Finite Group Databasebeta

← C31×#18689 #21475448

Subgroups — C31×#18689 #21475448

Order 7936.

1006 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3543C211yesmaximal, characteristic
2C31×#3239C211yesmaximal, characteristic, complemented
3C31×#3218C211yesmaximal, characteristic, complemented
4C2×C31×#486C211yesmaximal, characteristic, complemented
5C31×#1642C212yesmaximal, complemented
6C31×#1641C212yesmaximal, complemented
7C31×#1640C212yesmaximal, complemented
8C31×#1639C212yesmaximal, complemented
9C31×#1555C211yesmaximal, characteristic, complemented
10C31×#1494C211yesmaximal, characteristic, complemented
11C31×#1494C211yesmaximal, characteristic, complemented
12C31×#560—22no—
13C31×#559C2211yescharacteristic
14C31×#550C2211yescharacteristic
15C31×#550C2211yescharacteristic
16C31×#546—21no—
17C31×#546—21no—
18D42×C31—21no—
19D42×C31—21no—
20D42×C31C2211yescharacteristic
21C31×#540—21no—
22C31×#540—21no—
23C31×#535—21no—
24C31×#535—21no—
25C31×#534—22no—
26C2×C31×C4⋊Q8C2211yescharacteristic, complemented
27C2×C31×#127C2211yescharacteristic
28C31×#529C2211yescharacteristic
29C31×#529C2211yescharacteristic
30C31×#501—21no—
31C31×#497—21no—
32C31×#491—21no—
33C31×#489—21no—
34C31×#486—22no—
35C31×#486C2212yes—
36C31×#486C2212yescomplemented
37C31×#484—21no—
38C31×#481—21no—
39C31×#475C2212yescomplemented
40C31×#474C2212yescomplemented
41C31×#462C2212yescomplemented
42C31×#461C2212yescomplemented
43C31×#451—21no—
44C31×#448—21no—
45C31×#432C2211yescharacteristic, complemented
46C31×#432C2211yescharacteristic, complemented
47C2×C31×#102—22no—
48C2×C4×C8×C31C2211yescharacteristic, complemented
49C31×#331C2212yes—
50C31×#331C2212yescomplemented
51C31×Q8⋊C8C2212yescomplemented
52C31×Q8⋊C8C2212yescomplemented
53C31×D4⋊C8C2212yescomplemented
54C31×D4⋊C8C2212yescomplemented
55C2×#41×C31—21no—
56C2×#41×C31—21no—
57C22×Q8×C31—21no—
58C22×D4×C31—22no—
59C22×D4×C31—21no—
60C22×D4×C31—21no—
61C22×D4×C31—21no—
62C22×D4×C31—21no—
63C22×D4×C31D412yes—
64C2×Q16×C31—22no—
65C2×SD16×C31—22no—
66C22×C8×C31—21no—
67C31×C4⋊Q8—21no—
68C31×C4⋊Q8—21no—
69C31×C4⋊Q8—21no—
70C31×C4⋊Q8C2312yes—
71C31×#127—21no—
72C31×#127—21no—
73C31×#127—21no—
74C31×#127—21no—
75C31×#127D412yes—
76C31×#127C2311yescharacteristic
77C31×#127C2311yescharacteristic
78C31×#126—21no—
79C31×#126—21no—
80C31×#125D411yescharacteristic
81C31×#125D411yescharacteristic
82C31×#124—21no—
83C31×#124—21no—
84C31×#124D411yescharacteristic
85C31×#124D411yescharacteristic
86C31×#123—22no—
87C31×#123—22no—
88C31×#123—22no—
89C31×#123—22no—
90C31×#123—21no—
91C31×#123—21no—
