Finite Group Databasebeta

← C31×#19042 #21475801

Subgroups — C31×#19042 #21475801

Order 7936.

1036 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3558C211yesmaximal, characteristic, complemented
2C31×#3213C211yesmaximal, characteristic, complemented
3C31×#3180C211yesmaximal, characteristic, complemented
4C31×#3150C211yesmaximal, characteristic, complemented
5C31×#1661C212yesmaximal, complemented
6C31×#1659C212yesmaximal, complemented
7C31×#1656C212yesmaximal, complemented
8C31×#1655C212yesmaximal, complemented
9C31×#1532C211yesmaximal, characteristic, complemented
10C31×#1530C211yesmaximal, characteristic, complemented
11C2×C31×#329C211yesmaximal, characteristic, complemented
12C2×C31×#135—22no—
13C31×#562C2212yes—
14C31×H(3,𝔽4)C2212yes—
15C31×#559—21no—
16C31×#548—21no—
17C31×#546—22no—
18C31×#546—21no—
19D42×C31—21no—
20C31×#542—21no—
21C31×#538—22no—
22C31×#538—21no—
23C31×#535—21no—
24C31×#534—21no—
25C31×#532C2211yescharacteristic
26C31×#532C2211yescharacteristic, complemented
27C31×#529C2211yescharacteristic
28C2×C31×#124C2211yescharacteristic
29C31×#501C2211yescharacteristic, complemented
30C31×#497—21no—
31C31×#497C2212yescomplemented
32C31×#496—21no—
33C31×#496C2212yescomplemented
34C31×#496C2211yescharacteristic, complemented
35C31×#496C2212yescomplemented
36C31×#490C2211yescharacteristic, complemented
37C31×#489C2211yescharacteristic, complemented
38C31×#489C2212yescomplemented
39C31×#466—21no—
40C31×#465—21no—
41C31×#463—22no—
42C31×#448—21no—
43C31×#447—21no—
44C31×#442—21no—
45C31×#440—21no—
46C31×#432C2211yescharacteristic, complemented
47C2×C31×#97C2211yescharacteristic, complemented
48C2×C31×#97C2211yescharacteristic, complemented
49C31×#330C2212yescomplemented
50C31×#329C2212yescomplemented
51C31×#329C2212yescomplemented
52C31×#329C2212yescomplemented
53C31×#329C2212yescomplemented
54C31×#329C2212yescomplemented
55C2×#41×C31—21no—
56C2×#41×C31—21no—
57C2×#41×C31D412yes—
58C22×Q8×C31D411yescharacteristic
59C22×D4×C31—22no—
60C22×D4×C31—21no—
61C22×D4×C31D411yescharacteristic
62C31×#135—48no—
63C2×Q16×C31—22no—
64C2×SD16×C31—21no—
65C2×SD16×C31—22no—
66C2×SD16×C31—22no—
67C2×SD16×C31—22no—
68C2×D8×C31—21no—
69C2×D8×C31—22no—
70C2×D8×C31—22no—
71C22×C8×C31—21no—
72C22×C8×C31—21no—
73C31×C4⋊Q8D411yescharacteristic
74C31×C4⋊Q8C2311yescharacteristic
75C31×#127—21no—
76C31×#127D411yescharacteristic
77C31×#127C2311yescharacteristic
78C31×#126—22no—
79C31×#126—21no—
80C31×#125C2311yescharacteristic
81C31×#124—21no—
82C31×#124—22no—
83C31×#124—22no—
84C31×#124—22no—
85C31×#124—21no—
86C31×#124—21no—
87C31×#124—21no—
88C31×#124D412yes—
89C31×#124C2312yes—
90C31×#124C2311yescharacteristic
91C31×#124C2312yes—
