Finite Group Databasebeta

← C31×#18461 #21475220

Subgroups — C31×#18461 #21475220

Order 7936.

880 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3563C211yesmaximal, characteristic, complemented
2C31×#3215C211yesmaximal, characteristic, complemented
3C31×#3211C211yesmaximal, characteristic
4C31×#2820C211yesmaximal, characteristic, complemented
5C31×#1605C212yesmaximal, complemented
6C31×#1603C212yesmaximal, complemented
7C31×#1599C212yesmaximal, complemented
8C31×#1597C212yesmaximal
9C31×#1518C211yesmaximal, characteristic, complemented
10C31×#1512C211yesmaximal, characteristic, complemented
11C31×#1470C211yesmaximal, characteristic, complemented
12C31×#564—21no—
13C31×#563—21no—
14C31×#557—21no—
15C31×#554—21no—
16C31×#552—21no—
17C31×#552—22no—
18C31×#552—21no—
19C31×#544—21no—
20C31×#544C2212yes—
21C31×#541—21no—
22C31×#541—22no—
23C31×#539C2212yes—
24C31×#533C2211yescharacteristic
25C31×#529C2211yescharacteristic
26C2×C31×#125C2211yescharacteristic
27C2×C31×#124C2211yescharacteristic
28C31×#497—21no—
29C31×#495—21no—
30C31×#489—21no—
31C31×#488—21no—
32C31×#464—21no—
33C31×#464—22no—
34C31×#463—21no—
35C31×#451C2212yes—
36C31×#451C2211yescharacteristic
37C31×#450C2212yes—
38C31×#450C2211yescharacteristic
39C31×#448C2211yescharacteristic
40C31×#447C2211yescharacteristic
41C31×#432C2211yescharacteristic
42C31×#431C2211yescharacteristic
43C31×#399—21no—
44C31×#397—21no—
45C31×#396—22no—
46C31×#395C2212yes—
47C31×#394C2212yes—
48C31×#382C2211yescharacteristic
49C31×#328C2212yes—
50C31×#328C2212yes—
51C31×#328C2212yes—
52C31×#327C2212yes—
53C31×#324C2212yes—
54C31×#324C2212yes—
55C22×Q8×C31C2311yescharacteristic
56C22×D4×C31C2311yescharacteristic
57C2×Q16×C31—21no—
58C2×Q16×C31—22no—
59C2×SD16×C31—21no—
60C2×SD16×C31—22no—
61C2×SD16×C31—21no—
62C2×D8×C31—21no—
63C31×C4⋊Q8—21no—
64C31×C4⋊Q8—22no—
65C31×C4⋊Q8—22no—
66C31×C4⋊Q8D411yescharacteristic
67C31×C4⋊Q8D411yescharacteristic
68C31×#126—22no—
69C31×#126—22no—
70C31×#126—22no—
71C31×#126—22no—
72C31×#126—21no—
73C31×#125—22no—
74C31×#125—22no—
75C31×#125—22no—
76C31×#125—22no—
77C31×#125—21no—
78C31×#125—22no—
79C31×#125—22no—
80C31×#125—22no—
81C31×#125—22no—
82C31×#125—21no—
83C31×#125—21no—
84C31×#125—21no—
85C31×#125—21no—
86C31×#125—21no—
87C31×#125D411yescharacteristic
88C31×#124—21no—
89C31×#124—21no—
90C31×#124—21no—
91C31×#124—21no—
92C31×#124D411yescharacteristic
93C31×#123—22no—
94C31×#123—21no—
95C31×#123—22no—
96C31×C22⋊Q8—21no—
97C31×C22⋊Q8—21no—
98C31×C22⋊Q8—22no—
99C31×C22⋊Q8—22no—
100C31×C22⋊Q8—22no—
101C31×C22⋊Q8—21no—
102C31×C22⋊Q8—22no—
103C31×C22⋊Q8C2311yescharacteristic
104C31×C22⋊Q8C2312yes—
105C31×#121C2311yescharacteristic
