Finite Group Databasebeta

← C31×#24416 #21481175

Subgroups — C31×#24416 #21481175

Order 7936.

1448 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3487C211yesmaximal, characteristic, complemented
2C31×#3055C212yesmaximal, complemented
3C2×C31×#454C211yesmaximal, characteristic, complemented
4C2×C31×#413C211yesmaximal, characteristic, complemented
5C31×#2431C211yesmaximal, characteristic, complemented
6C31×#2037C212yesmaximal, complemented
7C31×#2002C212yesmaximal, complemented
8C31×#1888C211yesmaximal, characteristic, complemented
9C31×#1871C212yesmaximal, complemented
10C31×#1857C211yesmaximal, characteristic, complemented
11C31×#1793C211yesmaximal, characteristic, complemented
12C2×C31×2−1+4C2211yescharacteristic
13C22×#41×C31C2211yescharacteristic
14C2×C31×AGL(1,ℤ/8)C2212yes—
15C22×M16×C31C2212yes—
16C31×#550—22no—
17C31×#550—22no—
18C31×#548—22no—
19C31×#548—22no—
20C31×#535—22no—
21C31×#535C2212yes—
22C2×C31×#127C2211yescharacteristic
23C2×C31×#124C2211yescharacteristic
24C31×#525—22no—
25C31×#525C2212yes—
26C2×C31×#121C2211yescharacteristic
27C31×#519C2211yescharacteristic
28C31×#469—22no—
29C31×#468—22no—
30C31×#466—22no—
31C31×#465—22no—
32C31×#454—24no—
33C31×#428C2212yes—
34C2×C31×#104C2212yes—
35C2×C31×#104C2212yes—
36C31×#418C2212yes—
37C31×#417C2212yes—
38C31×#413—24no—
39C2×C31×#101C2211yescharacteristic
40C2×C31×#101C2211yescharacteristic
41C2×C31×#100C2212yes—
42C31×#392—22no—
43C31×#391C2211yescharacteristic
44C31×#390C2211yescharacteristic
45C31×#386—22no—
46C4×C22⋊C4×C31C2211yescharacteristic
47C31×#339C2212yes—
48C31×#339C2212yes—
49C31×2−1+4—22no—
50C31×2−1+4—22no—
51C31×2−1+4—42no—
52C2×#41×C31—22no—
53C2×#41×C31—22no—
54C2×#41×C31—22no—
55C2×#41×C31—22no—
56C2×#41×C31—22no—
57C2×#41×C31—22no—
58C2×#41×C31—42no—
59C2×#41×C31C2312yes—
60C22×Q8×C31—22no—
61C22×Q8×C31C2311yescharacteristic
62C22×D4×C31—22no—
63C22×D4×C31—21no—
64C22×D4×C31D412yes—
65C22×D4×C31C2311yescharacteristic
66C22×D4×C31C2311yescharacteristic
67C23×C4×C31D412yes—
68C23×C4×C31C2311yescharacteristic
69C31×AGL(1,ℤ/8)—24no—
70C31×AGL(1,ℤ/8)—42no—
71C2×SD16×C31—22no—
72C2×D8×C31—22no—
73C2×M16×C31—22no—
74C2×M16×C31—24no—
75C2×M16×C31—42no—
76C2×M16×C31C2312yes—
77C2×M16×C31C2312yes—
78C22×C8×C31—22no—
79C31×#127—24no—
80C31×#127—22no—
81C31×#127—42no—
82C31×#124—24no—
83C31×#124—22no—
84C31×#124—22no—
85C31×#124—22no—
86C31×#123—22no—
87C31×C22⋊Q8—22no—
88C31×C22⋊Q8—22no—
89C31×C22⋊Q8D412yescomplemented
