Finite Group Databasebeta

← C31×#18586 #21475345

Subgroups — C31×#18586 #21475345

Order 7936.

964 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3520C211yesmaximal, characteristic
2C31×#3239C211yesmaximal, characteristic, complemented
3C31×#3165C211yesmaximal, characteristic, complemented
4C2×C31×#461C211yesmaximal, characteristic, complemented
5C31×#1654C212yesmaximal, complemented
6C31×#1651C212yesmaximal, complemented
7C31×#1650C212yesmaximal, complemented
8C31×#1647C212yesmaximal, complemented
9C31×#1552C211yesmaximal, characteristic, complemented
10C31×#1490C211yesmaximal, characteristic, complemented
11C31×#1483C211yesmaximal, characteristic, complemented
12C22×SD16×C31—21no—
13C31×#556C2211yescharacteristic
14C31×#554—21no—
15C31×#552—21no—
16C31×#551—22no—
17D4×Q8×C31C2211yescharacteristic
18C31×#548—21no—
19C31×#547C2211yescharacteristic
20C31×#547C2211yescharacteristic
21C31×#546—21no—
22C31×#542—21no—
23C31×#539—22no—
24C31×#533C2211yescharacteristic
25C2×C31×C4⋊Q8C2211yescharacteristic
26C31×#529C2211yescharacteristic
27C31×#525—21no—
28C2×C31×C22⋊Q8—21no—
29C31×#520—21no—
30C2×C4×D4×C31C2211yescharacteristic
31C31×#501—22no—
32C31×#501—21no—
33C31×#500C2212yescomplemented
34C31×#498C2212yescomplemented
35C31×#489—21no—
36C31×#484—21no—
37C31×#481—21no—
38C31×#475C2212yescomplemented
39C31×#474C2212yescomplemented
40C31×#461—21no—
41C31×#461—21no—
42C31×#461C2212yes—
43C31×#461C2212yes—
44C31×#432C2211yescharacteristic
45C31×#428—22no—
46C2×C31×C4⋊C8C2211yescharacteristic
47C2×C31×#103—21no—
48C4×M16×C31C2211yescharacteristic
49C31×#332C2212yescomplemented
50C31×#332C2212yescomplemented
51C31×Q8⋊C8C2212yescomplemented
52C31×D4⋊C8C2212yes—
53C31×D4⋊C8C2212yes—
54C31×D4⋊C8C2212yescomplemented
55C2×#41×C31—21no—
56C2×#41×C31—21no—
57C22×Q8×C31—21no—
58C22×Q8×C31—21no—
59C22×D4×C31D411yescharacteristic
60C23×C4×C31—21no—
61C2×SD16×C31—22no—
62C2×SD16×C31—22no—
63C2×SD16×C31—44no—
64C2×M16×C31—21no—
65C22×C8×C31—21no—
66C31×C4⋊Q8—21no—
67C31×C4⋊Q8—21no—
68C31×C4⋊Q8—21no—
69C31×C4⋊Q8—21no—
70C31×C4⋊Q8C2312yes—
71C31×C4⋊Q8C2311yescharacteristic
72C31×C4⋊Q8C2311yescharacteristic
73C31×#126—22no—
74C31×#126—22no—
75C31×#126—21no—
76C31×#125—22no—
77C31×#125D411yescharacteristic
78C31×#125D412yes—
79C31×#124—22no—
80C31×#124—21no—
81C31×#124D411yescharacteristic
82C31×#123—22no—
83C31×#123—22no—
84C31×#123—21no—
85C31×#123—22no—
86C31×#123—22no—
87C31×C22⋊Q8—22no—
88C31×C22⋊Q8—22no—
89C31×C22⋊Q8—22no—
90C31×C22⋊Q8—21no—
91C31×C22⋊Q8—21no—
