Finite Group Databasebeta

← C31×#18845 #21475604

Subgroups — C31×#18845 #21475604

Order 7936.

876 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×#3515C211yesmaximal, characteristic
2C31×#3229C211yesmaximal, characteristic, complemented
3C31×#3218C211yesmaximal, characteristic, complemented
4C31×#3079C211yesmaximal, characteristic, complemented
5C31×#1654C212yesmaximal, complemented
6C31×#1653C212yesmaximal, complemented
7C31×#1644C212yesmaximal, complemented
8C31×#1643C212yesmaximal, complemented
9C31×#1559C211yesmaximal, characteristic, complemented
10C31×#1495C211yesmaximal, characteristic, complemented
11C31×#1488C211yesmaximal, characteristic, complemented
12C31×#557C2211yescharacteristic
13C31×#554C2211yescharacteristic
14C31×#554C2211yescharacteristic
15C31×#553—21no—
16C31×#552—21no—
17C31×#552—22no—
18C31×#552—21no—
19C31×#547—21no—
20C31×#547C2211yescharacteristic
21C31×#541—21no—
22C31×#539—22no—
23C31×#533C2211yescharacteristic
24C2×C31×C4⋊Q8C2211yescharacteristic
25C31×#529—21no—
26C31×#529C2211yescharacteristic
27C2×C31×#125—21no—
28C31×#520—21no—
29C4×#41×C31C2211yescharacteristic
30C31×#498C2212yes—
31C31×#497—21no—
32C31×#491—21no—
33C31×#487C2212yes—
34C31×#483—22no—
35C31×#477—21no—
36C31×#476—21no—
37C31×#475—22no—
38C31×#475C2212yes—
39C31×#474C2212yes—
40C31×#462C2212yes—
41C31×#461C2212yes—
42C31×#451—21no—
43C31×#448—21no—
44C31×#432C2211yescharacteristic
45C31×#431C2211yescharacteristic
46C31×#428—21no—
47C31×#423C2211yescharacteristic
48C31×#417—21no—
49C31×#333C2212yes—
50C31×#332C2212yes—
51C31×Q8⋊C8C2212yes—
52C31×Q8⋊C8C2212yes—
53C31×D4⋊C8C2212yes—
54C31×D4⋊C8C2212yes—
55C2×#41×C31—21no—
56C2×#41×C31D411yescharacteristic
57C22×Q8×C31—21no—
58C2×Q16×C31—22no—
59C2×SD16×C31—22no—
60C2×M16×C31—21no—
61C31×C4⋊Q8—21no—
62C31×C4⋊Q8—21no—
63C31×C4⋊Q8—21no—
64C31×C4⋊Q8—21no—
65C31×C4⋊Q8C2312yes—
66C31×C4⋊Q8C2311yescharacteristic
67C31×C4⋊Q8C2311yescharacteristic
68C31×#126—22no—
69C31×#126—22no—
70C31×#126—22no—
71C31×#126—21no—
72C31×#125—22no—
73C31×#125—22no—
74C31×#125—22no—
75C31×#125—21no—
76C31×#125—21no—
77C31×#125—22no—
78C31×#125—21no—
79C31×#125—21no—
80C31×#125D411yescharacteristic
81C31×#124D411yescharacteristic
82C31×#123—22no—
83C31×#123—22no—
84C31×#123—22no—
85C31×#123—21no—
86C31×#123—22no—
87C31×C22⋊Q8—21no—
88C31×C22⋊Q8—22no—
89C31×C22⋊Q8—22no—
90C31×C22⋊Q8—21no—
91C31×C22⋊Q8—21no—
92C31×C22⋊Q8—21no—
93C31×C22⋊Q8D411yescharacteristic
94C31×C22⋊Q8D412yes—
95C31×#121—22no—
96C31×#121—21no—
97C31×#121D411yescharacteristic
98C4×Q8×C31—21no—
99C4×Q8×C31—21no—
100C4×Q8×C31—21no—
101C4×Q8×C31D412yes—
102C4×Q8×C31D411yescharacteristic
103C4×Q8×C31C2311yescharacteristic