92C31×C22⋊Q8D411yescharacteristic
93C31×C22⋊Q8D411yescharacteristic
94C31×#121—22no—
95C31×#121—22no—
96C31×#121—22no—
97C31×#121—22no—
98C31×#121—22no—
99C31×#121—22no—
100C31×#121—22no—
101C31×#121—22no—
102C31×#121—21no—
103C31×#121—21no—
104C31×#121—21no—
105C31×#121—21no—
106C31×#121D411yescharacteristic
107C31×#121D411yescharacteristic
108C31×#120—22no—
109C31×#120—22no—
110C31×#120—22no—
111C31×#120—22no—
112C31×#120—22no—
113C31×#120—22no—
114C4×Q8×C31C2311yescharacteristic
115C4×Q8×C31C2311yescharacteristic
116C4×D4×C31—21no—
117C4×D4×C31—21no—
118C4×D4×C31—21no—
119C4×D4×C31—21no—
120C4×D4×C31—21no—
121C4×D4×C31—21no—
122C4×D4×C31C2311yescharacteristic
123C4×D4×C31C2311yescharacteristic
124C31×#117—21no—
125C31×#117—21no—
126C2×C4⋊C4×C31—21no—
127C2×C4⋊C4×C31—21no—
128C2×C22⋊C4×C31—22no—
129C2×C42×C31C2311yescharacteristic
130C31×#107—22no—
131C31×#106—22no—
132C31×C4⋊C8C2312yes—
133C31×C4⋊C8C2312yes—
134C31×#103—22no—
135C31×#103—22no—
136C31×#103—22no—
137C31×#102—24no—
138C31×#102—24no—
139C31×#102—22no—
140C31×C22⋊C8—21no—
141C31×C22⋊C8—21no—
142C31×#97—21no—
143C31×#97—21no—
144C4×C8×C31—21no—
145C4×C8×C31C2312yes—
146C24×C31—22no—
147C24×C31—22no—
148#41×C31—82no—
149#41×C31—82no—
150C2×Q8×C31—22no—
151C2×Q8×C31—41no—
152C2×Q8×C31—82no—
153C2×Q8×C31C2×D411yescharacteristic
154C2×Q8×C31C2×D411yescharacteristic
155C2×D4×C31—21no—
156C2×D4×C31—21no—
157C2×D4×C31—22no—
158C2×D4×C31—22no—
159C2×D4×C31—22no—
160C2×D4×C31—22no—
161C2×D4×C31—22no—
162C2×D4×C31—22no—
163C2×D4×C31—22no—
164C2×D4×C31—22no—
165C2×D4×C31—22no—
166C2×D4×C31—21no—
167C2×D4×C31—21no—
168C2×D4×C31—21no—
169C2×D4×C31—21no—
170C2×D4×C31—44no—
171C2×D4×C31—82no—
172C2×D4×C31—82no—
173C2×D4×C31—82no—
174C2×D4×C31—82no—
175C2×D4×C31—84no—
176C2×D4×C31C2×D411yescharacteristic
177C2×D4×C31C2×D411yescharacteristic
178C2×D4×C31C2×D412yes—
179C2×D4×C31C2×D412yes—
180C22×C4×C31—22no—
181C22×C4×C31—22no—
182C22×C4×C31—22no—
183C22×C4×C31—21no—
184C22×C4×C31—21no—
185C22×C4×C31—41no—
186C22×C4×C31C2×D411yescharacteristic
187C22×C4×C31C2×D411yescharacteristic
188C22×C4×C31C2×D411yescharacteristic
189Q16×C31—82no—
190SD16×C31—82no—
191C2×C8×C31—22no—
192C2×C8×C31—22no—
193C2×C8×C31—22no—
194C2×C8×C31—42no—
195C4⋊C4×C31—21no—
196C4⋊C4×C31—21no—
197C4⋊C4×C31—21no—
198C4⋊C4×C31—21no—
199C4⋊C4×C31—21no—
200C4⋊C4×C31—21no—
201C4⋊C4×C31—21no—
202C4⋊C4×C31—21no—
203C4⋊C4×C31—22no—
204C4⋊C4×C31—22no—
205C4⋊C4×C31—41no—
206C4⋊C4×C31—41no—
207C4⋊C4×C31C2×D411yescharacteristic
208C4⋊C4×C31C2×D411yescharacteristic
209C4⋊C4×C31C2×D411yescharacteristic
210C4⋊C4×C31C2×D411yescharacteristic
