Finite Group Databasebeta

← C31×#24571 #21481330

Subgroups — C31×#24571 #21481330

Order 7936.

910 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#483C211yesmaximal, characteristic, complemented
2C31×#3007C211yesmaximal, characteristic
3C2×C31×#417C211yesmaximal, characteristic
4C31×#2880C211yesmaximal, characteristic
5C31×#2079C212yesmaximal, complemented
6C31×#2052C212yesmaximal, complemented
7C31×#1908C212yesmaximal, complemented
8C31×#1901C211yesmaximal, characteristic, complemented
9C31×#1855C211yesmaximal, characteristic, complemented
10C31×#1821C211yesmaximal, characteristic, complemented
11C31×#1815C212yesmaximal, complemented
12C22×#41×C31C2211yescharacteristic
13C2×C31×#137C2211yescharacteristic
14C2×C31×#137C2211yescharacteristic
15C2×C31×#131C2212yes—
16C22×M16×C31C2211yescharacteristic
17C2×C31×C22⋊Q8C2211yescharacteristic
18C22×C4⋊C4×C31C2211yescharacteristic
19C31×#483—21no—
20C31×#483—21no—
21C31×#483—22no—
22C31×#452—21no—
23C31×#452—21no—
24C31×#451—21no—
25C31×#450—21no—
26C2×C31×#108C2212yescomplemented
27C2×C31×#106C2212yescomplemented
28C31×#417—21no—
29C31×#417—21no—
30C31×#417C2212yes—
31C31×#417C2211yescharacteristic
32C31×#417C2212yes—
33C31×#417C2211yescharacteristic
34C2×C31×#103—21no—
35C2×C31×#103C2212yescomplemented
36C2×C31×#102—21no—
37C2×C31×#101C2212yescomplemented
38C31×#408—22no—
39C31×#408C2212yes—
40C31×#407—21no—
41C31×#407C2211yescharacteristic
42C31×#407C2211yescharacteristic
43C2×C31×C22⋊C8—21no—
44C2×C31×C22⋊C8C2211yescharacteristic
45C31×#343C2212yescomplemented
46C31×#343C2212yescomplemented
47C31×#343C2212yescomplemented
48C31×#341C2212yescomplemented
49C2×#41×C31—21no—
50C2×#41×C31—21no—
51C2×#41×C31—24no—
52C2×#41×C31—42no—
53C2×#41×C31D411yescharacteristic
54C2×#41×C31D411yescharacteristic
55C2×#41×C31C2311yescharacteristic
56C2×#41×C31C2311yescharacteristic
57C22×Q8×C31D411yescharacteristic
58C22×Q8×C31C2311yescharacteristic
59C22×D4×C31—21no—
60C22×D4×C31D411yescharacteristic
61C23×C4×C31—21no—
62C23×C4×C31—21no—
63C23×C4×C31C2311yescharacteristic
64C31×#137—24no—
65C31×#137—42no—
66C2×Q16×C31—21no—
67C2×Q16×C31—21no—
68C2×SD16×C31—21no—
69C2×SD16×C31—21no—
70C31×#131—24no—
71C31×#131—42no—
72C2×M16×C31—21no—
73C2×M16×C31—22no—
74C2×M16×C31—42no—
75C2×M16×C31C2312yes—
76C2×M16×C31C2311yescharacteristic
77C2×M16×C31C2311yescharacteristic
78C2×M16×C31C2312yes—
79C22×C8×C31—21no—
80C22×C8×C31—22no—
81C22×C8×C31C2312yes—
82C31×C22⋊Q8—21no—
83C31×C22⋊Q8—21no—
84C31×C22⋊Q8—21no—
85C31×C22⋊Q8—21no—
86C2×C4⋊C4×C31—21no—
87C2×C4⋊C4×C31—21no—
88C2×C4⋊C4×C31—21no—
89C2×C4⋊C4×C31—21no—
90C2×C4⋊C4×C31—21no—
91C2×C4⋊C4×C31—21no—
92C2×C4⋊C4×C31C2312yes—
93C2×C4⋊C4×C31C2311yescharacteristic
94C2×C22⋊C4×C31—21no—
95C31×#108—42no—
