Finite Group Databasebeta

← C31×#18121 #21474880

Subgroups — C31×#18121 #21474880

Order 7936.

990 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3460C211yesmaximal, characteristic
2C31×#3091C212yesmaximal, complemented
3C31×#2395C211yesmaximal, characteristic, complemented
4C31×#1618C212yesmaximal, complemented
5C31×#1617C211yesmaximal, characteristic, complemented
6C31×#1616C211yesmaximal, characteristic, complemented
7C31×#1612C212yesmaximal, complemented
8C31×#1611C211yesmaximal, characteristic, complemented
9C31×#1610C211yesmaximal, characteristic, complemented
10C31×#1501C212yesmaximal, complemented
11C31×#1459C211yesmaximal, characteristic, complemented
12C31×#544C2211yescharacteristic
13C31×#542—22no—
14C31×#540—22no—
15C31×#538—21no—
16C31×#538C2212yes—
17C31×#537C2211yescharacteristic
18C31×#535—21no—
19C31×#532—22no—
20C31×#529—22no—
21C31×#525—21no—
22C31×#525C2212yes—
23C4×#41×C31—21no—
24C2×C4×D4×C31C2211yescharacteristic
25C2×C31×#117C2211yescharacteristic
26C31×#485—22no—
27C31×#484C2212yes—
28C31×#483C2212yes—
29C31×#482—22no—
30C31×#481C2212yes—
31C31×#480C2212yes—
32C31×#427—22no—
33C31×#416—22no—
34C2×C31×C22⋊C8C2212yes—
35C31×#400C2211yescharacteristic
36C31×#399C2211yescharacteristic
37C31×#398C2211yescharacteristic
38C31×#397—21no—
39C31×#397C2211yescharacteristic
40C31×#394—21no—
41C4×C4⋊C4×C31—21no—
42C4×C22⋊C4×C31—21no—
43C4×C22⋊C4×C31C2211yescharacteristic
44C31×#328C2212yes—
45C31×#328C2212yes—
46C31×#327C2212yes—
47C31×#327C2212yes—
48C31×#324C2211yescharacteristic
49C31×#324C2211yescharacteristic
50C31×#323C2212yes—
51C2×#41×C31—21no—
52C2×#41×C31—21no—
53C2×#41×C31D412yes—
54C22×D4×C31D411yescharacteristic
55C23×C4×C31D411yescharacteristic
56C23×C4×C31C2311yescharacteristic
57C22×C8×C31—22no—
58C31×C4⋊Q8—22no—
59C31×#127—22no—
60C31×#126—22no—
61C31×#126—22no—
62C31×#125—22no—
63C31×#124—22no—
64C31×#124—22no—
65C31×#124—22no—
66C31×#123—22no—
67C31×#123—22no—
68C31×#123—22no—
69C31×#123—22no—
70C31×C22⋊Q8—21no—
71C31×C22⋊Q8—21no—
72C31×C22⋊Q8—22no—
73C31×C22⋊Q8D412yes—
74C31×C22⋊Q8C2312yes—
75C31×#121—21no—
76C31×#121—22no—
77C31×#121—22no—
78C31×#121—22no—
79C31×#121—22no—
80C31×#121—22no—
81C31×#121—21no—
82C31×#121D412yes—
83C31×#121C2312yes—
84C31×#120—22no—
85C31×#120—22no—
86C4×Q8×C31—21no—
87C4×Q8×C31—21no—
88C4×D4×C31—21no—
89C4×D4×C31—21no—
90C4×D4×C31—22no—
91C4×D4×C31—22no—
92C4×D4×C31—22no—
93C4×D4×C31—22no—
94C4×D4×C31—22no—
95C4×D4×C31—22no—
96C31×#117—22no—
97C31×#117—21no—
98C31×#117—21no—
99C31×#117—22no—
100C31×#117—22no—
101C31×#117D412yes—
102C2×C4⋊C4×C31—21no—
103C2×C4⋊C4×C31—21no—
104C2×C4⋊C4×C31C2311yescharacteristic
105C2×C4⋊C4×C31C2311yescharacteristic