92C31×C22⋊Q8—21no—
93C31×C22⋊Q8—21no—
94C31×C22⋊Q8—21no—
95C31×C22⋊Q8—22no—
96C31×C22⋊Q8—21no—
97C31×C22⋊Q8D411yescharacteristic
98C31×#121—22no—
99C31×#121—21no—
100C31×#121—21no—
101C31×#121D411yescharacteristic
102C31×#120—22no—
103C4×Q8×C31—21no—
104C4×Q8×C31—21no—
105C4×Q8×C31—21no—
106C4×Q8×C31C2311yescharacteristic
107C4×D4×C31—22no—
108C4×D4×C31—22no—
109C4×D4×C31—22no—
110C4×D4×C31—21no—
111C4×D4×C31—21no—
112C4×D4×C31—21no—
113C4×D4×C31—21no—
114C4×D4×C31D411yescharacteristic
115C4×D4×C31D411yescharacteristic
116C4×D4×C31C2311yescharacteristic
117C4×D4×C31C2311yescharacteristic
118C4×D4×C31C2311yescharacteristic
119C31×#117—22no—
120C31×#117—21no—
121C31×#117—21no—
122C31×#117D412yes—
123C2×C4⋊C4×C31—22no—
124C2×C4⋊C4×C31—21no—
125C2×C4⋊C4×C31—21no—
126C2×C4⋊C4×C31—21no—
127C2×C4⋊C4×C31—21no—
128C2×C4⋊C4×C31D411yescharacteristic
129C2×C22⋊C4×C31—22no—
130C2×C22⋊C4×C31—21no—
131C2×C42×C31C2311yescharacteristic
132C31×#107—24no—
133C31×#107—22no—
134C31×#106—24no—
135C31×#106—22no—
136C31×C4⋊C8—21no—
137C31×C4⋊C8C2312yes—
138C31×C4⋊C8C2312yes—
139C31×#103—22no—
140C31×#103—22no—
141C31×#103—22no—
142C31×#102—22no—
143C31×C22⋊C8—21no—
144C31×#97—21no—
145C31×#97—21no—
146C4×C8×C31C2312yes—
147C24×C31—21no—
148#41×C31—42no—
149#41×C31—42no—
150#41×C31—82no—
151C2×Q8×C31—22no—
152C2×Q8×C31—21no—
153C2×Q8×C31—21no—
154C2×Q8×C31—22no—
155C2×Q8×C31—42no—
156C2×Q8×C31—42no—
157C2×Q8×C31—41no—
158C2×Q8×C31—82no—
159C2×Q8×C31C2×D411yescharacteristic
160C2×D4×C31—22no—
161C2×D4×C31—22no—
162C2×D4×C31—21no—
163C2×D4×C31—22no—
164C2×D4×C31—21no—
165C2×D4×C31—41no—
166C2×D4×C31C2×D411yescharacteristic
167C2×D4×C31C2×D411yescharacteristic
168C2×D4×C31C2×D411yescharacteristic
169C22×C4×C31—22no—
170C22×C4×C31—22no—
171C22×C4×C31—22no—
172C22×C4×C31—21no—
173C22×C4×C31—21no—
174C22×C4×C31—21no—
175C22×C4×C31—21no—
176C22×C4×C31—21no—
177C22×C4×C31—21no—
178C22×C4×C31—41no—
179C22×C4×C31—82no—
180C22×C4×C31C2×D411yescharacteristic
181C22×C4×C31C2×D411yescharacteristic
182C22×C4×C31C2×D411yescharacteristic
183SD16×C31—44no—
184SD16×C31—44no—
185M16×C31—24no—
186C2×C8×C31—22no—
187C2×C8×C31—22no—
188C2×C8×C31—22no—
189C2×C8×C31—42no—
190C4⋊C4×C31—22no—
191C4⋊C4×C31—22no—
192C4⋊C4×C31—22no—
193C4⋊C4×C31—22no—
194C4⋊C4×C31—21no—
195C4⋊C4×C31—22no—
196C4⋊C4×C31—21no—
197C4⋊C4×C31—21no—
198C4⋊C4×C31—21no—
199C4⋊C4×C31—22no—
200C4⋊C4×C31—21no—
201C4⋊C4×C31—21no—
202C4⋊C4×C31—21no—
203C4⋊C4×C31—22no—
204C4⋊C4×C31—22no—
205C4⋊C4×C31—41no—
206C4⋊C4×C31—41no—
