Finite Group Databasebeta

← C31×#18460 #21475219

Subgroups — C31×#18460 #21475219

Order 7936.

936 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3559C211yesmaximal, characteristic, complemented
2C31×#3213C211yesmaximal, characteristic, complemented
3C31×#3211C211yesmaximal, characteristic, complemented
4C31×#2820C211yesmaximal, characteristic, complemented
5C31×#1604C212yesmaximal, complemented
6C31×#1603C212yesmaximal, complemented
7C31×#1598C212yesmaximal, complemented
8C31×#1597C212yesmaximal, complemented
9C31×#1517C211yesmaximal, characteristic, complemented
10C31×#1512C211yesmaximal, characteristic, complemented
11C31×#1470C211yesmaximal, characteristic, complemented
12C31×#563—21no—
13C31×#562—21no—
14C31×#554—21no—
15C31×#553—21no—
16C31×#553—21no—
17C31×#547—22no—
18C31×#547—21no—
19C31×#544—21no—
20C31×#540—21no—
21C31×#539—22no—
22C31×#539C2212yes—
23C31×#537C2212yes—
24C31×#532C2211yescharacteristic
25C31×#529C2211yescharacteristic
26C2×C31×#125C2211yescharacteristic
27C2×C31×#124C2211yescharacteristic
28C31×#496—21no—
29C31×#495—21no—
30C31×#489—21no—
31C31×#488—21no—
32C31×#464—21no—
33C31×#463—21no—
34C31×#463—22no—
35C31×#451C2211yescharacteristic
36C31×#450C2211yescharacteristic
37C31×#448C2212yes—
38C31×#448C2211yescharacteristic
39C31×#447C2212yes—
40C31×#447C2211yescharacteristic
41C31×#432C2211yescharacteristic
42C31×#431C2211yescharacteristic
43C31×#399—21no—
44C31×#397—21no—
45C31×#396—22no—
46C31×#395C2212yes—
47C31×#394C2212yes—
48C31×#382C2211yescharacteristic
49C31×#328C2212yes—
50C31×#327C2212yes—
51C31×#327C2212yes—
52C31×#327C2212yes—
53C31×#324C2212yes—
54C31×#324C2212yes—
55C2×#41×C31—21no—
56C2×#41×C31—22no—
57C22×Q8×C31C2311yescharacteristic
58C22×D4×C31C2311yescharacteristic
59C2×Q16×C31—21no—
60C2×SD16×C31—21no—
61C2×SD16×C31—21no—
62C2×SD16×C31—22no—
63C2×D8×C31—21no—
64C2×D8×C31—22no—
65C31×C4⋊Q8—21no—
66C31×C4⋊Q8—22no—
67C31×C4⋊Q8D411yescharacteristic
68C31×#127D411yescharacteristic
69C31×#126—22no—
70C31×#126—22no—
71C31×#126—21no—
72C31×#125—21no—
73C31×#125—22no—
74C31×#125—21no—
75C31×#125—21no—
76C31×#125—21no—
77C31×#125—21no—
78C31×#125D411yescharacteristic
79C31×#124—21no—
80C31×#124—21no—
81C31×#124—21no—
82C31×#124—21no—
83C31×#124—21no—
84C31×#124D411yescharacteristic
85C31×#123—22no—
86C31×#123—22no—
87C31×#123—22no—
88C31×#123—22no—
89C31×#123—22no—
90C31×#123—21no—
91C31×#123—22no—
92C31×C22⋊Q8—22no—
93C31×C22⋊Q8—22no—
94C31×C22⋊Q8—22no—
95C31×C22⋊Q8—22no—
96C31×C22⋊Q8—22no—
97C31×C22⋊Q8—22no—
98C31×C22⋊Q8C2311yescharacteristic
99C31×#121—21no—
100C31×#121—22no—
101C31×#121—21no—
102C31×#121—22no—
103C31×#121—22no—
104C31×#121—21no—
105C31×#121—22no—
106C31×#121C2311yescharacteristic
107C31×#121C2312yes—
108C4×Q8×C31D411yescharacteristic
109C4×D4×C31—22no—
110C4×D4×C31—21no—
111C4×D4×C31—21no—
