Finite Group Databasebeta

← C31×#18241 #21475000

Subgroups — C31×#18241 #21475000

Order 7936.

992 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3558C211yesmaximal, characteristic
2C31×#3117C211yesmaximal, characteristic, complemented
3C31×#3103C211yesmaximal, characteristic, complemented
4C31×#2632C211yesmaximal, characteristic, complemented
5C31×#1618C212yesmaximal, complemented
6C31×#1613C212yesmaximal, complemented
7C31×#1606C212yesmaximal, complemented
8C31×#1601C212yesmaximal, complemented
9C31×#1521C211yesmaximal, characteristic, complemented
10C31×#1514C211yesmaximal, characteristic, complemented
11C31×#1469C211yesmaximal, characteristic, complemented
12C31×#562—21no—
13C31×H(3,𝔽4)—21no—
14C31×#559—21no—
15C31×#548—21no—
16C31×#546—21no—
17C31×#546—22no—
18D42×C31—21no—
19C31×#542—21no—
20C31×#538—22no—
21C31×#538C2212yes—
22C31×#535—21no—
23C31×#534C2212yes—
24C31×#532C2211yescharacteristic
25C31×#529C2211yescharacteristic
26C2×C31×#124C2211yescharacteristic
27C2×C4×Q8×C31C2211yescharacteristic, complemented
28C31×#484C2211yescharacteristic, complemented
29C31×#481C2211yescharacteristic
30C31×#481C2212yescomplemented
31C31×#479—21no—
32C31×#475—21no—
33C31×#464—21no—
34C31×#460—21no—
35C31×#451C2211yescharacteristic, complemented
36C31×#448C2211yescharacteristic
37C31×#448C2212yescomplemented
38C31×#441—21no—
39C31×#440—21no—
40C4×SD16×C31—22no—
41C31×#432C2211yescharacteristic, complemented
42C31×#431C2211yescharacteristic, complemented
43C31×#399C2212yescomplemented
44C31×#395—21no—
45C31×#394—21no—
46C31×#393C2212yescomplemented
47C31×#388—22no—
48C31×#383C2211yescharacteristic, complemented
49C31×#328C2212yescomplemented
50C31×#327C2212yes—
51C31×#327C2212yescomplemented
52C31×#327C2212yescomplemented
53C31×#323C2212yescomplemented
54C31×#323C2212yescomplemented
55C2×#41×C31—21no—
56C2×#41×C31—21no—
57C22×Q8×C31D411yescharacteristic
58C22×Q8×C31C2311yescharacteristic
59C22×D4×C31—21no—
60C22×D4×C31—22no—
61C22×D4×C31D411yescharacteristic
62C2×Q16×C31—21no—
63C2×SD16×C31—22no—
64C2×SD16×C31—21no—
65C31×C4⋊Q8D411yescharacteristic
66C31×#127—21no—
67C31×#127D411yescharacteristic
68C31×#126—22no—
69C31×#126—21no—
70C31×#125D411yescharacteristic
71C31×#124—21no—
72C31×#124—22no—
73C31×#124—22no—
74C31×#124—22no—
75C31×#124—21no—
76C31×#124—21no—
77C31×#124—21no—
78C31×#124—21no—
79C31×#124—21no—
80C31×#124D411yescharacteristic
81C31×#123—22no—
82C31×#123—21no—
83C31×#123—22no—
84C31×#123—22no—
85C31×#123—22no—
86C31×C22⋊Q8—22no—
87C31×C22⋊Q8—21no—
88C31×C22⋊Q8—22no—
89C31×C22⋊Q8C2311yescharacteristic
