Finite Group Databasebeta

← C31×#18466 #21475225

Subgroups — C31×#18466 #21475225

Order 7936.

954 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3558C211yesmaximal, characteristic
2C31×#3234C211yesmaximal, characteristic, complemented
3C31×#3232C211yesmaximal, characteristic, complemented
4C31×#2820C211yesmaximal, characteristic, complemented
5C31×#1618C212yesmaximal, complemented
6C31×#1613C212yesmaximal, complemented
7C31×#1612C212yesmaximal, complemented
8C31×#1607C212yesmaximal, complemented
9C31×#1517C211yesmaximal, characteristic, complemented
10C31×#1512C211yesmaximal, 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×#125C2211yescharacteristic, complemented
27C2×C31×#124C2211yescharacteristic
28C31×#501—21no—
29C31×#499—21no—
30C31×#490—21no—
31C31×#488—21no—
32C31×#484C2211yescharacteristic, complemented
33C31×#483C2211yescharacteristic, complemented
34C31×#481C2211yescharacteristic
35C31×#481C2212yescomplemented
36C31×#480C2211yescharacteristic
37C31×#480C2212yescomplemented
38C31×#479—21no—
39C31×#478—21no—
40C31×#478—22no—
41C31×#432C2211yescharacteristic, complemented
42C31×#431C2211yescharacteristic, complemented
43C31×#399C2212yescomplemented
44C31×#397C2212yescomplemented
45C31×#396—22no—
46C31×#395—21no—
47C31×#394—21no—
48C31×#382C2211yescharacteristic, 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×D4×C31—21no—
59C22×D4×C31—22no—
60C22×D4×C31D411yescharacteristic
61C31×C4⋊Q8D411yescharacteristic
62C31×#127—21no—
63C31×#127D411yescharacteristic
64C31×#126—22no—
65C31×#126—21no—
66C31×#125—21no—
67C31×#125—21no—
68C31×#125—21no—
69C31×#125—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×C31D411yescharacteristic
106C4×D4×C31—22no—
107C4×D4×C31—21no—
108C4×D4×C31—21no—
109C4×D4×C31—22no—
110C4×D4×C31—21no—
111C4×D4×C31D411yescharacteristic
112C4×D4×C31D412yes—
113C31×#117—21no—
114C2×C4⋊C4×C31—21no—
115C2×C4⋊C4×C31—21no—
116C2×C4⋊C4×C31—22no—
117C2×C4⋊C4×C31—21no—
118C2×C4⋊C4×C31C2311yescharacteristic
119C2×C4⋊C4×C31C2311yescharacteristic
120C2×C22⋊C4×C31—22no—
121C2×C22⋊C4×C31—22no—
122C2×C22⋊C4×C31—22no—
123C2×C22⋊C4×C31D412yes—
124C2×C22⋊C4×C31C2312yes—
125C2×C42×C31C2311yescharacteristic
126C31×#107—22no—
127C31×#107—21no—
128C31×#107—21no—
129C31×#106—22no—
130C31×#106—21no—
131C31×#106—21no—
132C31×C4⋊C8—21no—
133C31×C4⋊C8—21no—
134C31×#103—21no—
135C31×#103—21no—
136C31×#102—21no—
137C31×#102—22no—
138C31×#102—21no—
139C31×#102—22no—
140C31×C22⋊C8C2311yescharacteristic
141C31×C22⋊C8C2311yescharacteristic
142C31×C22⋊C8C2312yes—
143C31×#97—21no—
144C4×C8×C31—21no—
145C31×#95—22no—
146C31×#95—21no—
147C31×#95C2312yes—
148C24×C31—22no—
149C24×C31C2×D412yes—
150#41×C31—81no—
151#41×C31—81no—
152#41×C31—82no—
153C2×Q8×C31—21no—
154C2×Q8×C31—21no—
155C2×Q8×C31—21no—
156C2×Q8×C31—81no—
157C2×Q8×C31C2×D411yescharacteristic
