Finite Group Databasebeta

← C31×#20867 #21477626

Subgroups — C31×#20867 #21477626

Order 7936.

1312 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#568C211yesmaximal, characteristic
2C2×C31×#430C211yesmaximal, characteristic, complemented
3C31×#2905C211yesmaximal, characteristic, complemented
4C2×C31×#421C211yesmaximal, characteristic
5C2×C31×#421C211yesmaximal, characteristic
6C31×#1861C218yesmaximal, complemented
7C2×C31×#344C211yesmaximal, characteristic, complemented
8C31×#1733C211yesmaximal, characteristic, complemented
9C22×#41×C31C2211yescharacteristic
10C31×#568—22no—
11C31×#568—22no—
12C31×#568—22no—
13C31×#568C2214yes—
14C2×C31×#131—22no—
15C2×C31×#131—22no—
16C22×M16×C31—22no—
17C22×M16×C31C2211yescharacteristic
18C22×M16×C31C2211yescharacteristic
19C22×M16×C31C2211yescharacteristic
20C22×M16×C31C2211yescharacteristic
21C2×C31×#117C2211yescharacteristic
22C2×C31×#117C2211yescharacteristic
23C31×#430—22no—
24C31×#430C2214yescomplemented
25C2×C31×#108—22no—
26C31×#424—22no—
27C31×#424—22no—
28C31×#423C2214yescomplemented
29C31×#421—22no—
30C31×#421C414yescomplemented
31C31×#421C414yescomplemented
32C31×#421C2214yes—
33C2×C31×#104—22no—
34C2×C31×#104C2214yes—
35C31×#344C2214yescomplemented
36C31×#339C2214yescomplemented
37C2×#41×C31—21no—
38C2×#41×C31—21no—
39C2×#41×C31—21no—
40C2×#41×C31—21no—
41C2×#41×C31—21no—
42C2×#41×C31—21no—
43C2×#41×C31C2×C412yes—
44C2×#41×C31C2×C412yes—
45C2×#41×C31D412yes—
46C2×#41×C31D412yes—
47C2×#41×C31Q812yes—
48C2×#41×C31C2312yes—
49C22×Q8×C31Q811yescharacteristic
50C22×D4×C31—21no—
51C22×D4×C31Q811yescharacteristic
52C23×C4×C31—21no—
53C23×C4×C31C2311yescharacteristic
54C31×#131—28no—
55C31×#131—28no—
56C31×#131—28no—
57C31×#131—28no—
58C2×M16×C31—24no—
59C2×M16×C31—28no—
60C2×M16×C31—21no—
61C2×M16×C31—21no—
62C2×M16×C31—21no—
63C2×M16×C31—21no—
64C2×M16×C31—21no—
65C2×M16×C31—21no—
66C2×M16×C31—21no—
67C2×M16×C31—21no—
68C2×M16×C31—21no—
69C2×M16×C31—21no—
70C2×M16×C31—28no—
71C2×M16×C31—21no—
72C2×M16×C31—21no—
73C2×M16×C31—21no—
74C2×M16×C31—21no—
75C2×M16×C31—21no—
76C2×M16×C31C2×C412yes—
77C2×M16×C31C2×C412yes—
78C2×M16×C31D412yes—
79C2×M16×C31D412yes—
80C2×M16×C31D412yes—
81C2×M16×C31C2312yes—
82C2×M16×C31C2312yes—
83C2×M16×C31C2312yes—
84C2×M16×C31C2312yes—
85C22×C8×C31—24no—
86C22×C8×C31—21no—
87C22×C8×C31—21no—
88C22×C8×C31—21no—
89C22×C8×C31D412yes—
90C31×#117—21no—
91C31×#117—21no—
92C31×#117—21no—
93C31×#117—21no—
94C31×#117—21no—
95C31×#117C2×C412yes—
96C31×#117C2×C412yes—
97C31×#117C2312yes—
98C2×C4⋊C4×C31—21no—
99C2×C4⋊C4×C31—21no—
100C2×C22⋊C4×C31—21no—