90C31×#121—22no—
91C31×#121—22no—
92C31×#121—22no—
93C31×#121—21no—
94C31×#121—22no—
95C31×#121—21no—
96C31×#121—22no—
97C31×#121C2311yescharacteristic
98C31×#121C2312yes—
99C31×#120—22no—
100C31×#120—22no—
101C31×#120—22no—
102C31×#120—22no—
103C31×#120—22no—
104C4×Q8×C31—21no—
105C4×Q8×C31—21no—
106C4×Q8×C31—21no—
107C4×Q8×C31—21no—
108C4×Q8×C31—21no—
109C4×Q8×C31D411yescharacteristic
110C4×D4×C31—22no—
111C4×D4×C31—21no—
112C4×D4×C31—21no—
113C4×D4×C31—22no—
114C4×D4×C31—21no—
115C4×D4×C31D412yes—
116C4×D4×C31D411yescharacteristic
117C31×#117—21no—
118C2×C4⋊C4×C31—21no—
119C2×C4⋊C4×C31—22no—
120C2×C4⋊C4×C31C2311yescharacteristic
121C2×C22⋊C4×C31—22no—
122C2×C22⋊C4×C31—22no—
123C2×C22⋊C4×C31—22no—
124C2×C22⋊C4×C31D412yes—
125C2×C22⋊C4×C31C2312yes—
126C2×C42×C31—21no—
127C2×C42×C31C2311yescharacteristic
128C31×#107—21no—
129C31×#106—22no—
130C31×#106—21no—
131C31×C4⋊C8—21no—
132C31×C4⋊C8—21no—
133C31×#103—21no—
134C31×#103—22no—
135C31×#103—21no—
136C31×#103—21no—
137C31×#102—22no—
138C31×#102—21no—
139C31×C22⋊C8C2312yes—
140C31×C22⋊C8C2311yescharacteristic
141C31×C22⋊C8C2311yescharacteristic
142C31×#97—21no—
143C4×C8×C31—21no—
144C31×#95—21no—
145C31×#95—22no—
146C31×#95—21no—
147C31×#95C2312yes—
148C24×C31—22no—
149C24×C31C2×D412yes—
150#41×C31—82no—
151#41×C31—82no—
152C2×Q8×C31—21no—
153C2×Q8×C31—21no—
154C2×Q8×C31—21no—
155C2×Q8×C31—21no—
156C2×Q8×C31—21no—
157C2×Q8×C31—42no—
158C2×Q8×C31—81no—
159C2×Q8×C31C2×D411yescharacteristic
160C2×D4×C31—22no—
161C2×D4×C31—21no—
162C2×D4×C31—22no—
163C2×D4×C31—22no—
164C2×D4×C31—22no—
165C2×D4×C31—22no—
166C2×D4×C31—22no—
167C2×D4×C31—21no—
168C2×D4×C31—21no—
169C2×D4×C31—21no—
170C2×D4×C31—21no—
171C2×D4×C31—82no—
172C2×D4×C31—81no—
173C2×D4×C31—84no—
174C2×D4×C31C2×D411yescharacteristic
175C2×D4×C31C2×D412yes—
176C22×C4×C31—21no—
177C22×C4×C31—22no—
178C22×C4×C31—22no—
179C22×C4×C31—21no—
180C22×C4×C31—22no—
181C22×C4×C31—22no—
182C22×C4×C31—21no—
183C22×C4×C31C2×D411yescharacteristic
184C22×C4×C31C2×D412yes—
185C22×C4×C31C2×D411yescharacteristic
186C22×C4×C31C2411yescharacteristic
187Q16×C31—42no—
188SD16×C31—44no—
189SD16×C31—42no—
190C2×C8×C31—22no—
191C2×C8×C31—21no—
192C2×C8×C31—21no—
193C4⋊C4×C31—22no—
194C4⋊C4×C31—22no—
195C4⋊C4×C31—21no—
196C4⋊C4×C31—21no—
197C4⋊C4×C31—21no—
198C4⋊C4×C31—21no—
199C4⋊C4×C31—21no—
200C4⋊C4×C31—21no—
201C4⋊C4×C31—21no—
202C4⋊C4×C31—41no—
203C4⋊C4×C31—41no—
204C4⋊C4×C31—82no—
205C4⋊C4×C31C2×D412yes—
206C4⋊C4×C31C2×D411yescharacteristic
207C4⋊C4×C31C2×D411yescharacteristic
208C22⋊C4×C31—22no—
209C22⋊C4×C31—22no—