112C4×D4×C31—21no—
113C4×D4×C31—22no—
114C4×D4×C31—21no—
115C4×D4×C31D411yescharacteristic
116C4×D4×C31D412yes—
117C31×#117—21no—
118C31×#117—21no—
119C2×C4⋊C4×C31—22no—
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—22no—
135C31×#103—21no—
136C31×#103—21no—
137C31×#102—22no—
138C31×#102—21no—
139C31×#102—21no—
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—
149#41×C31—81no—
150#41×C31—81no—
151#41×C31—82no—
152#41×C31—82no—
153C2×Q8×C31—22no—
154C2×Q8×C31—21no—
155C2×Q8×C31—21no—
156C2×Q8×C31—21no—
157C2×Q8×C31—81no—
158C2×Q8×C31C2×D411yescharacteristic
159C2×D4×C31—21no—
160C2×D4×C31—22no—
161C2×D4×C31—22no—
162C2×D4×C31—22no—
163C2×D4×C31—21no—
164C2×D4×C31—21no—
165C2×D4×C31—21no—
166C2×D4×C31—21no—
167C2×D4×C31—81no—
168C2×D4×C31C2×D411yescharacteristic
169C2×D4×C31C2×D412yes—
170C22×C4×C31—22no—
171C22×C4×C31—22no—
172C22×C4×C31—22no—
173C22×C4×C31—21no—
174C22×C4×C31—22no—
175C22×C4×C31—22no—
176C22×C4×C31—21no—
177C22×C4×C31C2×D412yes—
178C22×C4×C31C2×D411yescharacteristic
179C22×C4×C31C2×D412yes—
180C22×C4×C31C2×D411yescharacteristic
181C22×C4×C31C2411yescharacteristic
182Q16×C31—81no—
183SD16×C31—82no—
184SD16×C31—81no—
185SD16×C31—81no—
186D8×C31—82no—
187D8×C31—81no—
188C2×C8×C31—21no—
189C2×C8×C31—21no—
190C2×C8×C31—22no—
191C4⋊C4×C31—22no—
192C4⋊C4×C31—22no—
193C4⋊C4×C31—22no—
194C4⋊C4×C31—22no—
195C4⋊C4×C31—21no—
196C4⋊C4×C31—21no—
197C4⋊C4×C31—22no—
198C4⋊C4×C31—21no—
199C4⋊C4×C31—21no—
200C4⋊C4×C31—22no—
201C4⋊C4×C31—21no—
202C4⋊C4×C31—21no—
203C4⋊C4×C31—21no—
204C4⋊C4×C31—21no—
205C4⋊C4×C31—22no—
206C4⋊C4×C31—22no—
207C4⋊C4×C31—21no—
208C4⋊C4×C31—81no—
209C4⋊C4×C31—82no—
210C4⋊C4×C31C2×D412yes—
211C4⋊C4×C31C2×D411yescharacteristic
212C4⋊C4×C31C2×D411yescharacteristic
213C22⋊C4×C31—22no—
214C22⋊C4×C31—22no—
215C22⋊C4×C31—22no—
216C22⋊C4×C31—22no—
217C22⋊C4×C31—22no—
218C22⋊C4×C31—21no—
219C22⋊C4×C31—21no—
220C22⋊C4×C31—21no—
221C22⋊C4×C31—21no—
222C22⋊C4×C31—22no—
223C22⋊C4×C31—21no—
224C22⋊C4×C31—22no—
225C22⋊C4×C31—41no—
226C22⋊C4×C31—41no—
227C22⋊C4×C31—41no—
228C22⋊C4×C31—41no—
229C22⋊C4×C31—82no—
230C22⋊C4×C31—82no—
231C42×C31—22no—
232C42×C31—21no—
233C42×C31C2×D411yescharacteristic
234C42×C31C2×D411yescharacteristic
235C42×C31C2×D411yescharacteristic
236C42×C31C2×D411yescharacteristic
237#18460C3111yesmaximal, characteristic, direct product factor, Sylow
238C23×C31—21no—
239C23×C31—22no—
240C23×C31—42no—
241C23×C31—42no—
242C23×C31—41no—
243C23×C31—41no—
244C23×C31—41no—
245C23×C31C22×D411yescharacteristic
246Q8×C31—41no—
247Q8×C31—42no—
248Q8×C31—41no—
249Q8×C31—41no—
250Q8×C31—81no—
251Q8×C31—82no—
252D4×C31—42no—
253D4×C31—41no—
254D4×C31—42no—
255D4×C31—42no—
256D4×C31—41no—
257D4×C31—41no—
258D4×C31—42no—
259D4×C31—41no—
260D4×C31—42no—
261D4×C31—41no—
262D4×C31—81no—
263D4×C31—82no—
264D4×C31—82no—
265C2×C4×C31—21no—
