Finite Group Databasebeta

← C31×#18290 #21475049

Subgroups — C31×#18290 #21475049

Order 7936.

1034 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3459C211yesmaximal, characteristic, complemented
2C31×#3188C212yesmaximal, complemented
3C31×#2635C211yesmaximal, characteristic, complemented
4C31×#1615C212yesmaximal, complemented
5C31×#1614C211yesmaximal, characteristic, complemented
6C31×#1613C211yesmaximal, characteristic, complemented
7C31×#1597C212yesmaximal, complemented
8C31×#1596C211yesmaximal, characteristic, complemented
9C31×#1595C211yesmaximal, characteristic, complemented
10C31×#1516C212yesmaximal, complemented
11C31×#1459C211yesmaximal, characteristic, complemented
12C31×#543C2211yescharacteristic
13C31×#539—22no—
14C31×#539C2212yes—
15C31×#538—22no—
16C31×#538—21no—
17C31×#534—21no—
18C31×#534C2211yescharacteristic
19C31×#529—22no—
20C31×#529—21no—
21C31×#525—22no—
22C31×#525C2212yes—
23C2×C31×#121C2211yescharacteristic, complemented
24C2×C4×D4×C31—21no—
25C2×C4×D4×C31C2211yescharacteristic
26C31×#484C2212yescomplemented
27C31×#482—22no—
28C31×#481C2212yes—
29C31×#466—22no—
30C31×#465—22no—
31C31×#450C2212yescomplemented
32C31×#449—22no—
33C31×#447C2212yes—
34C2×C31×C22⋊C8C2212yescomplemented
35C31×#400C2211yescharacteristic, complemented
36C31×#399C2211yescharacteristic
37C31×#394—21no—
38C31×#394C2211yescharacteristic, complemented
39C31×#392—21no—
40C31×#392C2211yescharacteristic
41C31×#388—21no—
42C31×#386—21no—
43C31×#381C2211yescharacteristic, complemented
44C31×#328C2212yescomplemented
45C31×#328C2212yescomplemented
46C31×#327C2212yes—
47C31×#327C2212yes—
48C31×#324C2212yescomplemented
49C31×#323C2211yescharacteristic
50C31×#323C2211yescharacteristic
51C2×#41×C31D412yescomplemented
52C22×D4×C31—21no—
53C22×D4×C31—21no—
54C22×D4×C31—21no—
55C22×D4×C31D411yescharacteristic
56C22×D4×C31C2311yescharacteristic
57C23×C4×C31—21no—
58C23×C4×C31D411yescharacteristic
59C23×C4×C31C2311yescharacteristic
60C2×SD16×C31—22no—
61C2×D8×C31—22no—
62C22×C8×C31—22no—
63C31×#126—22no—
64C31×#126—22no—
65C31×#126—22no—
66C31×#125—21no—
67C31×#125—22no—
68C31×#124—22no—
69C31×#124—21no—
70C31×#123—22no—
71C31×#123—22no—
72C31×#123—22no—
73C31×#123—22no—
74C31×#123—22no—
75C31×#123—22no—
76C31×C22⋊Q8—22no—
77C31×C22⋊Q8—22no—
78C31×C22⋊Q8—21no—
79C31×C22⋊Q8—22no—
80C31×C22⋊Q8D412yescomplemented
81C31×C22⋊Q8C2312yes—
82C31×#121—22no—
83C31×#121—22no—
84C31×#121—22no—
85C31×#121—21no—
86C31×#121—22no—
87C31×#121—22no—
88C31×#121D412yescomplemented
89C31×#121C2312yes—
90C31×#120—22no—