210C22⋊C4×C31—22no—
211C22⋊C4×C31—22no—
212C22⋊C4×C31—22no—
213C22⋊C4×C31—21no—
214C22⋊C4×C31—21no—
215C22⋊C4×C31—22no—
216C22⋊C4×C31—22no—
217C22⋊C4×C31—21no—
218C22⋊C4×C31—21no—
219C22⋊C4×C31—22no—
220C22⋊C4×C31—22no—
221C22⋊C4×C31—22no—
222C22⋊C4×C31—21no—
223C22⋊C4×C31—22no—
224C22⋊C4×C31—44no—
225C22⋊C4×C31—82no—
226C42×C31—22no—
227C42×C31—21no—
228C42×C31—41no—
229C42×C31—41no—
230C42×C31C2×D411yescharacteristic
231C42×C31C2×D411yescharacteristic
232C42×C31C2×D411yescharacteristic
233C42×C31C2×D411yescharacteristic
234#18241C3111yesmaximal, characteristic, direct product factor, Sylow
235C23×C31—22no—
236C23×C31—21no—
237C23×C31—22no—
238C23×C31—24no—
239C23×C31—42no—
240C23×C31—42no—
241C23×C31—82no—
242C23×C31—82no—
243C23×C31—82no—
244C23×C31C22×D411yescharacteristic
245Q8×C31—22no—
246Q8×C31—22no—
247Q8×C31—41no—
248Q8×C31—41no—
249Q8×C31—41no—
250Q8×C31—42no—
251Q8×C31—81no—
252Q8×C31—81no—
253D4×C31—42no—
254D4×C31—41no—
255D4×C31—42no—
256D4×C31—42no—
257D4×C31—42no—
258D4×C31—44no—
259D4×C31—81no—
260D4×C31—81no—
261D4×C31—82no—
262D4×C31—82no—
263D4×C31—82no—
264D4×C31—82no—
265D4×C31—82no—
266C2×C4×C31—22no—
267C2×C4×C31—22no—
268C2×C4×C31—21no—
269C2×C4×C31—22no—
270C2×C4×C31—22no—
271C2×C4×C31—21no—
272C2×C4×C31—21no—
273C2×C4×C31—41no—
274C2×C4×C31—41no—
275C2×C4×C31—42no—
276C2×C4×C31—41no—
277C2×C4×C31—41no—
278C2×C4×C31—42no—
279C2×C4×C31—42no—
280C2×C4×C31—42no—
281C2×C4×C31—42no—
282C2×C4×C31—41no—
283C2×C4×C31—42no—
284C2×C4×C31—82no—
285C2×C4×C31—82no—
286C2×C4×C31—82no—
287C2×C4×C31#12011yescharacteristic
288C2×C4×C31#12011yescharacteristic
289C2×C4×C31#12011yescharacteristic
290C2×C4×C31#12011yescharacteristic
291C2×C4×C31C22×D411yescharacteristic
292C2×C4×C31C22×D411yescharacteristic
293C8×C31—42no—
294C8×C31—41no—
295C8×C31—41no—
296#3558C2×C3111yescharacteristic
297#3117C2×C3111yescharacteristic, complemented
298#3103C2×C3111yescharacteristic, complemented
299#2632C2×C3111yescharacteristic, complemented
300#1618C2×C3112yescomplemented
301#1613C2×C3112yescomplemented
302#1606C2×C3112yescomplemented
303#1601C2×C3112yescomplemented
304#1521C2×C3111yescharacteristic, complemented
305#1514C2×C3111yescharacteristic, complemented
306#1469C2×C3111yescharacteristic, complemented
307C22×C31—21no—
308C22×C31—21no—
309C22×C31—42no—
310C22×C31—42no—
311C22×C31—41no—
312C22×C31—41no—
313C22×C31—42no—
314C22×C31—42no—
315C22×C31—42no—
316C22×C31—41no—
317C22×C31—42no—
318C22×C31—84no—
319C22×C31—164no—
320C22×C31C2×#12011yescharacteristic
321C22×C31#57712yes—
322C4×C31—21no—
323C4×C31—21no—
324C4×C31—21no—
