Finite Group Databasebeta

← C31×#19043 #21475802

Subgroups — C31×#19043 #21475802

Order 7936.

950 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3558C211yesmaximal, characteristic
2C31×#3215C211yesmaximal, characteristic, complemented
3C31×#3181C211yesmaximal, characteristic, complemented
4C31×#3135C211yesmaximal, characteristic, complemented
5C31×#1662C212yesmaximal, complemented
6C31×#1661C212yesmaximal, complemented
7C31×#1659C212yesmaximal, complemented
8C31×#1659C212yesmaximal, complemented
9C31×#1532C211yesmaximal, characteristic, complemented
10C31×#1530C211yesmaximal, characteristic, complemented
11C2×C31×#329C211yesmaximal, characteristic, complemented
12C31×#562C2212yes—
13C31×H(3,𝔽4)C2212yes—
14C31×#559—21no—
15C31×#548—21no—
16C31×#546—22no—
17C31×#546—21no—
18D42×C31—21no—
19C31×#542—21no—
20C31×#538—22no—
21C31×#538—21no—
22C31×#535—21no—
23C31×#534—21no—
24C31×#533C2211yescharacteristic, complemented
25C31×#532C2211yescharacteristic
26C31×#529C2211yescharacteristic
27C2×C31×#124C2211yescharacteristic
28C31×#501C2211yescharacteristic, complemented
29C31×#497C2211yescharacteristic, complemented
30C31×#497C2212yescomplemented
31C31×#491—21no—
32C31×#491C2211yescharacteristic, complemented
33C31×#490—21no—
34C31×#489C2212yescomplemented
35C31×#489C2212yescomplemented
36C31×#489C2211yescharacteristic, complemented
37C31×#489C2212yescomplemented
38C31×#467—21no—
39C31×#465—21no—
40C31×#464—22no—
41C31×#451—21no—
42C31×#450—21no—
43C31×#441—21no—
44C31×#440—21no—
45C31×#432C2211yescharacteristic, complemented
46C31×#416—22no—
47C2×C31×#97C2211yescharacteristic, complemented
48C2×C31×#97C2211yescharacteristic, complemented
49C31×#330C2212yescomplemented
50C31×#329C2212yescomplemented
51C31×#329C2212yescomplemented
52C31×#329C2212yescomplemented
53C31×#329C2212yescomplemented
54C31×#329C2212yescomplemented
55C2×#41×C31—21no—
56C2×#41×C31D412yes—
57C22×Q8×C31D411yescharacteristic
58C22×D4×C31—22no—
59C22×D4×C31—21no—
60C22×D4×C31D411yescharacteristic
61C2×Q16×C31—21no—
62C2×Q16×C31—22no—
63C2×SD16×C31—21no—
64C2×SD16×C31—22no—
65C22×C8×C31—21no—
66C22×C8×C31—21no—
67C31×C4⋊Q8D411yescharacteristic
68C31×C4⋊Q8C2311yescharacteristic
69C31×C4⋊Q8C2311yescharacteristic
70C31×#127—21no—
71C31×#127D411yescharacteristic
72C31×#126—22no—
73C31×#126—21no—
74C31×#125—21no—
75C31×#125C2311yescharacteristic
76C31×#124—21no—
77C31×#124—22no—
78C31×#124—22no—
79C31×#124—22no—
80C31×#124—21no—
81C31×#124—21no—
82C31×#124D412yes—
83C31×#124C2312yes—
84C31×#124C2312yes—
85C31×#124C2311yescharacteristic
86C31×#123—22no—
87C31×#123—22no—
88C31×#123—21no—
89C31×#123—22no—
90C31×#123—22no—
91C31×C22⋊Q8—21no—
92C31×C22⋊Q8—22no—
93C31×C22⋊Q8—22no—
94C31×C22⋊Q8—21no—
95C31×C22⋊Q8—21no—
96C31×C22⋊Q8D411yescharacteristic
97C31×#121—22no—
98C31×#121—22no—
99C31×#121—22no—
100C31×#121—21no—
101C31×#121—22no—
102C31×#121—22no—
103C31×#121—21no—
104C31×#121—21no—
105C31×#121D411yescharacteristic
106C31×#120—22no—
107C31×#120—22no—
108C31×#120—22no—
109C31×#120—22no—