92C31×#123—22no—
93C31×#123—22no—
94C31×#123—21no—
95C31×#123—22no—
96C31×#123—22no—
97C31×C22⋊Q8—21no—
98C31×C22⋊Q8—22no—
99C31×C22⋊Q8—22no—
100C31×C22⋊Q8D411yescharacteristic
101C31×#121—22no—
102C31×#121—22no—
103C31×#121—22no—
104C31×#121—21no—
105C31×#121—22no—
106C31×#121—22no—
107C31×#121—21no—
108C31×#121—21no—
109C31×#121—21no—
110C31×#121—21no—
111C31×#121D411yescharacteristic
112C31×#120—22no—
113C31×#120—22no—
114C31×#120—22no—
115C31×#120—22no—
116C31×#120—22no—
117C4×Q8×C31—21no—
118C4×Q8×C31D411yescharacteristic
119C4×D4×C31—22no—
120C4×D4×C31—22no—
121C4×D4×C31—21no—
122C4×D4×C31—21no—
123C4×D4×C31—21no—
124C4×D4×C31—21no—
125C4×D4×C31—21no—
126C4×D4×C31—21no—
127C4×D4×C31D411yescharacteristic
128C31×#117—21no—
129C2×C22⋊C4×C31—22no—
130C2×C22⋊C4×C31—21no—
131C2×C22⋊C4×C31—21no—
132C2×C42×C31C2311yescharacteristic
133C31×#107—21no—
134C31×#106—21no—
135C31×C4⋊C8—21no—
136C31×#103—21no—
137C31×#103—22no—
138C31×#102—21no—
139C31×#102—22no—
140C31×#102—21no—
141C31×#102—21no—
142C31×C22⋊C8—21no—
143C31×#97—21no—
144C31×#97—21no—
145C31×#97C2311yescharacteristic
146C31×#97C2312yes—
147C31×#97C2311yescharacteristic
148C31×#97C2312yes—
149C24×C31—22no—
150C24×C31—21no—
151#41×C31—44no—
152#41×C31—44no—
153#41×C31—82no—
154C2×Q8×C31—21no—
155C2×Q8×C31—21no—
156C2×Q8×C31—81no—
157C2×Q8×C31C2×D411yescharacteristic
158C2×Q8×C31C2×D412yes—
159C2×Q8×C31C2×D412yes—
160C2×D4×C31—22no—
161C2×D4×C31—21no—
162C2×D4×C31—22no—
163C2×D4×C31—22no—
164C2×D4×C31—22no—
165C2×D4×C31—22no—
166C2×D4×C31—22no—
167C2×D4×C31—21no—
168C2×D4×C31—21no—
169C2×D4×C31—21no—
170C2×D4×C31—21no—
171C2×D4×C31—21no—
172C2×D4×C31—21no—
173C2×D4×C31—41no—
174C2×D4×C31—81no—
175C2×D4×C31—82no—
176C2×D4×C31—84no—
177C2×D4×C31C2×D411yescharacteristic
178C2×D4×C31C2×D412yes—
179C2×D4×C31C2×D412yes—
180C22×C4×C31—22no—
181C22×C4×C31—22no—
182C22×C4×C31—21no—
183C22×C4×C31—21no—
184C22×C4×C31—22no—
185C22×C4×C31—41no—
186C22×C4×C31C2×D411yescharacteristic
187C22×C4×C31C2×D411yescharacteristic
188C22×C4×C31C2×D411yescharacteristic
189Q16×C31—44no—
190SD16×C31—42no—
191SD16×C31—44no—
192SD16×C31—44no—
193SD16×C31—82no—
194D8×C31—42no—
195D8×C31—44no—
196D8×C31—82no—
197C2×C8×C31—24no—
198C2×C8×C31—21no—
199C2×C8×C31—22no—
200C2×C8×C31—21no—
201C2×C8×C31—22no—
202C2×C8×C31—42no—
203C4⋊C4×C31—22no—
204C4⋊C4×C31—22no—
205C4⋊C4×C31—21no—
206C4⋊C4×C31—21no—
207C4⋊C4×C31—21no—
208C4⋊C4×C31—21no—
209C4⋊C4×C31—21no—
210C4⋊C4×C31—21no—
211C4⋊C4×C31—21no—
212C4⋊C4×C31—21no—
213C4⋊C4×C31C2×D411yescharacteristic
214C4⋊C4×C31C2×D411yescharacteristic
215C22⋊C4×C31—22no—
216C22⋊C4×C31—22no—
217C22⋊C4×C31—22no—
218C22⋊C4×C31—22no—
219C22⋊C4×C31—22no—
