31D411yescharacteristic
105C4×D4×C31—22no—
106C4×D4×C31—21no—
107C4×D4×C31—21no—
108C4×D4×C31—21no—
109C4×D4×C31—22no—
110C4×D4×C31—21no—
111C4×D4×C31D412yes—
112C4×D4×C31D411yescharacteristic
113C31×#117—21no—
114C31×#117—21no—
115C2×C4⋊C4×C31—22no—
116C2×C4⋊C4×C31—21no—
117C2×C4⋊C4×C31—22no—
118C2×C4⋊C4×C31—21no—
119C2×C4⋊C4×C31D411yescharacteristic
120C2×C4⋊C4×C31D411yescharacteristic
121C2×C4⋊C4×C31D412yes—
122C2×C4⋊C4×C31C2311yescharacteristic
123C2×C4⋊C4×C31C2312yes—
124C2×C42×C31—21no—
125C2×C42×C31C2311yescharacteristic
126C31×#107—21no—
127C31×#106—22no—
128C31×#106—21no—
129C31×C4⋊C8—21no—
130C31×C4⋊C8—21no—
131C31×#103—21no—
132C31×#103—22no—
133C31×#103—21no—
134C31×#103—21no—
135C31×#102—22no—
136C31×#102—21no—
137C31×C22⋊C8C2312yes—
138C31×C22⋊C8C2311yescharacteristic
139C31×C22⋊C8C2311yescharacteristic
140C31×#97—21no—
141C4×C8×C31—21no—
142C31×#95—22no—
143C31×#95—21no—
144C31×#95—22no—
145C31×#95—22no—
146C31×#95—21no—
147C31×#95C2312yes—
148#41×C31—82no—
149#41×C31—84no—
150C2×Q8×C31—22no—
151C2×Q8×C31—21no—
152C2×Q8×C31—21no—
153C2×Q8×C31—21no—
154C2×Q8×C31—42no—
155C2×Q8×C31C2×D411yescharacteristic
156C2×D4×C31—21no—
157C2×D4×C31—22no—
158C2×D4×C31—22no—
159C2×D4×C31—22no—
160C2×D4×C31—21no—
161C2×D4×C31—21no—
162C2×D4×C31C2×D411yescharacteristic
163C2×D4×C31C2×D412yes—
164C22×C4×C31—21no—
165C22×C4×C31—22no—
166C22×C4×C31—22no—
167C22×C4×C31—22no—
168C22×C4×C31—21no—
169C22×C4×C31—22no—
170C22×C4×C31—22no—
171C22×C4×C31—21no—
172C22×C4×C31C2×D412yes—
173C22×C4×C31C2×D411yescharacteristic
174C22×C4×C31C2×D412yes—
175C22×C4×C31C2×D411yescharacteristic
176C22×C4×C31C2411yescharacteristic
177Q16×C31—42no—
178SD16×C31—44no—
179SD16×C31—42no—
180C2×C8×C31—22no—
181C2×C8×C31—21no—
182C2×C8×C31—21no—
183C4⋊C4×C31—22no—
184C4⋊C4×C31—22no—
185C4⋊C4×C31—22no—
186C4⋊C4×C31—22no—
187C4⋊C4×C31—21no—
188C4⋊C4×C31—21no—
189C4⋊C4×C31—22no—
190C4⋊C4×C31—21no—
191C4⋊C4×C31—21no—
192C4⋊C4×C31—22no—
193C4⋊C4×C31—21no—
194C4⋊C4×C31—21no—
195C4⋊C4×C31—21no—
196C4⋊C4×C31—21no—
197C4⋊C4×C31—22no—
198C4⋊C4×C31—22no—
199C4⋊C4×C31—21no—
200C4⋊C4×C31—21no—
201C4⋊C4×C31—21no—
202C4⋊C4×C31—44no—
203C4⋊C4×C31—41no—
204C4⋊C4×C31—41no—
205C4⋊C4×C31C2×D412yes—
206C4⋊C4×C31C2×D411yescharacteristic
207C4⋊C4×C31C2×D411yescharacteristic
208C22⋊C4×C31—22no—
209C22⋊C4×C31—22no—
210C22⋊C4×C31—22no—
211C22⋊C4×C31—22no—
212C22⋊C4×C31—22no—
213C22⋊C4×C31—21no—
214C22⋊C4×C31—21no—
215C22⋊C4×C31—21no—
216C22⋊C4×C31—21no—
217C22⋊C4×C31—22no—
218C22⋊C4×C31—21no—
219C22⋊C4×C31—22no—
220C42×C31—22no—
221C42×C31—21no—
222C42×C31—41no—
223C42×C31—41no—
224C42×C31C2×D411yescharacteristic
225C42×C31C2×D411yescharacteristic
226C42×C31C2×D411yescharacteristic
227C42×C31C2×D411yescharacteristic