104C4×Q8×C31C2311yescharacteristic
105C4×D4×C31—22no—
106C4×D4×C31—22no—
107C4×D4×C31—21no—
108C4×D4×C31—21no—
109C4×D4×C31—21no—
110C4×D4×C31—21no—
111C4×D4×C31—21no—
112C4×D4×C31D411yescharacteristic
113C4×D4×C31C2311yescharacteristic
114C4×D4×C31C2311yescharacteristic
115C31×#117—22no—
116C31×#117—22no—
117C31×#117—21no—
118C31×#117—21no—
119C31×#117—21no—
120C31×#117D411yescharacteristic
121C2×C4⋊C4×C31—22no—
122C2×C4⋊C4×C31—22no—
123C2×C4⋊C4×C31—21no—
124C2×C4⋊C4×C31—21no—
125C2×C4⋊C4×C31—21no—
126C2×C4⋊C4×C31—21no—
127C2×C4⋊C4×C31—21no—
128C2×C42×C31—21no—
129C2×C42×C31C2311yescharacteristic
130C31×#107—22no—
131C31×#106—22no—
132C31×#106—24no—
133C31×C4⋊C8—21no—
134C31×C4⋊C8—21no—
135C31×C4⋊C8C2312yes—
136C31×C4⋊C8C2312yes—
137C31×#103—22no—
138C31×#103—24no—
139C31×#103—22no—
140C31×#103—22no—
141C31×#102—22no—
142C31×C22⋊C8—21no—
143C31×C22⋊C8—21no—
144C31×#97—21no—
145C4×C8×C31C2312yes—
146#41×C31—22no—
147#41×C31—41no—
148#41×C31—42no—
149#41×C31—42no—
150C2×Q8×C31—21no—
151C2×Q8×C31—22no—
152C2×Q8×C31—41no—
153C2×Q8×C31—82no—
154C2×Q8×C31C2×D411yescharacteristic
155C2×Q8×C31C2×D411yescharacteristic
156C2×Q8×C31C2×D412yes—
157C2×D4×C31—22no—
158C2×D4×C31—21no—
159C2×D4×C31—21no—
160C2×D4×C31C2×D411yescharacteristic
161C2×D4×C31C2×D411yescharacteristic
162C22×C4×C31—22no—
163C22×C4×C31—22no—
164C22×C4×C31—22no—
165C22×C4×C31—21no—
166C22×C4×C31—21no—
167C22×C4×C31—21no—
168C22×C4×C31—21no—
169C22×C4×C31—41no—
170C22×C4×C31C2×D411yescharacteristic
171C22×C4×C31C2×D411yescharacteristic
172C22×C4×C31C2×D411yescharacteristic
173Q16×C31—82no—
174SD16×C31—82no—
175M16×C31—42no—
176C2×C8×C31—22no—
177C2×C8×C31—22no—
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—22no—
190C4⋊C4×C31—22no—
191C4⋊C4×C31—21no—
192C4⋊C4×C31—21no—
193C4⋊C4×C31—21no—
194C4⋊C4×C31—21no—
195C4⋊C4×C31—21no—
196C4⋊C4×C31—21no—
197C4⋊C4×C31—21no—
198C4⋊C4×C31—22no—
199C4⋊C4×C31—22no—
200C4⋊C4×C31—41no—
201C4⋊C4×C31—41no—
202C4⋊C4×C31C2×D411yescharacteristic
203C4⋊C4×C31C2×D411yescharacteristic
204C4⋊C4×C31C2×D411yescharacteristic
205C4⋊C4×C31C2×D412yes—
206C4⋊C4×C31C2×D411yescharacteristic
207C22⋊C4×C31—22no—
208C22⋊C4×C31—22no—
209C22⋊C4×C31—22no—
210C22⋊C4×C31—22no—
211C22⋊C4×C31—21no—
212C22⋊C4×C31—21no—
213C22⋊C4×C31—21no—
214C22⋊C4×C31—21no—
215C22⋊C4×C31—21no—
216C22⋊C4×C31—21no—
217C22⋊C4×C31—21no—
218C22⋊C4×C31—21no—
219C42×C31—22no—
220C42×C31—21no—
221C42×C31—21no—
222C42×C31—22no—
223C42×C31—21no—
224C42×C31—21no—
225C42×C31C2×D411yescharacteristic
226C42×C31C2×D411yescharacteristic
227C42×C31C2×D411yescharacteristic
228C42×C31C2411yescharacteristic
229#18845C3111yesmaximal, characteristic, direct product factor, Sylow