101C2×C22⋊C4×C31—21no—
102C2×C42×C31—21no—
103C2×C42×C31C2312yes—
104C31×#108—28no—
105C31×C4⋊C8—28no—
106C31×#104—28no—
107C31×#104C2×C4116yescomplemented
108C24×C31—21no—
109C24×C31#4111yescharacteristic
110#41×C31—22no—
111#41×C31—22no—
112#41×C31—22no—
113#41×C31—22no—
114#41×C31—22no—
115#41×C31—22no—
116#41×C31—22no—
117#41×C31—22no—
118#41×C31—22no—
119#41×C31—22no—
120#41×C31—22no—
121#41×C31—22no—
122#41×C31C4⋊C414yes—
123#41×C31C4⋊C414yes—
124#41×C31C22×C414yes—
125#41×C31C2×D414yes—
126C2×Q8×C31—21no—
127C2×Q8×C31—21no—
128C2×Q8×C31—21no—
129C2×Q8×C31C4⋊C412yes—
130C2×Q8×C31C4⋊C412yes—
131C2×Q8×C31C2×Q812yes—
132C2×D4×C31—22no—
133C2×D4×C31—22no—
134C2×D4×C31—22no—
135C2×D4×C31—22no—
136C2×D4×C31—22no—
137C2×D4×C31—22no—
138C2×D4×C31—21no—
139C2×D4×C31—21no—
140C2×D4×C31—21no—
141C2×D4×C31C4⋊C412yes—
142C2×D4×C31C4⋊C412yes—
143C2×D4×C31C2×Q812yes—
144C22×C4×C31—21no—
145C22×C4×C31—22no—
146C22×C4×C31—22no—
147C22×C4×C31—22no—
148C22×C4×C31—22no—
149C22×C4×C31—22no—
150C22×C4×C31—22no—
151C22×C4×C31—22no—
152C22×C4×C31—41no—
153C22×C4×C31C22×C411yescharacteristic
154C22×C4×C31C22×C411yescharacteristic
155C22×C4×C31C2×D411yescharacteristic
156C22×C4×C31C2×D411yescharacteristic
157C22×C4×C31C2×D411yescharacteristic
158C22×C4×C31C2×D411yescharacteristic
159C22×C4×C31C2×D411yescharacteristic
160C22×C4×C31C2×D411yescharacteristic
161C22×C4×C31C2×Q811yescharacteristic
162C22×C4×C31C2×Q811yescharacteristic
163C22×C4×C31#4111yescharacteristic
164C22×C4×C31#4111yescharacteristic
165C22×C4×C31#4111yescharacteristic
166C22×C4×C31C2411yescharacteristic
167M16×C31—216no—
168M16×C31—22no—
169M16×C31—22no—
170M16×C31—22no—
171M16×C31—22no—
172M16×C31—22no—
173M16×C31—22no—
174M16×C31—22no—
175M16×C31—22no—
176M16×C31—22no—
177M16×C31—22no—
178M16×C31—22no—
179M16×C31—22no—
180M16×C31—41no—
181M16×C31—41no—
182M16×C31C22×C414yes—
183M16×C31C2×D414yes—
184C2×C8×C31—216no—
185C2×C8×C31—22no—
186C2×C8×C31—22no—
187C2×C8×C31—22no—
188C2×C8×C31—22no—
189C2×C8×C31—22no—
190C2×C8×C31—22no—
191C2×C8×C31—22no—
192C2×C8×C31—22no—
193C2×C8×C31—22no—
194C2×C8×C31—41no—
195C2×C8×C31—41no—
196C2×C8×C31C2×D414yes—
197C4⋊C4×C31—22no—
198C4⋊C4×C31—22no—
199C4⋊C4×C31—22no—
200C4⋊C4×C31—41no—
201C22⋊C4×C31—22no—
202C22⋊C4×C31—22no—
203C22⋊C4×C31—22no—
204C22⋊C4×C31—41no—
205C42×C31—22no—
206C42×C31—22no—
207C42×C31—41no—
208C42×C31C22×C414yes—
209#20867C3111yesmaximal, characteristic, direct product factor, Sylow
210C23×C31—22no—
211C23×C31—22no—
212C23×C31—22no—
213C23×C31—81no—
214C23×C31—82no—
215C23×C31C4×D411yescharacteristic
216C23×C31C4×D411yescharacteristic