207C4⋊C4×C31C2×D411yescharacteristic
208C4⋊C4×C31C2×D411yescharacteristic
209C4⋊C4×C31C2×D411yescharacteristic
210C4⋊C4×C31C2×D412yes—
211C4⋊C4×C31C2×D411yescharacteristic
212C4⋊C4×C31C2×D412yes—
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—22no—
220C22⋊C4×C31—21no—
221C22⋊C4×C31—21no—
222C22⋊C4×C31—21no—
223C22⋊C4×C31—21no—
224C22⋊C4×C31—21no—
225C22⋊C4×C31—21no—
226C22⋊C4×C31—21no—
227C22⋊C4×C31—21no—
228C42×C31—22no—
229C42×C31—22no—
230C42×C31—21no—
231C42×C31C2×D411yescharacteristic
232C42×C31C2×D411yescharacteristic
233C42×C31C2×D411yescharacteristic
234C42×C31C2411yescharacteristic
235#18586C3111yesmaximal, characteristic, direct product factor, Sylow
236C23×C31—21no—
237C23×C31—21no—
238C23×C31—21no—
239C23×C31—42no—
240C23×C31—42no—
241C23×C31—81no—
242C23×C31#12011yescharacteristic
243Q8×C31—22no—
244Q8×C31—41no—
245Q8×C31—41no—
246Q8×C31—42no—
247Q8×C31—42no—
248Q8×C31—42no—
249Q8×C31—82no—
250Q8×C31—82no—
251D4×C31—21no—
252D4×C31—21no—
253D4×C31—21no—
254D4×C31—22no—
255D4×C31—42no—
256D4×C31—44no—
257D4×C31—42no—
258D4×C31—82no—
259D4×C31AGL(1,ℤ/8)12yes—
260C2×C4×C31—22no—
261C2×C4×C31—21no—
262C2×C4×C31—21no—
263C2×C4×C31—22no—
264C2×C4×C31—21no—
265C2×C4×C31—22no—
266C2×C4×C31—21no—
267C2×C4×C31—21no—
268C2×C4×C31—22no—
269C2×C4×C31—21no—
270C2×C4×C31—21no—
271C2×C4×C31—44no—
272C2×C4×C31—41no—
273C2×C4×C31—41no—
274C2×C4×C31—42no—
275C2×C4×C31—42no—
276C2×C4×C31—42no—
277C2×C4×C31—81no—
278C2×C4×C31—81no—
279C2×C4×C31—81no—
280C2×C4×C31—82no—
281C2×C4×C31—82no—
282C2×C4×C31—82no—
283C2×C4×C31#12011yescharacteristic
284C2×C4×C31#12011yescharacteristic
285C2×C4×C31#12011yescharacteristic
286C2×C4×C31C22×D411yescharacteristic
287C2×C4×C31C22×D411yescharacteristic
288C2×C4×C31C22×D411yescharacteristic
289C8×C31—24no—
290C8×C31—42no—
291C8×C31—42no—
292#3520C2×C3111yescharacteristic
293#3239C2×C3111yescharacteristic, complemented
294#3165C2×C3111yescharacteristic, complemented
295C2×#461C2×C3111yescharacteristic, complemented
296#1654C2×C3112yescomplemented
297#1651C2×C3112yescomplemented
298#1650C2×C3112yescomplemented
299#1647C2×C3112yescomplemented
300#1552C2×C3111yescharacteristic, complemented
301#1490C2×C3111yescharacteristic, complemented
302#1483C2×C3111yescharacteristic, complemented
303C22×C31—21no—
304C22×C31—21no—
305C22×C31—22no—
306C22×C31—41no—
307C22×C31—41no—
308C22×C31—41no—
309C22×C31—41no—
310C22×C31—41no—
311C22×C31—41no—
312C22×C31—41no—
313C22×C31—41no—
314C22×C31—84no—
315C22×C31#45612yes—
316C22×C31C2×#12011yescharacteristic