228#18245C3111yesmaximal, characteristic, direct product factor, Sylow
229C23×C31—21no—
230C23×C31—22no—
231C23×C31C22×D411yescharacteristic
232Q8×C31—22no—
233Q8×C31—22no—
234Q8×C31—41no—
235Q8×C31—41no—
236Q8×C31—41no—
237Q8×C31—81no—
238Q8×C31—82no—
239D4×C31—42no—
240D4×C31—41no—
241D4×C31—42no—
242D4×C31—44no—
243D4×C31—42no—
244D4×C31—81no—
245D4×C31—82no—
246D4×C31—82no—
247C2×C4×C31—22no—
248C2×C4×C31—22no—
249C2×C4×C31—21no—
250C2×C4×C31—21no—
251C2×C4×C31—22no—
252C2×C4×C31—22no—
253C2×C4×C31—21no—
254C2×C4×C31—22no—
255C2×C4×C31—24no—
256C2×C4×C31—41no—
257C2×C4×C31—41no—
258C2×C4×C31—42no—
259C2×C4×C31—41no—
260C2×C4×C31—41no—
261C2×C4×C31—42no—
262C2×C4×C31—42no—
263C2×C4×C31—42no—
264C2×C4×C31—42no—
265C2×C4×C31—41no—
266C2×C4×C31—42no—
267C2×C4×C31—82no—
268C2×C4×C31—82no—
269C2×C4×C31—82no—
270C2×C4×C31—82no—
271C2×C4×C31#12011yescharacteristic
272C2×C4×C31#12011yescharacteristic
273C2×C4×C31#12011yescharacteristic
274C2×C4×C31#12011yescharacteristic
275C2×C4×C31C22×D411yescharacteristic
276C2×C4×C31C22×D411yescharacteristic
277C8×C31—42no—
278C8×C31—41no—
279C8×C31—41no—
280#3559C2×C3111yescharacteristic
281#3117C2×C3111yescharacteristic, complemented
282#3103C2×C3111yescharacteristic, complemented
283#2634C2×C3111yescharacteristic, complemented
284#1616C2×C3112yescomplemented
285#1615C2×C3112yescomplemented
286#1604C2×C3112yescomplemented
287#1603C2×C3112yescomplemented
288#1521C2×C3111yescharacteristic, complemented
289#1514C2×C3111yescharacteristic, complemented
290#1470C2×C3111yescharacteristic, complemented
291C22×C31—21no—
292C22×C31—21no—
293C22×C31—42no—
294C22×C31—41no—
295C22×C31—41no—
296C22×C31—42no—
297C22×C31—42no—
298C22×C31—41no—
299C22×C31C2×#12011yescharacteristic
300C22×C31#57712yes—
301C4×C31—21no—
302C4×C31—21no—
303C4×C31—21no—
304C4×C31—21no—
305C4×C31—21no—
306C4×C31—21no—
307C4×C31—42no—
308C4×C31—42no—
309C4×C31—41no—
310C4×C31—41no—
311C4×C31—42no—
312C4×C31—42no—
313C4×C31—41no—
314C4×C31—82no—
315C4×C31—81no—
316#563—21no—
317#562—21no—
318#554—21no—
319#553—21no—
320#553—21no—
321#547—22no—
322#547—21no—
323#544—21no—
324#540—21no—
325#539—22no—
326#539C22×C3112yes—
327#537C22×C3112yes—
328#532C22×C3111yescharacteristic
329#529C22×C3111yescharacteristic
330C2×#125C22×C3111yescharacteristic
331C2×C4×Q8C22×C3111yescharacteristic
332#484C22×C3111yescharacteristic
333#481C22×C3112yes—
334#481C22×C3111yescharacteristic
335#479—21no—
336#475—21no—
337#464—21no—
338#460—21no—
339#451C22×C3111yescharacteristic
340#448C22×C3112yes—
341#448C22×C3111yescharacteristic
342#441—21no—
343#440—21no—
344C4×SD16—22no—
345#432C22×C3111yescharacteristic
346#431C22×C3111yescharacteristic
347#400—21no—
348#400C22×C3112yes—