96C31×#106—22no—
97C31×#106—22no—
98C31×#103—22no—
99C31×#103—22no—
100C31×#103—21no—
101C31×#103—21no—
102C31×#103—22no—
103C31×#103—22no—
104C31×#102—22no—
105C31×#102—22no—
106C31×#102—21no—
107C31×#102—21no—
108C31×#101—42no—
109C31×C22⋊C8—21no—
110C31×C22⋊C8—21no—
111C31×C22⋊C8—21no—
112C31×C22⋊C8—21no—
113C31×C22⋊C8—22no—
114C31×C22⋊C8—22no—
115C31×C22⋊C8D411yescharacteristic
116C31×C22⋊C8D411yescharacteristic
117C31×C22⋊C8D412yes—
118C24×C31—21no—
119C24×C31#4111yescharacteristic
120#41×C31—22no—
121#41×C31—22no—
122#41×C31—22no—
123#41×C31—22no—
124#41×C31—22no—
125#41×C31—44no—
126#41×C31—41no—
127#41×C31—41no—
128#41×C31—44no—
129#41×C31#4114yes—
130C2×Q8×C31—21no—
131C2×Q8×C31—22no—
132C2×Q8×C31—21no—
133C2×Q8×C31—21no—
134C2×Q8×C31—22no—
135C2×Q8×C31—22no—
136C2×Q8×C31—41no—
137C2×Q8×C31C2×D411yescharacteristic
138C2×Q8×C31C2×D411yescharacteristic
139C2×D4×C31—21no—
140C2×D4×C31—22no—
141C2×D4×C31—21no—
142C2×D4×C31—21no—
143C2×D4×C31—21no—
144C2×D4×C31—21no—
145C2×D4×C31—41no—
146C2×D4×C31—44no—
147C2×D4×C31C2×D411yescharacteristic
148C2×D4×C31C2×D411yescharacteristic
149C22×C4×C31—21no—
150C22×C4×C31—22no—
151C22×C4×C31—22no—
152C22×C4×C31—21no—
153C22×C4×C31—21no—
154C22×C4×C31—21no—
155C22×C4×C31—21no—
156C22×C4×C31—21no—
157C22×C4×C31—21no—
158C22×C4×C31—22no—
159C22×C4×C31—44no—
160C22×C4×C31—41no—
161C22×C4×C31—41no—
162C22×C4×C31—82no—
163C22×C4×C31C2×D411yescharacteristic
164C22×C4×C31C2×D411yescharacteristic
165C22×C4×C31C2×D411yescharacteristic
166C22×C4×C31C2×D411yescharacteristic
167C22×C4×C31#4111yescharacteristic
168C22×C4×C31C2411yescharacteristic
169Q16×C31—42no—
170Q16×C31—42no—
171SD16×C31—42no—
172SD16×C31—42no—
173M16×C31—22no—
174M16×C31—22no—
175M16×C31—22no—
176M16×C31—22no—
177M16×C31—22no—
178M16×C31—22no—
179M16×C31—44no—
180M16×C31—42no—
181C2×C8×C31—22no—
182C2×C8×C31—22no—
183C2×C8×C31—22no—
184C2×C8×C31—21no—
185C2×C8×C31—21no—
186C2×C8×C31—22no—
187C2×C8×C31—44no—
188C2×C8×C31—42no—
189C2×C8×C31C2×D412yes—
190C2×C8×C31C2×D412yes—
191C4⋊C4×C31—22no—
192C4⋊C4×C31—21no—
193C4⋊C4×C31—22no—
194C4⋊C4×C31—21no—
195C4⋊C4×C31—41no—
196C4⋊C4×C31—41no—
197C4⋊C4×C31—41no—
198C4⋊C4×C31—41no—
199C22⋊C4×C31—41no—
200C22⋊C4×C31—41no—
201#24571C3111yesmaximal, characteristic, direct product factor, Sylow
202C23×C31—21no—
203C23×C31—22no—
204C23×C31—22no—
205C23×C31—21no—
206C23×C31—41no—
207C23×C31—42no—
208C23×C31—42no—
209C23×C31—81no—
210C23×C31#12111yescharacteristic
211C23×C31#12111yescharacteristic
212C23×C31C2×#4111yescharacteristic
213Q8×C31—21no—
214Q8×C31—21no—
215Q8×C31—21no—
216Q8×C31—42no—
217Q8×C31—41no—
218Q8×C31—41no—
219Q8×C31—42no—
220Q8×C31#12312yes—
221D4×C31—24no—
222D4×C31—21no—
223D4×C31—21no—
224D4×C31—21no—
225D4×C31—42no—