91C31×#120—22no—
92C31×#120—22no—
93C4×Q8×C31—21no—
94C4×D4×C31—22no—
95C4×D4×C31—22no—
96C4×D4×C31—21no—
97C4×D4×C31—22no—
98C4×D4×C31—22no—
99C4×D4×C31—22no—
100C4×D4×C31—22no—
101C4×D4×C31—22no—
102C31×#117—22no—
103C31×#117D412yescomplemented
104C2×C4⋊C4×C31—21no—
105C2×C4⋊C4×C31—21no—
106C2×C4⋊C4×C31C2311yescharacteristic
107C2×C4⋊C4×C31C2311yescharacteristic
108C2×C22⋊C4×C31—21no—
109C2×C22⋊C4×C31—21no—
110C2×C22⋊C4×C31—21no—
111C2×C22⋊C4×C31—21no—
112C2×C22⋊C4×C31—21no—
113C2×C22⋊C4×C31—21no—
114C2×C22⋊C4×C31—21no—
115C2×C22⋊C4×C31—21no—
116C2×C22⋊C4×C31D411yescharacteristic
117C2×C22⋊C4×C31C2311yescharacteristic
118C2×C42×C31—21no—
119C2×C42×C31D411yescharacteristic
120C31×#107—22no—
121C31×#106—22no—
122C31×#103—22no—
123C31×#102—22no—
124C31×#102—22no—
125C31×#102—22no—
126C31×C22⋊C8—22no—
127C31×C22⋊C8C2312yes—
128C31×C22⋊C8C2312yes—
129C31×#95—21no—
130C31×#95—21no—
131C31×#95—21no—
132C31×#95C2311yescharacteristic
133C31×#95C2311yescharacteristic
134C24×C31—21no—
135C24×C31—21no—
136C24×C31C2×D411yescharacteristic
137C24×C31C2×D411yescharacteristic
138#41×C31—82no—
139C2×Q8×C31—21no—
140C2×Q8×C31C2×D412yes—
141C2×D4×C31—22no—
142C2×D4×C31—22no—
143C2×D4×C31—22no—
144C2×D4×C31—22no—
145C2×D4×C31—21no—
146C2×D4×C31—22no—
147C2×D4×C31—22no—
148C2×D4×C31—22no—
149C2×D4×C31—41no—
150C2×D4×C31—42no—
151C2×D4×C31—42no—
152C2×D4×C31—42no—
153C2×D4×C31—42no—
154C2×D4×C31—81no—
155C2×D4×C31—81no—
156C2×D4×C31—81no—
157C2×D4×C31—81no—
158C2×D4×C31—81no—
159C2×D4×C31C2×D412yes—
160C22×C4×C31—21no—
161C22×C4×C31—21no—
162C22×C4×C31—21no—
163C22×C4×C31—22no—
164C22×C4×C31—21no—
165C22×C4×C31—22no—
166C22×C4×C31—21no—
167C22×C4×C31—21no—
168C22×C4×C31—21no—
169C22×C4×C31—21no—
170C22×C4×C31—81no—
171C22×C4×C31—81no—
172C22×C4×C31—82no—
173C22×C4×C31C2×D411yescharacteristic
174C22×C4×C31C2×D411yescharacteristic
175C22×C4×C31C2×D411yescharacteristic
176C22×C4×C31C2×D411yescharacteristic
177C22×C4×C31C2×D412yes—
178C22×C4×C31C2×D412yes—
179C22×C4×C31C2411yescharacteristic
180SD16×C31—82no—
181D8×C31—82no—
182C2×C8×C31—22no—
183C2×C8×C31—22no—
184C2×C8×C31—82no—
185C4⋊C4×C31—22no—
186C4⋊C4×C31—22no—
187C4⋊C4×C31—22no—
188C4⋊C4×C31—21no—
189C4⋊C4×C31—21no—
190C4⋊C4×C31—22no—
191C4⋊C4×C31—22no—
192C4⋊C4×C31—22no—
193C4⋊C4×C31—22no—
194C4⋊C4×C31—22no—
195C4⋊C4×C31—81no—
196C4⋊C4×C31C2×D412yes—
197C4⋊C4×C31C2×D412yes—
198C22⋊C4×C31—22no—
199C22⋊C4×C31—22no—
200C22⋊C4×C31—22no—
201C22⋊C4×C31—22no—
202C22⋊C4×C31—22no—
203C22⋊C4×C31—22no—
204C22⋊C4×C31—22no—
205C22⋊C4×C31—22no—
206C22⋊C4×C31—22no—
207C22⋊C4×C31—22no—