90C31×C22⋊Q8D412yescomplemented
91C31×#121—22no—
92C31×#121—22no—
93C31×#121—24no—
94C31×#121—24no—
95C31×#121—22no—
96C31×#121—22no—
97C31×#121—22no—
98C31×#121—22no—
99C31×#121D412yescomplemented
100C31×#121D412yescomplemented
101C31×#120—24no—
102C31×#120—22no—
103C4×Q8×C31—22no—
104C4×Q8×C31—22no—
105C4×D4×C31—24no—
106C4×D4×C31—24no—
107C4×D4×C31—22no—
108C4×D4×C31—22no—
109C31×#117—22no—
110C31×#117C2312yes—
111C2×C4⋊C4×C31C2311yescharacteristic
112C2×C22⋊C4×C31—21no—
113C2×C22⋊C4×C31—21no—
114C2×C22⋊C4×C31—22no—
115C2×C22⋊C4×C31—22no—
116C2×C42×C31C2311yescharacteristic
117C2×C42×C31C2311yescharacteristic
118C31×#107—22no—
119C31×#106—22no—
120C31×#104—24no—
121C31×#104—42no—
122C31×#103—22no—
123C31×#102—22no—
124C31×#102—22no—
125C31×#102—22no—
126C31×#101—22no—
127C31×#101D414yescomplemented
128C31×#100—24no—
129C31×#95—21no—
130C31×#95—22no—
131C24×C31—21no—
132C24×C31C2×D412yes—
133C24×C31C2×D411yescharacteristic
134#41×C31—22no—
135#41×C31—22no—
136#41×C31—22no—
137#41×C31—22no—
138#41×C31—44no—
139#41×C31—44no—
140#41×C31—44no—
141#41×C31—44no—
142#41×C31—44no—
143#41×C31—44no—
144#41×C31—44no—
145#41×C31—44no—
146C2×Q8×C31—22no—
147C2×Q8×C31—22no—
148C2×Q8×C31—22no—
149C2×Q8×C31—44no—
150C2×Q8×C31—44no—
151C2×Q8×C31C2×D412yes—
152C2×Q8×C31C2×D412yes—
153C2×Q8×C31#4112yes—
154C2×D4×C31—22no—
155C2×D4×C31—24no—
156C2×D4×C31—22no—
157C2×D4×C31—24no—
158C2×D4×C31—24no—
159C2×D4×C31—22no—
160C2×D4×C31—22no—
161C2×D4×C31—24no—
162C2×D4×C31—21no—
163C2×D4×C31—22no—
164C2×D4×C31—22no—
165C2×D4×C31—21no—
166C2×D4×C31—21no—
167C2×D4×C31—21no—
168C2×D4×C31—22no—
169C2×D4×C31—22no—
170C2×D4×C31—42no—
171C2×D4×C31—44no—
172C2×D4×C31—44no—
173C2×D4×C31—42no—
174C2×D4×C31—82no—
175C2×D4×C31—82no—
176C2×D4×C31C2×D412yes—
177C22×C4×C31—21no—
178C22×C4×C31—21no—
179C22×C4×C31—22no—
180C22×C4×C31—24no—
181C22×C4×C31—21no—
182C22×C4×C31—22no—
183C22×C4×C31—22no—
184C22×C4×C31—24no—
185C22×C4×C31—22no—
186C22×C4×C31—21no—
187C22×C4×C31—21no—
188C22×C4×C31—42no—
189C22×C4×C31C2×D411yescharacteristic
190C22×C4×C31C2×D412yes—
191C22×C4×C31C2×D412yes—
192C22×C4×C31C2×D412yes—
193C22×C4×C31C2411yescharacteristic
194SD16×C31—44no—
195D8×C31—44no—
196M16×C31—22no—
197M16×C31—22no—
198M16×C31—22no—
199M16×C31—44no—
200M16×C31C2×D414yes—
201C2×C8×C31—22no—
202C2×C8×C31—22no—
203C2×C8×C31—44no—
204C4⋊C4×C31—22no—
205C4⋊C4×C31—22no—
206C4⋊C4×C31—22no—
207C4⋊C4×C31—22no—
208C4⋊C4×C31—22no—
209C4⋊C4×C31C2×D412yes—
210C4⋊C4×C31C2×D412yes—