211C22⋊C4×C31—22no—
212C22⋊C4×C31—22no—
213C22⋊C4×C31—22no—
214C22⋊C4×C31—22no—
215C22⋊C4×C31—22no—
216C22⋊C4×C31—22no—
217C22⋊C4×C31—21no—
218C22⋊C4×C31—21no—
219C22⋊C4×C31—21no—
220C22⋊C4×C31—21no—
221C22⋊C4×C31—21no—
222C22⋊C4×C31—21no—
223C22⋊C4×C31—21no—
224C22⋊C4×C31—21no—
225C22⋊C4×C31—21no—
226C22⋊C4×C31—21no—
227C42×C31—21no—
228C42×C31—21no—
229C42×C31C2×D411yescharacteristic
230C42×C31C2×D411yescharacteristic
231C42×C31C2×D411yescharacteristic
232C42×C31C2411yescharacteristic
233#18689C3111yesmaximal, characteristic, direct product factor, Sylow
234C23×C31—21no—
235C23×C31—21no—
236C23×C31—22no—
237C23×C31—22no—
238C23×C31—44no—
239C23×C31—82no—
240C23×C31—82no—
241C23×C31—82no—
242C23×C31—82no—
243C23×C31—82no—
244C23×C31#12011yescharacteristic
245Q8×C31—22no—
246Q8×C31—22no—
247Q8×C31—42no—
248Q8×C31—42no—
249Q8×C31—82no—
250D4×C31—22no—
251D4×C31—22no—
252D4×C31—42no—
253D4×C31—42no—
254D4×C31—42no—
255D4×C31—42no—
256D4×C31—42no—
257D4×C31—42no—
258D4×C31—42no—
259D4×C31—42no—
260D4×C31—82no—
261D4×C31—82no—
262D4×C31—82no—
263D4×C31—82no—
264D4×C31—82no—
265D4×C31—82no—
266D4×C31—82no—
267D4×C31—82no—
268D4×C31—82no—
269C2×C4×C31—22no—
270C2×C4×C31—22no—
271C2×C4×C31—22no—
272C2×C4×C31—21no—
273C2×C4×C31—21no—
274C2×C4×C31—21no—
275C2×C4×C31—21no—
276C2×C4×C31—21no—
277C2×C4×C31—21no—
278C2×C4×C31—42no—
279C2×C4×C31—42no—
280C2×C4×C31—42no—
281C2×C4×C31—82no—
282C2×C4×C31—82no—
283C2×C4×C31—82no—
284C2×C4×C31—82no—
285C2×C4×C31#12011yescharacteristic
286C2×C4×C31#12011yescharacteristic
287C2×C4×C31#12011yescharacteristic
288C2×C4×C31C22×D411yescharacteristic
289C2×C4×C31C22×D411yescharacteristic
290C2×C4×C31C22×D411yescharacteristic
291C8×C31—42no—
292C8×C31—42no—
293C8×C31—42no—
294#3543C2×C3111yescharacteristic
295#3239C2×C3111yescharacteristic, complemented
296#3218C2×C3111yescharacteristic, complemented
297C2×#486C2×C3111yescharacteristic, complemented
298#1642C2×C3112yescomplemented
299#1641C2×C3112yescomplemented
300#1640C2×C3112yescomplemented
301#1639C2×C3112yescomplemented
302#1555C2×C3111yescharacteristic, complemented
303#1494C2×C3111yescharacteristic, complemented
304#1494C2×C3111yescharacteristic, complemented
305C22×C31—21no—
306C22×C31—21no—
307C22×C31—41no—
308C22×C31—42no—
309C22×C31—41no—
310C22×C31—42no—
311C22×C31—41no—
312C22×C31—42no—
313C22×C31—41no—
314C22×C31—42no—
315C22×C31—42no—
316C22×C31—41no—
317C22×C31—41no—
318C22×C31—42no—
319C22×C31—164no—
320C22×C31—164no—
321C22×C31#45512yes—
322C22×C31C2×#12011yescharacteristic
323C4×C31—21no—
324C4×C31—21no—
325C4×C31—21no—
326C4×C31—21no—
327C4×C31—41no—
