Finite Group Databasebeta

← C31×#24417 #21481176

Subgroups — C31×#24417 #21481176

Order 7936.

1390 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3487C211yesmaximal, characteristic
2C31×#3056C212yesmaximal, complemented
3C2×C31×#456C211yesmaximal, characteristic, complemented
4C2×C31×#413C211yesmaximal, characteristic, complemented
5C31×#2432C211yesmaximal, characteristic, complemented
6C31×#2037C212yesmaximal, complemented
7C31×#2004C212yesmaximal, complemented
8C31×#1886C211yesmaximal, characteristic, complemented
9C31×#1871C212yesmaximal, complemented
10C31×#1857C211yesmaximal, characteristic, complemented
11C31×#1793C211yesmaximal, characteristic, complemented
12C2×C31×2−1+4C2211yescharacteristic
13C22×#41×C31C2211yescharacteristic
14C2×C31×#137C2212yes—
15C22×M16×C31C2212yes—
16C31×#550—22no—
17C31×#550—22no—
18C31×#548—22no—
19C31×#548—22no—
20C31×#535—22no—
21C31×#535C2212yes—
22C2×C31×C4⋊Q8C2211yescharacteristic
23C2×C31×#124C2211yescharacteristic
24C31×#525—22no—
25C31×#525C2212yes—
26C2×C31×C22⋊Q8C2211yescharacteristic
27C31×#519C2211yescharacteristic
28C31×#470—22no—
29C31×#468—22no—
30C31×#467—22no—
31C31×#465—22no—
32C31×#456—24no—
33C31×#428C2212yes—
34C2×C31×#104C2212yes—
35C2×C31×#104C2212yes—
36C31×#419C2212yes—
37C31×#417C2212yes—
38C31×#413—24no—
39C2×C31×#101C2211yescharacteristic
40C2×C31×#101C2211yescharacteristic
41C2×C31×#100C2212yes—
42C31×#394—22no—
43C31×#391C2211yescharacteristic
44C31×#390C2211yescharacteristic
45C31×#385—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×Q8×C31C2311yescharacteristic
63C22×D4×C31D412yes—
64C22×D4×C31C2311yescharacteristic
65C23×C4×C31D412yes—
66C23×C4×C31C2311yescharacteristic
67C31×#137—24no—
68C31×#137—42no—
69C2×Q16×C31—22no—
70C2×SD16×C31—22no—
71C2×M16×C31—22no—
72C2×M16×C31—24no—
73C2×M16×C31—42no—
74C2×M16×C31C2312yes—
75C2×M16×C31C2312yes—
76C22×C8×C31—22no—
77C31×C4⋊Q8—22no—
78C31×C4⋊Q8—22no—
79C31×#127—24no—
80C31×#127—22no—
81C31×#124—24no—
82C31×#124—22no—
83C31×#124—42no—
84C31×#123—22no—
85C31×C22⋊Q8—22no—
86C31×C22⋊Q8—22no—
87C31×C22⋊Q8—22no—
88C31×C22⋊Q8—22no—
89C31×C22⋊Q8D412yes—
90C31×C22⋊Q8D412yes—
91C31×#121—22no—
92C31×#121—22no—
93C31×#121—24no—
94C31×#121—24no—
95C31×#121—22no—
96C31×#121—22no—
97C31×#121D412yes—
98C31×#121D412yes—
99C31×#120—24no—
100C31×#120—22no—
101C4×Q8×C31—22no—
102C4×Q8×C31—22no—
103C4×D4×C31—24no—
104C4×D4×C31—24no—
105C4×D4×C31—22no—
106C4×D4×C31—22no—
107C31×#117—22no—
108C31×#117C2312yes—
109C2×C4⋊C4×C31—21no—
110C2×C4⋊C4×C31—22no—
111C2×C4⋊C4×C31C2311yescharacteristic
112C2×C22⋊C4×C31—21no—
113C2×C22⋊C4×C31—22no—
114C2×C22⋊C4×C31—21no—
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×#103—22no—