106C2×C4⋊C4×C31C2311yescharacteristic
107C2×C22⋊C4×C31—21no—
108C2×C22⋊C4×C31—21no—
109C2×C22⋊C4×C31—21no—
110C2×C22⋊C4×C31—21no—
111C2×C22⋊C4×C31D411yescharacteristic
112C2×C22⋊C4×C31C2311yescharacteristic
113C2×C42×C31—22no—
114C2×C42×C31—21no—
115C2×C42×C31—21no—
116C2×C42×C31D411yescharacteristic
117C31×#107—22no—
118C31×#107—22no—
119C31×#106—22no—
120C31×#106—22no—
121C31×#103—22no—
122C31×#103—22no—
123C31×#102—22no—
124C31×#102—22no—
125C31×C22⋊C8—22no—
126C31×C22⋊C8C2312yes—
127C31×C22⋊C8C2312yes—
128C31×#95—21no—
129C31×#95—21no—
130C31×#95—21no—
131C31×#95—21no—
132C31×#95C2311yescharacteristic
133C31×#95C2311yescharacteristic
134C24×C31C2×D411yescharacteristic
135C24×C31C2×D411yescharacteristic
136#41×C31—42no—
137#41×C31—44no—
138#41×C31—42no—
139#41×C31—82no—
140C2×Q8×C31—21no—
141C2×Q8×C31—21no—
142C2×Q8×C31C2×D412yes—
143C2×D4×C31—21no—
144C2×D4×C31—22no—
145C2×D4×C31—22no—
146C2×D4×C31—22no—
147C2×D4×C31—22no—
148C2×D4×C31—21no—
149C2×D4×C31—22no—
150C2×D4×C31—22no—
151C2×D4×C31—22no—
152C2×D4×C31—41no—
153C2×D4×C31C2×D412yes—
154C22×C4×C31—22no—
155C22×C4×C31—21no—
156C22×C4×C31—22no—
157C22×C4×C31—22no—
158C22×C4×C31—21no—
159C22×C4×C31—21no—
160C22×C4×C31—21no—
161C22×C4×C31—21no—
162C22×C4×C31—21no—
163C22×C4×C31—42no—
164C22×C4×C31—42no—
165C22×C4×C31C2×D411yescharacteristic
166C22×C4×C31C2×D412yes—
167C22×C4×C31C2×D411yescharacteristic
168C22×C4×C31C2×D411yescharacteristic
169C22×C4×C31C2×D411yescharacteristic
170C22×C4×C31C2×D412yes—
171C22×C4×C31C2411yescharacteristic
172C2×C8×C31—22no—
173C2×C8×C31—22no—
174C2×C8×C31—44no—
175C4⋊C4×C31—21no—
176C4⋊C4×C31—21no—
177C4⋊C4×C31—22no—
178C4⋊C4×C31—21no—
179C4⋊C4×C31—21no—
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—42no—
187C4⋊C4×C31—42no—
188C4⋊C4×C31C2×D412yes—
189C4⋊C4×C31C2×D412yes—
190C22⋊C4×C31—22no—
191C22⋊C4×C31—22no—
192C22⋊C4×C31—22no—
193C22⋊C4×C31—22no—
194C22⋊C4×C31—22no—
195C22⋊C4×C31—22no—
196C22⋊C4×C31—22no—
197C22⋊C4×C31—22no—
198C22⋊C4×C31—22no—
199C22⋊C4×C31—22no—
200C22⋊C4×C31—22no—
201C22⋊C4×C31—42no—
202C22⋊C4×C31—42no—
203C22⋊C4×C31—42no—
204C22⋊C4×C31—42no—
205C42×C31—22no—
206C42×C31—22no—
207C42×C31—21no—
208C42×C31—21no—
209C42×C31—22no—
210C42×C31—42no—
211C42×C31—41no—
212C42×C31—41no—
213#18121C3111yesmaximal, characteristic, direct product factor, Sylow
214C23×C31—21no—
215C23×C31—21no—
216C23×C31—21no—
217C23×C31—22no—
218C23×C31—21no—
219C23×C31—21no—
220C23×C31—41no—
221C23×C31—41no—
222C23×C31—41no—
223C23×C31—41no—
224C23×C31#12012yes—
225C23×C31C22×D411yescharacteristic
226Q8×C31—42no—
227Q8×C31—42no—
228Q8×C31—81no—
229D4×C31—44no—
230D4×C31—42no—
231D4×C31—42no—
232D4×C31—42no—
233D4×C31—42no—