266C2×C4×C31—21no—
267C2×C4×C31—21no—
268C2×C4×C31—21no—
269C2×C4×C31—22no—
270C2×C4×C31—22no—
271C2×C4×C31—21no—
272C2×C4×C31—22no—
273C2×C4×C31—22no—
274C2×C4×C31—22no—
275C2×C4×C31—41no—
276C2×C4×C31—41no—
277C2×C4×C31—42no—
278C2×C4×C31—42no—
279C2×C4×C31—42no—
280C2×C4×C31—42no—
281C2×C4×C31—42no—
282C2×C4×C31—41no—
283C2×C4×C31—42no—
284C2×C4×C31—82no—
285C2×C4×C31—82no—
286C2×C4×C31—82no—
287C2×C4×C31—82no—
288C2×C4×C31#12011yescharacteristic
289C2×C4×C31#12011yescharacteristic
290C2×C4×C31#12011yescharacteristic
291C2×C4×C31#12011yescharacteristic
292C2×C4×C31C22×D411yescharacteristic
293C2×C4×C31C22×D411yescharacteristic
294C8×C31—42no—
295C8×C31—41no—
296C8×C31—41no—
297#3559C2×C3111yescharacteristic, complemented
298#3213C2×C3111yescharacteristic, complemented
299#3211C2×C3111yescharacteristic, complemented
300#2820C2×C3111yescharacteristic, complemented
301#1604C2×C3112yescomplemented
302#1603C2×C3112yescomplemented
303#1598C2×C3112yescomplemented
304#1597C2×C3112yescomplemented
305#1517C2×C3111yescharacteristic, complemented
306#1512C2×C3111yescharacteristic, complemented
307#1470C2×C3111yescharacteristic, complemented
308C22×C31—21no—
309C22×C31—21no—
310C22×C31—42no—
311C22×C31—41no—
312C22×C31—41no—
313C22×C31—42no—
314C22×C31—42no—
315C22×C31—41no—
316C22×C31—82no—
317C22×C31—82no—
318C22×C31—81no—
319C22×C31—81no—
320C22×C31—81no—
321C22×C31C2×#12011yescharacteristic
322C22×C31#57611yescharacteristic
323C22×C31#57711yescharacteristic
324C4×C31—21no—
325C4×C31—21no—
326C4×C31—21no—
327C4×C31—21no—
328C4×C31—21no—
329C4×C31—21no—
330C4×C31—42no—
331C4×C31—42no—
332C4×C31—41no—
333C4×C31—41no—
334C4×C31—42no—
335C4×C31—41no—
336C4×C31—82no—
337C4×C31—81no—
338#563—21no—
339#562—21no—
340#554—21no—
341#553—21no—
342#553—21no—
343#547—22no—
344#547—21no—
345#544—21no—
346#540—21no—
347#539—22no—
348#539C22×C3112yes—
349#537C22×C3112yes—
350#532C22×C3111yescharacteristic
351#529C22×C3111yescharacteristic
352C2×#125C22×C3111yescharacteristic
353C2×#124C22×C3111yescharacteristic
354#496—21no—
355#495—21no—
356#489—21no—
357#488—21no—
358#464—21no—
359#463—21no—
360#463—22no—
361#451C22×C3111yescharacteristic
362#450C22×C3111yescharacteristic
363#448C22×C3112yes—
364#448C22×C3111yescharacteristic
365#447C22×C3112yes—
366#447C22×C3111yescharacteristic
367#432C22×C3111yescharacteristic
368#431C22×C3111yescharacteristic
369#399—21no—
370#397—21no—
371#396—22no—
372#395C22×C3112yes—
373#394C22×C3112yes—
374#382C22×C3111yescharacteristic
375#328C22×C3112yes—
376#327C22×C3112yes—
377#327C22×C3112yes—
378#327C22×C3112yes—
379#324C22×C3112yes—
380#324C22×C3112yes—
381C2×C31—21no—
382C2×C31—21no—
383C2×C31—81no—
384C2×C31—82no—
385C2×C31—161no—
386C2×C31#301211yescharacteristic
387C2×C31#302011yescharacteristic
388C2×C31#302711yescharacteristic
389C2×#41—21no—