110C31×#120—22no—
111C4×Q8×C31—21no—
112C4×Q8×C31—21no—
113C4×Q8×C31—21no—
114C4×Q8×C31D411yescharacteristic
115C4×D4×C31—22no—
116C4×D4×C31—22no—
117C4×D4×C31—21no—
118C4×D4×C31—21no—
119C4×D4×C31—21no—
120C4×D4×C31—21no—
121C4×D4×C31D411yescharacteristic
122C31×#117—21no—
123C31×#117—21no—
124C2×C22⋊C4×C31—22no—
125C2×C22⋊C4×C31—21no—
126C2×C22⋊C4×C31—21no—
127C2×C42×C31C2311yescharacteristic
128C31×#107—21no—
129C31×#106—21no—
130C31×C4⋊C8—21no—
131C31×#103—22no—
132C31×#103—22no—
133C31×#103—21no—
134C31×#103—22no—
135C31×#103—21no—
136C31×#103—21no—
137C31×#102—22no—
138C31×#102—22no—
139C31×#102—21no—
140C31×#102—22no—
141C31×C22⋊C8—21no—
142C31×#97—21no—
143C31×#97—21no—
144C31×#97C2312yes—
145C31×#97C2311yescharacteristic
146C31×#97C2311yescharacteristic
147C31×#97C2312yes—
148C24×C31—22no—
149C24×C31—21no—
150#41×C31—44no—
151#41×C31—82no—
152C2×Q8×C31—21no—
153C2×Q8×C31—21no—
154C2×Q8×C31—21no—
155C2×Q8×C31—81no—
156C2×Q8×C31C2×D412yes—
157C2×Q8×C31C2×D411yescharacteristic
158C2×Q8×C31C2×D412yes—
159C2×D4×C31—22no—
160C2×D4×C31—21no—
161C2×D4×C31—22no—
162C2×D4×C31—22no—
163C2×D4×C31—22no—
164C2×D4×C31—22no—
165C2×D4×C31—22no—
166C2×D4×C31—21no—
167C2×D4×C31—21no—
168C2×D4×C31—21no—
169C2×D4×C31—81no—
170C2×D4×C31—82no—
171C2×D4×C31—84no—
172C2×D4×C31C2×D412yes—
173C2×D4×C31C2×D411yescharacteristic
174C2×D4×C31C2×D412yes—
175C22×C4×C31—22no—
176C22×C4×C31—22no—
177C22×C4×C31—21no—
178C22×C4×C31—21no—
179C22×C4×C31—22no—
180C22×C4×C31C2×D411yescharacteristic
181C22×C4×C31C2×D411yescharacteristic
182C22×C4×C31C2×D411yescharacteristic
183Q16×C31—42no—
184Q16×C31—82no—
185SD16×C31—42no—
186SD16×C31—82no—
187C2×C8×C31—21no—
188C2×C8×C31—22no—
189C2×C8×C31—21no—
190C2×C8×C31—22no—
191C2×C8×C31—42no—
192C2×C8×C31—42no—
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—21no—
203C4⋊C4×C31—21no—
204C4⋊C4×C31—21no—
205C4⋊C4×C31—41no—
206C4⋊C4×C31—41no—
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—22no—
216C22⋊C4×C31—21no—
217C22⋊C4×C31—22no—
218C22⋊C4×C31—21no—
219C22⋊C4×C31—22no—
220C22⋊C4×C31—22no—
221C22⋊C4×C31—21no—
222C22⋊C4×C31—21no—
223C22⋊C4×C31—22no—
224C22⋊C4×C31—22no—
225C22⋊C4×C31—41no—
226C42×C31—22no—
227C42×C31—21no—
228C42×C31—41no—
229C42×C31C2×D411yescharacteristic
230C42×C31C2×D411yescharacteristic
231C42×C31C2×D411yescharacteristic
232C42×C31C2411yescharacteristic
233#19043C3111yesmaximal, characteristic, direct product factor, Sylow
234C23×C31—22no—
235C23×C31—21no—
236C23×C31—22no—
237C23×C31—42no—
238C23×C31—81no—
239C23×C31—81no—
240C23×C31—82no—
241C23×C31—82no—
242C23×C31—82no—
243C23×C31#12011yescharacteristic
244Q8×C31—41no—
245Q8×C31—42no—
246Q8×C31—42no—
247Q8×C31—42no—
248Q8×C31—41no—
249Q8×C31—42no—
250Q8×C31—41no—
251D4×C31—41no—
252D4×C31—42no—
253D4×C31—42no—
254D4×C31—42no—
255D4×C31—42no—
256D4×C31—42no—
257D4×C31—44no—
258D4×C31—81no—
259D4×C31—82no—
260D4×C31—82no—
261D4×C31—82no—
262D4×C31—82no—
263D4×C31—82no—
264C2×C4×C31—22no—
265C2×C4×C31—21no—
266C2×C4×C31—22no—
267C2×C4×C31—22no—
268C2×C4×C31—21no—