328C4×C31—41no—
329C4×C31—41no—
330C4×C31—41no—
331C4×C31—41no—
332C4×C31—41no—
333C4×C31—82no—
334C4×C31#57612yes—
335C4×C31#57712yes—
336#560—22no—
337#559C22×C3111yescharacteristic
338#550C22×C3111yescharacteristic
339#550C22×C3111yescharacteristic
340#546—21no—
341#546—21no—
342D42—21no—
343D42—21no—
344D42C22×C3111yescharacteristic
345#540—21no—
346#540—21no—
347#535—21no—
348#535—21no—
349#534—22no—
350C2×C4⋊Q8C22×C3111yescharacteristic, complemented
351C2×#127C22×C3111yescharacteristic
352#529C22×C3111yescharacteristic
353#529C22×C3111yescharacteristic
354#501—21no—
355#497—21no—
356#491—21no—
357#489—21no—
358#486—22no—
359#486C22×C3112yes—
360#486C22×C3112yescomplemented
361#484—21no—
362#481—21no—
363#475C22×C3112yescomplemented
364#474C22×C3112yescomplemented
365#462C22×C3112yescomplemented
366#461C22×C3112yescomplemented
367#451—21no—
368#448—21no—
369#432C22×C3111yescharacteristic, complemented
370#432C22×C3111yescharacteristic, complemented
371C2×#102—22no—
372C2×C4×C8C22×C3111yescharacteristic, complemented
373#331C22×C3112yes—
374#331C22×C3112yescomplemented
375Q8⋊C8C22×C3112yescomplemented
376Q8⋊C8C22×C3112yescomplemented
377D4⋊C8C22×C3112yescomplemented
378D4⋊C8C22×C3112yescomplemented
379C2×C31—22no—
380C2×C31—82no—
381C2×C31—81no—
382C2×C31—81no—
383C2×C31—82no—
384C2×C31#300811yescharacteristic
385C2×C31#300911yescharacteristic
386C2×C31C2×#45511yescharacteristic
387C2×#41—21no—
388C2×#41—21no—
389C22×Q8—21no—
390C22×D4—22no—
391C22×D4—21no—
392C22×D4—21no—
393C22×D4—21no—
394C22×D4—21no—
395C22×D4D4×C3112yes—
396C2×Q16—22no—
397C2×SD16—22no—
398C22×C8—21no—
399C4⋊Q8—21no—
400C4⋊Q8—21no—
401C4⋊Q8—21no—
402C4⋊Q8C23×C3112yes—
403#127—21no—
404#127—21no—
405#127—21no—
406#127—21no—
407#127D4×C3112yes—
408#127C23×C3111yescharacteristic
409#127C23×C3111yescharacteristic
410#126—21no—
411#126—21no—
412#125D4×C3111yescharacteristic
413#125D4×C3111yescharacteristic
414#124—21no—
415#124—21no—
416#124D4×C3111yescharacteristic
417#124D4×C3111yescharacteristic
418#123—22no—
419#123—22no—
420#123—22no—
421#123—22no—
422#123—21no—
423#123—21no—
424C22⋊Q8D4×C3111yescharacteristic
425C22⋊Q8D4×C3111yescharacteristic
426#121—22no—
427#121—22no—
428#121—22no—
429#121—22no—
430#121—22no—
431#121—22no—
432#121—22no—
433#121—22no—
434#121—21no—
435#121—21no—
436#121—21no—
437#121—21no—
438#121D4×C3111yescharacteristic
439#121D4×C3111yescharacteristic
440#120—22no—
441#120—22no—
442#120—22no—
443#120—22no—
444#120—22no—
445#120—22no—
446C4×Q8C23×C3111yescharacteristic
447C4×Q8C23×C3111yescharacteristic
448C4×D4—21no—
449C4×D4—21no—