349#398—21no—
350#395C22×C3112yes—
351#387—22no—
352#383C22×C3111yescharacteristic
353#328C22×C3112yes—
354#327C22×C3112yes—
355#327C22×C3112yes—
356#327C22×C3112yes—
357#324C22×C3112yes—
358#324C22×C3112yes—
359C2×C31—22no—
360C2×C31—81no—
361C2×C31—82no—
362C2×C31#300911yescharacteristic
363C2×C31#302011yescharacteristic
364C2×C31#302511yescharacteristic
365C2×#41—21no—
366C2×#41—22no—
367C22×Q8C23×C3111yescharacteristic
368C2×Q16—21no—
369C2×SD16—22no—
370C2×SD16—21no—
371C4⋊Q8—21no—
372C4⋊Q8—22no—
373C4⋊Q8D4×C3111yescharacteristic
374#127D4×C3111yescharacteristic
375#126—22no—
376#126—22no—
377#126—21no—
378#125—21no—
379#125—22no—
380#125—21no—
381#125—21no—
382#125—21no—
383#125—21no—
384#125D4×C3111yescharacteristic
385#124—21no—
386#124D4×C3111yescharacteristic
387#123—22no—
388#123—22no—
389#123—22no—
390#123—22no—
391#123—22no—
392#123—21no—
393#123—22no—
394C22⋊Q8—22no—
395C22⋊Q8—22no—
396C22⋊Q8—22no—
397C22⋊Q8—22no—
398C22⋊Q8—22no—
399C22⋊Q8—22no—
400C22⋊Q8C23×C3111yescharacteristic
401#121—21no—
402#121—22no—
403#121—21no—
404#121—22no—
405#121—22no—
406#121—21no—
407#121—22no—
408#121C23×C3111yescharacteristic
409#121C23×C3112yes—
410C4×Q8—21no—
411C4×Q8—21no—
412C4×Q8—21no—
413C4×Q8—21no—
414C4×Q8D4×C3111yescharacteristic
415C4×D4—22no—
416C4×D4—21no—
417C4×D4—21no—
418C4×D4—21no—
419C4×D4—22no—
420C4×D4—21no—
421C4×D4D4×C3111yescharacteristic
422C4×D4D4×C3112yes—
423#117—21no—
424#117—21no—
425C2×C4⋊C4—22no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—21no—
428C2×C4⋊C4—22no—
429C2×C4⋊C4D4×C3112yes—
430C2×C4⋊C4D4×C3111yescharacteristic
431C2×C4⋊C4D4×C3111yescharacteristic
432C2×C4⋊C4C23×C3111yescharacteristic
433C2×C4⋊C4C23×C3112yes—
434C2×C42—21no—
435C2×C42C23×C3111yescharacteristic
436#107—21no—
437#106—22no—
438#106—21no—
439C4⋊C8—21no—
440C4⋊C8—21no—
441#103—21no—
442#103—22no—
443#103—21no—
444#103—21no—
445#102—22no—
446#102—21no—
447C22⋊C8C23×C3111yescharacteristic
448C22⋊C8C23×C3111yescharacteristic
449C22⋊C8C23×C3112yes—
450#97—21no—
451C4×C8—21no—
452#95—21no—
453#95—21no—
454#95—22no—
455#95—22no—
456#95—22no—
457#95C23×C3112yes—
458C31#1824511yescharacteristic, direct product factor, Sylow
459#41—82no—
460#41—84no—
461C2×Q8—22no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—42no—
466C2×Q8C2×D4×C3111yescharacteristic
467C2×D4—21no—
468C2×D4—22no—
469C2×D4—22no—
470C2×D4—22no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4C2×D4×C3111yescharacteristic
474C2×D4C2×D4×C3112yes—
475C22×C4—21no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—21no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—21no—
483C22×C4C2×D4×C3112yes—
484C22×C4C2×D4×C3111yescharacteristic
485C22×C4C2×D4×C3112yes—
486C22×C4C2×D4×C3111yescharacteristic