211C22⋊C4×C31—24no—
212C22⋊C4×C31—24no—
213C22⋊C4×C31—22no—
214C22⋊C4×C31—22no—
215C22⋊C4×C31—22no—
216C22⋊C4×C31—22no—
217C22⋊C4×C31—22no—
218C22⋊C4×C31—42no—
219C22⋊C4×C31—41no—
220C22⋊C4×C31—41no—
221C22⋊C4×C31—44no—
222C22⋊C4×C31—82no—
223C42×C31—24no—
224C42×C31—22no—
225C42×C31—21no—
226C42×C31—21no—
227C42×C31—22no—
228#24416C3111yesmaximal, characteristic, direct product factor, Sylow
229C23×C31—21no—
230C23×C31—21no—
231C23×C31—22no—
232C23×C31—22no—
233C23×C31—22no—
234C23×C31—22no—
235C23×C31—41no—
236C23×C31—41no—
237C23×C31—42no—
238C23×C31—42no—
239C23×C31—42no—
240C23×C31—42no—
241C23×C31—41no—
242C23×C31—41no—
243C23×C31—41no—
244C23×C31#12012yes—
245C23×C31C22×D411yescharacteristic
246Q8×C31—24no—
247Q8×C31—24no—
248Q8×C31—22no—
249Q8×C31—22no—
250Q8×C31—44no—
251D4×C31—22no—
252D4×C31—22no—
253D4×C31—48no—
254D4×C31—44no—
255D4×C31—42no—
256D4×C31—42no—
257D4×C31—42no—
258D4×C31—42no—
259D4×C31—42no—
260D4×C31—42no—
261D4×C31—82no—
262D4×C31—82no—
263D4×C31—84no—
264D4×C31—84no—
265D4×C31—88no—
266D4×C31—84no—
267C2×C4×C31—22no—
268C2×C4×C31—21no—
269C2×C4×C31—21no—
270C2×C4×C31—22no—
271C2×C4×C31—22no—
272C2×C4×C31—22no—
273C2×C4×C31—22no—
274C2×C4×C31—22no—
275C2×C4×C31—22no—
276C2×C4×C31—22no—
277C2×C4×C31—22no—
278C2×C4×C31—22no—
279C2×C4×C31—44no—
280C2×C4×C31—44no—
281C2×C4×C31—48no—
282C2×C4×C31—41no—
283C2×C4×C31—41no—
284C2×C4×C31—42no—
285C2×C4×C31—42no—
286C2×C4×C31—41no—
287C2×C4×C31—41no—
288C2×C4×C31—42no—
289C2×C4×C31—41no—
290C2×C4×C31—41no—
291C2×C4×C31#12011yescharacteristic
292C2×C4×C31#12011yescharacteristic
293C2×C4×C31#12114yes—
294C2×C4×C31C22×D411yescharacteristic
295C2×C4×C31C22×D412yes—
296C2×C4×C31C2×#4111yescharacteristic
297C2×C4×C31H(5,𝔽2)12yes—
298C8×C31—24no—
299C8×C31—42no—
300#3487C2×C3111yescharacteristic, complemented
301#3055C2×C3112yescomplemented
302C2×#454C2×C3111yescharacteristic, complemented
303C2×#413C2×C3111yescharacteristic, complemented
304#2431C2×C3111yescharacteristic, complemented
305#2037C2×C3112yescomplemented
306#2002C2×C3112yescomplemented
307#1888C2×C3111yescharacteristic, complemented
308#1871C2×C3112yescomplemented
309#1857C2×C3111yescharacteristic, complemented
310#1793C2×C3111yescharacteristic, complemented
311C22×C31—21no—
312C22×C31—21no—
313C22×C31—22no—
314C22×C31—22no—
315C22×C31—22no—
316C22×C31—44no—
317C22×C31—42no—
318C22×C31—42no—
319C22×C31—42no—
320C22×C31—42no—
321C22×C31—42no—
322C22×C31—42no—
323C22×C31—81no—