217C23×C31C22⋊Q811yescharacteristic
218C23×C31C22⋊Q811yescharacteristic
219C23×C31#12311yescharacteristic
220C23×C31#12311yescharacteristic
221C23×C31C2×#4111yescharacteristic
222Q8×C31—22no—
223Q8×C31—22no—
224Q8×C31—22no—
225Q8×C31C2×C4⋊C414yes—
226D4×C31—24no—
227D4×C31—24no—
228D4×C31—24no—
229D4×C31—24no—
230D4×C31—22no—
231D4×C31—22no—
232D4×C31—22no—
233D4×C31C2×C4⋊C414yes—
234C2×C4×C31—22no—
235C2×C4×C31—22no—
236C2×C4×C31—22no—
237C2×C4×C31—24no—
238C2×C4×C31—24no—
239C2×C4×C31—24no—
240C2×C4×C31—24no—
241C2×C4×C31—24no—
242C2×C4×C31—42no—
243C2×C4×C31—82no—
244C2×C4×C31—82no—
245C2×C4×C31C2×C4⋊C411yescharacteristic
246C2×C4×C31C2×C4⋊C411yescharacteristic
247C2×C4×C31C2×C4⋊C411yescharacteristic
248C2×C4×C31C2×C4⋊C411yescharacteristic
249C2×C4×C31C4×D411yescharacteristic
250C2×C4×C31C4×D411yescharacteristic
251C2×C4×C31C4×D411yescharacteristic
252C2×C4×C31C4×D411yescharacteristic
253C2×C4×C31C4×D411yescharacteristic
254C2×C4×C31C4×D411yescharacteristic
255C2×C4×C31#12011yescharacteristic
256C2×C4×C31#12011yescharacteristic
257C2×C4×C31#12011yescharacteristic
258C2×C4×C31#12011yescharacteristic
259C2×C4×C31C22⋊Q811yescharacteristic
260C2×C4×C31C22⋊Q811yescharacteristic
261C2×C4×C31C22⋊Q811yescharacteristic
262C2×C4×C31C22⋊Q811yescharacteristic
263C2×C4×C31C22⋊Q811yescharacteristic
264C2×C4×C31C22⋊Q811yescharacteristic
265C2×C4×C31#12311yescharacteristic
266C2×C4×C31#12311yescharacteristic
267C2×C4×C31C23×C411yescharacteristic
268C2×C4×C31C22×D411yescharacteristic
269C2×C4×C31C22×D411yescharacteristic
270C2×C4×C31C22×D411yescharacteristic
271C2×C4×C31C22×Q811yescharacteristic
272C2×C4×C31C2×#4111yescharacteristic
273C8×C31—24no—
274C8×C31—24no—
275C8×C31—24no—
276C8×C31—42no—
277C2×#568C2×C3111yescharacteristic
278C2×#430C2×C3111yescharacteristic, complemented
279#2905C2×C3111yescharacteristic, complemented
280C2×#421C2×C3111yescharacteristic
281C2×#421C2×C3111yescharacteristic
282#1861C2×C3118yescomplemented
283C2×#344C2×C3111yescharacteristic, complemented
284#1733C2×C3111yescharacteristic, complemented
285C22×C31—24no—
286C22×C31—41no—
287C22×C31—41no—
288C22×C31—41no—
289C22×C31—41no—
290C22×C31—41no—
291C22×C31—41no—
292C22×C31—41no—
293C22×C31—44no—
294C22×C31—82no—
295C22×C31—82no—
296C22×C31#38511yescharacteristic
297C22×C31#38511yescharacteristic
298C22×C31#38511yescharacteristic
299C22×C31#38511yescharacteristic
300C22×C31C2×C4×D411yescharacteristic
301C22×C31C2×C22⋊Q811yescharacteristic
302C22×C31C2×#12311yescharacteristic
303C4×C31—24no—
304C4×C31—44no—
305C4×C31—82no—
306C4×C31#38511yescharacteristic