487C22×C4C24×C3111yescharacteristic
488Q16—42no—
489SD16—44no—
490SD16—42no—
491C2×C8—22no—
492C2×C8—21no—
493C2×C8—21no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—22no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—22no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—44no—
514C4⋊C4—41no—
515C4⋊C4—41no—
516C4⋊C4C2×D4×C3112yes—
517C4⋊C4C2×D4×C3111yescharacteristic
518C4⋊C4C2×D4×C3111yescharacteristic
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—22no—
522C22⋊C4—22no—
523C22⋊C4—22no—
524C22⋊C4—21no—
525C22⋊C4—21no—
526C22⋊C4—21no—
527C22⋊C4—21no—
528C22⋊C4—22no—
529C22⋊C4—21no—
530C22⋊C4—22no—
531C42—22no—
532C42—21no—
533C42—41no—
534C42—41no—
535C42C2×D4×C3111yescharacteristic
536C42C2×D4×C3111yescharacteristic
537C42C2×D4×C3111yescharacteristic
538C42C2×D4×C3111yescharacteristic
539C23—21no—
540C23—22no—
541C23C22×D4×C3111yescharacteristic
542Q8—22no—
543Q8—22no—
544Q8—41no—
545Q8—41no—
546Q8—41no—
547Q8—81no—
548Q8—82no—
549D4—42no—
550D4—41no—
551D4—42no—
552D4—44no—
553D4—42no—
554D4—81no—
555D4—82no—
556D4—82no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4—82no—
581C2×C4C31×#12011yescharacteristic
582C2×C4C31×#12011yescharacteristic
583C2×C4C31×#12011yescharacteristic
584C2×C4C31×#12011yescharacteristic
585C2×C4C22×D4×C3111yescharacteristic
586C2×C4C22×D4×C3111yescharacteristic
587C8—42no—
588C8—41no—
589C8—41no—
590C22—21no—
591C22—21no—
592C22—42no—
593C22—41no—
594C22—41no—
595C22—42no—
596C22—42no—
597C22—41no—
598C22C2×C31×#12011yescharacteristic
599C22C31×#57712yes—
600C4—21no—
601C4—21no—
602C4—21no—
603C4—21no—
604C4—21no—
605C4—21no—
606C4—42no—
607C4—42no—
608C4—41no—
609C4—41no—
610C4—42no—
611C4—42no—
612C4—41no—
613C4—82no—
614C4—81no—
615C2—22no—
616C2—81no—
617C2—82no—
618C2C31×#300911yescharacteristic
619C2C31×#302011yescharacteristic
620C2C31×#302511yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#3559C2
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×#3117C2semidirect
3. C31×#3103C2semidirect
4. C31×#2634C2semidirect
5. C31×#1616C2semidirect
6. C31×#1615C2semidirect
7. C31×#1604C2semidirect
8. C31×#1603C2semidirect
9. C31×#1521C2semidirect
10. C31×#1514C2semidirect
11. C31×#1470C2semidirect
228. #18245C31direct (N × H)
281. #3117C2×C31semidirect
282. #3103C2×C31semidirect
283. #2634C2×C31semidirect
284. #1616C2×C31semidirect
285. #1615C2×C31semidirect
286. #1604C2×C31semidirect
287. #1603C2×C31semidirect
288. #1521C2×C31semidirect
289. #1514C2×C31semidirect
290. #1470C2×C31semidirect
458. C31#18245direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35592
C31×#31172
C31×#31032
C31×#26342
C31×#16162
C31×#16152
C31×#16042
C31×#16032
C31×#15212
C31×#15142
C31×#14702
#1824531