208C22⋊C4×C31—22no—
209C22⋊C4×C31—22no—
210C22⋊C4×C31—42no—
211C22⋊C4×C31—41no—
212C22⋊C4×C31—41no—
213C22⋊C4×C31—81no—
214C22⋊C4×C31—81no—
215C22⋊C4×C31—81no—
216C22⋊C4×C31—81no—
217C42×C31—22no—
218C42×C31—22no—
219C42×C31—21no—
220C42×C31—21no—
221#18290C3111yesmaximal, characteristic, direct product factor, Sylow
222C23×C31—21no—
223C23×C31—21no—
224C23×C31—21no—
225C23×C31—21no—
226C23×C31—22no—
227C23×C31—41no—
228C23×C31—41no—
229C23×C31—41no—
230C23×C31—41no—
231C23×C31—42no—
232C23×C31—41no—
233C23×C31—41no—
234C23×C31—41no—
235C23×C31—41no—
236C23×C31—41no—
237C23×C31—41no—
238C23×C31—41no—
239C23×C31—81no—
240C23×C31—81no—
241C23×C31#12012yes—
242C23×C31#13512yes—
243C23×C31C22×D411yescharacteristic
244Q8×C31—42no—
245Q8×C31—81no—
246D4×C31—42no—
247D4×C31—42no—
248D4×C31—42no—
249D4×C31—44no—
250D4×C31—44no—
251D4×C31—42no—
252D4×C31—42no—
253D4×C31—82no—
254D4×C31—82no—
255D4×C31—81no—
256D4×C31—84no—
257D4×C31—84no—
258C2×C4×C31—21no—
259C2×C4×C31—22no—
260C2×C4×C31—21no—
261C2×C4×C31—22no—
262C2×C4×C31—21no—
263C2×C4×C31—22no—
264C2×C4×C31—21no—
265C2×C4×C31—21no—
266C2×C4×C31—42no—
267C2×C4×C31—41no—
268C2×C4×C31—42no—
269C2×C4×C31—41no—
270C2×C4×C31—42no—
271C2×C4×C31—44no—
272C2×C4×C31—42no—
273C2×C4×C31—41no—
274C2×C4×C31—41no—
275C2×C4×C31—41no—
276C2×C4×C31—41no—
277C2×C4×C31—41no—
278C2×C4×C31—41no—
279C2×C4×C31—41no—
280C2×C4×C31—41no—
281C2×C4×C31—41no—
282C2×C4×C31—41no—
283C2×C4×C31—41no—
284C2×C4×C31—81no—
285C2×C4×C31—81no—
286C2×C4×C31—82no—
287C2×C4×C31—81no—
288C2×C4×C31#12011yescharacteristic
289C2×C4×C31#12011yescharacteristic
290C2×C4×C31C22×D412yes—
291C8×C31—42no—
292C8×C31—42no—
293#3459C2×C3111yescharacteristic, complemented
294#3188C2×C3112yescomplemented
295#2635C2×C3111yescharacteristic, complemented
296#1615C2×C3112yescomplemented
297#1614C2×C3111yescharacteristic, complemented
298#1613C2×C3111yescharacteristic, complemented
299#1597C2×C3112yescomplemented
300#1596C2×C3111yescharacteristic, complemented
301#1595C2×C3111yescharacteristic, complemented
302#1516C2×C3112yescomplemented
303#1459C2×C3111yescharacteristic, complemented
304C22×C31—21no—
305C22×C31—21no—
306C22×C31—22no—
307C22×C31—22no—
308C22×C31—22no—
309C22×C31—22no—
310C22×C31—42no—
311C22×C31—41no—
312C22×C31—41no—
313C22×C31—42no—
314C22×C31—42no—
315C22×C31—42no—
316C22×C31—81no—
317C22×C31—81no—
318C22×C31—81no—
319C22×C31—81no—
320C22×C31—81no—
321C22×C31—81no—
322C22×C31—81no—
323C22×C31—84no—
324C22×C31—81no—
325C22×C31—81no—
326C22×C31—81no—
327C22×C31C2×#12011yescharacteristic