230C23×C31—21no—
231C23×C31—21no—
232C23×C31#12011yescharacteristic
233Q8×C31—21no—
234Q8×C31—21no—
235Q8×C31—22no—
236Q8×C31—42no—
237Q8×C31—41no—
238Q8×C31—41no—
239Q8×C31—42no—
240Q8×C31—42no—
241Q8×C31—82no—
242D4×C31—21no—
243D4×C31—22no—
244D4×C31—42no—
245D4×C31—44no—
246D4×C31—81no—
247D4×C31#13712yes—
248C2×C4×C31—21no—
249C2×C4×C31—21no—
250C2×C4×C31—22no—
251C2×C4×C31—21no—
252C2×C4×C31—21no—
253C2×C4×C31—21no—
254C2×C4×C31—21no—
255C2×C4×C31—22no—
256C2×C4×C31—22no—
257C2×C4×C31—21no—
258C2×C4×C31—21no—
259C2×C4×C31—41no—
260C2×C4×C31—42no—
261C2×C4×C31—44no—
262C2×C4×C31—41no—
263C2×C4×C31—41no—
264C2×C4×C31—42no—
265C2×C4×C31—42no—
266C2×C4×C31—81no—
267C2×C4×C31—82no—
268C2×C4×C31—82no—
269C2×C4×C31—82no—
270C2×C4×C31#12011yescharacteristic
271C2×C4×C31#12011yescharacteristic
272C2×C4×C31#12011yescharacteristic
273C2×C4×C31C22×D411yescharacteristic
274C2×C4×C31C22×D411yescharacteristic
275C2×C4×C31C22×D411yescharacteristic
276C8×C31—42no—
277C8×C31—42no—
278C8×C31—42no—
279#3515C2×C3111yescharacteristic
280#3229C2×C3111yescharacteristic, complemented
281#3218C2×C3111yescharacteristic, complemented
282#3079C2×C3111yescharacteristic, complemented
283#1654C2×C3112yescomplemented
284#1653C2×C3112yescomplemented
285#1644C2×C3112yescomplemented
286#1643C2×C3112yescomplemented
287#1559C2×C3111yescharacteristic, complemented
288#1495C2×C3111yescharacteristic, complemented
289#1488C2×C3111yescharacteristic, complemented
290C22×C31—21no—
291C22×C31—22no—
292C22×C31—21no—
293C22×C31—21no—
294C22×C31—41no—
295C22×C31—41no—
296C22×C31—41no—
297C22×C31—41no—
298C22×C31—41no—
299C22×C31C2×#12011yescharacteristic
300C4×C31—21no—
301C4×C31—21no—
302C4×C31—21no—
303C4×C31—21no—
304C4×C31—41no—
305C4×C31—41no—
306C4×C31—42no—
307C4×C31—41no—
308C4×C31—41no—
309C4×C31—41no—
310C4×C31—41no—
311C4×C31—42no—
312C4×C31—82no—
313C4×C31C2×#13711yescharacteristic
314C4×C31#57511yescharacteristic
315C4×C31#57712yes—
316#557C22×C3111yescharacteristic
317#554C22×C3111yescharacteristic
318#554C22×C3111yescharacteristic
319#553—21no—
320#552—21no—
321#552—22no—
322#552—21no—
323#547—21no—
324#547C22×C3111yescharacteristic
325#541—21no—
326#539—22no—
327#533C22×C3111yescharacteristic
328C2×C4⋊Q8C22×C3111yescharacteristic
329#529—21no—
330#529C22×C3111yescharacteristic
331C2×#125—21no—
332#520—21no—
333C4×#41C22×C3111yescharacteristic
334#498C22×C3112yes—
335#497—21no—
336#491—21no—
337#487C22×C3112yes—
338#483—22no—
339#477—21no—
340#476—21no—
341#475—22no—
342#475C22×C3112yes—
343#474C22×C3112yes—
344#462C22×C3112yes—
345#461C22×C3112yes—
346#451—21no—
347#448—21no—
348#432C22×C3111yescharacteristic
349#431C22×C3111yescharacteristic