269C2×C4×C31—21no—
270C2×C4×C31—41no—
271C2×C4×C31—41no—
272C2×C4×C31—42no—
273C2×C4×C31—41no—
274C2×C4×C31—41no—
275C2×C4×C31—82no—
276C2×C4×C31—81no—
277C2×C4×C31—82no—
278C2×C4×C31—82no—
279C2×C4×C31#12011yescharacteristic
280C2×C4×C31#12011yescharacteristic
281C2×C4×C31#12011yescharacteristic
282C2×C4×C31C22×D411yescharacteristic
283C2×C4×C31C22×D411yescharacteristic
284C2×C4×C31C22×D411yescharacteristic
285C8×C31—22no—
286C8×C31—42no—
287C8×C31—41no—
288C8×C31—42no—
289#3558C2×C3111yescharacteristic
290#3215C2×C3111yescharacteristic, complemented
291#3181C2×C3111yescharacteristic, complemented
292#3135C2×C3111yescharacteristic, complemented
293#1662C2×C3112yescomplemented
294#1661C2×C3112yescomplemented
295#1659C2×C3112yescomplemented
296#1659C2×C3112yescomplemented
297#1532C2×C3111yescharacteristic, complemented
298#1530C2×C3111yescharacteristic, complemented
299C2×#329C2×C3111yescharacteristic, complemented
300C22×C31—21no—
301C22×C31—21no—
302C22×C31—42no—
303C22×C31—42no—
304C22×C31—41no—
305C22×C31—41no—
306C22×C31—42no—
307C22×C31—42no—
308C22×C31—41no—
309C22×C31—42no—
310C22×C31—42no—
311C22×C31—162no—
312C22×C31—164no—
313C22×C31#45512yes—
314C22×C31C2×#12011yescharacteristic
315C4×C31—21no—
316C4×C31—21no—
317C4×C31—21no—
318C4×C31—21no—
319C4×C31—21no—
320C4×C31—21no—
321C4×C31—41no—
322C4×C31—42no—
323C4×C31—41no—
324C4×C31—41no—
325C4×C31—42no—
326C4×C31—81no—
327C4×C31—81no—
328#562C22×C3112yes—
329H(3,𝔽4)C22×C3112yes—
330#559—21no—
331#548—21no—
332#546—22no—
333#546—21no—
334D42—21no—
335#542—21no—
336#538—22no—
337#538—21no—
338#535—21no—
339#534—21no—
340#533C22×C3111yescharacteristic, complemented
341#532C22×C3111yescharacteristic
342#529C22×C3111yescharacteristic
343C2×#124C22×C3111yescharacteristic
344#501C22×C3111yescharacteristic, complemented
345#497C22×C3111yescharacteristic, complemented
346#497C22×C3112yescomplemented
347#491—21no—
348#491C22×C3111yescharacteristic, complemented
349#490—21no—
350#489C22×C3112yescomplemented
351#489C22×C3112yescomplemented
352#489C22×C3111yescharacteristic, complemented
353#489C22×C3112yescomplemented
354#467—21no—
355#465—21no—
356#464—22no—
357#451—21no—
358#450—21no—
359#441—21no—
360#440—21no—
361#432C22×C3111yescharacteristic, complemented
362#416—22no—
363C2×#97C22×C3111yescharacteristic, complemented
364C2×#97C22×C3111yescharacteristic, complemented
365#330C22×C3112yescomplemented
366#329C22×C3112yescomplemented
367#329C22×C3112yescomplemented
368#329C22×C3112yescomplemented
369#329C22×C3112yescomplemented
370#329C22×C3112yescomplemented
371C2×C31—22no—
372C2×C31—82no—
373C2×C31—82no—
374C2×C31—81no—
375C2×C31C2×#45511yescharacteristic
376C2×C31#301811yescharacteristic
377C2×C31#301911yescharacteristic
378C2×#41—21no—
379C2×#41D4×C3112yes—
380C22×Q8D4×C3111yescharacteristic
381C22×D4—22no—
382C22×D4—21no—
383C22×D4D4×C3111yescharacteristic
384C2×Q16—21no—
385C2×Q16—22no—
386C2×SD16—21no—
387C2×SD16—22no—
388C22×C8—21no—
389C22×C8—21no—
390C4⋊Q8D4×C3111yescharacteristic