124C31×#103—22no—
125C31×#102—22no—
126C31×#101—22no—
127C31×#101D414yescomplemented
128C31×#100—24no—
129C31×#95—21no—
130C31×#95—22no—
131C24×C31C2×D412yes—
132C24×C31C2×D411yescharacteristic
133#41×C31—22no—
134#41×C31—22no—
135#41×C31—22no—
136#41×C31—22no—
137#41×C31—44no—
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—
145C2×Q8×C31—22no—
146C2×Q8×C31—22no—
147C2×Q8×C31—22no—
148C2×Q8×C31—22no—
149C2×Q8×C31—21no—
150C2×Q8×C31—21no—
151C2×Q8×C31—22no—
152C2×Q8×C31—44no—
153C2×Q8×C31—44no—
154C2×Q8×C31C2×D412yes—
155C2×Q8×C31C2×D412yes—
156C2×Q8×C31#4112yes—
157C2×D4×C31—22no—
158C2×D4×C31—24no—
159C2×D4×C31—22no—
160C2×D4×C31—24no—
161C2×D4×C31—24no—
162C2×D4×C31—22no—
163C2×D4×C31—22no—
164C2×D4×C31—24no—
165C2×D4×C31—21no—
166C2×D4×C31—22no—
167C2×D4×C31—21no—
168C2×D4×C31—22no—
169C2×D4×C31—42no—
170C2×D4×C31C2×D412yes—
171C22×C4×C31—21no—
172C22×C4×C31—21no—
173C22×C4×C31—22no—
174C22×C4×C31—24no—
175C22×C4×C31—21no—
176C22×C4×C31—22no—
177C22×C4×C31—22no—
178C22×C4×C31—24no—
179C22×C4×C31—22no—
180C22×C4×C31—21no—
181C22×C4×C31—21no—
182C22×C4×C31—21no—
183C22×C4×C31—42no—
184C22×C4×C31C2×D411yescharacteristic
185C22×C4×C31C2×D412yes—
186C22×C4×C31C2×D412yes—
187C22×C4×C31C2×D412yes—
188C22×C4×C31C2411yescharacteristic
189Q16×C31—44no—
190SD16×C31—44no—
191M16×C31—22no—
192M16×C31—22no—
193M16×C31—22no—
194M16×C31—44no—
195M16×C31C2×D414yes—
196C2×C8×C31—22no—
197C2×C8×C31—22no—
198C2×C8×C31—44no—
199C4⋊C4×C31—22no—
200C4⋊C4×C31—22no—
201C4⋊C4×C31—22no—
202C4⋊C4×C31—22no—
203C4⋊C4×C31—22no—
204C4⋊C4×C31—41no—
205C4⋊C4×C31—41no—
206C4⋊C4×C31—44no—
207C4⋊C4×C31C2×D412yes—
208C4⋊C4×C31C2×D412yes—
209C22⋊C4×C31—24no—
210C22⋊C4×C31—24no—
211C22⋊C4×C31—22no—
212C22⋊C4×C31—22no—
213C22⋊C4×C31—22no—
214C22⋊C4×C31—22no—
215C22⋊C4×C31—22no—
216C22⋊C4×C31—42no—
217C22⋊C4×C31—44no—
218C22⋊C4×C31—44no—
219C22⋊C4×C31—42no—
220C42×C31—24no—
221C42×C31—22no—
222C42×C31—21no—
223C42×C31—21no—
224C42×C31—22no—
225#24417C3111yesmaximal, characteristic, direct product factor, Sylow
226C23×C31—21no—
227C23×C31—21no—
228C23×C31—22no—
229C23×C31—22no—
230C23×C31—22no—
231C23×C31—22no—
232C23×C31—41no—
233C23×C31—41no—
234C23×C31—42no—
235C23×C31—42no—
236C23×C31#12012yes—
237C23×C31C22×D411yescharacteristic
238Q8×C31—24no—
239Q8×C31—24no—
240Q8×C31—22no—
241Q8×C31—22no—
242Q8×C31—44no—
243Q8×C31—42no—
244Q8×C31—42no—
245D4×C31—22no—
246D4×C31—22no—
247D4×C31—48no—
248D4×C31—44no—
249D4×C31—42no—
250D4×C31—42no—
251D4×C31—42no—
252D4×C31—42no—
253D4×C31—82no—
254D4×C31—82no—
255D4×C31—84no—
256D4×C31—84no—
257C2×C4×C31—22no—
258C2×C4×C31—21no—
259C2×C4×C31—21no—
260C2×C4×C31—22no—
261C2×C4×C31—22no—
262C2×C4×C31—22no—