106C4×Q8×C31—21no—
107C4×Q8×C31—22no—
108C4×Q8×C31—21no—
109C4×Q8×C31—21no—
110C4×Q8×C31D411yescharacteristic
111C4×Q8×C31D412yes—
112C4×D4×C31—21no—
113C4×D4×C31—22no—
114C4×D4×C31D411yescharacteristic
115C31×#117—22no—
116C31×#117—21no—
117C31×#117—21no—
118C31×#117—22no—
119C31×#117—21no—
120C2×C4⋊C4×C31—21no—
121C2×C4⋊C4×C31—22no—
122C2×C4⋊C4×C31—21no—
123C2×C4⋊C4×C31D412yes—
124C2×C4⋊C4×C31D411yescharacteristic
125C2×C4⋊C4×C31D411yescharacteristic
126C2×C4⋊C4×C31C2312yes—
127C2×C22⋊C4×C31—21no—
128C2×C22⋊C4×C31—21no—
129C2×C22⋊C4×C31—22no—
130C2×C22⋊C4×C31—22no—
131C2×C42×C31C2311yescharacteristic
132C31×C4⋊C8—21no—
133C31×C4⋊C8—21no—
134C31×#103—21no—
135C31×#103—21no—
136C31×#103—22no—
137C31×#102—21no—
138C31×#102—21no—
139C31×#102—22no—
140C31×C22⋊C8C2311yescharacteristic
141C31×C22⋊C8C2311yescharacteristic
142C31×C22⋊C8C2312yes—
143C31×#97—21no—
144C4×C8×C31—21no—
145C31×#95—22no—
146C31×#95—21no—
147C31×#95C2312yes—
148C24×C31—21no—
149C2×Q8×C31—22no—
150C2×Q8×C31—21no—
151C2×Q8×C31—21no—
152C2×Q8×C31—21no—
153C2×Q8×C31—81no—
154C2×Q8×C31C2×D412yes—
155C2×Q8×C31C2×D411yescharacteristic
156C2×D4×C31—21no—
157C2×D4×C31—21no—
158C2×D4×C31—21no—
159C2×D4×C31—81no—
160C2×D4×C31C2×D411yescharacteristic
161C22×C4×C31—22no—
162C22×C4×C31—21no—
163C22×C4×C31—22no—
164C22×C4×C31—21no—
165C22×C4×C31C2×D412yes—
166C22×C4×C31C2×D411yescharacteristic
167C22×C4×C31C2×D412yes—
168C22×C4×C31C2×D411yescharacteristic
169C22×C4×C31C2411yescharacteristic
170Q16×C31—81no—
171Q16×C31—82no—
172SD16×C31—81no—
173SD16×C31—81no—
174SD16×C31—82no—
175D8×C31—81no—
176C2×C8×C31—21no—
177C2×C8×C31—21no—
178C2×C8×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—22no—
186C4⋊C4×C31—22no—
187C4⋊C4×C31—21no—
188C4⋊C4×C31—21no—
189C4⋊C4×C31—21no—
190C4⋊C4×C31—22no—
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—21no—
198C4⋊C4×C31—21no—
199C4⋊C4×C31—22no—
200C4⋊C4×C31—22no—
201C4⋊C4×C31—22no—
202C4⋊C4×C31—21no—
203C4⋊C4×C31—22no—
204C4⋊C4×C31—81no—
205C4⋊C4×C31—82no—
206C4⋊C4×C31C2×D411yescharacteristic
207C4⋊C4×C31C2×D411yescharacteristic
208C4⋊C4×C31C2×D412yes—
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—21no—
218C22⋊C4×C31—41no—
219C22⋊C4×C31—41no—
220C22⋊C4×C31—41no—
221C22⋊C4×C31—41no—
222C22⋊C4×C31—82no—
223C22⋊C4×C31—82no—
224C42×C31—22no—
225C42×C31—22no—
226C42×C31—22no—
227C42×C31—22no—
228C42×C31—21no—
229C42×C31C2×D411yescharacteristic
230C42×C31C2×D411yescharacteristic
231C42×C31C2×D411yescharacteristic
232C42×C31C2×D411yescharacteristic
233#18461C3111yesmaximal, characteristic, direct product factor, Sylow
234C23×C31—21no—
235C23×C31—42no—
236C23×C31—42no—
237C23×C31—41no—
238C23×C31—41no—
239C23×C31—41no—