324C22×C31—81no—
325C22×C31—82no—
326C22×C31—82no—
327C22×C31—82no—
328C22×C31—81no—
329C22×C31—81no—
330C22×C31—81no—
331C22×C31C2×#12011yescharacteristic
332C22×C31C2×#12111yescharacteristic
333C22×C31#53411yescharacteristic
334C4×C31—21no—
335C4×C31—21no—
336C4×C31—24no—
337C4×C31—42no—
338C4×C31—42no—
339C4×C31—42no—
340C4×C31—42no—
341C4×C31—81no—
342C4×C31—81no—
343C4×C31D4212yes—
344C4×C31#54612yes—
345C2×2−1+4C22×C3111yescharacteristic
346C22×#41C22×C3111yescharacteristic
347C2×AGL(1,ℤ/8)C22×C3112yes—
348C22×M16C22×C3112yes—
349#550—22no—
350#550—22no—
351#548—22no—
352#548—22no—
353#535—22no—
354#535C22×C3112yes—
355C2×#127C22×C3111yescharacteristic
356C2×#124C22×C3111yescharacteristic
357#525—22no—
358#525C22×C3112yes—
359C2×#121C22×C3111yescharacteristic
360#519C22×C3111yescharacteristic
361#469—22no—
362#468—22no—
363#466—22no—
364#465—22no—
365#454—24no—
366#428C22×C3112yes—
367C2×#104C22×C3112yes—
368C2×#104C22×C3112yes—
369#418C22×C3112yes—
370#417C22×C3112yes—
371#413—24no—
372C2×#101C22×C3111yescharacteristic
373C2×#101C22×C3111yescharacteristic
374C2×#100C22×C3112yes—
375#392—22no—
376#391C22×C3111yescharacteristic
377#390C22×C3111yescharacteristic
378#386—22no—
379C4×C22⋊C4C22×C3111yescharacteristic
380#339C22×C3112yes—
381#339C22×C3112yes—
382C2×C31—21no—
383C2×C31—21no—
384C2×C31—42no—
385C2×C31—82no—
386C2×C31—82no—
387C2×C31—161no—
388C2×C31#241011yescharacteristic
389C2×C31#301811yescharacteristic
390C2×C31#302111yescharacteristic
3912−1+4—22no—
3922−1+4—22no—
3932−1+4—42no—
394C2×#41—22no—
395C2×#41—22no—
396C2×#41—22no—
397C2×#41—22no—
398C2×#41—22no—
399C2×#41—22no—
400C2×#41—42no—
401C2×#41C23×C3112yes—
402C22×Q8—22no—
403C22×Q8C23×C3111yescharacteristic
404C22×D4—22no—
405C22×D4—21no—
406C22×D4D4×C3112yes—
407C22×D4C23×C3111yescharacteristic
408C22×D4C23×C3111yescharacteristic
409C23×C4D4×C3112yes—
410C23×C4C23×C3111yescharacteristic
411AGL(1,ℤ/8)—24no—
412AGL(1,ℤ/8)—42no—
413C2×SD16—22no—
414C2×D8—22no—
415C2×M16—22no—
416C2×M16—24no—
417C2×M16—42no—
418C2×M16C23×C3112yes—
419C2×M16C23×C3112yes—
420C22×C8—22no—
421#127—24no—
422#127—22no—
423#127—42no—
424#124—24no—
425#124—22no—
426#124—22no—
427#124—22no—
428#123—22no—
429C22⋊Q8—22no—
430C22⋊Q8—22no—
431C22⋊Q8D4×C3112yescomplemented
432C22⋊Q8D4×C3112yescomplemented
433#121—22no—
434#121—22no—
435#121—24no—
436#121—24no—
437#121—22no—
438#121—22no—
439#121—22no—
440#121—22no—
441#121D4×C3112yescomplemented
442#121D4×C3112yescomplemented
