Finite Group Databasebeta

← C31×#18114 #21474873

Subgroups — C31×#18114 #21474873

Order 7936.

1246 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3460C211yesmaximal, characteristic, complemented
2C31×#3002C212yesmaximal, complemented
3C31×#2388C211yesmaximal, characteristic, complemented
4C31×#1606C212yesmaximal, complemented
5C31×#1605C211yesmaximal, characteristic, complemented
6C31×#1604C211yesmaximal, characteristic, complemented
7C31×#1600C212yesmaximal, complemented
8C31×#1599C211yesmaximal, characteristic, complemented
9C31×#1598C211yesmaximal, characteristic, complemented
10C31×#1501C212yesmaximal, complemented
11C31×#1459C211yesmaximal, characteristic, complemented
12C22×#41×C31C2211yescharacteristic
13C2×C31×#135—22no—
14C31×#544C2211yescharacteristic
15C31×#542—22no—
16C31×#540—22no—
17C31×#538—21no—
18C31×#538C2212yes—
19C31×#537C2211yescharacteristic
20C31×#535—21no—
21C31×#532—22no—
22C31×#529—22no—
23C31×#525—21no—
24C31×#525C2212yes—
25C4×#41×C31—21no—
26C2×C4×D4×C31—21no—
27C2×C4×D4×C31C2211yescharacteristic
28C31×#452—22no—
29C31×#451C2212yes—
30C31×#450C2212yes—
31C31×#449—22no—
32C31×#448C2212yes—
33C31×#447C2212yes—
34C31×#416—22no—
35C2×C31×C22⋊C8C2212yes—
36C31×#395C2211yescharacteristic
37C31×#394—21no—
38C31×#394C2211yescharacteristic
39C31×#393C2211yescharacteristic
40C31×#392—21no—
41C31×#392C2211yescharacteristic
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×C31—22no—
54C2×#41×C31—22no—
55C2×#41×C31—44no—
56C2×#41×C31D412yescomplemented
57C22×Q8×C31C2311yescharacteristic
58C22×D4×C31—21no—
59C22×D4×C31—21no—
60C22×D4×C31D411yescharacteristic
61C22×D4×C31C2311yescharacteristic
62C23×C4×C31—21no—
63C23×C4×C31D411yescharacteristic
64C23×C4×C31C2311yescharacteristic
65C31×#135—48no—
66C2×Q16×C31—22no—
67C2×SD16×C31—22no—
68C2×SD16×C31—22no—
69C2×D8×C31—22no—
70C22×C8×C31—22no—
71C31×C4⋊Q8—22no—
72C31×#127—22no—
73C31×#126—22no—
74C31×#126—22no—
75C31×#125—22no—
76C31×#124—22no—
77C31×#124—22no—
78C31×#124—22no—
79C31×#123—22no—
80C31×#123—22no—
81C31×#123—22no—
82C31×#123—22no—
83C31×C22⋊Q8—21no—
84C31×C22⋊Q8—21no—
85C31×C22⋊Q8—22no—
86C31×C22⋊Q8D412yescomplemented
87C31×C22⋊Q8C2312yes—
88C31×#121—21no—
89C31×#121—22no—
90C31×#121—22no—
91C31×#121—22no—
92C31×#121—22no—
93C31×#121—22no—
94C31×#121—21no—
95C31×#121D412yescomplemented
96C31×#121C2312yes—
97C31×#120—22no—
98C31×#120—22no—
99C4×Q8×C31—21no—
100C4×Q8×C31—21no—
101C4×D4×C31—21no—
102C4×D4×C31—21no—
103C4×D4×C31—22no—
104C4×D4×C31—22no—
105C4×D4×C31—22no—
106C4×D4×C31—22no—
107C4×D4×C31—22no—
108C4×D4×C31—22no—