317C4×C31—21no—
318C4×C31—21no—
319C4×C31—21no—
320C4×C31—21no—
321C4×C31—41no—
322C4×C31—42no—
323C4×C31—41no—
324C4×C31—41no—
325C4×C31—42no—
326C4×C31—41no—
327C4×C31—41no—
328C4×C31—82no—
329C4×C31C2×AGL(1,ℤ/8)11yescharacteristic
330C4×C31#57511yescharacteristic
331C4×C31#57712yes—
332C22×SD16—21no—
333#556C22×C3111yescharacteristic
334#554—21no—
335#552—21no—
336#551—22no—
337D4×Q8C22×C3111yescharacteristic
338#548—21no—
339#547C22×C3111yescharacteristic
340#547C22×C3111yescharacteristic
341#546—21no—
342#542—21no—
343#539—22no—
344#533C22×C3111yescharacteristic
345C2×C4⋊Q8C22×C3111yescharacteristic
346#529C22×C3111yescharacteristic
347#525—21no—
348C2×C22⋊Q8—21no—
349#520—21no—
350C2×C4×D4C22×C3111yescharacteristic
351#501—22no—
352#501—21no—
353#500C22×C3112yescomplemented
354#498C22×C3112yescomplemented
355#489—21no—
356#484—21no—
357#481—21no—
358#475C22×C3112yescomplemented
359#474C22×C3112yescomplemented
360#461—21no—
361#461—21no—
362#461C22×C3112yes—
363#461C22×C3112yes—
364#432C22×C3111yescharacteristic
365#428—22no—
366C2×C4⋊C8C22×C3111yescharacteristic
367C2×#103—21no—
368C4×M16C22×C3111yescharacteristic
369#332C22×C3112yescomplemented
370#332C22×C3112yescomplemented
371Q8⋊C8C22×C3112yescomplemented
372D4⋊C8C22×C3112yes—
373D4⋊C8C22×C3112yes—
374D4⋊C8C22×C3112yescomplemented
375C2×C31—22no—
376C2×C31—42no—
377C2×C31—81no—
378C2×C31—81no—
379C2×C31#300611yescharacteristic
380C2×C31#300911yescharacteristic
381C2×C31C2×#45611yescharacteristic
382C2×#41—21no—
383C2×#41—21no—
384C22×Q8—21no—
385C22×Q8—21no—
386C22×D4D4×C3111yescharacteristic
387C23×C4—21no—
388C2×SD16—22no—
389C2×SD16—22no—
390C2×SD16—44no—
391C2×M16—21no—
392C22×C8—21no—
393C4⋊Q8—21no—
394C4⋊Q8—21no—
395C4⋊Q8—21no—
396C4⋊Q8—21no—
397C4⋊Q8C23×C3112yes—
398C4⋊Q8C23×C3111yescharacteristic
399C4⋊Q8C23×C3111yescharacteristic
400#126—22no—
401#126—22no—
402#126—21no—
403#125—22no—
404#125D4×C3111yescharacteristic
405#125D4×C3112yes—
406#124—22no—
407#124—21no—
408#124D4×C3111yescharacteristic
409#123—22no—
410#123—22no—
411#123—21no—
412#123—22no—
413#123—22no—
414C22⋊Q8—22no—
415C22⋊Q8—22no—
416C22⋊Q8—22no—
417C22⋊Q8—21no—
418C22⋊Q8—21no—
419C22⋊Q8—21no—
420C22⋊Q8—21no—
421C22⋊Q8—21no—
422C22⋊Q8—22no—
423C22⋊Q8—21no—
424C22⋊Q8D4×C3111yescharacteristic
425#121—22no—
426#121—21no—
427#121—21no—
428#121D4×C3111yescharacteristic
429#120—22no—
430C4×Q8—21no—
431C4×Q8—21no—
432C4×Q8—21no—
433C4×Q8C23×C3111yescharacteristic
434C4×D4—22no—
435C4×D4—22no—
436C4×D4—22no—
437C4×D4—21no—
