Finite Group Databasebeta

← C31×#18207 #21474966

Subgroups — C31×#18207 #21474966

Order 7936.

1008 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3505C211yesmaximal, characteristic
2C2×C31×#484C211yesmaximal, characteristic, complemented
3C2×C31×#481C211yesmaximal, characteristic, complemented
4C31×#2487C211yesmaximal, characteristic, complemented
5C31×#1618C214yesmaximal, complemented
6C31×#1615C214yesmaximal, complemented
7C31×#1520C211yesmaximal, characteristic, complemented
8C31×#1519C211yesmaximal, characteristic, complemented
9C31×#1459C211yesmaximal, characteristic, complemented
10C31×#553—22no—
11C31×#553—21no—
12C31×#552—21no—
13C31×#551—22no—
14C31×#548—21no—
15C31×#546—21no—
16C31×#539C2212yes—
17C31×#538C2212yes—
18C31×#534—21no—
19C31×#529—21no—
20C31×#525—21no—
21C2×C31×C22⋊Q8C2211yescharacteristic
22C2×C31×#121C2211yescharacteristic
23C31×#518—21no—
24C2×C4×D4×C31C2211yescharacteristic
25C22×C4⋊C4×C31C2211yescharacteristic
26C31×#484—22no—
27C31×#484C2214yescomplemented
28C31×#481—22no—
29C31×#481C2214yes—
30C2×C31×#107—21no—
31C2×C31×#106—21no—
32C2×C31×#103—21no—
33C2×C31×#102—21no—
34C2×C31×C22⋊C8C2211yescharacteristic
35C2×C31×C22⋊C8C2211yescharacteristic
36C31×#400C2212yescomplemented
37C31×#399C2212yescomplemented
38C31×#389—21no—
39C31×#386—21no—
40C31×#385—21no—
41C31×#380—21no—
42C31×#380C2211yescharacteristic
43C31×#328C2214yescomplemented
44C31×#327C2214yes—
45C31×#324C2212yescomplemented
46C31×#323C2212yescomplemented
47C2×#41×C31—21no—
48C22×Q8×C31D411yescharacteristic
49C22×D4×C31—21no—
50C22×D4×C31D411yescharacteristic
51C22×D4×C31D411yescharacteristic
52C23×C4×C31—21no—
53C23×C4×C31D411yescharacteristic
54C23×C4×C31C2311yescharacteristic
55C22×C8×C31—21no—
56C22×C8×C31—21no—
57C31×#126—22no—
58C31×#126—22no—
59C31×#126—22no—
60C31×#125—21no—
61C31×#125—22no—
62C31×#124—21no—
63C31×#124—22no—
64C31×#123—22no—
65C31×#123—22no—
66C31×#123—22no—
67C31×#123—22no—
68C31×#123—22no—
69C31×C22⋊Q8—21no—
70C31×C22⋊Q8—22no—
71C31×C22⋊Q8—21no—
72C31×C22⋊Q8—21no—
73C31×C22⋊Q8D412yes—
74C31×C22⋊Q8C2312yes—
75C31×#121—21no—
76C31×#121—22no—
77C31×#121—22no—
78C31×#121—22no—
79C31×#121—21no—
80C31×#121—21no—
81C31×#121D412yes—
82C31×#121C2312yes—
83C31×#120—22no—
84C31×#120—22no—
85C4×Q8×C31—21no—
86C4×D4×C31—22no—
87C4×D4×C31—22no—
88C4×D4×C31—22no—
89C4×D4×C31—21no—
90C4×D4×C31—21no—
91C4×D4×C31—21no—
92C4×D4×C31—21no—
93C4×D4×C31—21no—
94C4×D4×C31—22no—
95C4×D4×C31—22no—
96C31×#117—22no—
97C31×#117—21no—
98C31×#117—22no—
99C2×C4⋊C4×C31—21no—
100C2×C4⋊C4×C31—21no—
101C2×C4⋊C4×C31—21no—
102C2×C4⋊C4×C31—21no—
103C2×C4⋊C4×C31—21no—
104C2×C4⋊C4×C31—21no—
105C2×C4⋊C4×C31—21no—
106C2×C4⋊C4×C31—22no—
107C2×C4⋊C4×C31D411yescharacteristic
108C2×C4⋊C4×C31D411yescharacteristic
109C2×C4⋊C4×C31C2312yes—