240C23×C31C22×D411yescharacteristic
241Q8×C31—41no—
242Q8×C31—42no—
243Q8×C31—42no—
244Q8×C31—41no—
245Q8×C31—41no—
246Q8×C31—42no—
247Q8×C31—82no—
248D4×C31—41no—
249D4×C31—41no—
250D4×C31—41no—
251D4×C31—42no—
252D4×C31—41no—
253D4×C31—41no—
254C2×C4×C31—21no—
255C2×C4×C31—21no—
256C2×C4×C31—22no—
257C2×C4×C31—21no—
258C2×C4×C31—21no—
259C2×C4×C31—22no—
260C2×C4×C31—21no—
261C2×C4×C31—22no—
262C2×C4×C31—22no—
263C2×C4×C31—22no—
264C2×C4×C31—22no—
265C2×C4×C31—41no—
266C2×C4×C31—41no—
267C2×C4×C31—42no—
268C2×C4×C31—42no—
269C2×C4×C31—42no—
270C2×C4×C31—42no—
271C2×C4×C31—42no—
272C2×C4×C31—41no—
273C2×C4×C31—42no—
274C2×C4×C31—82no—
275C2×C4×C31#12011yescharacteristic
276C2×C4×C31#12011yescharacteristic
277C2×C4×C31#12011yescharacteristic
278C2×C4×C31#12011yescharacteristic
279C2×C4×C31C22×D411yescharacteristic
280C2×C4×C31C22×D411yescharacteristic
281C8×C31—41no—
282C8×C31—41no—
283C8×C31—42no—
284#3563C2×C3111yescharacteristic, complemented
285#3215C2×C3111yescharacteristic, complemented
286#3211C2×C3111yescharacteristic
287#2820C2×C3111yescharacteristic, complemented
288#1605C2×C3112yescomplemented
289#1603C2×C3112yescomplemented
290#1599C2×C3112yescomplemented
291#1597C2×C3112yes—
292#1518C2×C3111yescharacteristic, complemented
293#1512C2×C3111yescharacteristic, complemented
294#1470C2×C3111yescharacteristic, complemented
295C22×C31—21no—
296C22×C31—21no—
297C22×C31—41no—
298C22×C31—41no—
299C22×C31—41no—
300C22×C31—82no—
301C22×C31—82no—
302C22×C31—81no—
303C22×C31—81no—
304C22×C31—81no—
305C22×C31C2×#12011yescharacteristic
306C22×C31#57711yescharacteristic
307C22×C31#57811yescharacteristic
308C4×C31—21no—
309C4×C31—21no—
310C4×C31—21no—
311C4×C31—21no—
312C4×C31—21no—
313C4×C31—21no—
314C4×C31—42no—
315C4×C31—41no—
316C4×C31—41no—
317C4×C31—42no—
318C4×C31—42no—
319C4×C31—42no—
320C4×C31—41no—
321C4×C31—82no—
322C4×C31—81no—
323#564—21no—
324#563—21no—
325#557—21no—
326#554—21no—
327#552—21no—
328#552—22no—
329#552—21no—
330#544—21no—
331#544C22×C3112yes—
332#541—21no—
333#541—22no—
334#539C22×C3112yes—
335#533C22×C3111yescharacteristic
336#529C22×C3111yescharacteristic
337C2×#125C22×C3111yescharacteristic
338C2×#124C22×C3111yescharacteristic
339#497—21no—
340#495—21no—
341#489—21no—
342#488—21no—
343#464—21no—
344#464—22no—
345#463—21no—
346#451C22×C3112yes—
347#451C22×C3111yescharacteristic
348#450C22×C3112yes—
349#450C22×C3111yescharacteristic
350#448C22×C3111yescharacteristic
351#447C22×C3111yescharacteristic
352#432C22×C3111yescharacteristic
353#431C22×C3111yescharacteristic
354#399—21no—
355#397—21no—
356#396—22no—
357#395C22×C3112yes—
358#394C22×C3112yes—
359#382C22×C3111yescharacteristic
360#328C22×C3112yes—