450C4×D4—21no—
451C4×D4—21no—
452C4×D4—21no—
453C4×D4—21no—
454C4×D4C23×C3111yescharacteristic
455C4×D4C23×C3111yescharacteristic
456#117—21no—
457#117—21no—
458C2×C4⋊C4—21no—
459C2×C4⋊C4—21no—
460C2×C22⋊C4—22no—
461C2×C42C23×C3111yescharacteristic
462#107—22no—
463#106—22no—
464C4⋊C8C23×C3112yes—
465C4⋊C8C23×C3112yes—
466#103—22no—
467#103—22no—
468#103—22no—
469#102—24no—
470#102—24no—
471#102—22no—
472C22⋊C8—21no—
473C22⋊C8—21no—
474#97—21no—
475#97—21no—
476C4×C8—21no—
477C4×C8C23×C3112yes—
478C31#1868911yescharacteristic, direct product factor, Sylow
479C24—22no—
480C24—22no—
481#41—82no—
482#41—82no—
483C2×Q8—22no—
484C2×Q8—41no—
485C2×Q8—82no—
486C2×Q8C2×D4×C3111yescharacteristic
487C2×Q8C2×D4×C3111yescharacteristic
488C2×D4—21no—
489C2×D4—21no—
490C2×D4—22no—
491C2×D4—22no—
492C2×D4—22no—
493C2×D4—22no—
494C2×D4—22no—
495C2×D4—22no—
496C2×D4—22no—
497C2×D4—22no—
498C2×D4—22no—
499C2×D4—21no—
500C2×D4—21no—
501C2×D4—21no—
502C2×D4—21no—
503C2×D4—44no—
504C2×D4—82no—
505C2×D4—82no—
506C2×D4—82no—
507C2×D4—82no—
508C2×D4—84no—
509C2×D4C2×D4×C3111yescharacteristic
510C2×D4C2×D4×C3111yescharacteristic
511C2×D4C2×D4×C3112yes—
512C2×D4C2×D4×C3112yes—
513C22×C4—22no—
514C22×C4—22no—
515C22×C4—22no—
516C22×C4—21no—
517C22×C4—21no—
518C22×C4—41no—
519C22×C4C2×D4×C3111yescharacteristic
520C22×C4C2×D4×C3111yescharacteristic
521C22×C4C2×D4×C3111yescharacteristic
522Q16—82no—
523SD16—82no—
524C2×C8—22no—
525C2×C8—22no—
526C2×C8—22no—
527C2×C8—42no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—41no—
539C4⋊C4—41no—
540C4⋊C4C2×D4×C3111yescharacteristic
541C4⋊C4C2×D4×C3111yescharacteristic
542C4⋊C4C2×D4×C3111yescharacteristic
543C4⋊C4C2×D4×C3111yescharacteristic
544C22⋊C4—22no—
545C22⋊C4—22no—
546C22⋊C4—22no—
547C22⋊C4—22no—
548C22⋊C4—22no—
549C22⋊C4—22no—
550C22⋊C4—21no—
551C22⋊C4—21no—
552C22⋊C4—21no—
553C22⋊C4—21no—
554C22⋊C4—21no—
555C22⋊C4—21no—
556C22⋊C4—21no—
557C22⋊C4—21no—
558C22⋊C4—21no—
559C22⋊C4—21no—
560C42—21no—
561C42—21no—
562C42C2×D4×C3111yescharacteristic
563C42C2×D4×C3111yescharacteristic
564C42C2×D4×C3111yescharacteristic
565C42C24×C3111yescharacteristic
566C23—21no—
567C23—21no—
568C23—22no—
569C23—22no—
570C23—44no—
571C23—82no—
572C23—82no—
573C23—82no—
574C23—82no—
575C23—82no—
576C23C31×#12011yescharacteristic
577Q8—22no—
578Q8—22no—
579Q8—42no—
580Q8—42no—
581Q8—82no—
582D4—22no—
583D4—22no—
584D4—42no—
585D4—42no—
586D4—42no—
587D4—42no—
588D4—42no—
589D4—42no—
590D4—42no—
591D4—42no—
592D4—82no—
593D4—82no—
594D4—82no—
595D4—82no—
596D4—82no—
597D4—82no—
598D4—82no—
599D4—82no—