110C2×C4⋊C4×C31C2311yescharacteristic
111C2×C22⋊C4×C31—22no—
112C2×C22⋊C4×C31—21no—
113C2×C22⋊C4×C31—21no—
114C2×C22⋊C4×C31—21no—
115C2×C22⋊C4×C31—22no—
116C2×C22⋊C4×C31D411yescharacteristic
117C2×C22⋊C4×C31C2311yescharacteristic
118C2×C42×C31D411yescharacteristic
119C31×#107—24no—
120C31×#106—24no—
121C31×#103—24no—
122C31×#102—24no—
123C31×C22⋊C8—21no—
124C31×C22⋊C8—21no—
125C31×C22⋊C8C2312yes—
126C31×C22⋊C8C2312yes—
127C31×#95—22no—
128C31×#95—22no—
129C31×#95C2312yes—
130C24×C31—21no—
131C24×C31C2×D411yescharacteristic
132C24×C31C2×D411yescharacteristic
133#41×C31—82no—
134C2×Q8×C31—21no—
135C2×Q8×C31—42no—
136C2×Q8×C31C2×D412yes—
137C2×D4×C31—21no—
138C2×D4×C31—21no—
139C2×D4×C31—22no—
140C2×D4×C31—22no—
141C2×D4×C31—22no—
142C2×D4×C31—22no—
143C2×D4×C31—22no—
144C2×D4×C31—22no—
145C2×D4×C31—21no—
146C2×D4×C31—42no—
147C2×D4×C31—42no—
148C2×D4×C31—42no—
149C2×D4×C31C2×D412yes—
150C22×C4×C31—21no—
151C22×C4×C31—22no—
152C22×C4×C31—22no—
153C22×C4×C31—21no—
154C22×C4×C31—21no—
155C22×C4×C31—21no—
156C22×C4×C31—21no—
157C22×C4×C31—22no—
158C22×C4×C31—22no—
159C22×C4×C31—42no—
160C22×C4×C31—42no—
161C22×C4×C31—41no—
162C22×C4×C31—41no—
163C22×C4×C31—82no—
164C22×C4×C31C2×D411yescharacteristic
165C22×C4×C31C2×D411yescharacteristic
166C22×C4×C31C2×D411yescharacteristic
167C22×C4×C31C2×D411yescharacteristic
168C22×C4×C31C2×D411yescharacteristic
169C22×C4×C31C2×D411yescharacteristic
170C22×C4×C31C2×D411yescharacteristic
171C22×C4×C31C2×D411yescharacteristic
172C22×C4×C31C2411yescharacteristic
173C2×C8×C31—22no—
174C2×C8×C31—22no—
175C2×C8×C31—42no—
176C2×C8×C31—42no—
177C4⋊C4×C31—22no—
178C4⋊C4×C31—22no—
179C4⋊C4×C31—22no—
180C4⋊C4×C31—22no—
181C4⋊C4×C31—22no—
182C4⋊C4×C31—21no—
183C4⋊C4×C31—21no—
184C4⋊C4×C31—22no—
185C4⋊C4×C31—21no—
186C4⋊C4×C31—21no—
187C4⋊C4×C31—22no—
188C4⋊C4×C31—22no—
189C4⋊C4×C31—42no—
190C4⋊C4×C31—41no—
191C4⋊C4×C31—41no—
192C4⋊C4×C31—82no—
193C4⋊C4×C31C2×D412yes—
194C4⋊C4×C31C2×D412yes—
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—21no—
202C22⋊C4×C31—22no—
203C22⋊C4×C31—21no—
204C22⋊C4×C31—22no—
205C22⋊C4×C31—22no—
206C22⋊C4×C31—22no—
207C22⋊C4×C31—21no—
208C22⋊C4×C31—21no—
209C22⋊C4×C31—42no—
210C22⋊C4×C31—82no—
211C42×C31—22no—
212C42×C31—22no—
213C42×C31—21no—
214C42×C31—21no—
215#18207C3111yesmaximal, characteristic, direct product factor, Sylow
216C23×C31—21no—
217C23×C31—22no—
218C23×C31—21no—
219C23×C31—21no—
220C23×C31—41no—
221C23×C31—41no—
222C23×C31—41no—
223C23×C31—41no—
224C23×C31—42no—
225C23×C31—81no—
226C23×C31—81no—
227C23×C31#12011yescharacteristic
228C23×C31#12011yescharacteristic
229C23×C31#13512yes—
230C23×C31#13712yes—
231C23×C31C22×D411yescharacteristic
232Q8×C31—42no—
233Q8×C31—42no—
234D4×C31—42no—
235D4×C31—42no—
236D4×C31—42no—
237D4×C31—42no—
238D4×C31—42no—
239D4×C31—44no—