158C2×D4×C31—22no—
159C2×D4×C31—21no—
160C2×D4×C31—22no—
161C2×D4×C31—22no—
162C2×D4×C31—22no—
163C2×D4×C31—22no—
164C2×D4×C31—22no—
165C2×D4×C31—21no—
166C2×D4×C31—21no—
167C2×D4×C31—21no—
168C2×D4×C31—21no—
169C2×D4×C31—82no—
170C2×D4×C31—81no—
171C2×D4×C31—82no—
172C2×D4×C31—82no—
173C2×D4×C31C2×D411yescharacteristic
174C2×D4×C31C2×D412yes—
175C22×C4×C31—22no—
176C22×C4×C31—22no—
177C22×C4×C31—21no—
178C22×C4×C31—22no—
179C22×C4×C31—22no—
180C22×C4×C31—21no—
181C22×C4×C31—21no—
182C22×C4×C31C2×D411yescharacteristic
183C22×C4×C31C2×D412yes—
184C22×C4×C31C2×D411yescharacteristic
185C22×C4×C31C2411yescharacteristic
186C2×C8×C31—21no—
187C2×C8×C31—21no—
188C2×C8×C31—22no—
189C4⋊C4×C31—22no—
190C4⋊C4×C31—22no—
191C4⋊C4×C31—21no—
192C4⋊C4×C31—21no—
193C4⋊C4×C31—21no—
194C4⋊C4×C31—21no—
195C4⋊C4×C31—21no—
196C4⋊C4×C31—21no—
197C4⋊C4×C31—21no—
198C4⋊C4×C31—21no—
199C4⋊C4×C31—21no—
200C4⋊C4×C31—41no—
201C4⋊C4×C31—41no—
202C4⋊C4×C31—41no—
203C4⋊C4×C31—41no—
204C4⋊C4×C31—81no—
205C4⋊C4×C31—82no—
206C4⋊C4×C31C2×D412yes—
207C4⋊C4×C31C2×D411yescharacteristic
208C4⋊C4×C31C2×D411yescharacteristic
209C22⋊C4×C31—22no—
210C22⋊C4×C31—22no—
211C22⋊C4×C31—22no—
212C22⋊C4×C31—22no—
213C22⋊C4×C31—22no—
214C22⋊C4×C31—21no—
215C22⋊C4×C31—21no—
216C22⋊C4×C31—22no—
217C22⋊C4×C31—22no—
218C22⋊C4×C31—21no—
219C22⋊C4×C31—21no—
220C22⋊C4×C31—22no—
221C22⋊C4×C31—22no—
222C22⋊C4×C31—22no—
223C22⋊C4×C31—21no—
224C22⋊C4×C31—22no—
225C22⋊C4×C31—82no—
226C22⋊C4×C31—82no—
227C42×C31—22no—
228C42×C31—21no—
229C42×C31C2×D411yescharacteristic
230C42×C31C2×D411yescharacteristic
231C42×C31C2×D411yescharacteristic
232C42×C31C2×D411yescharacteristic
233#18466C3111yesmaximal, characteristic, direct product factor, Sylow
234C23×C31—22no—
235C23×C31—21no—
236C23×C31—22no—
237C23×C31—22no—
238C23×C31—22no—
239C23×C31—42no—
240C23×C31—42no—
241C23×C31—82no—
242C23×C31—82no—
243C23×C31—82no—
244C23×C31C22×D411yescharacteristic
245Q8×C31—41no—
246Q8×C31—42no—
247Q8×C31—81no—
248Q8×C31—81no—
249D4×C31—42no—
250D4×C31—41no—
251D4×C31—42no—
252D4×C31—42no—
253D4×C31—41no—
254D4×C31—42no—
255D4×C31—41no—
256D4×C31—42no—
257D4×C31—81no—
258D4×C31—81no—
259D4×C31—82no—
260D4×C31—82no—
261D4×C31—82no—
262D4×C31—82no—
263D4×C31—82no—
264C2×C4×C31—21no—
265C2×C4×C31—21no—
266C2×C4×C31—21no—
267C2×C4×C31—22no—
268C2×C4×C31—22no—
269C2×C4×C31—21no—
270C2×C4×C31—21no—
271C2×C4×C31—41no—
272C2×C4×C31—41no—
273C2×C4×C31—42no—
274C2×C4×C31—42no—
275C2×C4×C31—42no—
276C2×C4×C31—42no—
277C2×C4×C31—42no—
278C2×C4×C31—41no—
279C2×C4×C31—41no—
280C2×C4×C31—41no—
281C2×C4×C31—42no—
282C2×C4×C31—41no—
283C2×C4×C31—82no—
284C2×C4×C31—82no—
285C2×C4×C31—82no—
286C2×C4×C31#12011yescharacteristic
287C2×C4×C31#12011yescharacteristic
288C2×C4×C31#12011yescharacteristic
289C2×C4×C31#12011yescharacteristic
290C2×C4×C31C22×D411yescharacteristic