438C4×D4—21no—
439C4×D4—21no—
440C4×D4—21no—
441C4×D4D4×C3111yescharacteristic
442C4×D4D4×C3111yescharacteristic
443C4×D4C23×C3111yescharacteristic
444C4×D4C23×C3111yescharacteristic
445C4×D4C23×C3111yescharacteristic
446#117—22no—
447#117—21no—
448#117—21no—
449#117D4×C3112yes—
450C2×C4⋊C4—22no—
451C2×C4⋊C4—21no—
452C2×C4⋊C4—21no—
453C2×C4⋊C4—21no—
454C2×C4⋊C4—21no—
455C2×C4⋊C4D4×C3111yescharacteristic
456C2×C22⋊C4—22no—
457C2×C22⋊C4—21no—
458C2×C42C23×C3111yescharacteristic
459#107—24no—
460#107—22no—
461#106—24no—
462#106—22no—
463C4⋊C8—21no—
464C4⋊C8C23×C3112yes—
465C4⋊C8C23×C3112yes—
466#103—22no—
467#103—22no—
468#103—22no—
469#102—22no—
470C22⋊C8—21no—
471#97—21no—
472#97—21no—
473C4×C8C23×C3112yes—
474C31#1858611yescharacteristic, direct product factor, Sylow
475C24—21no—
476#41—42no—
477#41—42no—
478#41—82no—
479C2×Q8—22no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—22no—
483C2×Q8—42no—
484C2×Q8—42no—
485C2×Q8—41no—
486C2×Q8—82no—
487C2×Q8C2×D4×C3111yescharacteristic
488C2×D4—22no—
489C2×D4—22no—
490C2×D4—21no—
491C2×D4—22no—
492C2×D4—21no—
493C2×D4—41no—
494C2×D4C2×D4×C3111yescharacteristic
495C2×D4C2×D4×C3111yescharacteristic
496C2×D4C2×D4×C3111yescharacteristic
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—41no—
507C22×C4—82no—
508C22×C4C2×D4×C3111yescharacteristic
509C22×C4C2×D4×C3111yescharacteristic
510C22×C4C2×D4×C3111yescharacteristic
511SD16—44no—
512SD16—44no—
513M16—24no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—42no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—21no—
523C4⋊C4—22no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—22no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—41no—
534C4⋊C4—41no—
535C4⋊C4C2×D4×C3111yescharacteristic
536C4⋊C4C2×D4×C3111yescharacteristic
537C4⋊C4C2×D4×C3111yescharacteristic
538C4⋊C4C2×D4×C3112yes—
539C4⋊C4C2×D4×C3111yescharacteristic
540C4⋊C4C2×D4×C3112yes—
541C22⋊C4—22no—
542C22⋊C4—22no—
543C22⋊C4—22no—
544C22⋊C4—22no—
545C22⋊C4—22no—
546C22⋊C4—21no—
547C22⋊C4—22no—
548C22⋊C4—21no—
549C22⋊C4—21no—
550C22⋊C4—21no—
551C22⋊C4—21no—
552C22⋊C4—21no—
553C22⋊C4—21no—
554C22⋊C4—21no—
555C22⋊C4—21no—
556C42—22no—
557C42—22no—
558C42—21no—
559C42C2×D4×C3111yescharacteristic
560C42C2×D4×C3111yescharacteristic
561C42C2×D4×C3111yescharacteristic
562C42C24×C3111yescharacteristic
563C23—21no—
564C23—21no—
565C23—21no—
566C23—42no—
567C23—42no—
568C23—81no—
569C23C31×#12011yescharacteristic
570Q8—22no—
571Q8—41no—
572Q8—41no—
573Q8—42no—
574Q8—42no—
575Q8—42no—