391C4⋊Q8C23×C3111yescharacteristic
392C4⋊Q8C23×C3111yescharacteristic
393#127—21no—
394#127D4×C3111yescharacteristic
395#126—22no—
396#126—21no—
397#125—21no—
398#125C23×C3111yescharacteristic
399#124—21no—
400#124—22no—
401#124—22no—
402#124—22no—
403#124—21no—
404#124—21no—
405#124D4×C3112yes—
406#124C23×C3112yes—
407#124C23×C3111yescharacteristic
408#124C23×C3112yes—
409#123—22no—
410#123—22no—
411#123—21no—
412#123—22no—
413#123—22no—
414C22⋊Q8—21no—
415C22⋊Q8—22no—
416C22⋊Q8—22no—
417C22⋊Q8—21no—
418C22⋊Q8—21no—
419C22⋊Q8D4×C3111yescharacteristic
420#121—22no—
421#121—22no—
422#121—22no—
423#121—21no—
424#121—22no—
425#121—22no—
426#121—21no—
427#121—21no—
428#121D4×C3111yescharacteristic
429#120—22no—
430#120—22no—
431#120—22no—
432#120—22no—
433#120—22no—
434C4×Q8—21no—
435C4×Q8—21no—
436C4×Q8—21no—
437C4×Q8D4×C3111yescharacteristic
438C4×D4—22no—
439C4×D4—22no—
440C4×D4—21no—
441C4×D4—21no—
442C4×D4—21no—
443C4×D4—21no—
444C4×D4D4×C3111yescharacteristic
445#117—21no—
446#117—21no—
447C2×C22⋊C4—22no—
448C2×C22⋊C4—21no—
449C2×C22⋊C4—21no—
450C2×C42C23×C3111yescharacteristic
451#107—21no—
452#106—21no—
453C4⋊C8—21no—
454#103—22no—
455#103—22no—
456#103—21no—
457#103—22no—
458#103—21no—
459#103—21no—
460#102—22no—
461#102—22no—
462#102—21no—
463#102—22no—
464C22⋊C8—21no—
465#97—21no—
466#97—21no—
467#97C23×C3111yescharacteristic
468#97C23×C3111yescharacteristic
469#97C23×C3112yes—
470#97C23×C3112yes—
471C31#1904311yescharacteristic, direct product factor, Sylow
472C24—22no—
473C24—21no—
474#41—44no—
475#41—82no—
476C2×Q8—21no—
477C2×Q8—21no—
478C2×Q8—21no—
479C2×Q8—81no—
480C2×Q8C2×D4×C3112yes—
481C2×Q8C2×D4×C3111yescharacteristic
482C2×Q8C2×D4×C3112yes—
483C2×D4—22no—
484C2×D4—21no—
485C2×D4—22no—
486C2×D4—22no—
487C2×D4—22no—
488C2×D4—22no—
489C2×D4—22no—
490C2×D4—21no—
491C2×D4—21no—
492C2×D4—21no—
493C2×D4—81no—
494C2×D4—82no—
495C2×D4—84no—
496C2×D4C2×D4×C3112yes—
497C2×D4C2×D4×C3111yescharacteristic
498C2×D4C2×D4×C3112yes—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—22no—
504C22×C4C2×D4×C3111yescharacteristic
505C22×C4C2×D4×C3111yescharacteristic
506C22×C4C2×D4×C3111yescharacteristic
507Q16—42no—
508Q16—82no—
509SD16—42no—
510SD16—82no—
511C2×C8—21no—
512C2×C8—22no—
513C2×C8—21no—
514C2×C8—22no—
515C2×C8—42no—
516C2×C8—42no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—41no—
530C4⋊C4—41no—
531C4⋊C4C2×D4×C3111yescharacteristic
532C4⋊C4C2×D4×C3111yescharacteristic
533C22⋊C4—22no—
534C22⋊C4—22no—
535C22⋊C4—22no—
536C22⋊C4—22no—
537C22⋊C4—22no—
538C22⋊C4—21no—
539C22⋊C4—22no—
540C22⋊C4—21no—
541C22⋊C4—22no—
542C22⋊C4—21no—
543C22⋊C4—22no—
544C22⋊C4—22no—
545C22⋊C4—21no—
546C22⋊C4—21no—
547C22⋊C4—22no—
548C22⋊C4—22no—
549C22⋊C4—41no—
550C42—22no—
551C42—21no—
552C42—41no—
553C42C2×D4×C3111yescharacteristic
554C42C2×D4×C3111yescharacteristic
555C42C2×D4×C3111yescharacteristic
556C42C24×C3111yescharacteristic
557C23—22no—
558C23—21no—
559C23—22no—
560C23—42no—
561C23—81no—
562C23—81no—
563C23—82no—
564C23—82no—
565C23—82no—