220C22⋊C4×C31—21no—
221C22⋊C4×C31—22no—
222C22⋊C4×C31—21no—
223C22⋊C4×C31—22no—
224C22⋊C4×C31—21no—
225C22⋊C4×C31—22no—
226C22⋊C4×C31—22no—
227C22⋊C4×C31—21no—
228C22⋊C4×C31—21no—
229C22⋊C4×C31—22no—
230C22⋊C4×C31—22no—
231C22⋊C4×C31—41no—
232C22⋊C4×C31—41no—
233C42×C31—22no—
234C42×C31—21no—
235C42×C31C2×D411yescharacteristic
236C42×C31C2×D411yescharacteristic
237C42×C31C2×D411yescharacteristic
238C42×C31C2411yescharacteristic
239#19042C3111yesmaximal, characteristic, direct product factor, Sylow
240C23×C31—22no—
241C23×C31—21no—
242C23×C31—22no—
243C23×C31—42no—
244C23×C31—41no—
245C23×C31—81no—
246C23×C31—81no—
247C23×C31—82no—
248C23×C31—82no—
249C23×C31—82no—
250C23×C31#12011yescharacteristic
251Q8×C31—42no—
252Q8×C31—41no—
253Q8×C31—42no—
254Q8×C31—42no—
255Q8×C31—42no—
256D4×C31—42no—
257D4×C31—41no—
258D4×C31—42no—
259D4×C31—42no—
260D4×C31—42no—
261D4×C31—42no—
262D4×C31—41no—
263D4×C31—42no—
264D4×C31—41no—
265D4×C31—42no—
266D4×C31—44no—
267D4×C31—81no—
268D4×C31—82no—
269D4×C31—82no—
270D4×C31—82no—
271D4×C31—82no—
272D4×C31—82no—
273D4×C31—82no—
274C2×C4×C31—22no—
275C2×C4×C31—21no—
276C2×C4×C31—22no—
277C2×C4×C31—22no—
278C2×C4×C31—21no—
279C2×C4×C31—21no—
280C2×C4×C31—41no—
281C2×C4×C31—41no—
282C2×C4×C31—42no—
283C2×C4×C31—41no—
284C2×C4×C31—82no—
285C2×C4×C31—82no—
286C2×C4×C31—81no—
287C2×C4×C31—82no—
288C2×C4×C31—82no—
289C2×C4×C31#12011yescharacteristic
290C2×C4×C31#12011yescharacteristic
291C2×C4×C31#12011yescharacteristic
292C2×C4×C31C22×D411yescharacteristic
293C2×C4×C31C22×D411yescharacteristic
294C2×C4×C31C22×D411yescharacteristic
295C8×C31—24no—
296C8×C31—22no—
297C8×C31—41no—
298C8×C31—42no—
299#3558C2×C3111yescharacteristic, complemented
300#3213C2×C3111yescharacteristic, complemented
301#3180C2×C3111yescharacteristic, complemented
302#3150C2×C3111yescharacteristic, complemented
303#1661C2×C3112yescomplemented
304#1659C2×C3112yescomplemented
305#1656C2×C3112yescomplemented
306#1655C2×C3112yescomplemented
307#1532C2×C3111yescharacteristic, complemented
308#1530C2×C3111yescharacteristic, complemented
309C2×#329C2×C3111yescharacteristic, complemented
310C22×C31—21no—
311C22×C31—21no—
312C22×C31—42no—
313C22×C31—42no—
314C22×C31—41no—
315C22×C31—41no—
316C22×C31—42no—
317C22×C31—42no—
318C22×C31—41no—
319C22×C31—42no—
320C22×C31—42no—
321C22×C31—81no—
322C22×C31—81no—
323C22×C31—81no—
324C22×C31—162no—
325C22×C31—164no—
326C22×C31#45312yes—
327C22×C31C2×#12011yescharacteristic
328C4×C31—21no—
329C4×C31—21no—
330C4×C31—21no—
331C4×C31—21no—
332C4×C31—21no—
333C4×C31—21no—
334C4×C31—41no—
335C4×C31—42no—
336C4×C31—41no—
337C4×C31—41no—
338C4×C31—42no—
339C4×C31—81no—
340C2×#135—22no—
341#562C22×C3112yes—