226D4×C31—42no—
227D4×C31—42no—
228D4×C31—42no—
229D4×C31#12312yes—
230C2×C4×C31—21no—
231C2×C4×C31—22no—
232C2×C4×C31—21no—
233C2×C4×C31—21no—
234C2×C4×C31—21no—
235C2×C4×C31—21no—
236C2×C4×C31—24no—
237C2×C4×C31—22no—
238C2×C4×C31—22no—
239C2×C4×C31—21no—
240C2×C4×C31—42no—
241C2×C4×C31—42no—
242C2×C4×C31—42no—
243C2×C4×C31—42no—
244C2×C4×C31—41no—
245C2×C4×C31—41no—
246C2×C4×C31—42no—
247C2×C4×C31—42no—
248C2×C4×C31—82no—
249C2×C4×C31—82no—
250C2×C4×C31—81no—
251C2×C4×C31—82no—
252C2×C4×C31#12111yescharacteristic
253C2×C4×C31#12111yescharacteristic
254C2×C4×C31C22×D411yescharacteristic
255C2×C4×C31C22×D411yescharacteristic
256C2×C4×C31C2×#4112yes—
257C2×C4×C31H(5,𝔽2)12yes—
258C8×C31—22no—
259C8×C31—22no—
260C8×C31—41no—
261C8×C31—42no—
262C8×C31—41no—
263C8×C31—42no—
264C2×#483C2×C3111yescharacteristic, complemented
265#3007C2×C3111yescharacteristic
266C2×#417C2×C3111yescharacteristic
267#2880C2×C3111yescharacteristic
268#2079C2×C3112yescomplemented
269#2052C2×C3112yescomplemented
270#1908C2×C3112yescomplemented
271#1901C2×C3111yescharacteristic, complemented
272#1855C2×C3111yescharacteristic, complemented
273#1821C2×C3111yescharacteristic, complemented
274#1815C2×C3112yescomplemented
275C22×C31—21no—
276C22×C31—21no—
277C22×C31—21no—
278C22×C31—21no—
279C22×C31—21no—
280C22×C31—22no—
281C22×C31—21no—
282C22×C31—21no—
283C22×C31—44no—
284C22×C31—41no—
285C22×C31—41no—
286C22×C31—41no—
287C22×C31—41no—
288C22×C31—41no—
289C22×C31—84no—
290C22×C31#47112yes—
291C22×C31C2×#12111yescharacteristic
292C22×C31#55112yes—
293C4×C31—21no—
294C4×C31—21no—
295C4×C31—22no—
296C4×C31—41no—
297C4×C31—81no—
298C4×C31—82no—
299C4×C31—81no—
300C4×C31C2×#12311yescharacteristic
301C4×C31#53411yescharacteristic
302C4×C31#54612yes—
303C22×#41C22×C3111yescharacteristic
304C2×#137C22×C3111yescharacteristic
305C2×#137C22×C3111yescharacteristic
306C2×#131C22×C3112yes—
307C22×M16C22×C3111yescharacteristic
308C2×C22⋊Q8C22×C3111yescharacteristic
309C22×C4⋊C4C22×C3111yescharacteristic
310#483—21no—
311#483—21no—
312#483—22no—
313#452—21no—
314#452—21no—
315#451—21no—
316#450—21no—
317C2×#108C22×C3112yescomplemented
318C2×#106C22×C3112yescomplemented
319#417—21no—
320#417—21no—
321#417C22×C3112yes—
322#417C22×C3111yescharacteristic
323#417C22×C3112yes—
324#417C22×C3111yescharacteristic
325C2×#103—21no—
326C2×#103C22×C3112yescomplemented
327C2×#102—21no—
328C2×#101C22×C3112yescomplemented
329#408—22no—
330#408C22×C3112yes—
331#407—21no—
332#407C22×C3111yescharacteristic
333#407C22×C3111yescharacteristic
334C2×C22⋊C8—21no—
335C2×C22⋊C8C22×C3111yescharacteristic
336#343C22×C3112yescomplemented
337#343C22×C3112yescomplemented
338#343C22×C3112yescomplemented
339#341C22×C3112yescomplemented
340C2×C31—22no—