328C22×C31C2×#13511yescharacteristic
329C22×C31#57511yescharacteristic
330C4×C31—22no—
331C4×C31—21no—
332C4×C31—21no—
333C4×C31—41no—
334C4×C31—42no—
335C4×C31—41no—
336C4×C31—42no—
337C4×C31—41no—
338C4×C31—42no—
339C4×C31—81no—
340C4×C31—81no—
341C4×C31—81no—
342#543C22×C3111yescharacteristic
343#539—22no—
344#539C22×C3112yes—
345#538—22no—
346#538—21no—
347#534—21no—
348#534C22×C3111yescharacteristic
349#529—22no—
350#529—21no—
351#525—22no—
352#525C22×C3112yes—
353C2×#121C22×C3111yescharacteristic, complemented
354C2×C4×D4—21no—
355C2×C4×D4C22×C3111yescharacteristic
356#484C22×C3112yescomplemented
357#482—22no—
358#481C22×C3112yes—
359#466—22no—
360#465—22no—
361#450C22×C3112yescomplemented
362#449—22no—
363#447C22×C3112yes—
364C2×C22⋊C8C22×C3112yescomplemented
365#400C22×C3111yescharacteristic, complemented
366#399C22×C3111yescharacteristic
367#394—21no—
368#394C22×C3111yescharacteristic, complemented
369#392—21no—
370#392C22×C3111yescharacteristic
371#388—21no—
372#386—21no—
373#381C22×C3111yescharacteristic, complemented
374#328C22×C3112yescomplemented
375#328C22×C3112yescomplemented
376#327C22×C3112yes—
377#327C22×C3112yes—
378#324C22×C3112yescomplemented
379#323C22×C3111yescharacteristic
380#323C22×C3111yescharacteristic
381C2×C31—21no—
382C2×C31—21no—
383C2×C31—42no—
384C2×C31—42no—
385C2×C31—82no—
386C2×C31—161no—
387C2×C31#300211yescharacteristic
388C2×C31#302111yescharacteristic
389C2×C31#302611yescharacteristic
390C2×#41D4×C3112yescomplemented
391C22×D4—21no—
392C22×D4—21no—
393C22×D4—21no—
394C22×D4D4×C3111yescharacteristic
395C22×D4C23×C3111yescharacteristic
396C23×C4—21no—
397C23×C4D4×C3111yescharacteristic
398C23×C4C23×C3111yescharacteristic
399C2×SD16—22no—
400C2×D8—22no—
401C22×C8—22no—
402#126—22no—
403#126—22no—
404#126—22no—
405#125—21no—
406#125—22no—
407#124—22no—
408#124—21no—
409#123—22no—
410#123—22no—
411#123—22no—
412#123—22no—
413#123—22no—
414#123—22no—
415C22⋊Q8—22no—
416C22⋊Q8—22no—
417C22⋊Q8—21no—
418C22⋊Q8—22no—
419C22⋊Q8D4×C3112yescomplemented
420C22⋊Q8C23×C3112yes—
421#121—22no—
422#121—22no—
423#121—22no—
424#121—21no—
425#121—22no—
426#121—22no—
427#121D4×C3112yescomplemented
428#121C23×C3112yes—
429#120—22no—
430#120—22no—
431#120—22no—
432C4×Q8—21no—
433C4×D4—22no—
434C4×D4—22no—
435C4×D4—21no—
436C4×D4—22no—
437C4×D4—22no—
438C4×D4—22no—
439C4×D4—22no—
440C4×D4—22no—
441#117—22no—
442#117D4×C3112yescomplemented
443C2×C4⋊C4—21no—
444C2×C4⋊C4—21no—
445C2×C4⋊C4C23×C3111yescharacteristic
446C2×C4⋊C4C23×C3111yescharacteristic
447C2×C22⋊C4—21no—