234D4×C31—82no—
235D4×C31—82no—
236D4×C31—82no—
237D4×C31—81no—
238D4×C31—82no—
239C2×C4×C31—21no—
240C2×C4×C31—21no—
241C2×C4×C31—22no—
242C2×C4×C31—24no—
243C2×C4×C31—22no—
244C2×C4×C31—21no—
245C2×C4×C31—22no—
246C2×C4×C31—22no—
247C2×C4×C31—22no—
248C2×C4×C31—21no—
249C2×C4×C31—42no—
250C2×C4×C31—41no—
251C2×C4×C31—44no—
252C2×C4×C31—41no—
253C2×C4×C31—41no—
254C2×C4×C31—42no—
255C2×C4×C31—44no—
256C2×C4×C31—41no—
257C2×C4×C31—41no—
258C2×C4×C31—41no—
259C2×C4×C31—41no—
260C2×C4×C31—41no—
261C2×C4×C31—41no—
262C2×C4×C31—41no—
263C2×C4×C31—41no—
264C2×C4×C31—41no—
265C2×C4×C31—41no—
266C2×C4×C31—41no—
267C2×C4×C31—81no—
268C2×C4×C31—82no—
269C2×C4×C31#12011yescharacteristic
270C2×C4×C31#12011yescharacteristic
271C2×C4×C31#13512yes—
272C2×C4×C31C22×D412yes—
273C8×C31—42no—
274C8×C31—42no—
275#3460C2×C3111yescharacteristic
276#3091C2×C3112yescomplemented
277#2395C2×C3111yescharacteristic, complemented
278#1618C2×C3112yescomplemented
279#1617C2×C3111yescharacteristic, complemented
280#1616C2×C3111yescharacteristic, complemented
281#1612C2×C3112yescomplemented
282#1611C2×C3111yescharacteristic, complemented
283#1610C2×C3111yescharacteristic, complemented
284#1501C2×C3112yescomplemented
285#1459C2×C3111yescharacteristic, complemented
286C22×C31—21no—
287C22×C31—21no—
288C22×C31—22no—
289C22×C31—22no—
290C22×C31—22no—
291C22×C31—42no—
292C22×C31—42no—
293C22×C31—41no—
294C22×C31—42no—
295C22×C31—41no—
296C22×C31—41no—
297C22×C31—42no—
298C22×C31—81no—
299C22×C31—81no—
300C22×C31—81no—
301C22×C31C2×#12011yescharacteristic
302C22×C31C2×#13511yescharacteristic
303C22×C31#57511yescharacteristic
304C4×C31—22no—
305C4×C31—22no—
306C4×C31—21no—
307C4×C31—42no—
308C4×C31—42no—
309C4×C31—41no—
310C4×C31—42no—
311C4×C31—41no—
312C4×C31—42no—
313C4×C31—81no—
314C4×C31—81no—
315C4×C31—81no—
316C4×C31#45812yes—
317#544C22×C3111yescharacteristic
318#542—22no—
319#540—22no—
320#538—21no—
321#538C22×C3112yes—
322#537C22×C3111yescharacteristic
323#535—21no—
324#532—22no—
325#529—22no—
326#525—21no—
327#525C22×C3112yes—
328C4×#41—21no—
329C2×C4×D4C22×C3111yescharacteristic
330C2×#117C22×C3111yescharacteristic
331#485—22no—
332#484C22×C3112yes—
333#483C22×C3112yes—
334#482—22no—
335#481C22×C3112yes—
336#480C22×C3112yes—
337#427—22no—
338#416—22no—
339C2×C22⋊C8C22×C3112yes—
340#400C22×C3111yescharacteristic
341#399C22×C3111yescharacteristic
342#398C22×C3111yescharacteristic
343#397—21no—
344#397C22×C3111yescharacteristic
345#394—21no—
346C4×C4⋊C4—21no—
347C4×C22⋊C4—21no—
348C4×C22⋊C4C22×C3111yescharacteristic
349#328C22×C3112yes—
350#328C22×C3112yes—
351#327C22×C3112yes—
352#327C22×C3112yes—
353#324C22×C3111yescharacteristic
354#324C22×C3111yescharacteristic