390C2×#41—22no—
391C22×Q8C23×C3111yescharacteristic
392C22×D4C23×C3111yescharacteristic
393C2×Q16—21no—
394C2×SD16—21no—
395C2×SD16—21no—
396C2×SD16—22no—
397C2×D8—21no—
398C2×D8—22no—
399C4⋊Q8—21no—
400C4⋊Q8—22no—
401C4⋊Q8D4×C3111yescharacteristic
402#127D4×C3111yescharacteristic
403#126—22no—
404#126—22no—
405#126—21no—
406#125—21no—
407#125—22no—
408#125—21no—
409#125—21no—
410#125—21no—
411#125—21no—
412#125D4×C3111yescharacteristic
413#124—21no—
414#124—21no—
415#124—21no—
416#124—21no—
417#124—21no—
418#124D4×C3111yescharacteristic
419#123—22no—
420#123—22no—
421#123—22no—
422#123—22no—
423#123—22no—
424#123—21no—
425#123—22no—
426C22⋊Q8—22no—
427C22⋊Q8—22no—
428C22⋊Q8—22no—
429C22⋊Q8—22no—
430C22⋊Q8—22no—
431C22⋊Q8—22no—
432C22⋊Q8C23×C3111yescharacteristic
433#121—21no—
434#121—22no—
435#121—21no—
436#121—22no—
437#121—22no—
438#121—21no—
439#121—22no—
440#121C23×C3111yescharacteristic
441#121C23×C3112yes—
442C4×Q8D4×C3111yescharacteristic
443C4×D4—22no—
444C4×D4—21no—
445C4×D4—21no—
446C4×D4—21no—
447C4×D4—22no—
448C4×D4—21no—
449C4×D4D4×C3111yescharacteristic
450C4×D4D4×C3112yes—
451#117—21no—
452#117—21no—
453C2×C4⋊C4—22no—
454C2×C4⋊C4—21no—
455C2×C4⋊C4—21no—
456C2×C4⋊C4—22no—
457C2×C4⋊C4D4×C3112yes—
458C2×C4⋊C4D4×C3111yescharacteristic
459C2×C4⋊C4D4×C3111yescharacteristic
460C2×C4⋊C4C23×C3112yes—
461C2×C22⋊C4—21no—
462C2×C22⋊C4—21no—
463C2×C22⋊C4—22no—
464C2×C22⋊C4—22no—
465C2×C42C23×C3111yescharacteristic
466C4⋊C8—21no—
467C4⋊C8—21no—
468#103—22no—
469#103—21no—
470#103—21no—
471#102—22no—
472#102—21no—
473#102—21no—
474C22⋊C8C23×C3111yescharacteristic
475C22⋊C8C23×C3111yescharacteristic
476C22⋊C8C23×C3112yes—
477#97—21no—
478C4×C8—21no—
479#95—21no—
480#95—22no—
481#95C23×C3112yes—
482C31#1846011yescharacteristic, direct product factor, Sylow
483C24—21no—
484#41—81no—
485#41—81no—
486#41—82no—
487#41—82no—
488C2×Q8—22no—
489C2×Q8—21no—
490C2×Q8—21no—
491C2×Q8—21no—
492C2×Q8—81no—
493C2×Q8C2×D4×C3111yescharacteristic
494C2×D4—21no—
495C2×D4—22no—
496C2×D4—22no—
497C2×D4—22no—
498C2×D4—21no—
499C2×D4—21no—
500C2×D4—21no—
501C2×D4—21no—
502C2×D4—81no—
503C2×D4C2×D4×C3111yescharacteristic
504C2×D4C2×D4×C3112yes—
505C22×C4—22no—
506C22×C4—22no—
507C22×C4—22no—
508C22×C4—21no—
509C22×C4—22no—
510C22×C4—22no—
511C22×C4—21no—
512C22×C4C2×D4×C3112yes—
513C22×C4C2×D4×C3111yescharacteristic
514C22×C4C2×D4×C3112yes—
515C22×C4C2×D4×C3111yescharacteristic
516C22×C4C24×C3111yescharacteristic
517Q16—81no—
518SD16—82no—
519SD16—81no—
520SD16—81no—
521D8—82no—
522D8—81no—
523C2×C8—21no—
524C2×C8—21no—
525C2×C8—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—22no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—22no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—22no—
541C4⋊C4—22no—
542C4⋊C4—21no—
543C4⋊C4—81no—
544C4⋊C4—82no—
545C4⋊C4C2×D4×C3112yes—