325C4×C31—21no—
326C4×C31—21no—
327C4×C31—21no—
328C4×C31—41no—
329C4×C31—42no—
330C4×C31—41no—
331C4×C31—42no—
332C4×C31—42no—
333C4×C31—41no—
334C4×C31—82no—
335C4×C31—81no—
336#562—21no—
337H(3,𝔽4)—21no—
338#559—21no—
339#548—21no—
340#546—21no—
341#546—22no—
342D42—21no—
343#542—21no—
344#538—22no—
345#538C22×C3112yes—
346#535—21no—
347#534C22×C3112yes—
348#532C22×C3111yescharacteristic
349#529C22×C3111yescharacteristic
350C2×#124C22×C3111yescharacteristic
351C2×C4×Q8C22×C3111yescharacteristic, complemented
352#484C22×C3111yescharacteristic, complemented
353#481C22×C3111yescharacteristic
354#481C22×C3112yescomplemented
355#479—21no—
356#475—21no—
357#464—21no—
358#460—21no—
359#451C22×C3111yescharacteristic, complemented
360#448C22×C3111yescharacteristic
361#448C22×C3112yescomplemented
362#441—21no—
363#440—21no—
364C4×SD16—22no—
365#432C22×C3111yescharacteristic, complemented
366#431C22×C3111yescharacteristic, complemented
367#399C22×C3112yescomplemented
368#395—21no—
369#394—21no—
370#393C22×C3112yescomplemented
371#388—22no—
372#383C22×C3111yescharacteristic, complemented
373#328C22×C3112yescomplemented
374#327C22×C3112yes—
375#327C22×C3112yescomplemented
376#327C22×C3112yescomplemented
377#323C22×C3112yescomplemented
378#323C22×C3112yescomplemented
379C2×C31—22no—
380C2×C31—82no—
381C2×C31—82no—
382C2×C31—81no—
383C2×C31#300911yescharacteristic
384C2×C31#301911yescharacteristic
385C2×C31#302511yescharacteristic
386C2×#41—21no—
387C2×#41—21no—
388C22×Q8D4×C3111yescharacteristic
389C22×Q8C23×C3111yescharacteristic
390C22×D4—21no—
391C22×D4—22no—
392C22×D4D4×C3111yescharacteristic
393C2×Q16—21no—
394C2×SD16—22no—
395C2×SD16—21no—
396C4⋊Q8D4×C3111yescharacteristic
397#127—21no—
398#127D4×C3111yescharacteristic
399#126—22no—
400#126—21no—
401#125D4×C3111yescharacteristic
402#124—21no—
403#124—22no—
404#124—22no—
405#124—22no—
406#124—21no—
407#124—21no—
408#124—21no—
409#124—21no—
410#124—21no—
411#124D4×C3111yescharacteristic
412#123—22no—
413#123—21no—
414#123—22no—
415#123—22no—
416#123—22no—
417C22⋊Q8—22no—
418C22⋊Q8—21no—
419C22⋊Q8—22no—
420C22⋊Q8C23×C3111yescharacteristic
421#121—22no—
422#121—22no—
423#121—22no—
424#121—21no—
425#121—22no—
426#121—21no—
427#121—22no—
428#121C23×C3111yescharacteristic
429#121C23×C3112yes—
430#120—22no—
431#120—22no—
432#120—22no—
433#120—22no—
434#120—22no—
435C4×Q8—21no—
436C4×Q8—21no—
437C4×Q8—21no—
438C4×Q8—21no—
439C4×Q8—21no—
440C4×Q8D4×C3111yescharacteristic
441C4×D4—22no—
442C4×D4—21no—
443C4×D4—21no—
444C4×D4—22no—
445C4×D4—21no—
446C4×D4D4×C3111yescharacteristic