361#328C22×C3112yes—
362#328C22×C3112yes—
363#327C22×C3112yes—
364#324C22×C3112yes—
365#324C22×C3112yes—
366C2×C31—21no—
367C2×C31—21no—
368C2×C31—81no—
369C2×C31—161no—
370C2×C31#301311yescharacteristic
371C2×C31#302011yescharacteristic
372C2×C31#302711yescharacteristic
373C22×Q8C23×C3111yescharacteristic
374C22×D4C23×C3111yescharacteristic
375C2×Q16—21no—
376C2×Q16—22no—
377C2×SD16—21no—
378C2×SD16—22no—
379C2×SD16—21no—
380C2×D8—21no—
381C4⋊Q8—21no—
382C4⋊Q8—22no—
383C4⋊Q8—22no—
384C4⋊Q8D4×C3111yescharacteristic
385C4⋊Q8D4×C3111yescharacteristic
386#126—22no—
387#126—22no—
388#126—22no—
389#126—22no—
390#126—21no—
391#125—22no—
392#125—22no—
393#125—22no—
394#125—22no—
395#125—21no—
396#125—22no—
397#125—22no—
398#125—22no—
399#125—22no—
400#125—21no—
401#125—21no—
402#125—21no—
403#125—21no—
404#125—21no—
405#125D4×C3111yescharacteristic
406#124—21no—
407#124—21no—
408#124—21no—
409#124—21no—
410#124D4×C3111yescharacteristic
411#123—22no—
412#123—21no—
413#123—22no—
414C22⋊Q8—21no—
415C22⋊Q8—21no—
416C22⋊Q8—22no—
417C22⋊Q8—22no—
418C22⋊Q8—22no—
419C22⋊Q8—21no—
420C22⋊Q8—22no—
421C22⋊Q8C23×C3111yescharacteristic
422C22⋊Q8C23×C3112yes—
423#121C23×C3111yescharacteristic
424C4×Q8—21no—
425C4×Q8—22no—
426C4×Q8—21no—
427C4×Q8—21no—
428C4×Q8D4×C3111yescharacteristic
429C4×Q8D4×C3112yes—
430C4×D4—21no—
431C4×D4—22no—
432C4×D4D4×C3111yescharacteristic
433#117—22no—
434#117—21no—
435#117—21no—
436#117—22no—
437#117—21no—
438C2×C4⋊C4—21no—
439C2×C4⋊C4—21no—
440C2×C4⋊C4—22no—
441C2×C4⋊C4D4×C3112yes—
442C2×C4⋊C4D4×C3111yescharacteristic
443C2×C4⋊C4D4×C3111yescharacteristic
444C2×C4⋊C4C23×C3112yes—
445C2×C22⋊C4—21no—
446C2×C22⋊C4—21no—
447C2×C22⋊C4—22no—
448C2×C22⋊C4—22no—
449C2×C42C23×C3111yescharacteristic
450C4⋊C8—21no—
451C4⋊C8—21no—
452#103—21no—
453#103—21no—
454#103—22no—
455#102—21no—
456#102—21no—
457#102—22no—
458C22⋊C8C23×C3112yes—
459C22⋊C8C23×C3111yescharacteristic
460C22⋊C8C23×C3111yescharacteristic
461#97—21no—
462C4×C8—21no—
463#95—21no—
464#95—22no—
465#95C23×C3112yes—
466C31#1846111yescharacteristic, direct product factor, Sylow
467C24—21no—
468C2×Q8—22no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—81no—
473C2×Q8C2×D4×C3112yes—
474C2×Q8C2×D4×C3111yescharacteristic
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4—81no—
479C2×D4C2×D4×C3111yescharacteristic
480C22×C4—22no—
481C22×C4—21no—
482C22×C4—22no—
483C22×C4—21no—
484C22×C4C2×D4×C3112yes—
485C22×C4C2×D4×C3111yescharacteristic
486C22×C4C2×D4×C3112yes—
487C22×C4C2×D4×C3111yescharacteristic
488C22×C4C24×C3111yescharacteristic
489Q16—81no—
490Q16—82no—
491SD16—81no—
492SD16—81no—
493SD16—82no—
494D8—81no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—22no—