307C4×C31#38511yescharacteristic
308C4×C31#38511yescharacteristic
309C4×C31#38511yescharacteristic
310C4×C31C22×C4⋊C411yescharacteristic
311C4×C31C2×C4×D411yescharacteristic
312C4×C31C2×#12011yescharacteristic
313C4×C31C2×C22⋊Q811yescharacteristic
314C22×#41C22×C3111yescharacteristic
315#568—22no—
316#568—22no—
317#568—22no—
318#568C22×C3114yes—
319C2×#131—22no—
320C2×#131—22no—
321C22×M16—22no—
322C22×M16C22×C3111yescharacteristic
323C22×M16C22×C3111yescharacteristic
324C22×M16C22×C3111yescharacteristic
325C22×M16C22×C3111yescharacteristic
326C2×#117C22×C3111yescharacteristic
327C2×#117C22×C3111yescharacteristic
328#430—22no—
329#430C22×C3114yescomplemented
330C2×#108—22no—
331#424—22no—
332#424—22no—
333#423C22×C3114yescomplemented
334#421—22no—
335#421C4×C3114yescomplemented
336#421C4×C3114yescomplemented
337#421C22×C3114yes—
338C2×#104—22no—
339C2×#104C22×C3114yes—
340#344C22×C3114yescomplemented
341#339C22×C3114yescomplemented
342C2×C31—21no—
343C2×C31—21no—
344C2×C31—21no—
345C2×C31—21no—
346C2×C31—21no—
347C2×C31—21no—
348C2×C31—21no—
349C2×C31—44no—
350C2×C31C2×#38511yescharacteristic
351C2×#41—21no—
352C2×#41—21no—
353C2×#41—21no—
354C2×#41—21no—
355C2×#41—21no—
356C2×#41—21no—
357C2×#41C2×C4×C3112yes—
358C2×#41C2×C4×C3112yes—
359C2×#41D4×C3112yes—
360C2×#41D4×C3112yes—
361C2×#41Q8×C3112yes—
362C2×#41C23×C3112yes—
363C22×Q8Q8×C3111yescharacteristic
364C22×D4—21no—
365C22×D4Q8×C3111yescharacteristic
366C23×C4—21no—
367C23×C4C23×C3111yescharacteristic
368#131—28no—
369#131—28no—
370#131—28no—
371#131—28no—
372C2×M16—24no—
373C2×M16—28no—
374C2×M16—21no—
375C2×M16—21no—
376C2×M16—21no—
377C2×M16—21no—
378C2×M16—21no—
379C2×M16—21no—
380C2×M16—21no—
381C2×M16—21no—
382C2×M16—21no—
383C2×M16—21no—
384C2×M16—28no—
385C2×M16—21no—
386C2×M16—21no—
387C2×M16—21no—
388C2×M16—21no—
389C2×M16—21no—
390C2×M16C2×C4×C3112yes—
391C2×M16C2×C4×C3112yes—
392C2×M16D4×C3112yes—
393C2×M16D4×C3112yes—
394C2×M16D4×C3112yes—
395C2×M16C23×C3112yes—
396C2×M16C23×C3112yes—
397C2×M16C23×C3112yes—
398C2×M16C23×C3112yes—
399C22×C8—24no—
400C22×C8—21no—
401C22×C8—21no—
402C22×C8—21no—
403C22×C8D4×C3112yes—
404#117—21no—
405#117—21no—
406#117—21no—
407#117—21no—
408#117—21no—
409#117C2×C4×C3112yes—
410#117C2×C4×C3112yes—
411#117C23×C3112yes—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—21no—
414C2×C22⋊C4—21no—
415C2×C22⋊C4—21no—
416C2×C42—21no—
417C2×C42C23×C3112yes—
418#108—28no—
419C4⋊C8—28no—
420#104—28no—
421#104C2×C4×C31116yescomplemented
422C31#2086711yescharacteristic, direct product factor, Sylow
423C24—21no—
424C24#41×C3111yescharacteristic
425#41—22no—