109C4×D4×C31—44no—
110C31×#117—22no—
111C31×#117—21no—
112C31×#117—21no—
113C31×#117D412yescomplemented
114C2×C4⋊C4×C31—21no—
115C2×C4⋊C4×C31C2311yescharacteristic
116C2×C22⋊C4×C31—21no—
117C2×C22⋊C4×C31—21no—
118C2×C22⋊C4×C31—21no—
119C2×C22⋊C4×C31—21no—
120C2×C22⋊C4×C31—21no—
121C2×C22⋊C4×C31D411yescharacteristic
122C2×C22⋊C4×C31C2311yescharacteristic
123C2×C42×C31—22no—
124C2×C42×C31—21no—
125C2×C42×C31D411yescharacteristic
126C31×#103—22no—
127C31×#103—22no—
128C31×#102—22no—
129C31×#102—22no—
130C31×C22⋊C8—22no—
131C31×C22⋊C8C2312yes—
132C31×C22⋊C8C2312yes—
133C31×#95—21no—
134C31×#95—21no—
135C31×#95C2311yescharacteristic
136C31×#95C2311yescharacteristic
137C24×C31—21no—
138C24×C31C2×D411yescharacteristic
139C24×C31C2×D411yescharacteristic
140#41×C31—28no—
141#41×C31—42no—
142#41×C31—44no—
143#41×C31—44no—
144#41×C31—44no—
145#41×C31—44no—
146#41×C31—42no—
147#41×C31—82no—
148C2×Q8×C31—21no—
149C2×Q8×C31—21no—
150C2×Q8×C31—22no—
151C2×Q8×C31—42no—
152C2×Q8×C31C2×D412yes—
153C2×D4×C31—21no—
154C2×D4×C31—22no—
155C2×D4×C31—22no—
156C2×D4×C31—22no—
157C2×D4×C31—22no—
158C2×D4×C31—21no—
159C2×D4×C31—22no—
160C2×D4×C31—22no—
161C2×D4×C31—22no—
162C2×D4×C31—22no—
163C2×D4×C31—41no—
164C2×D4×C31—42no—
165C2×D4×C31—42no—
166C2×D4×C31—42no—
167C2×D4×C31—42no—
168C2×D4×C31—42no—
169C2×D4×C31—42no—
170C2×D4×C31—42no—
171C2×D4×C31C2×D412yes—
172C22×C4×C31—22no—
173C22×C4×C31—24no—
174C22×C4×C31—21no—
175C22×C4×C31—21no—
176C22×C4×C31—22no—
177C22×C4×C31—22no—
178C22×C4×C31—21no—
179C22×C4×C31—21no—
180C22×C4×C31—21no—
181C22×C4×C31—42no—
182C22×C4×C31—42no—
183C22×C4×C31—42no—
184C22×C4×C31—41no—
185C22×C4×C31—41no—
186C22×C4×C31C2×D411yescharacteristic
187C22×C4×C31C2×D412yes—
188C22×C4×C31C2×D411yescharacteristic
189C22×C4×C31C2×D411yescharacteristic
190C22×C4×C31C2×D411yescharacteristic
191C22×C4×C31C2×D412yes—
192C22×C4×C31C2411yescharacteristic
193Q16×C31—44no—
194SD16×C31—44no—
195SD16×C31—44no—
196D8×C31—44no—
197C2×C8×C31—22no—
198C2×C8×C31—22no—
199C2×C8×C31—44no—
200C4⋊C4×C31—21no—
201C4⋊C4×C31—21no—
202C4⋊C4×C31—22no—
203C4⋊C4×C31—21no—
204C4⋊C4×C31—21no—
205C4⋊C4×C31—22no—
206C4⋊C4×C31—22no—
207C4⋊C4×C31—22no—
208C4⋊C4×C31—22no—
209C4⋊C4×C31—42no—
210C4⋊C4×C31C2×D412yes—
211C4⋊C4×C31C2×D412yes—
212C22⋊C4×C31—22no—
213C22⋊C4×C31—22no—
214C22⋊C4×C31—22no—
215C22⋊C4×C31—22no—
216C22⋊C4×C31—22no—
217C22⋊C4×C31—22no—
218C22⋊C4×C31—22no—
219C22⋊C4×C31—22no—
220C22⋊C4×C31—22no—
221C22⋊C4×C31—22no—
222C22⋊C4×C31—22no—
223C22⋊C4×C31—42no—
224C22⋊C4×C31—42no—
225C22⋊C4×C31—42no—
226C22⋊C4×C31—42no—