342H(3,𝔽4)C22×C3112yes—
343#559—21no—
344#548—21no—
345#546—22no—
346#546—21no—
347D42—21no—
348#542—21no—
349#538—22no—
350#538—21no—
351#535—21no—
352#534—21no—
353#532C22×C3111yescharacteristic
354#532C22×C3111yescharacteristic, complemented
355#529C22×C3111yescharacteristic
356C2×#124C22×C3111yescharacteristic
357#501C22×C3111yescharacteristic, complemented
358#497—21no—
359#497C22×C3112yescomplemented
360#496—21no—
361#496C22×C3112yescomplemented
362#496C22×C3111yescharacteristic, complemented
363#496C22×C3112yescomplemented
364#490C22×C3111yescharacteristic, complemented
365#489C22×C3111yescharacteristic, complemented
366#489C22×C3112yescomplemented
367#466—21no—
368#465—21no—
369#463—22no—
370#448—21no—
371#447—21no—
372#442—21no—
373#440—21no—
374#432C22×C3111yescharacteristic, complemented
375C2×#97C22×C3111yescharacteristic, complemented
376C2×#97C22×C3111yescharacteristic, complemented
377#330C22×C3112yescomplemented
378#329C22×C3112yescomplemented
379#329C22×C3112yescomplemented
380#329C22×C3112yescomplemented
381#329C22×C3112yescomplemented
382#329C22×C3112yescomplemented
383C2×C31—22no—
384C2×C31—82no—
385C2×C31—82no—
386C2×C31—81no—
387C2×C31—161no—
388C2×C31C2×#45311yescharacteristic
389C2×C31#301811yescharacteristic
390C2×C31#301911yescharacteristic
391C2×#41—21no—
392C2×#41—21no—
393C2×#41D4×C3112yes—
394C22×Q8D4×C3111yescharacteristic
395C22×D4—22no—
396C22×D4—21no—
397C22×D4D4×C3111yescharacteristic
398#135—48no—
399C2×Q16—22no—
400C2×SD16—22no—
401C2×SD16—21no—
402C2×SD16—22no—
403C2×SD16—22no—
404C2×D8—22no—
405C2×D8—21no—
406C2×D8—22no—
407C22×C8—21no—
408C22×C8—21no—
409C4⋊Q8D4×C3111yescharacteristic
410C4⋊Q8C23×C3111yescharacteristic
411#127—21no—
412#127D4×C3111yescharacteristic
413#127C23×C3111yescharacteristic
414#126—22no—
415#126—21no—
416#125C23×C3111yescharacteristic
417#124—21no—
418#124—22no—
419#124—22no—
420#124—22no—
421#124—21no—
422#124—21no—
423#124—21no—
424#124D4×C3112yes—
425#124C23×C3112yes—
426#124C23×C3111yescharacteristic
427#124C23×C3112yes—
428#123—22no—
429#123—22no—
430#123—21no—
431#123—22no—
432#123—22no—
433C22⋊Q8—21no—
434C22⋊Q8—22no—
435C22⋊Q8—22no—
436C22⋊Q8D4×C3111yescharacteristic
437#121—22no—
438#121—22no—
439#121—22no—
440#121—21no—
441#121—22no—
442#121—22no—
443#121—21no—
444#121—21no—
445#121—21no—
446#121—21no—
447#121D4×C3111yescharacteristic
448#120—22no—
449#120—22no—
450#120—22no—
451#120—22no—
452#120—22no—
453C4×Q8—21no—
454C4×Q8D4×C3111yescharacteristic
455C4×D4—22no—
456C4×D4—22no—
457C4×D4—21no—
458C4×D4—21no—
459C4×D4—21no—
460C4×D4—21no—
461C4×D4—21no—
462C4×D4—21no—
463C4×D4D4×C3111yescharacteristic
464#117—21no—
465C2×C22⋊C4—22no—