240D4×C31—42no—
241D4×C31—82no—
242D4×C31—82no—
243C2×C4×C31—22no—
244C2×C4×C31—22no—
245C2×C4×C31—22no—
246C2×C4×C31—21no—
247C2×C4×C31—21no—
248C2×C4×C31—22no—
249C2×C4×C31—21no—
250C2×C4×C31—22no—
251C2×C4×C31—22no—
252C2×C4×C31—21no—
253C2×C4×C31—21no—
254C2×C4×C31—41no—
255C2×C4×C31—41no—
256C2×C4×C31—41no—
257C2×C4×C31—42no—
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—42no—
266C2×C4×C31—42no—
267C2×C4×C31—81no—
268C2×C4×C31—81no—
269C2×C4×C31—81no—
270C2×C4×C31—82no—
271C2×C4×C31—82no—
272C2×C4×C31—82no—
273C2×C4×C31—82no—
274C2×C4×C31#12011yescharacteristic
275C2×C4×C31#12011yescharacteristic
276C2×C4×C31C22×D411yescharacteristic
277C2×C4×C31C22×D411yescharacteristic
278C8×C31—42no—
279C8×C31—42no—
280#3505C2×C3111yescharacteristic
281C2×#484C2×C3111yescharacteristic, complemented
282C2×#481C2×C3111yescharacteristic, complemented
283#2487C2×C3111yescharacteristic, complemented
284#1618C2×C3114yescomplemented
285#1615C2×C3114yescomplemented
286#1520C2×C3111yescharacteristic, complemented
287#1519C2×C3111yescharacteristic, complemented
288#1459C2×C3111yescharacteristic, complemented
289C22×C31—21no—
290C22×C31—21no—
291C22×C31—21no—
292C22×C31—21no—
293C22×C31—21no—
294C22×C31—21no—
295C22×C31—42no—
296C22×C31—42no—
297C22×C31—42no—
298C22×C31—41no—
299C22×C31—42no—
300C22×C31—42no—
301C22×C31—41no—
302C22×C31—41no—
303C22×C31—42no—
304C22×C31—81no—
305C22×C31—162no—
306C22×C31#45212yes—
307C22×C31#45212yes—
308C22×C31C2×#12011yescharacteristic
309C22×C31C2×#13511yescharacteristic
310C22×C31C2×#13711yescharacteristic
311C4×C31—21no—
312C4×C31—21no—
313C4×C31—21no—
314C4×C31—41no—
315C4×C31—42no—
316C4×C31—41no—
317C4×C31—42no—
318C4×C31—41no—
319C4×C31—42no—
320C4×C31—82no—
321C4×C31—82no—
322C4×C31#45812yes—
323#553—22no—
324#553—21no—
325#552—21no—
326#551—22no—
327#548—21no—
328#546—21no—
329#539C22×C3112yes—
330#538C22×C3112yes—
331#534—21no—
332#529—21no—
333#525—21no—
334C2×C22⋊Q8C22×C3111yescharacteristic
335C2×#121C22×C3111yescharacteristic
336#518—21no—
337C2×C4×D4C22×C3111yescharacteristic
338C22×C4⋊C4C22×C3111yescharacteristic
339#484—22no—
340#484C22×C3114yescomplemented
341#481—22no—
342#481C22×C3114yes—
343C2×#107—21no—
344C2×#106—21no—
345C2×#103—21no—
346C2×#102—21no—
347C2×C22⋊C8C22×C3111yescharacteristic
348C2×C22⋊C8C22×C3111yescharacteristic
349#400C22×C3112yescomplemented
350#399C22×C3112yescomplemented
351#389—21no—
352#386—21no—
353#385—21no—
354#380—21no—
355#380C22×C3111yescharacteristic
356#328C22×C3114yescomplemented
357#327C22×C3114yes—
358#324C22×C3112yescomplemented
359#323C22×C3112yescomplemented
360C2×C31—21no—
361C2×C31—21no—
362C2×C31—22no—
363C2×C31—22no—
364C2×C31—81no—
365C2×C31—82no—
366C2×C31C2×#45211yescharacteristic
367C2×C31#302111yescharacteristic