291C2×C4×C31C22×D411yescharacteristic
292C8×C31—42no—
293C8×C31—41no—
294C8×C31—41no—
295#3558C2×C3111yescharacteristic
296#3234C2×C3111yescharacteristic, complemented
297#3232C2×C3111yescharacteristic, complemented
298#2820C2×C3111yescharacteristic, complemented
299#1618C2×C3112yescomplemented
300#1613C2×C3112yescomplemented
301#1612C2×C3112yescomplemented
302#1607C2×C3112yescomplemented
303#1517C2×C3111yescharacteristic, complemented
304#1512C2×C3111yescharacteristic, complemented
305#1469C2×C3111yescharacteristic, complemented
306C22×C31—21no—
307C22×C31—21no—
308C22×C31—42no—
309C22×C31—42no—
310C22×C31—41no—
311C22×C31—41no—
312C22×C31—42no—
313C22×C31—42no—
314C22×C31—42no—
315C22×C31—41no—
316C22×C31—42no—
317C22×C31—82no—
318C22×C31—82no—
319C22×C31—164no—
320C22×C31C2×#12011yescharacteristic
321C22×C31#57611yescharacteristic
322C22×C31#57711yescharacteristic
323C4×C31—21no—
324C4×C31—21no—
325C4×C31—21no—
326C4×C31—21no—
327C4×C31—21no—
328C4×C31—21no—
329C4×C31—41no—
330C4×C31—42no—
331C4×C31—41no—
332C4×C31—42no—
333C4×C31—41no—
334C4×C31—82no—
335C4×C31—81no—
336C4×C31—81no—
337#562—21no—
338H(3,𝔽4)—21no—
339#559—21no—
340#548—21no—
341#546—21no—
342#546—22no—
343D42—21no—
344#542—21no—
345#538—22no—
346#538C22×C3112yes—
347#535—21no—
348#534C22×C3112yes—
349#532C22×C3111yescharacteristic
350#529C22×C3111yescharacteristic
351C2×#125C22×C3111yescharacteristic, complemented
352C2×#124C22×C3111yescharacteristic
353#501—21no—
354#499—21no—
355#490—21no—
356#488—21no—
357#484C22×C3111yescharacteristic, complemented
358#483C22×C3111yescharacteristic, complemented
359#481C22×C3111yescharacteristic
360#481C22×C3112yescomplemented
361#480C22×C3111yescharacteristic
362#480C22×C3112yescomplemented
363#479—21no—
364#478—21no—
365#478—22no—
366#432C22×C3111yescharacteristic, complemented
367#431C22×C3111yescharacteristic, complemented
368#399C22×C3112yescomplemented
369#397C22×C3112yescomplemented
370#396—22no—
371#395—21no—
372#394—21no—
373#382C22×C3111yescharacteristic, complemented
374#328C22×C3112yescomplemented
375#327C22×C3112yes—
376#327C22×C3112yescomplemented
377#327C22×C3112yescomplemented
378#323C22×C3112yescomplemented
379#323C22×C3112yescomplemented
380C2×C31—21no—
381C2×C31—21no—
382C2×C31—82no—
383C2×C31—82no—
384C2×C31—81no—
385C2×C31#301211yescharacteristic
386C2×C31#301911yescharacteristic
387C2×C31#302811yescharacteristic
388C2×#41—21no—
389C2×#41—21no—
390C22×Q8D4×C3111yescharacteristic
391C22×D4—21no—
392C22×D4—22no—
393C22×D4D4×C3111yescharacteristic
394C4⋊Q8D4×C3111yescharacteristic
395#127—21no—
396#127D4×C3111yescharacteristic
397#126—22no—
398#126—21no—
399#125—21no—
400#125—21no—
401#125—21no—
402#125—21no—
403#125D4×C3111yescharacteristic
404#124—21no—
405#124—22no—
406#124—22no—
407#124—22no—
408#124—21no—
409#124—21no—
410#124—21no—
411#124—21no—
412#124—21no—