227C22⋊C4×C31—42no—
228C42×C31—22no—
229C42×C31—22no—
230C42×C31—21no—
231C42×C31—21no—
232C42×C31—22no—
233C42×C31—42no—
234#18114C3111yesmaximal, characteristic, direct product factor, Sylow
235C23×C31—21no—
236C23×C31—21no—
237C23×C31—22no—
238C23×C31—21no—
239C23×C31—22no—
240C23×C31—21no—
241C23×C31—21no—
242C23×C31—41no—
243C23×C31—41no—
244C23×C31—41no—
245C23×C31—41no—
246C23×C31—41no—
247C23×C31—41no—
248C23×C31—41no—
249C23×C31—41no—
250C23×C31—41no—
251C23×C31—41no—
252C23×C31#12012yes—
253C23×C31C22×D411yescharacteristic
254Q8×C31—24no—
255Q8×C31—42no—
256Q8×C31—42no—
257Q8×C31—42no—
258Q8×C31—81no—
259D4×C31—24no—
260D4×C31—44no—
261D4×C31—42no—
262D4×C31—42no—
263D4×C31—42no—
264D4×C31—48no—
265D4×C31—48no—
266D4×C31—42no—
267D4×C31—42no—
268D4×C31—82no—
269D4×C31—82no—
270D4×C31—82no—
271D4×C31—81no—
272D4×C31—82no—
273C2×C4×C31—21no—
274C2×C4×C31—21no—
275C2×C4×C31—22no—
276C2×C4×C31—24no—
277C2×C4×C31—22no—
278C2×C4×C31—22no—
279C2×C4×C31—21no—
280C2×C4×C31—22no—
281C2×C4×C31—22no—
282C2×C4×C31—22no—
283C2×C4×C31—21no—
284C2×C4×C31—42no—
285C2×C4×C31—41no—
286C2×C4×C31—44no—
287C2×C4×C31—44no—
288C2×C4×C31—41no—
289C2×C4×C31—41no—
290C2×C4×C31—41no—
291C2×C4×C31—41no—
292C2×C4×C31—44no—
293C2×C4×C31—42no—
294C2×C4×C31—44no—
295C2×C4×C31—41no—
296C2×C4×C31—41no—
297C2×C4×C31—41no—
298C2×C4×C31—41no—
299C2×C4×C31—41no—
300C2×C4×C31—81no—
301C2×C4×C31—82no—
302C2×C4×C31#12011yescharacteristic
303C2×C4×C31#12011yescharacteristic
304C2×C4×C31#13512yescomplemented
305C2×C4×C31C22×D412yes—
306C8×C31—42no—
307C8×C31—42no—
308#3460C2×C3111yescharacteristic, complemented
309#3002C2×C3112yescomplemented
310#2388C2×C3111yescharacteristic, complemented
311#1606C2×C3112yescomplemented
312#1605C2×C3111yescharacteristic, complemented
313#1604C2×C3111yescharacteristic, complemented
314#1600C2×C3112yescomplemented
315#1599C2×C3111yescharacteristic, complemented
316#1598C2×C3111yescharacteristic, complemented
317#1501C2×C3112yescomplemented
318#1459C2×C3111yescharacteristic, complemented
319C22×C31—21no—
320C22×C31—21no—
321C22×C31—22no—
322C22×C31—22no—
323C22×C31—22no—
324C22×C31—42no—
325C22×C31—42no—
326C22×C31—42no—
327C22×C31—42no—
328C22×C31—41no—
329C22×C31—42no—
330C22×C31—41no—
331C22×C31—42no—
332C22×C31—41no—
333C22×C31—42no—
334C22×C31—81no—
335C22×C31—81no—
336C22×C31—81no—
337C22×C31—81no—
338C22×C31—81no—
339C22×C31—81no—
340C22×C31—81no—
341C22×C31C2×#12011yescharacteristic
342C22×C31C2×#13511yescharacteristic
343C22×C31#57511yescharacteristic
344C4×C31—22no—
345C4×C31—22no—
346C4×C31—21no—