510C4⋊C4—21no—
511C4⋊C4—22no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—21no—
522C4⋊C4—22no—
523C4⋊C4—81no—
524C4⋊C4—82no—
525C4⋊C4C2×D4×C3111yescharacteristic
526C4⋊C4C2×D4×C3111yescharacteristic
527C4⋊C4C2×D4×C3112yes—
528C22⋊C4—22no—
529C22⋊C4—22no—
530C22⋊C4—22no—
531C22⋊C4—22no—
532C22⋊C4—21no—
533C22⋊C4—21no—
534C22⋊C4—21no—
535C22⋊C4—21no—
536C22⋊C4—21no—
537C22⋊C4—41no—
538C22⋊C4—41no—
539C22⋊C4—41no—
540C22⋊C4—41no—
541C22⋊C4—82no—
542C22⋊C4—82no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—21no—
548C42C2×D4×C3111yescharacteristic
549C42C2×D4×C3111yescharacteristic
550C42C2×D4×C3111yescharacteristic
551C42C2×D4×C3111yescharacteristic
552C23—21no—
553C23—42no—
554C23—42no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23C22×D4×C3111yescharacteristic
559Q8—41no—
560Q8—42no—
561Q8—42no—
562Q8—41no—
563Q8—41no—
564Q8—42no—
565Q8—82no—
566D4—41no—
567D4—41no—
568D4—41no—
569D4—42no—
570D4—41no—
571D4—41no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—82no—
593C2×C4C31×#12011yescharacteristic
594C2×C4C31×#12011yescharacteristic
595C2×C4C31×#12011yescharacteristic
596C2×C4C31×#12011yescharacteristic
597C2×C4C22×D4×C3111yescharacteristic
598C2×C4C22×D4×C3111yescharacteristic
599C8—41no—
600C8—41no—
601C8—42no—
602C22—21no—
603C22—21no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22—82no—
608C22—82no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22C2×C31×#12011yescharacteristic
613C22C31×#57711yescharacteristic
614C22C31×#57811yescharacteristic
615C4—21no—
616C4—21no—
617C4—21no—
618C4—21no—
619C4—21no—
620C4—21no—
621C4—42no—
622C4—41no—
623C4—41no—
624C4—42no—
625C4—42no—
626C4—42no—
627C4—41no—
628C4—82no—
629C4—81no—
630C2—21no—
631C2—21no—
632C2—81no—
633C2—161no—
634C2C31×#301311yescharacteristic
635C2C31×#302011yescharacteristic
636C2C31×#302711yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#447C22
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
1. C31×#3563C2semidirect
2. C31×#3215C2semidirect
4. C31×#2820C2semidirect
5. C31×#1605C2semidirect
6. C31×#1603C2semidirect
7. C31×#1599C2semidirect
9. C31×#1518C2semidirect
10. C31×#1512C2semidirect
11. C31×#1470C2semidirect
233. #18461C31direct (N × H)
284. #3563C2×C31semidirect
285. #3215C2×C31semidirect
287. #2820C2×C31semidirect
288. #1605C2×C31semidirect
289. #1603C2×C31semidirect
290. #1599C2×C31semidirect
292. #1518C2×C31semidirect
293. #1512C2×C31semidirect
294. #1470C2×C31semidirect
466. C31#18461direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35632
C31×#32152
C31×#32112
C31×#28202
C31×#16052
C31×#16032
C31×#15992
C31×#15972
C31×#15182
C31×#15122
C31×#14702
#1846131