341C2×C31—22no—
342C2×C31—41no—
343C2×C31—42no—
344C2×C31—81no—
345C2×C31#241211yescharacteristic
346C2×C31C2×#47111yescharacteristic
347C2×C31#306911yescharacteristic
348C2×#41—21no—
349C2×#41—21no—
350C2×#41—24no—
351C2×#41—42no—
352C2×#41D4×C3111yescharacteristic
353C2×#41D4×C3111yescharacteristic
354C2×#41C23×C3111yescharacteristic
355C2×#41C23×C3111yescharacteristic
356C22×Q8D4×C3111yescharacteristic
357C22×Q8C23×C3111yescharacteristic
358C22×D4—21no—
359C22×D4D4×C3111yescharacteristic
360C23×C4—21no—
361C23×C4—21no—
362C23×C4C23×C3111yescharacteristic
363#137—24no—
364#137—42no—
365C2×Q16—21no—
366C2×Q16—21no—
367C2×SD16—21no—
368C2×SD16—21no—
369#131—24no—
370#131—42no—
371C2×M16—21no—
372C2×M16—22no—
373C2×M16—42no—
374C2×M16C23×C3112yes—
375C2×M16C23×C3112yes—
376C2×M16C23×C3111yescharacteristic
377C2×M16C23×C3111yescharacteristic
378C22×C8—21no—
379C22×C8—22no—
380C22×C8C23×C3112yes—
381C22⋊Q8—21no—
382C22⋊Q8—21no—
383C22⋊Q8—21no—
384C22⋊Q8—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4C23×C3112yes—
392C2×C4⋊C4C23×C3111yescharacteristic
393C2×C22⋊C4—21no—
394#108—42no—
395#106—22no—
396#106—22no—
397#103—22no—
398#103—22no—
399#103—21no—
400#103—21no—
401#103—22no—
402#103—22no—
403#102—22no—
404#102—22no—
405#102—21no—
406#102—21no—
407#101—42no—
408C22⋊C8—21no—
409C22⋊C8—21no—
410C22⋊C8—21no—
411C22⋊C8—21no—
412C22⋊C8—22no—
413C22⋊C8—22no—
414C22⋊C8D4×C3111yescharacteristic
415C22⋊C8D4×C3111yescharacteristic
416C22⋊C8D4×C3112yes—
417C31#2457111yescharacteristic, direct product factor, Sylow
418C24—21no—
419C24#41×C3111yescharacteristic
420#41—22no—
421#41—22no—
422#41—22no—
423#41—22no—
424#41—22no—
425#41—44no—
426#41—41no—
427#41—41no—
428#41—44no—
429#41#41×C3114yes—
430C2×Q8—21no—
431C2×Q8—22no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—41no—
437C2×Q8C2×D4×C3111yescharacteristic
438C2×Q8C2×D4×C3111yescharacteristic
439C2×D4—21no—
440C2×D4—22no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—41no—
446C2×D4—44no—
447C2×D4C2×D4×C3111yescharacteristic
448C2×D4C2×D4×C3111yescharacteristic
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—44no—
460C22×C4—41no—
461C22×C4—41no—
462C22×C4—82no—
463C22×C4C2×D4×C3111yescharacteristic
464C22×C4C2×D4×C3111yescharacteristic
465C22×C4C2×D4×C3111yescharacteristic
466C22×C4C2×D4×C3111yescharacteristic
467C22×C4#41×C3111yescharacteristic
468C22×C4C24×C3111yescharacteristic
469Q16—42no—
470Q16—42no—
471SD16—42no—
472SD16—42no—
473M16—22no—
474M16—22no—
475M16—22no—
476M16—22no—
477M16—22no—
478M16—22no—
479M16—44no—
480M16—42no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—22no—
487C2×C8—44no—
488C2×C8—42no—
489C2×C8C2×D4×C3112yes—
490C2×C8C2×D4×C3112yes—
491C4⋊C4—22no—
492C4⋊C4—21no—
493C4⋊C4—22no—