576Q8—82no—
577Q8—82no—
578D4—21no—
579D4—21no—
580D4—21no—
581D4—22no—
582D4—42no—
583D4—44no—
584D4—42no—
585D4—82no—
586D4C31×AGL(1,ℤ/8)12yes—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—44no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—81no—
605C2×C4—81no—
606C2×C4—81no—
607C2×C4—82no—
608C2×C4—82no—
609C2×C4—82no—
610C2×C4C31×#12011yescharacteristic
611C2×C4C31×#12011yescharacteristic
612C2×C4C31×#12011yescharacteristic
613C2×C4C22×D4×C3111yescharacteristic
614C2×C4C22×D4×C3111yescharacteristic
615C2×C4C22×D4×C3111yescharacteristic
616C8—24no—
617C8—42no—
618C8—42no—
619C22—21no—
620C22—21no—
621C22—22no—
622C22—41no—
623C22—41no—
624C22—41no—
625C22—41no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22—84no—
631C22C31×#45612yes—
632C22C2×C31×#12011yescharacteristic
633C4—21no—
634C4—21no—
635C4—21no—
636C4—21no—
637C4—41no—
638C4—42no—
639C4—41no—
640C4—41no—
641C4—42no—
642C4—41no—
643C4—41no—
644C4—82no—
645C4C2×C31×AGL(1,ℤ/8)11yescharacteristic
646C4C31×#57511yescharacteristic
647C4C31×#57712yes—
648C2—22no—
649C2—42no—
650C2—81no—
651C2—81no—
652C2C31×#300611yescharacteristic
653C2C31×#300911yescharacteristic
654C2C2×C31×#45611yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC42C24×C31
prime radicalC31×#3520C2
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
2. C31×#3239C2semidirect
3. C31×#3165C2semidirect
4. C2×C31×#461C2semidirect
5. C31×#1654C2semidirect
6. C31×#1651C2semidirect
7. C31×#1650C2semidirect
8. C31×#1647C2semidirect
9. C31×#1552C2semidirect
10. C31×#1490C2semidirect
11. C31×#1483C2semidirect
33. C31×#500C22semidirect
34. C31×#498C22semidirect
38. C31×#475C22semidirect
39. C31×#474C22semidirect
49. C31×#332C22semidirect
50. C31×#332C22semidirect
51. C31×Q8⋊C8C22semidirect
54. C31×D4⋊C8C22semidirect
235. #18586C31direct (N × H)
293. #3239C2×C31semidirect
294. #3165C2×C31semidirect
295. C2×#461C2×C31semidirect
296. #1654C2×C31semidirect
297. #1651C2×C31semidirect
298. #1650C2×C31semidirect
299. #1647C2×C31semidirect
300. #1552C2×C31semidirect
301. #1490C2×C31semidirect
302. #1483C2×C31semidirect
353. #500C22×C31semidirect
354. #498C22×C31semidirect
358. #475C22×C31semidirect
359. #474C22×C31semidirect
369. #332C22×C31semidirect
370. #332C22×C31semidirect
371. Q8⋊C8C22×C31semidirect
374. D4⋊C8C22×C31semidirect
474. C31#18586direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35202
C31×#32392
C31×#31652
C2×C31×#4612
C31×#16542
C31×#16512
C31×#16502
C31×#16472
C31×#15522
C31×#14902
C31×#14832
#1858631