368C2×C31C2×#45811yescharacteristic
369C2×#41—21no—
370C22×Q8D4×C3111yescharacteristic
371C22×D4—21no—
372C22×D4D4×C3111yescharacteristic
373C22×D4D4×C3111yescharacteristic
374C23×C4—21no—
375C23×C4D4×C3111yescharacteristic
376C23×C4C23×C3111yescharacteristic
377C22×C8—21no—
378C22×C8—21no—
379#126—22no—
380#126—22no—
381#126—22no—
382#125—21no—
383#125—22no—
384#124—21no—
385#124—22no—
386#123—22no—
387#123—22no—
388#123—22no—
389#123—22no—
390#123—22no—
391C22⋊Q8—21no—
392C22⋊Q8—22no—
393C22⋊Q8—21no—
394C22⋊Q8—21no—
395C22⋊Q8D4×C3112yes—
396C22⋊Q8C23×C3112yes—
397#121—21no—
398#121—22no—
399#121—22no—
400#121—22no—
401#121—21no—
402#121—21no—
403#121D4×C3112yes—
404#121C23×C3112yes—
405#120—22no—
406#120—22no—
407C4×Q8—21no—
408C4×D4—22no—
409C4×D4—22no—
410C4×D4—22no—
411C4×D4—21no—
412C4×D4—21no—
413C4×D4—21no—
414C4×D4—21no—
415C4×D4—21no—
416C4×D4—22no—
417C4×D4—22no—
418#117—22no—
419#117—21no—
420#117—22no—
421C2×C4⋊C4—21no—
422C2×C4⋊C4—21no—
423C2×C4⋊C4—21no—
424C2×C4⋊C4—21no—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—21no—
428C2×C4⋊C4—22no—
429C2×C4⋊C4D4×C3111yescharacteristic
430C2×C4⋊C4D4×C3111yescharacteristic
431C2×C4⋊C4C23×C3112yes—
432C2×C4⋊C4C23×C3111yescharacteristic
433C2×C22⋊C4—22no—
434C2×C22⋊C4—21no—
435C2×C22⋊C4—21no—
436C2×C22⋊C4—21no—
437C2×C22⋊C4—22no—
438C2×C22⋊C4D4×C3111yescharacteristic
439C2×C22⋊C4C23×C3111yescharacteristic
440C2×C42D4×C3111yescharacteristic
441#107—24no—
442#106—24no—
443#103—24no—
444#102—24no—
445C22⋊C8—21no—
446C22⋊C8—21no—
447C22⋊C8C23×C3112yes—
448C22⋊C8C23×C3112yes—
449#95—22no—
450#95—22no—
451#95C23×C3112yes—
452C31#1820711yescharacteristic, direct product factor, Sylow
453C24—21no—
454C24C2×D4×C3111yescharacteristic
455C24C2×D4×C3111yescharacteristic
456#41—82no—
457C2×Q8—21no—
458C2×Q8—42no—
459C2×Q8C2×D4×C3112yes—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—22no—
463C2×D4—22no—
464C2×D4—22no—
465C2×D4—22no—
466C2×D4—22no—
467C2×D4—22no—
468C2×D4—21no—
469C2×D4—42no—
470C2×D4—42no—
471C2×D4—42no—
472C2×D4C2×D4×C3112yes—
473C22×C4—21no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—41no—
485C22×C4—41no—
486C22×C4—82no—
487C22×C4C2×D4×C3111yescharacteristic
488C22×C4C2×D4×C3111yescharacteristic
489C22×C4C2×D4×C3111yescharacteristic
490C22×C4C2×D4×C3111yescharacteristic
491C22×C4C2×D4×C3111yescharacteristic
492C22×C4C2×D4×C3111yescharacteristic
493C22×C4C2×D4×C3111yescharacteristic
494C22×C4C2×D4×C3111yescharacteristic
495C22×C4C24×C3111yescharacteristic
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—42no—
499C2×C8—42no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—22no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—42no—
513C4⋊C4—41no—
514C4⋊C4—41no—
515C4⋊C4—82no—
516C4⋊C4C2×D4×C3112yes—
517C4⋊C4C2×D4×C3112yes—