350#428—21no—
351#423C22×C3111yescharacteristic
352#417—21no—
353#333C22×C3112yes—
354#332C22×C3112yes—
355Q8⋊C8C22×C3112yes—
356Q8⋊C8C22×C3112yes—
357D4⋊C8C22×C3112yes—
358D4⋊C8C22×C3112yes—
359C2×C31—41no—
360C2×C31—42no—
361C2×C31—81no—
362C2×C31#300711yescharacteristic
363C2×C31#301011yescharacteristic
364C2×C31#302211yescharacteristic
365C2×#41—21no—
366C2×#41D4×C3111yescharacteristic
367C22×Q8—21no—
368C2×Q16—22no—
369C2×SD16—22no—
370C2×M16—21no—
371C4⋊Q8—21no—
372C4⋊Q8—21no—
373C4⋊Q8—21no—
374C4⋊Q8—21no—
375C4⋊Q8C23×C3112yes—
376C4⋊Q8C23×C3111yescharacteristic
377C4⋊Q8C23×C3111yescharacteristic
378#126—22no—
379#126—22no—
380#126—22no—
381#126—21no—
382#125—22no—
383#125—22no—
384#125—22no—
385#125—21no—
386#125—21no—
387#125—22no—
388#125—21no—
389#125—21no—
390#125D4×C3111yescharacteristic
391#124D4×C3111yescharacteristic
392#123—22no—
393#123—22no—
394#123—22no—
395#123—21no—
396#123—22no—
397C22⋊Q8—21no—
398C22⋊Q8—22no—
399C22⋊Q8—22no—
400C22⋊Q8—21no—
401C22⋊Q8—21no—
402C22⋊Q8—21no—
403C22⋊Q8D4×C3111yescharacteristic
404C22⋊Q8D4×C3112yes—
405#121—22no—
406#121—21no—
407#121D4×C3111yescharacteristic
408C4×Q8—21no—
409C4×Q8—21no—
410C4×Q8—21no—
411C4×Q8D4×C3111yescharacteristic
412C4×Q8D4×C3112yes—
413C4×Q8C23×C3111yescharacteristic
414C4×Q8C23×C3111yescharacteristic
415C4×D4—22no—
416C4×D4—22no—
417C4×D4—21no—
418C4×D4—21no—
419C4×D4—21no—
420C4×D4—21no—
421C4×D4—21no—
422C4×D4D4×C3111yescharacteristic
423C4×D4C23×C3111yescharacteristic
424C4×D4C23×C3111yescharacteristic
425#117—22no—
426#117—22no—
427#117—21no—
428#117—21no—
429#117—21no—
430#117D4×C3111yescharacteristic
431C2×C4⋊C4—22no—
432C2×C4⋊C4—22no—
433C2×C4⋊C4—21no—
434C2×C4⋊C4—21no—
435C2×C4⋊C4—21no—
436C2×C4⋊C4—21no—
437C2×C4⋊C4—21no—
438C2×C42—21no—
439C2×C42C23×C3111yescharacteristic
440#107—22no—
441#106—22no—
442#106—24no—
443C4⋊C8—21no—
444C4⋊C8—21no—
445C4⋊C8C23×C3112yes—
446C4⋊C8C23×C3112yes—
447#103—22no—
448#103—24no—
449#103—22no—
450#103—22no—
451#102—22no—
452C22⋊C8—21no—
453C22⋊C8—21no—
454#97—21no—
455C4×C8C23×C3112yes—
456C31#1884511yescharacteristic, direct product factor, Sylow
457#41—22no—
458#41—41no—
459#41—42no—
460#41—42no—
461C2×Q8—21no—
462C2×Q8—22no—
463C2×Q8—41no—
464C2×Q8—82no—
465C2×Q8C2×D4×C3111yescharacteristic
466C2×Q8C2×D4×C3111yescharacteristic
467C2×Q8C2×D4×C3112yes—
468C2×D4—22no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4C2×D4×C3111yescharacteristic
472C2×D4C2×D4×C3111yescharacteristic
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—41no—
481C22×C4C2×D4×C3111yescharacteristic
482C22×C4C2×D4×C3111yescharacteristic
483C22×C4C2×D4×C3111yescharacteristic
484Q16—82no—
485SD16—82no—