443#120—24no—
444#120—22no—
445C4×Q8—22no—
446C4×Q8—22no—
447C4×D4—24no—
448C4×D4—24no—
449C4×D4—22no—
450C4×D4—22no—
451#117—22no—
452#117C23×C3112yes—
453C2×C4⋊C4C23×C3111yescharacteristic
454C2×C22⋊C4—21no—
455C2×C22⋊C4—21no—
456C2×C22⋊C4—22no—
457C2×C22⋊C4—22no—
458C2×C42C23×C3111yescharacteristic
459C2×C42C23×C3111yescharacteristic
460#107—22no—
461#106—22no—
462#104—24no—
463#104—42no—
464#103—22no—
465#102—22no—
466#102—22no—
467#102—22no—
468#101—22no—
469#101D4×C3114yescomplemented
470#100—24no—
471#95—21no—
472#95—22no—
473C31#2441611yescharacteristic, direct product factor, Sylow
474C24—21no—
475C24C2×D4×C3112yes—
476C24C2×D4×C3111yescharacteristic
477#41—22no—
478#41—22no—
479#41—22no—
480#41—22no—
481#41—44no—
482#41—44no—
483#41—44no—
484#41—44no—
485#41—44no—
486#41—44no—
487#41—44no—
488#41—44no—
489C2×Q8—22no—
490C2×Q8—22no—
491C2×Q8—22no—
492C2×Q8—44no—
493C2×Q8—44no—
494C2×Q8C2×D4×C3112yes—
495C2×Q8C2×D4×C3112yes—
496C2×Q8#41×C3112yes—
497C2×D4—22no—
498C2×D4—24no—
499C2×D4—22no—
500C2×D4—24no—
501C2×D4—24no—
502C2×D4—22no—
503C2×D4—22no—
504C2×D4—24no—
505C2×D4—21no—
506C2×D4—22no—
507C2×D4—22no—
508C2×D4—21no—
509C2×D4—21no—
510C2×D4—21no—
511C2×D4—22no—
512C2×D4—22no—
513C2×D4—42no—
514C2×D4—44no—
515C2×D4—44no—
516C2×D4—42no—
517C2×D4—82no—
518C2×D4—82no—
519C2×D4C2×D4×C3112yes—
520C22×C4—21no—
521C22×C4—21no—
522C22×C4—22no—
523C22×C4—24no—
524C22×C4—21no—
525C22×C4—22no—
526C22×C4—22no—
527C22×C4—24no—
528C22×C4—22no—
529C22×C4—21no—
530C22×C4—21no—
531C22×C4—42no—
532C22×C4C2×D4×C3111yescharacteristic
533C22×C4C2×D4×C3112yes—
534C22×C4C2×D4×C3112yes—
535C22×C4C2×D4×C3112yes—
536C22×C4C24×C3111yescharacteristic
537SD16—44no—
538D8—44no—
539M16—22no—
540M16—22no—
541M16—22no—
542M16—44no—
543M16C2×D4×C3114yes—
544C2×C8—22no—
545C2×C8—22no—
546C2×C8—44no—
547C4⋊C4—22no—
548C4⋊C4—22no—
549C4⋊C4—22no—
550C4⋊C4—22no—
551C4⋊C4—22no—
552C4⋊C4C2×D4×C3112yes—
553C4⋊C4C2×D4×C3112yes—
554C22⋊C4—24no—
555C22⋊C4—24no—
556C22⋊C4—22no—
557C22⋊C4—22no—
558C22⋊C4—22no—
559C22⋊C4—22no—
560C22⋊C4—22no—
561C22⋊C4—42no—
562C22⋊C4—41no—
563C22⋊C4—41no—
564C22⋊C4—44no—
565C22⋊C4—82no—
566C42—24no—
567C42—22no—
568C42—21no—
569C42—21no—
570C42—22no—
571C23—21no—
572C23—21no—
573C23—22no—
574C23—22no—
575C23—22no—
576C23—22no—
577C23—41no—
578C23—41no—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—41no—
584C23—41no—
585C23—41no—
586C23C31×#12012yes—
587C23C22×D4×C3111yescharacteristic