447C4×D4D4×C3112yes—
448#117—21no—
449C2×C4⋊C4—21no—
450C2×C4⋊C4—22no—
451C2×C4⋊C4C23×C3111yescharacteristic
452C2×C22⋊C4—22no—
453C2×C22⋊C4—22no—
454C2×C22⋊C4—22no—
455C2×C22⋊C4D4×C3112yes—
456C2×C22⋊C4C23×C3112yes—
457C2×C42—21no—
458C2×C42C23×C3111yescharacteristic
459#107—21no—
460#106—22no—
461#106—21no—
462C4⋊C8—21no—
463C4⋊C8—21no—
464#103—21no—
465#103—22no—
466#103—21no—
467#103—21no—
468#102—22no—
469#102—21no—
470C22⋊C8C23×C3112yes—
471C22⋊C8C23×C3111yescharacteristic
472C22⋊C8C23×C3111yescharacteristic
473#97—21no—
474C4×C8—21no—
475#95—21no—
476#95—21no—
477#95—22no—
478#95C23×C3112yes—
479C31#1824111yescharacteristic, direct product factor, Sylow
480C24—22no—
481C24C2×D4×C3112yes—
482#41—82no—
483#41—82no—
484C2×Q8—21no—
485C2×Q8—21no—
486C2×Q8—21no—
487C2×Q8—21no—
488C2×Q8—21no—
489C2×Q8—42no—
490C2×Q8—81no—
491C2×Q8C2×D4×C3111yescharacteristic
492C2×D4—22no—
493C2×D4—21no—
494C2×D4—22no—
495C2×D4—22no—
496C2×D4—22no—
497C2×D4—22no—
498C2×D4—22no—
499C2×D4—21no—
500C2×D4—21no—
501C2×D4—21no—
502C2×D4—21no—
503C2×D4—82no—
504C2×D4—81no—
505C2×D4—84no—
506C2×D4C2×D4×C3111yescharacteristic
507C2×D4C2×D4×C3112yes—
508C22×C4—21no—
509C22×C4—22no—
510C22×C4—22no—
511C22×C4—21no—
512C22×C4—22no—
513C22×C4—22no—
514C22×C4—21no—
515C22×C4C2×D4×C3111yescharacteristic
516C22×C4C2×D4×C3112yes—
517C22×C4C2×D4×C3111yescharacteristic
518C22×C4C24×C3111yescharacteristic
519Q16—42no—
520SD16—44no—
521SD16—42no—
522C2×C8—22no—
523C2×C8—21no—
524C2×C8—21no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—41no—
535C4⋊C4—41no—
536C4⋊C4—82no—
537C4⋊C4C2×D4×C3112yes—
538C4⋊C4C2×D4×C3111yescharacteristic
539C4⋊C4C2×D4×C3111yescharacteristic
540C22⋊C4—22no—
541C22⋊C4—22no—
542C22⋊C4—22no—
543C22⋊C4—22no—
544C22⋊C4—22no—
545C22⋊C4—21no—
546C22⋊C4—21no—
547C22⋊C4—22no—
548C22⋊C4—22no—
549C22⋊C4—21no—
550C22⋊C4—21no—
551C22⋊C4—22no—
552C22⋊C4—22no—
553C22⋊C4—22no—
554C22⋊C4—21no—
555C22⋊C4—22no—
556C22⋊C4—44no—
557C22⋊C4—82no—
558C42—22no—
559C42—21no—
560C42—41no—
561C42—41no—
562C42C2×D4×C3111yescharacteristic
563C42C2×D4×C3111yescharacteristic
564C42C2×D4×C3111yescharacteristic
565C42C2×D4×C3111yescharacteristic
566C23—22no—
567C23—21no—
568C23—22no—
569C23—24no—
570C23—42no—
571C23—42no—
572C23—82no—
573C23—82no—
574C23—82no—
575C23C22×D4×C3111yescharacteristic
576Q8—22no—
577Q8—22no—
578Q8—41no—
579Q8—41no—
580Q8—41no—
581Q8—42no—
582Q8—81no—
583Q8—81no—
584D4—42no—
585D4—41no—
586D4—42no—
587D4—42no—
588D4—42no—
589D4—44no—
590D4—81no—
591D4—81no—
592D4—82no—
593D4—82no—
594D4—82no—
595D4—82no—