347C4×C31—42no—
348C4×C31—42no—
349C4×C31—41no—
350C4×C31—42no—
351C4×C31—42no—
352C4×C31—41no—
353C4×C31—42no—
354C4×C31—81no—
355C4×C31B(4,𝔽2)12yes—
356C22×#41C22×C3111yescharacteristic
357C2×#135—22no—
358#544C22×C3111yescharacteristic
359#542—22no—
360#540—22no—
361#538—21no—
362#538C22×C3112yes—
363#537C22×C3111yescharacteristic
364#535—21no—
365#532—22no—
366#529—22no—
367#525—21no—
368#525C22×C3112yes—
369C4×#41—21no—
370C2×C4×D4—21no—
371C2×C4×D4C22×C3111yescharacteristic
372#452—22no—
373#451C22×C3112yes—
374#450C22×C3112yes—
375#449—22no—
376#448C22×C3112yes—
377#447C22×C3112yes—
378#416—22no—
379C2×C22⋊C8C22×C3112yes—
380#395C22×C3111yescharacteristic
381#394—21no—
382#394C22×C3111yescharacteristic
383#393C22×C3111yescharacteristic
384#392—21no—
385#392C22×C3111yescharacteristic
386C4×C22⋊C4—21no—
387C4×C22⋊C4C22×C3111yescharacteristic
388#328C22×C3112yes—
389#328C22×C3112yes—
390#327C22×C3112yes—
391#327C22×C3112yes—
392#324C22×C3111yescharacteristic
393#324C22×C3111yescharacteristic
394#323C22×C3112yes—
395C2×C31—21no—
396C2×C31—21no—
397C2×C31—42no—
398C2×C31—82no—
399C2×C31—81no—
400C2×C31—82no—
401C2×C31#300211yescharacteristic
402C2×C31#301811yescharacteristic
403C2×C31C2×B(4,𝔽2)11yescharacteristic
404C2×#41—21no—
405C2×#41—21no—
406C2×#41—22no—
407C2×#41—22no—
408C2×#41—44no—
409C2×#41D4×C3112yescomplemented
410C22×Q8C23×C3111yescharacteristic
411C22×D4—21no—
412C22×D4—21no—
413C22×D4D4×C3111yescharacteristic
414C22×D4C23×C3111yescharacteristic
415C23×C4—21no—
416C23×C4D4×C3111yescharacteristic
417C23×C4C23×C3111yescharacteristic
418#135—48no—
419C2×Q16—22no—
420C2×SD16—22no—
421C2×SD16—22no—
422C2×D8—22no—
423C22×C8—22no—
424C4⋊Q8—22no—
425#127—22no—
426#126—22no—
427#126—22no—
428#125—22no—
429#124—22no—
430#124—22no—
431#124—22no—
432#123—22no—
433#123—22no—
434#123—22no—
435#123—22no—
436C22⋊Q8—21no—
437C22⋊Q8—21no—
438C22⋊Q8—22no—
439C22⋊Q8D4×C3112yescomplemented
440C22⋊Q8C23×C3112yes—
441#121—21no—
442#121—22no—
443#121—22no—
444#121—22no—
445#121—22no—
446#121—22no—
447#121—21no—
448#121D4×C3112yescomplemented
449#121C23×C3112yes—
450#120—22no—
451#120—22no—
452C4×Q8—21no—
453C4×Q8—21no—
454C4×D4—21no—
455C4×D4—21no—
456C4×D4—22no—
457C4×D4—22no—
458C4×D4—22no—
459C4×D4—22no—
460C4×D4—22no—
461C4×D4—22no—
462C4×D4—44no—
463#117—22no—
464#117—21no—
465#117—21no—
466#117D4×C3112yescomplemented
467C2×C4⋊C4—21no—
468C2×C4⋊C4C23×C3111yescharacteristic
469C2×C22⋊C4—21no—
470C2×C22⋊C4—21no—
471C2×C22⋊C4—21no—
472C2×C22⋊C4—21no—
473C2×C22⋊C4—21no—
474C2×C22⋊C4D4×C3111yescharacteristic