588Q8—24no—
589Q8—24no—
590Q8—22no—
591Q8—22no—
592Q8—44no—
593D4—22no—
594D4—22no—
595D4—48no—
596D4—44no—
597D4—42no—
598D4—42no—
599D4—42no—
600D4—42no—
601D4—42no—
602D4—42no—
603D4—82no—
604D4—82no—
605D4—84no—
606D4—84no—
607D4—88no—
608D4—84no—
609C2×C4—22no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—48no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4C31×#12011yescharacteristic
634C2×C4C31×#12011yescharacteristic
635C2×C4C31×#12114yes—
636C2×C4C22×D4×C3111yescharacteristic
637C2×C4C22×D4×C3112yes—
638C2×C4C2×#41×C3111yescharacteristic
639C2×C4C31×H(5,𝔽2)12yes—
640C8—24no—
641C8—42no—
642C22—21no—
643C22—21no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—44no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—42no—
654C22—81no—
655C22—81no—
656C22—82no—
657C22—82no—
658C22—82no—
659C22—81no—
660C22—81no—
661C22—81no—
662C22C2×C31×#12011yescharacteristic
663C22C2×C31×#12111yescharacteristic
664C22C31×#53411yescharacteristic
665C4—21no—
666C4—21no—
667C4—24no—
668C4—42no—
669C4—42no—
670C4—42no—
671C4—42no—
672C4—81no—
673C4—81no—
674C4D42×C3112yes—
675C4C31×#54612yes—
676C2—21no—
677C2—21no—
678C2—42no—
679C2—82no—
680C2—82no—
681C2—161no—
682C2C31×#241011yescharacteristic
683C2C31×#301811yescharacteristic
684C2C31×#302111yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
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. C31×#3487C2semidirect
2. C31×#3055C2semidirect
3. C2×C31×#454C2semidirect
4. C2×C31×#413C2semidirect
5. C31×#2431C2semidirect
6. C31×#2037C2semidirect
7. C31×#2002C2semidirect
8. C31×#1888C2semidirect
9. C31×#1871C2semidirect
10. C31×#1857C2semidirect
11. C31×#1793C2semidirect
89. C31×C22⋊Q8D4semidirect
90. C31×C22⋊Q8D4semidirect
99. C31×#121D4semidirect
100. C31×#121D4semidirect
127. C31×#101D4semidirect
228. #24416C31direct (N × H)
300. #3487C2×C31semidirect
301. #3055C2×C31semidirect
302. C2×#454C2×C31semidirect
303. C2×#413C2×C31semidirect
304. #2431C2×C31semidirect
305. #2037C2×C31semidirect
306. #2002C2×C31semidirect
307. #1888C2×C31semidirect
308. #1871C2×C31semidirect
309. #1857C2×C31semidirect
310. #1793C2×C31semidirect
431. C22⋊Q8D4×C31semidirect
432. C22⋊Q8D4×C31semidirect
441. #121D4×C31semidirect
442. #121D4×C31semidirect
469. #101D4×C31semidirect
473. C31#24416direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#34872
C31×#30552
C2×C31×#4542
C2×C31×#4132
C31×#24312
C31×#20372
C31×#20022
C31×#18882
C31×#18712
C31×#18572
C31×#17932
#2441631