466C2×C22⋊C4—21no—
467C2×C22⋊C4—21no—
468C2×C42C23×C3111yescharacteristic
469#107—21no—
470#106—21no—
471C4⋊C8—21no—
472#103—21no—
473#103—22no—
474#102—21no—
475#102—22no—
476#102—21no—
477#102—21no—
478C22⋊C8—21no—
479#97—21no—
480#97—21no—
481#97C23×C3111yescharacteristic
482#97C23×C3111yescharacteristic
483#97C23×C3112yes—
484#97C23×C3112yes—
485C31#1904211yescharacteristic, direct product factor, Sylow
486C24—22no—
487C24—21no—
488#41—44no—
489#41—44no—
490#41—82no—
491C2×Q8—21no—
492C2×Q8—21no—
493C2×Q8—81no—
494C2×Q8C2×D4×C3111yescharacteristic
495C2×Q8C2×D4×C3112yes—
496C2×Q8C2×D4×C3112yes—
497C2×D4—22no—
498C2×D4—21no—
499C2×D4—22no—
500C2×D4—22no—
501C2×D4—22no—
502C2×D4—22no—
503C2×D4—22no—
504C2×D4—21no—
505C2×D4—21no—
506C2×D4—21no—
507C2×D4—21no—
508C2×D4—21no—
509C2×D4—21no—
510C2×D4—41no—
511C2×D4—81no—
512C2×D4—82no—
513C2×D4—84no—
514C2×D4C2×D4×C3111yescharacteristic
515C2×D4C2×D4×C3112yes—
516C2×D4C2×D4×C3112yes—
517C22×C4—22no—
518C22×C4—22no—
519C22×C4—21no—
520C22×C4—21no—
521C22×C4—22no—
522C22×C4—41no—
523C22×C4C2×D4×C3111yescharacteristic
524C22×C4C2×D4×C3111yescharacteristic
525C22×C4C2×D4×C3111yescharacteristic
526Q16—44no—
527SD16—42no—
528SD16—44no—
529SD16—44no—
530SD16—82no—
531D8—42no—
532D8—44no—
533D8—82no—
534C2×C8—24no—
535C2×C8—21no—
536C2×C8—22no—
537C2×C8—21no—
538C2×C8—22no—
539C2×C8—42no—
540C4⋊C4—22no—
541C4⋊C4—22no—
542C4⋊C4—21no—
543C4⋊C4—21no—
544C4⋊C4—21no—
545C4⋊C4—21no—
546C4⋊C4—21no—
547C4⋊C4—21no—
548C4⋊C4—21no—
549C4⋊C4—21no—
550C4⋊C4C2×D4×C3111yescharacteristic
551C4⋊C4C2×D4×C3111yescharacteristic
552C22⋊C4—22no—
553C22⋊C4—22no—
554C22⋊C4—22no—
555C22⋊C4—22no—
556C22⋊C4—22no—
557C22⋊C4—21no—
558C22⋊C4—22no—
559C22⋊C4—21no—
560C22⋊C4—22no—
561C22⋊C4—21no—
562C22⋊C4—22no—
563C22⋊C4—22no—
564C22⋊C4—21no—
565C22⋊C4—21no—
566C22⋊C4—22no—
567C22⋊C4—22no—
568C22⋊C4—41no—
569C22⋊C4—41no—
570C42—22no—
571C42—21no—
572C42C2×D4×C3111yescharacteristic
573C42C2×D4×C3111yescharacteristic
574C42C2×D4×C3111yescharacteristic
575C42C24×C3111yescharacteristic
576C23—22no—
577C23—21no—
578C23—22no—
579C23—42no—
580C23—41no—
581C23—81no—
582C23—81no—
583C23—82no—
584C23—82no—
585C23—82no—
586C23C31×#12011yescharacteristic
587Q8—42no—
588Q8—41no—
589Q8—42no—
590Q8—42no—
591Q8—42no—
592D4—42no—
593D4—41no—
594D4—42no—
595D4—42no—
596D4—42no—
597D4—42no—
598D4—41no—
599D4—42no—
600D4—41no—
601D4—42no—
602D4—44no—
603D4—81no—
604D4—82no—
605D4—82no—
606D4—82no—
607D4—82no—
608D4—82no—
609D4—82no—
610C2×C4—22no—
611C2×C4—21no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—82no—
621C2×C4—82no—
622C2×C4—81no—
623C2×C4—82no—
624C2×C4—82no—
625C2×C4C31×#12011yescharacteristic