600D4—82no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—82no—
614C2×C4—82no—
615C2×C4—82no—
616C2×C4—82no—
617C2×C4C31×#12011yescharacteristic
618C2×C4C31×#12011yescharacteristic
619C2×C4C31×#12011yescharacteristic
620C2×C4C22×D4×C3111yescharacteristic
621C2×C4C22×D4×C3111yescharacteristic
622C2×C4C22×D4×C3111yescharacteristic
623C8—42no—
624C8—42no—
625C8—42no—
626C22—21no—
627C22—21no—
628C22—41no—
629C22—42no—
630C22—41no—
631C22—42no—
632C22—41no—
633C22—42no—
634C22—41no—
635C22—42no—
636C22—42no—
637C22—41no—
638C22—41no—
639C22—42no—
640C22—164no—
641C22—164no—
642C22C31×#45512yes—
643C22C2×C31×#12011yescharacteristic
644C4—21no—
645C4—21no—
646C4—21no—
647C4—21no—
648C4—41no—
649C4—41no—
650C4—41no—
651C4—41no—
652C4—41no—
653C4—41no—
654C4—82no—
655C4C31×#57612yes—
656C4C31×#57712yes—
657C2—22no—
658C2—81no—
659C2—81no—
660C2—82no—
661C2—82no—
662C2C31×#300811yescharacteristic
663C2C31×#300911yescharacteristic
664C2C2×C31×#45511yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC42C24×C31
prime radicalC31×#3543C2
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×#3239C2semidirect
3. C31×#3218C2semidirect
4. C2×C31×#486C2semidirect
5. C31×#1642C2semidirect
6. C31×#1641C2semidirect
7. C31×#1640C2semidirect
8. C31×#1639C2semidirect
9. C31×#1555C2semidirect
10. C31×#1494C2semidirect
11. C31×#1494C2semidirect
26. C2×C31×C4⋊Q8C22semidirect
36. C31×#486C22semidirect
39. C31×#475C22semidirect
40. C31×#474C22semidirect
41. C31×#462C22semidirect
42. C31×#461C22semidirect
45. C31×#432C22semidirect
46. C31×#432C22semidirect
48. C2×C4×C8×C31C22semidirect
50. C31×#331C22semidirect
51. C31×Q8⋊C8C22semidirect
52. C31×Q8⋊C8C22semidirect
53. C31×D4⋊C8C22semidirect
54. C31×D4⋊C8C22semidirect
233. #18689C31direct (N × H)
295. #3239C2×C31semidirect
296. #3218C2×C31semidirect
297. C2×#486C2×C31semidirect
298. #1642C2×C31semidirect
299. #1641C2×C31semidirect
300. #1640C2×C31semidirect
301. #1639C2×C31semidirect
302. #1555C2×C31semidirect
303. #1494C2×C31semidirect
304. #1494C2×C31semidirect
350. C2×C4⋊Q8C22×C31semidirect
360. #486C22×C31semidirect
363. #475C22×C31semidirect
364. #474C22×C31semidirect
365. #462C22×C31semidirect
366. #461C22×C31semidirect
369. #432C22×C31semidirect
370. #432C22×C31semidirect
372. C2×C4×C8C22×C31semidirect
374. #331C22×C31semidirect
375. Q8⋊C8C22×C31semidirect
376. Q8⋊C8C22×C31semidirect
377. D4⋊C8C22×C31semidirect
378. D4⋊C8C22×C31semidirect
478. C31#18689direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35432
C31×#32392
C31×#32182
C2×C31×#4862
C31×#16422
C31×#16412
C31×#16402
C31×#16392
C31×#15552
C31×#14942
#1868931