494C4⋊C4—21no—
495C4⋊C4—41no—
496C4⋊C4—41no—
497C4⋊C4—41no—
498C4⋊C4—41no—
499C22⋊C4—41no—
500C22⋊C4—41no—
501C23—22no—
502C23—21no—
503C23—22no—
504C23—21no—
505C23—41no—
506C23—42no—
507C23—42no—
508C23—81no—
509C23C31×#12111yescharacteristic
510C23C31×#12111yescharacteristic
511C23C2×#41×C3111yescharacteristic
512Q8—21no—
513Q8—21no—
514Q8—21no—
515Q8—42no—
516Q8—41no—
517Q8—41no—
518Q8—42no—
519Q8C31×#12312yes—
520D4—24no—
521D4—21no—
522D4—21no—
523D4—21no—
524D4—42no—
525D4—42no—
526D4—42no—
527D4—42no—
528D4C31×#12312yes—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4—81no—
550C2×C4—82no—
551C2×C4C31×#12111yescharacteristic
552C2×C4C31×#12111yescharacteristic
553C2×C4C22×D4×C3111yescharacteristic
554C2×C4C22×D4×C3111yescharacteristic
555C2×C4C2×#41×C3112yes—
556C2×C4C31×H(5,𝔽2)12yes—
557C8—22no—
558C8—22no—
559C8—41no—
560C8—42no—
561C8—41no—
562C8—42no—
563C22—21no—
564C22—21no—
565C22—21no—
566C22—21no—
567C22—21no—
568C22—22no—
569C22—21no—
570C22—21no—
571C22—44no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—84no—
578C22C31×#47112yes—
579C22C2×C31×#12111yescharacteristic
580C22C31×#55112yes—
581C4—21no—
582C4—21no—
583C4—22no—
584C4—41no—
585C4—81no—
586C4—82no—
587C4—81no—
588C4C2×C31×#12311yescharacteristic
589C4C31×#53411yescharacteristic
590C4C31×#54612yes—
591C2—22no—
592C2—22no—
593C2—41no—
594C2—42no—
595C2—81no—
596C2C31×#241211yescharacteristic
597C2C2×C31×#47111yescharacteristic
598C2C31×#306911yescharacteristic
subgroupG/H
center, socleC22×C31C2×#121
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC22×#41×C31C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C31×#483C2semidirect
5. C31×#2079C2semidirect
6. C31×#2052C2semidirect
7. C31×#1908C2semidirect
8. C31×#1901C2semidirect
9. C31×#1855C2semidirect
10. C31×#1821C2semidirect
11. C31×#1815C2semidirect
26. C2×C31×#108C22semidirect
27. C2×C31×#106C22semidirect
35. C2×C31×#103C22semidirect
37. C2×C31×#101C22semidirect
45. C31×#343C22semidirect
46. C31×#343C22semidirect
47. C31×#343C22semidirect
48. C31×#341C22semidirect
201. #24571C31direct (N × H)
264. C2×#483C2×C31semidirect
268. #2079C2×C31semidirect
269. #2052C2×C31semidirect
270. #1908C2×C31semidirect
271. #1901C2×C31semidirect
272. #1855C2×C31semidirect
273. #1821C2×C31semidirect
274. #1815C2×C31semidirect
317. C2×#108C22×C31semidirect
318. C2×#106C22×C31semidirect
326. C2×#103C22×C31semidirect
328. C2×#101C22×C31semidirect
336. #343C22×C31semidirect
337. #343C22×C31semidirect
338. #343C22×C31semidirect
339. #341C22×C31semidirect
417. C31#24571direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#4832
C31×#30072
C2×C31×#4172
C31×#28802
C31×#20792
C31×#20522
C31×#19082
C31×#19012
C31×#18552
C31×#18212
C31×#18152
#2457131