596D4—82no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—82no—
616C2×C4—82no—
617C2×C4—82no—
618C2×C4C31×#12011yescharacteristic
619C2×C4C31×#12011yescharacteristic
620C2×C4C31×#12011yescharacteristic
621C2×C4C31×#12011yescharacteristic
622C2×C4C22×D4×C3111yescharacteristic
623C2×C4C22×D4×C3111yescharacteristic
624C8—42no—
625C8—41no—
626C8—41no—
627C22—21no—
628C22—21no—
629C22—42no—
630C22—42no—
631C22—41no—
632C22—41no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—41no—
637C22—42no—
638C22—84no—
639C22—164no—
640C22C2×C31×#12011yescharacteristic
641C22C31×#57712yes—
642C4—21no—
643C4—21no—
644C4—21no—
645C4—21no—
646C4—21no—
647C4—21no—
648C4—41no—
649C4—42no—
650C4—41no—
651C4—42no—
652C4—42no—
653C4—41no—
654C4—82no—
655C4—81no—
656C2—22no—
657C2—82no—
658C2—81no—
659C2—82no—
660C2C31×#300911yescharacteristic
661C2C31×#301911yescharacteristic
662C2C31×#302511yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#3558C2
LCSG ▷ C22×C4 ▷ 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×#3117C2semidirect
3. C31×#3103C2semidirect
4. C31×#2632C2semidirect
5. C31×#1618C2semidirect
6. C31×#1613C2semidirect
7. C31×#1606C2semidirect
8. C31×#1601C2semidirect
9. C31×#1521C2semidirect
10. C31×#1514C2semidirect
11. C31×#1469C2semidirect
27. C2×C4×Q8×C31C22semidirect
28. C31×#484C22semidirect
30. C31×#481C22semidirect
35. C31×#451C22semidirect
37. C31×#448C22semidirect
41. C31×#432C22semidirect
42. C31×#431C22semidirect
43. C31×#399C22semidirect
46. C31×#393C22semidirect
48. C31×#383C22semidirect
49. C31×#328C22semidirect
51. C31×#327C22semidirect
52. C31×#327C22semidirect
53. C31×#323C22semidirect
54. C31×#323C22semidirect
234. #18241C31direct (N × H)
297. #3117C2×C31semidirect
298. #3103C2×C31semidirect
299. #2632C2×C31semidirect
300. #1618C2×C31semidirect
301. #1613C2×C31semidirect
302. #1606C2×C31semidirect
303. #1601C2×C31semidirect
304. #1521C2×C31semidirect
305. #1514C2×C31semidirect
306. #1469C2×C31semidirect
351. C2×C4×Q8C22×C31semidirect
352. #484C22×C31semidirect
354. #481C22×C31semidirect
359. #451C22×C31semidirect
361. #448C22×C31semidirect
365. #432C22×C31semidirect
366. #431C22×C31semidirect
367. #399C22×C31semidirect
370. #393C22×C31semidirect
372. #383C22×C31semidirect
373. #328C22×C31semidirect
375. #327C22×C31semidirect
376. #327C22×C31semidirect
377. #323C22×C31semidirect
378. #323C22×C31semidirect
479. C31#18241direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35582
C31×#31172
C31×#31032
C31×#26322
C31×#16182
C31×#16132
C31×#16062
C31×#16012
C31×#15212
C31×#15142
C31×#14692
#1824131