626C2×C4C31×#12011yescharacteristic
627C2×C4C31×#12011yescharacteristic
628C2×C4C22×D4×C3111yescharacteristic
629C2×C4C22×D4×C3111yescharacteristic
630C2×C4C22×D4×C3111yescharacteristic
631C8—24no—
632C8—22no—
633C8—41no—
634C8—42no—
635C22—21no—
636C22—21no—
637C22—42no—
638C22—42no—
639C22—41no—
640C22—41no—
641C22—42no—
642C22—42no—
643C22—41no—
644C22—42no—
645C22—42no—
646C22—81no—
647C22—81no—
648C22—81no—
649C22—162no—
650C22—164no—
651C22C31×#45312yes—
652C22C2×C31×#12011yescharacteristic
653C4—21no—
654C4—21no—
655C4—21no—
656C4—21no—
657C4—21no—
658C4—21no—
659C4—41no—
660C4—42no—
661C4—41no—
662C4—41no—
663C4—42no—
664C4—81no—
665C2—22no—
666C2—82no—
667C2—81no—
668C2—82no—
669C2—161no—
670C2C2×C31×#45311yescharacteristic
671C2C31×#301811yescharacteristic
672C2C31×#301911yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC42C24×C31
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
1. C31×#3558C2semidirect
2. C31×#3213C2semidirect
3. C31×#3180C2semidirect
4. C31×#3150C2semidirect
5. C31×#1661C2semidirect
6. C31×#1659C2semidirect
7. C31×#1656C2semidirect
8. C31×#1655C2semidirect
9. C31×#1532C2semidirect
10. C31×#1530C2semidirect
11. C2×C31×#329C2semidirect
26. C31×#532C22semidirect
29. C31×#501C22semidirect
31. C31×#497C22semidirect
33. C31×#496C22semidirect
34. C31×#496C22semidirect
35. C31×#496C22semidirect
36. C31×#490C22semidirect
37. C31×#489C22semidirect
38. C31×#489C22semidirect
46. C31×#432C22semidirect
47. C2×C31×#97C22semidirect
48. C2×C31×#97C22semidirect
49. C31×#330C22semidirect
50. C31×#329C22semidirect
51. C31×#329C22semidirect
52. C31×#329C22semidirect
53. C31×#329C22semidirect
54. C31×#329C22semidirect
239. #19042C31direct (N × H)
299. #3558C2×C31semidirect
300. #3213C2×C31semidirect
301. #3180C2×C31semidirect
302. #3150C2×C31semidirect
303. #1661C2×C31semidirect
304. #1659C2×C31semidirect
305. #1656C2×C31semidirect
306. #1655C2×C31semidirect
307. #1532C2×C31semidirect
308. #1530C2×C31semidirect
309. C2×#329C2×C31semidirect
354. #532C22×C31semidirect
357. #501C22×C31semidirect
359. #497C22×C31semidirect
361. #496C22×C31semidirect
362. #496C22×C31semidirect
363. #496C22×C31semidirect
364. #490C22×C31semidirect
365. #489C22×C31semidirect
366. #489C22×C31semidirect
374. #432C22×C31semidirect
375. C2×#97C22×C31semidirect
376. C2×#97C22×C31semidirect
377. #330C22×C31semidirect
378. #329C22×C31semidirect
379. #329C22×C31semidirect
380. #329C22×C31semidirect
381. #329C22×C31semidirect
382. #329C22×C31semidirect
485. C31#19042direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35582
C31×#32132
C31×#31802
C31×#31502
C31×#16612
C31×#16592
C31×#16562
C31×#16552
C31×#15322
C31×#15302
C2×C31×#3292
#1904231