426#41—22no—
427#41—22no—
428#41—22no—
429#41—22no—
430#41—22no—
431#41—22no—
432#41—22no—
433#41—22no—
434#41—22no—
435#41—22no—
436#41—22no—
437#41C4⋊C4×C3114yes—
438#41C4⋊C4×C3114yes—
439#41C22×C4×C3114yes—
440#41C2×D4×C3114yes—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8C4⋊C4×C3112yes—
445C2×Q8C4⋊C4×C3112yes—
446C2×Q8C2×Q8×C3112yes—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4C4⋊C4×C3112yes—
457C2×D4C4⋊C4×C3112yes—
458C2×D4C2×Q8×C3112yes—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—41no—
468C22×C4C22×C4×C3111yescharacteristic
469C22×C4C22×C4×C3111yescharacteristic
470C22×C4C2×D4×C3111yescharacteristic
471C22×C4C2×D4×C3111yescharacteristic
472C22×C4C2×D4×C3111yescharacteristic
473C22×C4C2×D4×C3111yescharacteristic
474C22×C4C2×D4×C3111yescharacteristic
475C22×C4C2×D4×C3111yescharacteristic
476C22×C4C2×Q8×C3111yescharacteristic
477C22×C4C2×Q8×C3111yescharacteristic
478C22×C4#41×C3111yescharacteristic
479C22×C4#41×C3111yescharacteristic
480C22×C4#41×C3111yescharacteristic
481C22×C4C24×C3111yescharacteristic
482M16—216no—
483M16—22no—
484M16—22no—
485M16—22no—
486M16—22no—
487M16—22no—
488M16—22no—
489M16—22no—
490M16—22no—
491M16—22no—
492M16—22no—
493M16—22no—
494M16—22no—
495M16—41no—
496M16—41no—
497M16C22×C4×C3114yes—
498M16C2×D4×C3114yes—
499C2×C8—216no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—41no—
510C2×C8—41no—
511C2×C8C2×D4×C3114yes—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—41no—
516C22⋊C4—22no—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—41no—
520C42—22no—
521C42—22no—
522C42—41no—
523C42C22×C4×C3114yes—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—81no—
528C23—82no—
529C23C4×D4×C3111yescharacteristic
530C23C4×D4×C3111yescharacteristic
531C23C31×C22⋊Q811yescharacteristic
532C23C31×C22⋊Q811yescharacteristic
533C23C31×#12311yescharacteristic
534C23C31×#12311yescharacteristic
535C23C2×#41×C3111yescharacteristic
536Q8—22no—
537Q8—22no—
538Q8—22no—
539Q8C2×C4⋊C4×C3114yes—
540D4—24no—
541D4—24no—
542D4—24no—
543D4—24no—
544D4—22no—
545D4—22no—
546D4—22no—
547D4C2×C4⋊C4×C3114yes—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—42no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4C2×C4⋊C4×C3111yescharacteristic
560C2×C4C2×C4⋊C4×C3111yescharacteristic
561C2×C4C2×C4⋊C4×C3111yescharacteristic
562C2×C4C2×C4⋊C4×C3111yescharacteristic
563C2×C4C4×D4×C3111yescharacteristic
564C2×C4C4×D4×C3111yescharacteristic
565C2×C4C4×D4×C3111yescharacteristic
566C2×C4C4×D4×C3111yescharacteristic
567C2×C4C4×D4×C3111yescharacteristic
568C2×C4C4×D4×C3111yescharacteristic