475C2×C22⋊C4C23×C3111yescharacteristic
476C2×C42—22no—
477C2×C42—21no—
478C2×C42D4×C3111yescharacteristic
479#103—22no—
480#103—22no—
481#102—22no—
482#102—22no—
483C22⋊C8—22no—
484C22⋊C8C23×C3112yes—
485C22⋊C8C23×C3112yes—
486#95—21no—
487#95—21no—
488#95C23×C3111yescharacteristic
489#95C23×C3111yescharacteristic
490C31#1811411yescharacteristic, direct product factor, Sylow
491C24—21no—
492C24C2×D4×C3111yescharacteristic
493C24C2×D4×C3111yescharacteristic
494#41—28no—
495#41—42no—
496#41—44no—
497#41—44no—
498#41—44no—
499#41—44no—
500#41—42no—
501#41—82no—
502C2×Q8—21no—
503C2×Q8—21no—
504C2×Q8—22no—
505C2×Q8—42no—
506C2×Q8C2×D4×C3112yes—
507C2×D4—21no—
508C2×D4—22no—
509C2×D4—22no—
510C2×D4—22no—
511C2×D4—22no—
512C2×D4—21no—
513C2×D4—22no—
514C2×D4—22no—
515C2×D4—22no—
516C2×D4—22no—
517C2×D4—41no—
518C2×D4—42no—
519C2×D4—42no—
520C2×D4—42no—
521C2×D4—42no—
522C2×D4—42no—
523C2×D4—42no—
524C2×D4—42no—
525C2×D4C2×D4×C3112yes—
526C22×C4—22no—
527C22×C4—24no—
528C22×C4—21no—
529C22×C4—21no—
530C22×C4—22no—
531C22×C4—22no—
532C22×C4—21no—
533C22×C4—21no—
534C22×C4—21no—
535C22×C4—42no—
536C22×C4—42no—
537C22×C4—42no—
538C22×C4—41no—
539C22×C4—41no—
540C22×C4C2×D4×C3111yescharacteristic
541C22×C4C2×D4×C3112yes—
542C22×C4C2×D4×C3111yescharacteristic
543C22×C4C2×D4×C3111yescharacteristic
544C22×C4C2×D4×C3111yescharacteristic
545C22×C4C2×D4×C3112yes—
546C22×C4C24×C3111yescharacteristic
547Q16—44no—
548SD16—44no—
549SD16—44no—
550D8—44no—
551C2×C8—22no—
552C2×C8—22no—
553C2×C8—44no—
554C4⋊C4—21no—
555C4⋊C4—21no—
556C4⋊C4—22no—
557C4⋊C4—21no—
558C4⋊C4—21no—
559C4⋊C4—22no—
560C4⋊C4—22no—
561C4⋊C4—22no—
562C4⋊C4—22no—
563C4⋊C4—42no—
564C4⋊C4C2×D4×C3112yes—
565C4⋊C4C2×D4×C3112yes—
566C22⋊C4—22no—
567C22⋊C4—22no—
568C22⋊C4—22no—
569C22⋊C4—22no—
570C22⋊C4—22no—
571C22⋊C4—22no—
572C22⋊C4—22no—
573C22⋊C4—22no—
574C22⋊C4—22no—
575C22⋊C4—22no—
576C22⋊C4—22no—
577C22⋊C4—42no—
578C22⋊C4—42no—
579C22⋊C4—42no—
580C22⋊C4—42no—
581C22⋊C4—42no—
582C42—22no—
583C42—22no—
584C42—21no—
585C42—22no—
586C42—21no—
587C42—42no—
588C23—21no—
589C23—21no—
590C23—22no—
591C23—21no—
592C23—22no—
593C23—21no—
594C23—21no—
595C23—41no—
596C23—41no—
597C23—41no—
598C23—41no—
599C23—41no—
600C23—41no—
601C23—41no—
602C23—41no—
603C23—41no—
604C23—41no—
605C23C31×#12012yes—
606C23C22×D4×C3111yescharacteristic
607Q8—24no—
608Q8—42no—
609Q8—42no—
610Q8—42no—
611Q8—81no—
612D4—24no—
613D4—44no—
614D4—42no—
615D4—42no—
616D4—42no—
617D4—48no—
618D4—48no—
619D4—42no—
620D4—42no—
621D4—82no—
622D4—82no—
623D4—82no—
624D4—81no—