448C2×C22⋊C4—21no—
449C2×C22⋊C4—21no—
450C2×C22⋊C4—21no—
451C2×C22⋊C4—21no—
452C2×C22⋊C4—21no—
453C2×C22⋊C4—21no—
454C2×C22⋊C4—21no—
455C2×C22⋊C4D4×C3111yescharacteristic
456C2×C22⋊C4C23×C3111yescharacteristic
457C2×C42—21no—
458C2×C42D4×C3111yescharacteristic
459#107—22no—
460#106—22no—
461#103—22no—
462#102—22no—
463#102—22no—
464#102—22no—
465C22⋊C8—22no—
466C22⋊C8C23×C3112yes—
467C22⋊C8C23×C3112yes—
468#95—21no—
469#95—21no—
470#95—21no—
471#95C23×C3111yescharacteristic
472#95C23×C3111yescharacteristic
473C31#1829011yescharacteristic, direct product factor, Sylow
474C24—21no—
475C24—21no—
476C24C2×D4×C3111yescharacteristic
477C24C2×D4×C3111yescharacteristic
478#41—82no—
479C2×Q8—21no—
480C2×Q8C2×D4×C3112yes—
481C2×D4—22no—
482C2×D4—22no—
483C2×D4—22no—
484C2×D4—22no—
485C2×D4—21no—
486C2×D4—22no—
487C2×D4—22no—
488C2×D4—22no—
489C2×D4—41no—
490C2×D4—42no—
491C2×D4—42no—
492C2×D4—42no—
493C2×D4—42no—
494C2×D4—81no—
495C2×D4—81no—
496C2×D4—81no—
497C2×D4—81no—
498C2×D4—81no—
499C2×D4C2×D4×C3112yes—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—22no—
504C22×C4—21no—
505C22×C4—22no—
506C22×C4—21no—
507C22×C4—21no—
508C22×C4—21no—
509C22×C4—21no—
510C22×C4—81no—
511C22×C4—81no—
512C22×C4—82no—
513C22×C4C2×D4×C3111yescharacteristic
514C22×C4C2×D4×C3111yescharacteristic
515C22×C4C2×D4×C3111yescharacteristic
516C22×C4C2×D4×C3111yescharacteristic
517C22×C4C2×D4×C3112yes—
518C22×C4C2×D4×C3112yes—
519C22×C4C24×C3111yescharacteristic
520SD16—82no—
521D8—82no—
522C2×C8—22no—
523C2×C8—22no—
524C2×C8—82no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—81no—
536C4⋊C4C2×D4×C3112yes—
537C4⋊C4C2×D4×C3112yes—
538C22⋊C4—22no—
539C22⋊C4—22no—
540C22⋊C4—22no—
541C22⋊C4—22no—
542C22⋊C4—22no—
543C22⋊C4—22no—
544C22⋊C4—22no—
545C22⋊C4—22no—
546C22⋊C4—22no—
547C22⋊C4—22no—
548C22⋊C4—22no—
549C22⋊C4—22no—
550C22⋊C4—42no—
551C22⋊C4—41no—
552C22⋊C4—41no—
553C22⋊C4—81no—
554C22⋊C4—81no—
555C22⋊C4—81no—
556C22⋊C4—81no—
557C42—22no—
558C42—22no—
559C42—21no—
560C42—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—22no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—42no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23—41no—
577C23—41no—
578C23—81no—
579C23—81no—
580C23C31×#12012yes—
581C23C31×#13512yes—
582C23C22×D4×C3111yescharacteristic
583Q8—42no—
584Q8—81no—
585D4—42no—
586D4—42no—
587D4—42no—
588D4—44no—
589D4—44no—
590D4—42no—
591D4—42no—
592D4—82no—
593D4—82no—
594D4—81no—
595D4—84no—
596D4—84no—
597C2×C4—21no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—21no—