566C23C31×#12011yescharacteristic
567Q8—41no—
568Q8—42no—
569Q8—42no—
570Q8—42no—
571Q8—41no—
572Q8—42no—
573Q8—41no—
574D4—41no—
575D4—42no—
576D4—42no—
577D4—42no—
578D4—42no—
579D4—42no—
580D4—44no—
581D4—81no—
582D4—82no—
583D4—82no—
584D4—82no—
585D4—82no—
586D4—82no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—82no—
599C2×C4—81no—
600C2×C4—82no—
601C2×C4—82no—
602C2×C4C31×#12011yescharacteristic
603C2×C4C31×#12011yescharacteristic
604C2×C4C31×#12011yescharacteristic
605C2×C4C22×D4×C3111yescharacteristic
606C2×C4C22×D4×C3111yescharacteristic
607C2×C4C22×D4×C3111yescharacteristic
608C8—22no—
609C8—42no—
610C8—41no—
611C8—42no—
612C22—21no—
613C22—21no—
614C22—42no—
615C22—42no—
616C22—41no—
617C22—41no—
618C22—42no—
619C22—42no—
620C22—41no—
621C22—42no—
622C22—42no—
623C22—162no—
624C22—164no—
625C22C31×#45512yes—
626C22C2×C31×#12011yescharacteristic
627C4—21no—
628C4—21no—
629C4—21no—
630C4—21no—
631C4—21no—
632C4—21no—
633C4—41no—
634C4—42no—
635C4—41no—
636C4—41no—
637C4—42no—
638C4—81no—
639C4—81no—
640C2—22no—
641C2—82no—
642C2—81no—
643C2—82no—
644C2C2×C31×#45511yescharacteristic
645C2C31×#301811yescharacteristic
646C2C31×#301911yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC42C24×C31
prime radicalC31×#3558C2
LCSG ▷ C42 ▷ 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×#3215C2semidirect
3. C31×#3181C2semidirect
4. C31×#3135C2semidirect
5. C31×#1662C2semidirect
6. C31×#1661C2semidirect
7. C31×#1659C2semidirect
8. C31×#1659C2semidirect
9. C31×#1532C2semidirect
10. C31×#1530C2semidirect
11. C2×C31×#329C2semidirect
24. C31×#533C22semidirect
28. C31×#501C22semidirect
29. C31×#497C22semidirect
30. C31×#497C22semidirect
32. C31×#491C22semidirect
34. C31×#489C22semidirect
35. C31×#489C22semidirect
36. C31×#489C22semidirect
37. C31×#489C22semidirect
45. C31×#432C22semidirect
47. C2×C31×#97C22semidirect
48. C2×C31×#97C22semidirect
49. C31×#330C22semidirect
50. C31×#329C22semidirect
51. C31×#329C22semidirect
52. C31×#329C22semidirect
53. C31×#329C22semidirect
54. C31×#329C22semidirect
233. #19043C31direct (N × H)
290. #3215C2×C31semidirect
291. #3181C2×C31semidirect
292. #3135C2×C31semidirect
293. #1662C2×C31semidirect
294. #1661C2×C31semidirect
295. #1659C2×C31semidirect
296. #1659C2×C31semidirect
297. #1532C2×C31semidirect
298. #1530C2×C31semidirect
299. C2×#329C2×C31semidirect
340. #533C22×C31semidirect
344. #501C22×C31semidirect
345. #497C22×C31semidirect
346. #497C22×C31semidirect
348. #491C22×C31semidirect
350. #489C22×C31semidirect
351. #489C22×C31semidirect
352. #489C22×C31semidirect
353. #489C22×C31semidirect
361. #432C22×C31semidirect
363. C2×#97C22×C31semidirect
364. C2×#97C22×C31semidirect
365. #330C22×C31semidirect
366. #329C22×C31semidirect
367. #329C22×C31semidirect
368. #329C22×C31semidirect
369. #329C22×C31semidirect
370. #329C22×C31semidirect
471. C31#19043direct (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×#32152
C31×#31812
C31×#31352
C31×#16622
C31×#16612
C31×#16592
C31×#15322
C31×#15302
C2×C31×#3292
#1904331