413#124D4×C3111yescharacteristic
414#123—22no—
415#123—21no—
416#123—22no—
417#123—22no—
418#123—22no—
419C22⋊Q8—22no—
420C22⋊Q8—21no—
421C22⋊Q8—22no—
422C22⋊Q8C23×C3111yescharacteristic
423#121—22no—
424#121—22no—
425#121—22no—
426#121—21no—
427#121—22no—
428#121—21no—
429#121—22no—
430#121C23×C3111yescharacteristic
431#121C23×C3112yes—
432#120—22no—
433#120—22no—
434#120—22no—
435#120—22no—
436#120—22no—
437C4×Q8—21no—
438C4×Q8D4×C3111yescharacteristic
439C4×D4—22no—
440C4×D4—21no—
441C4×D4—21no—
442C4×D4—22no—
443C4×D4—21no—
444C4×D4D4×C3111yescharacteristic
445C4×D4D4×C3112yes—
446#117—21no—
447C2×C4⋊C4—21no—
448C2×C4⋊C4—21no—
449C2×C4⋊C4—21no—
450C2×C4⋊C4—22no—
451C2×C4⋊C4C23×C3111yescharacteristic
452C2×C4⋊C4C23×C3111yescharacteristic
453C2×C22⋊C4—22no—
454C2×C22⋊C4—22no—
455C2×C22⋊C4—22no—
456C2×C22⋊C4D4×C3112yes—
457C2×C22⋊C4C23×C3112yes—
458C2×C42C23×C3111yescharacteristic
459#107—22no—
460#107—21no—
461#107—21no—
462#106—22no—
463#106—21no—
464#106—21no—
465C4⋊C8—21no—
466C4⋊C8—21no—
467#103—21no—
468#103—21no—
469#102—21no—
470#102—22no—
471#102—21no—
472#102—22no—
473C22⋊C8C23×C3112yes—
474C22⋊C8C23×C3111yescharacteristic
475C22⋊C8C23×C3111yescharacteristic
476#97—21no—
477C4×C8—21no—
478#95—21no—
479#95—22no—
480#95C23×C3112yes—
481C31#1846611yescharacteristic, direct product factor, Sylow
482C24—22no—
483C24C2×D4×C3112yes—
484#41—81no—
485#41—81no—
486#41—82no—
487C2×Q8—21no—
488C2×Q8—21no—
489C2×Q8—21no—
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—82no—
506C2×D4—82no—
507C2×D4C2×D4×C3111yescharacteristic
508C2×D4C2×D4×C3112yes—
509C22×C4—22no—
510C22×C4—22no—
511C22×C4—21no—
512C22×C4—22no—
513C22×C4—22no—
514C22×C4—21no—
515C22×C4—21no—
516C22×C4C2×D4×C3111yescharacteristic
517C22×C4C2×D4×C3112yes—
518C22×C4C2×D4×C3111yescharacteristic
519C22×C4C24×C3111yescharacteristic
520C2×C8—21no—
521C2×C8—21no—
522C2×C8—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—21no—
526C4⋊C4—21no—
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—41no—
537C4⋊C4—41no—
538C4⋊C4—81no—
539C4⋊C4—82no—
540C4⋊C4C2×D4×C3112yes—
541C4⋊C4C2×D4×C3111yescharacteristic
542C4⋊C4C2×D4×C3111yescharacteristic
543C22⋊C4—22no—
544C22⋊C4—22no—
545C22⋊C4—22no—
546C22⋊C4—22no—
547C22⋊C4—22no—
548C22⋊C4—21no—
549C22⋊C4—21no—
550C22⋊C4—22no—
551C22⋊C4—22no—
552C22⋊C4—21no—
553C22⋊C4—21no—
554C22⋊C4—22no—
555C22⋊C4—22no—
556C22⋊C4—22no—
557C22⋊C4—21no—
558C22⋊C4—22no—
559C22⋊C4—82no—
560C22⋊C4—82no—
561C42—22no—
562C42—21no—
563C42C2×D4×C3111yescharacteristic
564C42C2×D4×C3111yescharacteristic
565C42C2×D4×C3111yescharacteristic
566C42C2×D4×C3111yescharacteristic
567C23—22no—
568C23—21no—
569C23—22no—
570C23—22no—
571C23—22no—
572C23—42no—
573C23—42no—
574C23—82no—
575C23—82no—
576C23—82no—
577C23C22×D4×C3111yescharacteristic
578Q8—41no—
579Q8—42no—
580Q8—81no—
581Q8—81no—