486M16—42no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—41no—
512C4⋊C4—41no—
513C4⋊C4C2×D4×C3111yescharacteristic
514C4⋊C4C2×D4×C3111yescharacteristic
515C4⋊C4C2×D4×C3111yescharacteristic
516C4⋊C4C2×D4×C3112yes—
517C4⋊C4C2×D4×C3111yescharacteristic
518C22⋊C4—22no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—22no—
522C22⋊C4—21no—
523C22⋊C4—21no—
524C22⋊C4—21no—
525C22⋊C4—21no—
526C22⋊C4—21no—
527C22⋊C4—21no—
528C22⋊C4—21no—
529C22⋊C4—21no—
530C42—22no—
531C42—21no—
532C42—21no—
533C42—22no—
534C42—21no—
535C42—21no—
536C42C2×D4×C3111yescharacteristic
537C42C2×D4×C3111yescharacteristic
538C42C2×D4×C3111yescharacteristic
539C42C24×C3111yescharacteristic
540C23—21no—
541C23—21no—
542C23C31×#12011yescharacteristic
543Q8—21no—
544Q8—21no—
545Q8—22no—
546Q8—42no—
547Q8—41no—
548Q8—41no—
549Q8—42no—
550Q8—42no—
551Q8—82no—
552D4—21no—
553D4—22no—
554D4—42no—
555D4—44no—
556D4—81no—
557D4C31×#13712yes—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—81no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4C31×#12011yescharacteristic
581C2×C4C31×#12011yescharacteristic
582C2×C4C31×#12011yescharacteristic
583C2×C4C22×D4×C3111yescharacteristic
584C2×C4C22×D4×C3111yescharacteristic
585C2×C4C22×D4×C3111yescharacteristic
586C8—42no—
587C8—42no—
588C8—42no—
589C22—21no—
590C22—22no—
591C22—21no—
592C22—21no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22C2×C31×#12011yescharacteristic
599C4—21no—
600C4—21no—
601C4—21no—
602C4—21no—
603C4—41no—
604C4—41no—
605C4—42no—
606C4—41no—
607C4—41no—
608C4—41no—
609C4—41no—
610C4—42no—
611C4—82no—
612C4C2×C31×#13711yescharacteristic
613C4C31×#57511yescharacteristic
614C4C31×#57712yes—
615C2—41no—
616C2—42no—
617C2—81no—
618C2C31×#300711yescharacteristic
619C2C31×#301011yescharacteristic
620C2C31×#302211yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC42C24×C31
prime radicalC31×#3515C2
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×#3229C2semidirect
3. C31×#3218C2semidirect
4. C31×#3079C2semidirect
5. C31×#1654C2semidirect
6. C31×#1653C2semidirect
7. C31×#1644C2semidirect
8. C31×#1643C2semidirect
9. C31×#1559C2semidirect
10. C31×#1495C2semidirect
11. C31×#1488C2semidirect
229. #18845C31direct (N × H)
280. #3229C2×C31semidirect
281. #3218C2×C31semidirect
282. #3079C2×C31semidirect
283. #1654C2×C31semidirect
284. #1653C2×C31semidirect
285. #1644C2×C31semidirect
286. #1643C2×C31semidirect
287. #1559C2×C31semidirect
288. #1495C2×C31semidirect
289. #1488C2×C31semidirect
456. C31#18845direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35152
C31×#32292
C31×#32182
C31×#30792
C31×#16542
C31×#16532
C31×#16442
C31×#16432
C31×#15592
C31×#14952
C31×#14882
#1884531