569C2×C4C31×#12011yescharacteristic
570C2×C4C31×#12011yescharacteristic
571C2×C4C31×#12011yescharacteristic
572C2×C4C31×#12011yescharacteristic
573C2×C4C31×C22⋊Q811yescharacteristic
574C2×C4C31×C22⋊Q811yescharacteristic
575C2×C4C31×C22⋊Q811yescharacteristic
576C2×C4C31×C22⋊Q811yescharacteristic
577C2×C4C31×C22⋊Q811yescharacteristic
578C2×C4C31×C22⋊Q811yescharacteristic
579C2×C4C31×#12311yescharacteristic
580C2×C4C31×#12311yescharacteristic
581C2×C4C23×C4×C3111yescharacteristic
582C2×C4C22×D4×C3111yescharacteristic
583C2×C4C22×D4×C3111yescharacteristic
584C2×C4C22×D4×C3111yescharacteristic
585C2×C4C22×Q8×C3111yescharacteristic
586C2×C4C2×#41×C3111yescharacteristic
587C8—24no—
588C8—24no—
589C8—24no—
590C8—42no—
591C22—24no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—44no—
600C22—82no—
601C22—82no—
602C22C31×#38511yescharacteristic
603C22C31×#38511yescharacteristic
604C22C31×#38511yescharacteristic
605C22C31×#38511yescharacteristic
606C22C2×C4×D4×C3111yescharacteristic
607C22C2×C31×C22⋊Q811yescharacteristic
608C22C2×C31×#12311yescharacteristic
609C4—24no—
610C4—44no—
611C4—82no—
612C4C31×#38511yescharacteristic
613C4C31×#38511yescharacteristic
614C4C31×#38511yescharacteristic
615C4C31×#38511yescharacteristic
616C4C22×C4⋊C4×C3111yescharacteristic
617C4C2×C4×D4×C3111yescharacteristic
618C4C2×C31×#12011yescharacteristic
619C4C2×C31×C22⋊Q811yescharacteristic
620C2—21no—
621C2—21no—
622C2—21no—
623C2—21no—
624C2—21no—
625C2—21no—
626C2—21no—
627C2—44no—
628C2C2×C31×#38511yescharacteristic
subgroupG/H
centerC4×C31C2×#120
commutatorC2×C4C23×C4×C31
Frattini, cosocleC22×C4C24×C31
socleC2×C31C2×#385
prime radicalC22×#41×C31C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C4×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×#430C2semidirect
3. C31×#2905C2semidirect
6. C31×#1861C2semidirect
7. C2×C31×#344C2semidirect
8. C31×#1733C2semidirect
24. C31×#430C22semidirect
28. C31×#423C22semidirect
30. C31×#421C4semidirect
31. C31×#421C4semidirect
35. C31×#344C22semidirect
36. C31×#339C22semidirect
107. C31×#104C2×C4semidirect
209. #20867C31direct (N × H)
278. C2×#430C2×C31semidirect
279. #2905C2×C31semidirect
282. #1861C2×C31semidirect
283. C2×#344C2×C31semidirect
284. #1733C2×C31semidirect
329. #430C22×C31semidirect
333. #423C22×C31semidirect
335. #421C4×C31semidirect
336. #421C4×C31semidirect
340. #344C22×C31semidirect
341. #339C22×C31semidirect
421. #104C2×C4×C31semidirect
422. C31#20867direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5682
C2×C31×#4302
C31×#29052
C2×C31×#4212
C31×#18612
C2×C31×#3442
C31×#17332
#2086731