546C4⋊C4C2×D4×C3111yescharacteristic
547C4⋊C4C2×D4×C3111yescharacteristic
548C22⋊C4—22no—
549C22⋊C4—22no—
550C22⋊C4—22no—
551C22⋊C4—22no—
552C22⋊C4—22no—
553C22⋊C4—21no—
554C22⋊C4—21no—
555C22⋊C4—21no—
556C22⋊C4—21no—
557C22⋊C4—22no—
558C22⋊C4—21no—
559C22⋊C4—22no—
560C22⋊C4—41no—
561C22⋊C4—41no—
562C22⋊C4—41no—
563C22⋊C4—41no—
564C22⋊C4—82no—
565C22⋊C4—82no—
566C42—22no—
567C42—21no—
568C42C2×D4×C3111yescharacteristic
569C42C2×D4×C3111yescharacteristic
570C42C2×D4×C3111yescharacteristic
571C42C2×D4×C3111yescharacteristic
572C23—21no—
573C23—22no—
574C23—42no—
575C23—42no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23C22×D4×C3111yescharacteristic
580Q8—41no—
581Q8—42no—
582Q8—41no—
583Q8—41no—
584Q8—81no—
585Q8—82no—
586D4—42no—
587D4—41no—
588D4—42no—
589D4—42no—
590D4—41no—
591D4—41no—
592D4—42no—
593D4—41no—
594D4—42no—
595D4—41no—
596D4—81no—
597D4—82no—
598D4—82no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—21no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—82no—
619C2×C4—82no—
620C2×C4—82no—
621C2×C4—82no—
622C2×C4C31×#12011yescharacteristic
623C2×C4C31×#12011yescharacteristic
624C2×C4C31×#12011yescharacteristic
625C2×C4C31×#12011yescharacteristic
626C2×C4C22×D4×C3111yescharacteristic
627C2×C4C22×D4×C3111yescharacteristic
628C8—42no—
629C8—41no—
630C8—41no—
631C22—21no—
632C22—21no—
633C22—42no—
634C22—41no—
635C22—41no—
636C22—42no—
637C22—42no—
638C22—41no—
639C22—82no—
640C22—82no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22C2×C31×#12011yescharacteristic
645C22C31×#57611yescharacteristic
646C22C31×#57711yescharacteristic
647C4—21no—
648C4—21no—
649C4—21no—
650C4—21no—
651C4—21no—
652C4—21no—
653C4—42no—
654C4—42no—
655C4—41no—
656C4—41no—
657C4—42no—
658C4—41no—
659C4—82no—
660C4—81no—
661C2—21no—
662C2—21no—
663C2—81no—
664C2—82no—
665C2—161no—
666C2C31×#301211yescharacteristic
667C2C31×#302011yescharacteristic
668C2C31×#302711yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
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×#3559C2semidirect
2. C31×#3213C2semidirect
3. C31×#3211C2semidirect
4. C31×#2820C2semidirect
5. C31×#1604C2semidirect
6. C31×#1603C2semidirect
7. C31×#1598C2semidirect
8. C31×#1597C2semidirect
9. C31×#1517C2semidirect
10. C31×#1512C2semidirect
11. C31×#1470C2semidirect
237. #18460C31direct (N × H)
297. #3559C2×C31semidirect
298. #3213C2×C31semidirect
299. #3211C2×C31semidirect
300. #2820C2×C31semidirect
301. #1604C2×C31semidirect
302. #1603C2×C31semidirect
303. #1598C2×C31semidirect
304. #1597C2×C31semidirect
305. #1517C2×C31semidirect
306. #1512C2×C31semidirect
307. #1470C2×C31semidirect
482. C31#18460direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35592
C31×#32132
C31×#32112
C31×#28202
C31×#16042
C31×#16032
C31×#15982
C31×#15972
C31×#15172
C31×#15122
C31×#14702
#1846031