602C2×C4—22no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—81no—
624C2×C4—81no—
625C2×C4—82no—
626C2×C4—81no—
627C2×C4C31×#12011yescharacteristic
628C2×C4C31×#12011yescharacteristic
629C2×C4C22×D4×C3112yes—
630C8—42no—
631C8—42no—
632C22—21no—
633C22—21no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—42no—
639C22—41no—
640C22—41no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—81no—
645C22—81no—
646C22—81no—
647C22—81no—
648C22—81no—
649C22—81no—
650C22—81no—
651C22—84no—
652C22—81no—
653C22—81no—
654C22—81no—
655C22C2×C31×#12011yescharacteristic
656C22C2×C31×#13511yescharacteristic
657C22C31×#57511yescharacteristic
658C4—22no—
659C4—21no—
660C4—21no—
661C4—41no—
662C4—42no—
663C4—41no—
664C4—42no—
665C4—41no—
666C4—42no—
667C4—81no—
668C4—81no—
669C4—81no—
670C2—21no—
671C2—21no—
672C2—42no—
673C2—42no—
674C2—82no—
675C2—161no—
676C2C31×#300211yescharacteristic
677C2C31×#302111yescharacteristic
678C2C31×#302611yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#3459C2semidirect
2. C31×#3188C2semidirect
3. C31×#2635C2semidirect
4. C31×#1615C2semidirect
5. C31×#1614C2semidirect
6. C31×#1613C2semidirect
7. C31×#1597C2semidirect
8. C31×#1596C2semidirect
9. C31×#1595C2semidirect
10. C31×#1516C2semidirect
11. C31×#1459C2semidirect
23. C2×C31×#121C22semidirect
26. C31×#484C22semidirect
31. C31×#450C22semidirect
34. C2×C31×C22⋊C8C22semidirect
35. C31×#400C22semidirect
38. C31×#394C22semidirect
43. C31×#381C22semidirect
44. C31×#328C22semidirect
45. C31×#328C22semidirect
48. C31×#324C22semidirect
51. C2×#41×C31D4semidirect
80. C31×C22⋊Q8D4semidirect
88. C31×#121D4semidirect
103. C31×#117D4semidirect
221. #18290C31direct (N × H)
293. #3459C2×C31semidirect
294. #3188C2×C31semidirect
295. #2635C2×C31semidirect
296. #1615C2×C31semidirect
297. #1614C2×C31semidirect
298. #1613C2×C31semidirect
299. #1597C2×C31semidirect
300. #1596C2×C31semidirect
301. #1595C2×C31semidirect
302. #1516C2×C31semidirect
303. #1459C2×C31semidirect
353. C2×#121C22×C31semidirect
356. #484C22×C31semidirect
361. #450C22×C31semidirect
364. C2×C22⋊C8C22×C31semidirect
365. #400C22×C31semidirect
368. #394C22×C31semidirect
373. #381C22×C31semidirect
374. #328C22×C31semidirect
375. #328C22×C31semidirect
378. #324C22×C31semidirect
390. C2×#41D4×C31semidirect
419. C22⋊Q8D4×C31semidirect
427. #121D4×C31semidirect
442. #117D4×C31semidirect
473. C31#18290direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#34592
C31×#31882
C31×#26352
C31×#16152
C31×#16142
C31×#16132
C31×#15972
C31×#15962
C31×#15952
C31×#15162
C31×#14592
#1829031