263C2×C4×C31—22no—
264C2×C4×C31—22no—
265C2×C4×C31—22no—
266C2×C4×C31—22no—
267C2×C4×C31—22no—
268C2×C4×C31—22no—
269C2×C4×C31—44no—
270C2×C4×C31—44no—
271C2×C4×C31—48no—
272C2×C4×C31—41no—
273C2×C4×C31—41no—
274C2×C4×C31—42no—
275C2×C4×C31—42no—
276C2×C4×C31—41no—
277C2×C4×C31—42no—
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#12011yescharacteristic
285C2×C4×C31#12011yescharacteristic
286C2×C4×C31#12114yes—
287C2×C4×C31C22×D411yescharacteristic
288C2×C4×C31C22×D412yes—
289C2×C4×C31C2×#4111yescharacteristic
290C2×C4×C31H(5,𝔽2)12yes—
291C8×C31—24no—
292C8×C31—42no—
293#3487C2×C3111yescharacteristic
294#3056C2×C3112yescomplemented
295C2×#456C2×C3111yescharacteristic, complemented
296C2×#413C2×C3111yescharacteristic, complemented
297#2432C2×C3111yescharacteristic, complemented
298#2037C2×C3112yescomplemented
299#2004C2×C3112yescomplemented
300#1886C2×C3111yescharacteristic, complemented
301#1871C2×C3112yescomplemented
302#1857C2×C3111yescharacteristic, complemented
303#1793C2×C3111yescharacteristic, complemented
304C22×C31—21no—
305C22×C31—21no—
306C22×C31—22no—
307C22×C31—22no—
308C22×C31—22no—
309C22×C31—44no—
310C22×C31—42no—
311C22×C31—42no—
312C22×C31—42no—
313C22×C31—42no—
314C22×C31—42no—
315C22×C31—42no—
316C22×C31—81no—
317C22×C31—81no—
318C22×C31—82no—
319C22×C31C2×#12011yescharacteristic
320C22×C31C2×#12111yescharacteristic
321C22×C31#53411yescharacteristic
322C4×C31—21no—
323C4×C31—21no—
324C4×C31—24no—
325C4×C31—42no—
326C4×C31—42no—
327C4×C31—42no—
328C4×C31—42no—
329C4×C31—81no—
330C4×C31—81no—
331C4×C31—81no—
332C4×C31D4212yes—
333C4×C31#54612yes—
334C2×2−1+4C22×C3111yescharacteristic
335C22×#41C22×C3111yescharacteristic
336C2×#137C22×C3112yes—
337C22×M16C22×C3112yes—
338#550—22no—
339#550—22no—
340#548—22no—
341#548—22no—
342#535—22no—
343#535C22×C3112yes—
344C2×C4⋊Q8C22×C3111yescharacteristic
345C2×#124C22×C3111yescharacteristic
346#525—22no—
347#525C22×C3112yes—
348C2×C22⋊Q8C22×C3111yescharacteristic
349#519C22×C3111yescharacteristic
350#470—22no—
351#468—22no—
352#467—22no—
353#465—22no—
354#456—24no—
355#428C22×C3112yes—
356C2×#104C22×C3112yes—
357C2×#104C22×C3112yes—
358#419C22×C3112yes—
359#417C22×C3112yes—
360#413—24no—
361C2×#101C22×C3111yescharacteristic
362C2×#101C22×C3111yescharacteristic
363C2×#100C22×C3112yes—
364#394—22no—
365#391C22×C3111yescharacteristic
366#390C22×C3111yescharacteristic
367#385—22no—
368C4×C22⋊C4C22×C3111yescharacteristic
369#339C22×C3112yes—
370#339C22×C3112yes—
371C2×C31—21no—
372C2×C31—21no—
373C2×C31—42no—
374C2×C31—82no—
375C2×C31—82no—
376C2×C31#241011yescharacteristic
377C2×C31#301811yescharacteristic
378C2×C31#302211yescharacteristic
3792−1+4—22no—