582D4—42no—
583D4—41no—
584D4—42no—
585D4—42no—
586D4—41no—
587D4—42no—
588D4—41no—
589D4—42no—
590D4—81no—
591D4—81no—
592D4—82no—
593D4—82no—
594D4—82no—
595D4—82no—
596D4—82no—
597C2×C4—21no—
598C2×C4—21no—
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—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—82no—
617C2×C4—82no—
618C2×C4—82no—
619C2×C4C31×#12011yescharacteristic
620C2×C4C31×#12011yescharacteristic
621C2×C4C31×#12011yescharacteristic
622C2×C4C31×#12011yescharacteristic
623C2×C4C22×D4×C3111yescharacteristic
624C2×C4C22×D4×C3111yescharacteristic
625C8—42no—
626C8—41no—
627C8—41no—
628C22—21no—
629C22—21no—
630C22—42no—
631C22—42no—
632C22—41no—
633C22—41no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22—41no—
638C22—42no—
639C22—82no—
640C22—82no—
641C22—164no—
642C22C2×C31×#12011yescharacteristic
643C22C31×#57611yescharacteristic
644C22C31×#57711yescharacteristic
645C4—21no—
646C4—21no—
647C4—21no—
648C4—21no—
649C4—21no—
650C4—21no—
651C4—41no—
652C4—42no—
653C4—41no—
654C4—42no—
655C4—41no—
656C4—82no—
657C4—81no—
658C4—81no—
659C2—21no—
660C2—21no—
661C2—82no—
662C2—81no—
663C2—82no—
664C2C31×#301211yescharacteristic
665C2C31×#301911yescharacteristic
666C2C31×#302811yescharacteristic
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×#3234C2semidirect
3. C31×#3232C2semidirect
4. C31×#2820C2semidirect
5. C31×#1618C2semidirect
6. C31×#1613C2semidirect
7. C31×#1612C2semidirect
8. C31×#1607C2semidirect
9. C31×#1517C2semidirect
10. C31×#1512C2semidirect
11. C31×#1469C2semidirect
26. C2×C31×#125C22semidirect
32. C31×#484C22semidirect
33. C31×#483C22semidirect
35. C31×#481C22semidirect
37. C31×#480C22semidirect
41. C31×#432C22semidirect
42. C31×#431C22semidirect
43. C31×#399C22semidirect
44. C31×#397C22semidirect
48. C31×#382C22semidirect
49. C31×#328C22semidirect
51. C31×#327C22semidirect
52. C31×#327C22semidirect
53. C31×#323C22semidirect
54. C31×#323C22semidirect
233. #18466C31direct (N × H)
296. #3234C2×C31semidirect
297. #3232C2×C31semidirect
298. #2820C2×C31semidirect
299. #1618C2×C31semidirect
300. #1613C2×C31semidirect
301. #1612C2×C31semidirect
302. #1607C2×C31semidirect
303. #1517C2×C31semidirect
304. #1512C2×C31semidirect
305. #1469C2×C31semidirect
351. C2×#125C22×C31semidirect
357. #484C22×C31semidirect
358. #483C22×C31semidirect
360. #481C22×C31semidirect
362. #480C22×C31semidirect
366. #432C22×C31semidirect
367. #431C22×C31semidirect
368. #399C22×C31semidirect
369. #397C22×C31semidirect
373. #382C22×C31semidirect
374. #328C22×C31semidirect
376. #327C22×C31semidirect
377. #327C22×C31semidirect
378. #323C22×C31semidirect
379. #323C22×C31semidirect
481. C31#18466direct (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×#32342
C31×#32322
C31×#28202
C31×#16182
C31×#16132
C31×#16122
C31×#16072
C31×#15172
C31×#15122
C31×#14692
#1846631