518C22⋊C4—22no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—22no—
522C22⋊C4—22no—
523C22⋊C4—22no—
524C22⋊C4—21no—
525C22⋊C4—22no—
526C22⋊C4—21no—
527C22⋊C4—22no—
528C22⋊C4—22no—
529C22⋊C4—22no—
530C22⋊C4—21no—
531C22⋊C4—21no—
532C22⋊C4—42no—
533C22⋊C4—82no—
534C42—22no—
535C42—22no—
536C42—21no—
537C42—21no—
538C23—21no—
539C23—22no—
540C23—21no—
541C23—21no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—42no—
547C23—81no—
548C23—81no—
549C23C31×#12011yescharacteristic
550C23C31×#12011yescharacteristic
551C23C31×#13512yes—
552C23C31×#13712yes—
553C23C22×D4×C3111yescharacteristic
554Q8—42no—
555Q8—42no—
556D4—42no—
557D4—42no—
558D4—42no—
559D4—44no—
560D4—42no—
561D4—42no—
562D4—42no—
563D4—82no—
564D4—82no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—81no—
590C2×C4—81no—
591C2×C4—81no—
592C2×C4—82no—
593C2×C4—82no—
594C2×C4—82no—
595C2×C4—82no—
596C2×C4C31×#12011yescharacteristic
597C2×C4C31×#12011yescharacteristic
598C2×C4C22×D4×C3111yescharacteristic
599C2×C4C22×D4×C3111yescharacteristic
600C8—42no—
601C8—42no—
602C22—21no—
603C22—21no—
604C22—21no—
605C22—21no—
606C22—21no—
607C22—21no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—41no—
612C22—42no—
613C22—42no—
614C22—41no—
615C22—41no—
616C22—42no—
617C22—81no—
618C22—162no—
619C22C31×#45212yes—
620C22C31×#45212yes—
621C22C2×C31×#12011yescharacteristic
622C22C2×C31×#13511yescharacteristic
623C22C2×C31×#13711yescharacteristic
624C4—21no—
625C4—21no—
626C4—21no—
627C4—41no—
628C4—42no—
629C4—41no—
630C4—42no—
631C4—41no—
632C4—42no—
633C4—82no—
634C4—82no—
635C4C31×#45812yes—
636C2—21no—
637C2—21no—
638C2—22no—
639C2—22no—
640C2—81no—
641C2—82no—
642C2C2×C31×#45211yescharacteristic
643C2C31×#302111yescharacteristic
644C2C2×C31×#45811yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#3505C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×#484C2semidirect
3. C2×C31×#481C2semidirect
4. C31×#2487C2semidirect
5. C31×#1618C2semidirect
6. C31×#1615C2semidirect
7. C31×#1520C2semidirect
8. C31×#1519C2semidirect
9. C31×#1459C2semidirect
27. C31×#484C22semidirect
36. C31×#400C22semidirect
37. C31×#399C22semidirect
43. C31×#328C22semidirect
45. C31×#324C22semidirect
46. C31×#323C22semidirect
215. #18207C31direct (N × H)
281. C2×#484C2×C31semidirect
282. C2×#481C2×C31semidirect
283. #2487C2×C31semidirect
284. #1618C2×C31semidirect
285. #1615C2×C31semidirect
286. #1520C2×C31semidirect
287. #1519C2×C31semidirect
288. #1459C2×C31semidirect
340. #484C22×C31semidirect
349. #400C22×C31semidirect
350. #399C22×C31semidirect
356. #328C22×C31semidirect
358. #324C22×C31semidirect
359. #323C22×C31semidirect
452. C31#18207direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35052
C2×C31×#4842
C2×C31×#4812
C31×#24872
C31×#16182
C31×#16152
C31×#15202
C31×#15192
C31×#14592
#1820731