355#323C22×C3112yes—
356C2×C31—21no—
357C2×C31—21no—
358C2×C31—42no—
359C2×C31—82no—
360C2×C31—81no—
361C2×C31#300211yescharacteristic
362C2×C31#301811yescharacteristic
363C2×C31C2×#45811yescharacteristic
364C2×#41—21no—
365C2×#41—21no—
366C2×#41D4×C3112yes—
367C22×D4D4×C3111yescharacteristic
368C23×C4D4×C3111yescharacteristic
369C23×C4C23×C3111yescharacteristic
370C22×C8—22no—
371C4⋊Q8—22no—
372#127—22no—
373#126—22no—
374#126—22no—
375#125—22no—
376#124—22no—
377#124—22no—
378#124—22no—
379#123—22no—
380#123—22no—
381#123—22no—
382#123—22no—
383C22⋊Q8—21no—
384C22⋊Q8—21no—
385C22⋊Q8—22no—
386C22⋊Q8D4×C3112yes—
387C22⋊Q8C23×C3112yes—
388#121—21no—
389#121—22no—
390#121—22no—
391#121—22no—
392#121—22no—
393#121—22no—
394#121—21no—
395#121D4×C3112yes—
396#121C23×C3112yes—
397#120—22no—
398#120—22no—
399C4×Q8—21no—
400C4×Q8—21no—
401C4×D4—21no—
402C4×D4—21no—
403C4×D4—22no—
404C4×D4—22no—
405C4×D4—22no—
406C4×D4—22no—
407C4×D4—22no—
408C4×D4—22no—
409#117—22no—
410#117—21no—
411#117—21no—
412#117—22no—
413#117—22no—
414#117D4×C3112yes—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4C23×C3111yescharacteristic
418C2×C4⋊C4C23×C3111yescharacteristic
419C2×C4⋊C4C23×C3111yescharacteristic
420C2×C22⋊C4—21no—
421C2×C22⋊C4—21no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—21no—
424C2×C22⋊C4D4×C3111yescharacteristic
425C2×C22⋊C4C23×C3111yescharacteristic
426C2×C42—22no—
427C2×C42—21no—
428C2×C42—21no—
429C2×C42D4×C3111yescharacteristic
430#107—22no—
431#107—22no—
432#106—22no—
433#106—22no—
434#103—22no—
435#103—22no—
436#102—22no—
437#102—22no—
438C22⋊C8—22no—
439C22⋊C8C23×C3112yes—
440C22⋊C8C23×C3112yes—
441#95—21no—
442#95—21no—
443#95—21no—
444#95—21no—
445#95C23×C3111yescharacteristic
446#95C23×C3111yescharacteristic
447C31#1812111yescharacteristic, direct product factor, Sylow
448C24C2×D4×C3111yescharacteristic
449C24C2×D4×C3111yescharacteristic
450#41—42no—
451#41—44no—
452#41—42no—
453#41—82no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8C2×D4×C3112yes—
457C2×D4—21no—
458C2×D4—22no—
459C2×D4—22no—
460C2×D4—22no—
461C2×D4—22no—
462C2×D4—21no—
463C2×D4—22no—
464C2×D4—22no—
465C2×D4—22no—
466C2×D4—41no—
467C2×D4C2×D4×C3112yes—
468C22×C4—22no—
469C22×C4—21no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4C2×D4×C3111yescharacteristic
480C22×C4C2×D4×C3112yes—
481C22×C4C2×D4×C3111yescharacteristic
482C22×C4C2×D4×C3111yescharacteristic
483C22×C4C2×D4×C3111yescharacteristic
484C22×C4C2×D4×C3112yes—
485C22×C4C24×C3111yescharacteristic
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—44no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—22no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4C2×D4×C3112yes—