3802−1+4—22no—
3812−1+4—42no—
382C2×#41—22no—
383C2×#41—22no—
384C2×#41—22no—
385C2×#41—22no—
386C2×#41—22no—
387C2×#41—22no—
388C2×#41—42no—
389C2×#41C23×C3112yes—
390C22×Q8—22no—
391C22×Q8C23×C3111yescharacteristic
392C22×Q8C23×C3111yescharacteristic
393C22×D4D4×C3112yes—
394C22×D4C23×C3111yescharacteristic
395C23×C4D4×C3112yes—
396C23×C4C23×C3111yescharacteristic
397#137—24no—
398#137—42no—
399C2×Q16—22no—
400C2×SD16—22no—
401C2×M16—22no—
402C2×M16—24no—
403C2×M16—42no—
404C2×M16C23×C3112yes—
405C2×M16C23×C3112yes—
406C22×C8—22no—
407C4⋊Q8—22no—
408C4⋊Q8—22no—
409#127—24no—
410#127—22no—
411#124—24no—
412#124—22no—
413#124—42no—
414#123—22no—
415C22⋊Q8—22no—
416C22⋊Q8—22no—
417C22⋊Q8—22no—
418C22⋊Q8—22no—
419C22⋊Q8D4×C3112yes—
420C22⋊Q8D4×C3112yes—
421#121—22no—
422#121—22no—
423#121—24no—
424#121—24no—
425#121—22no—
426#121—22no—
427#121D4×C3112yes—
428#121D4×C3112yes—
429#120—24no—
430#120—22no—
431C4×Q8—22no—
432C4×Q8—22no—
433C4×D4—24no—
434C4×D4—24no—
435C4×D4—22no—
436C4×D4—22no—
437#117—22no—
438#117C23×C3112yes—
439C2×C4⋊C4—21no—
440C2×C4⋊C4—22no—
441C2×C4⋊C4C23×C3111yescharacteristic
442C2×C22⋊C4—21no—
443C2×C22⋊C4—22no—
444C2×C22⋊C4—21no—
445C2×C22⋊C4—22no—
446C2×C42C23×C3111yescharacteristic
447C2×C42C23×C3111yescharacteristic
448#107—22no—
449#106—22no—
450#104—24no—
451#104—42no—
452#103—22no—
453#103—22no—
454#103—22no—
455#102—22no—
456#101—22no—
457#101D4×C3114yescomplemented
458#100—24no—
459#95—21no—
460#95—22no—
461C31#2441711yescharacteristic, direct product factor, Sylow
462C24C2×D4×C3112yes—
463C24C2×D4×C3111yescharacteristic
464#41—22no—
465#41—22no—
466#41—22no—
467#41—22no—
468#41—44no—
469#41—44no—
470#41—44no—
471#41—44no—
472#41—44no—
473#41—44no—
474#41—44no—
475#41—44no—
476C2×Q8—22no—
477C2×Q8—22no—
478C2×Q8—22no—
479C2×Q8—22no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—22no—
483C2×Q8—44no—
484C2×Q8—44no—
485C2×Q8C2×D4×C3112yes—
486C2×Q8C2×D4×C3112yes—
487C2×Q8#41×C3112yes—
488C2×D4—22no—
489C2×D4—24no—
490C2×D4—22no—
491C2×D4—24no—
492C2×D4—24no—
493C2×D4—22no—
494C2×D4—22no—
495C2×D4—24no—
496C2×D4—21no—
497C2×D4—22no—
498C2×D4—21no—
499C2×D4—22no—
500C2×D4—42no—
501C2×D4C2×D4×C3112yes—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—22no—
505C22×C4—24no—
506C22×C4—21no—
507C22×C4—22no—
508C22×C4—22no—
509C22×C4—24no—
510C22×C4—22no—
511C22×C4—21no—
512C22×C4—21no—
513C22×C4—21no—
514C22×C4—42no—
515C22×C4C2×D4×C3111yescharacteristic
516C22×C4C2×D4×C3112yes—
517C22×C4C2×D4×C3112yes—
518C22×C4C2×D4×C3112yes—
519C22×C4C24×C3111yescharacteristic
520Q16—44no—
521SD16—44no—
522M16—22no—
523M16—22no—
524M16—22no—
525M16—44no—
526M16C2×D4×C3114yes—
527C2×C8—22no—
528C2×C8—22no—