625D4—82no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—22no—
629C2×C4—24no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—21no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—21no—
637C2×C4—42no—
638C2×C4—41no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—44no—
646C2×C4—42no—
647C2×C4—44no—
648C2×C4—41no—
649C2×C4—41no—
650C2×C4—41no—
651C2×C4—41no—
652C2×C4—41no—
653C2×C4—81no—
654C2×C4—82no—
655C2×C4C31×#12011yescharacteristic
656C2×C4C31×#12011yescharacteristic
657C2×C4C31×#13512yescomplemented
658C2×C4C22×D4×C3112yes—
659C8—42no—
660C8—42no—
661C22—21no—
662C22—21no—
663C22—22no—
664C22—22no—
665C22—22no—
666C22—42no—
667C22—42no—
668C22—42no—
669C22—42no—
670C22—41no—
671C22—42no—
672C22—41no—
673C22—42no—
674C22—41no—
675C22—42no—
676C22—81no—
677C22—81no—
678C22—81no—
679C22—81no—
680C22—81no—
681C22—81no—
682C22—81no—
683C22C2×C31×#12011yescharacteristic
684C22C2×C31×#13511yescharacteristic
685C22C31×#57511yescharacteristic
686C4—22no—
687C4—22no—
688C4—21no—
689C4—42no—
690C4—42no—
691C4—41no—
692C4—42no—
693C4—42no—
694C4—41no—
695C4—42no—
696C4—81no—
697C4C31×B(4,𝔽2)12yes—
698C2—21no—
699C2—21no—
700C2—42no—
701C2—81no—
702C2—82no—
703C2—82no—
704C2C31×#300211yescharacteristic
705C2C31×#301811yescharacteristic
706C2C2×C31×B(4,𝔽2)11yescharacteristic
subgroupG/H
centerC2×C4×C31#120
commutator, Frattini, cosocleC22×C4C24×C31
socleC22×C31C2×#120
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
1. C31×#3460C2semidirect
2. C31×#3002C2semidirect
3. C31×#2388C2semidirect
4. C31×#1606C2semidirect
5. C31×#1605C2semidirect
6. C31×#1604C2semidirect
7. C31×#1600C2semidirect
8. C31×#1599C2semidirect
9. C31×#1598C2semidirect
10. C31×#1501C2semidirect
11. C31×#1459C2semidirect
56. C2×#41×C31D4semidirect
86. C31×C22⋊Q8D4semidirect
95. C31×#121D4semidirect
113. C31×#117D4semidirect
234. #18114C31direct (N × H)
304. C2×C4×C31#135semidirect
308. #3460C2×C31semidirect
309. #3002C2×C31semidirect
310. #2388C2×C31semidirect
311. #1606C2×C31semidirect
312. #1605C2×C31semidirect
313. #1604C2×C31semidirect
314. #1600C2×C31semidirect
315. #1599C2×C31semidirect
316. #1598C2×C31semidirect
317. #1501C2×C31semidirect
318. #1459C2×C31semidirect
409. C2×#41D4×C31semidirect
439. C22⋊Q8D4×C31semidirect
448. #121D4×C31semidirect
466. #117D4×C31semidirect
490. C31#18114direct (N × H)
657. C2×C4C31×#135semidirect

Maximal subgroups

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

subgroupindex
C31×#34602
C31×#30022
C31×#23882
C31×#16062
C31×#16052
C31×#16042
C31×#16002
C31×#15992
C31×#15982
C31×#15012
C31×#14592
#1811431