503C4⋊C4C2×D4×C3112yes—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—22no—
507C22⋊C4—22no—
508C22⋊C4—22no—
509C22⋊C4—22no—
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—22no—
513C22⋊C4—22no—
514C22⋊C4—22no—
515C22⋊C4—42no—
516C22⋊C4—42no—
517C22⋊C4—42no—
518C22⋊C4—42no—
519C42—22no—
520C42—22no—
521C42—21no—
522C42—22no—
523C42—21no—
524C42—42no—
525C42—41no—
526C42—41no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—22no—
531C23—21no—
532C23—21no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23C31×#12012yes—
538C23C22×D4×C3111yescharacteristic
539Q8—42no—
540Q8—42no—
541Q8—81no—
542D4—44no—
543D4—42no—
544D4—42no—
545D4—42no—
546D4—42no—
547D4—82no—
548D4—82no—
549D4—82no—
550D4—81no—
551D4—82no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—44no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—81no—
581C2×C4—82no—
582C2×C4C31×#12011yescharacteristic
583C2×C4C31×#12011yescharacteristic
584C2×C4C31×#13512yes—
585C2×C4C22×D4×C3112yes—
586C8—42no—
587C8—42no—
588C22—21no—
589C22—21no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—42no—
594C22—42no—
595C22—41no—
596C22—42no—
597C22—41no—
598C22—41no—
599C22—42no—
600C22—81no—
601C22—81no—
602C22—81no—
603C22C2×C31×#12011yescharacteristic
604C22C2×C31×#13511yescharacteristic
605C22C31×#57511yescharacteristic
606C4—22no—
607C4—22no—
608C4—21no—
609C4—42no—
610C4—42no—
611C4—41no—
612C4—42no—
613C4—41no—
614C4—42no—
615C4—81no—
616C4—81no—
617C4—81no—
618C4C31×#45812yes—
619C2—21no—
620C2—21no—
621C2—42no—
622C2—81no—
623C2—82no—
624C2C31×#300211yescharacteristic
625C2C31×#301811yescharacteristic
626C2C2×C31×#45811yescharacteristic
subgroupG/H
centerC2×C4×C31#120
commutator, Frattini, cosocleC22×C4C24×C31
socleC22×C31C2×#120
prime radicalC31×#3460C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C31 ◁ C23×C4×C31 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C31×#3091C2semidirect
3. C31×#2395C2semidirect
4. C31×#1618C2semidirect
5. C31×#1617C2semidirect
6. C31×#1616C2semidirect
7. C31×#1612C2semidirect
8. C31×#1611C2semidirect
9. C31×#1610C2semidirect
10. C31×#1501C2semidirect
11. C31×#1459C2semidirect
213. #18121C31direct (N × H)
276. #3091C2×C31semidirect
277. #2395C2×C31semidirect
278. #1618C2×C31semidirect
279. #1617C2×C31semidirect
280. #1616C2×C31semidirect
281. #1612C2×C31semidirect
282. #1611C2×C31semidirect
283. #1610C2×C31semidirect
284. #1501C2×C31semidirect
285. #1459C2×C31semidirect
447. C31#18121direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#34602
C31×#30912
C31×#23952
C31×#16182
C31×#16172
C31×#16162
C31×#16122
C31×#16112
C31×#16102
C31×#15012
C31×#14592
#1812131