529C2×C8—44no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—41no—
536C4⋊C4—41no—
537C4⋊C4—44no—
538C4⋊C4C2×D4×C3112yes—
539C4⋊C4C2×D4×C3112yes—
540C22⋊C4—24no—
541C22⋊C4—24no—
542C22⋊C4—22no—
543C22⋊C4—22no—
544C22⋊C4—22no—
545C22⋊C4—22no—
546C22⋊C4—22no—
547C22⋊C4—42no—
548C22⋊C4—44no—
549C22⋊C4—44no—
550C22⋊C4—42no—
551C42—24no—
552C42—22no—
553C42—21no—
554C42—21no—
555C42—22no—
556C23—21no—
557C23—21no—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—22no—
562C23—41no—
563C23—41no—
564C23—42no—
565C23—42no—
566C23C31×#12012yes—
567C23C22×D4×C3111yescharacteristic
568Q8—24no—
569Q8—24no—
570Q8—22no—
571Q8—22no—
572Q8—44no—
573Q8—42no—
574Q8—42no—
575D4—22no—
576D4—22no—
577D4—48no—
578D4—44no—
579D4—42no—
580D4—42no—
581D4—42no—
582D4—42no—
583D4—82no—
584D4—82no—
585D4—84no—
586D4—84no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—48no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4C31×#12011yescharacteristic
615C2×C4C31×#12011yescharacteristic
616C2×C4C31×#12114yes—
617C2×C4C22×D4×C3111yescharacteristic
618C2×C4C22×D4×C3112yes—
619C2×C4C2×#41×C3111yescharacteristic
620C2×C4C31×H(5,𝔽2)12yes—
621C8—24no—
622C8—42no—
623C22—21no—
624C22—21no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—44no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—81no—
636C22—81no—
637C22—82no—
638C22C2×C31×#12011yescharacteristic
639C22C2×C31×#12111yescharacteristic
640C22C31×#53411yescharacteristic
641C4—21no—
642C4—21no—
643C4—24no—
644C4—42no—
645C4—42no—
646C4—42no—
647C4—42no—
648C4—81no—
649C4—81no—
650C4—81no—
651C4D42×C3112yes—
652C4C31×#54612yes—
653C2—21no—
654C2—21no—
655C2—42no—
656C2—82no—
657C2—82no—
658C2C31×#241011yescharacteristic
659C2C31×#301811yescharacteristic
660C2C31×#302211yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#3487C2
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
2. C31×#3056C2semidirect
3. C2×C31×#456C2semidirect
4. C2×C31×#413C2semidirect
5. C31×#2432C2semidirect
6. C31×#2037C2semidirect
7. C31×#2004C2semidirect
8. C31×#1886C2semidirect
9. C31×#1871C2semidirect
10. C31×#1857C2semidirect
11. C31×#1793C2semidirect
127. C31×#101D4semidirect
225. #24417C31direct (N × H)
294. #3056C2×C31semidirect
295. C2×#456C2×C31semidirect
296. C2×#413C2×C31semidirect
297. #2432C2×C31semidirect
298. #2037C2×C31semidirect
299. #2004C2×C31semidirect
300. #1886C2×C31semidirect
301. #1871C2×C31semidirect
302. #1857C2×C31semidirect
303. #1793C2×C31semidirect
457. #101D4×C31semidirect
461. C31#24417direct (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×#30562
C2×C31×#4562
C2×C31×#4132
C31×#24322
C31×#20372
C31×#20042
C31×#18862
C31×#18712
C31×#18572
C31×#17932
#2441731