Finite Group Databasebeta

← C31×#19019 #21475778

Subgroups — C31×#19019 #21475778

Order 7936.

958 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3561C211yesmaximal, characteristic, complemented
2C31×#3235C211yesmaximal, characteristic, complemented
3C31×#3171C211yesmaximal, characteristic, complemented
4C31×#3153C211yesmaximal, characteristic, complemented
5C31×#1661C212yesmaximal, complemented
6C31×#1660C212yesmaximal, complemented
7C31×#1657C212yesmaximal, complemented
8C31×#1656C212yesmaximal, complemented
9C31×#1529C211yesmaximal, characteristic, complemented
10C31×#1527C211yesmaximal, characteristic, complemented
11C31×#1526C211yesmaximal, characteristic, complemented
12C2×C31×#137—21no—
13C2×C31×AGL(1,ℤ/8)—21no—
14C31×#563C2212yes—
15C31×#562C2212yes—
16C31×#556—21no—
17C31×#551—22no—
18D4×Q8×C31—21no—
19C31×#548—21no—
20C31×#546—21no—
21C31×#543—21no—
22C31×#542—21no—
23C31×#542—22no—
24C31×#538—21no—
25C31×#536—21no—
26C31×#533C2211yescharacteristic
27C31×#532C2211yescharacteristic
28C31×#529C2211yescharacteristic
29C2×C31×#124C2211yescharacteristic
30C31×#501C2211yescharacteristic, complemented
31C31×#501C2212yescomplemented
32C31×#497C2212yes—
33C31×#496C2212yes—
34C31×#496C2211yescharacteristic, complemented
35C31×#495—22no—
36C31×#490C2211yescharacteristic, complemented
37C31×#490C2212yescomplemented
38C31×#489C2211yescharacteristic, complemented
39C31×#482—22no—
40C31×#476—21no—
41C31×#474—21no—
42C31×#469—21no—
43C31×#468—21no—
44C4×SD16×C31—21no—
45C4×D8×C31—21no—
46C31×#432C2211yescharacteristic
47C4×M16×C31C2211yescharacteristic
48C4×M16×C31C2211yescharacteristic
49C31×#330C2212yescomplemented
50C31×#330C2212yescomplemented
51C31×#330C2212yescomplemented
52C31×#329C2212yes—
53C31×#329C2212yes—
54C31×#329C2212yescomplemented
55C2×#41×C31—21no—
56C2×#41×C31—21no—
57C22×Q8×C31—21no—
58C22×Q8×C31D411yescharacteristic
59C22×D4×C31D411yescharacteristic
60C31×#137—44no—
61C31×AGL(1,ℤ/8)—44no—
62C2×Q16×C31—22no—
63C2×SD16×C31—21no—
64C2×SD16×C31—22no—
65C2×SD16×C31—22no—
66C2×D8×C31—21no—
67C2×D8×C31—22no—
68C2×M16×C31—21no—
69C2×M16×C31—21no—
70C31×C4⋊Q8—21no—
71C31×C4⋊Q8—22no—
72C31×C4⋊Q8D411yescharacteristic
73C31×C4⋊Q8D411yescharacteristic
74C31×C4⋊Q8C2311yescharacteristic
75C31×#127C2311yescharacteristic
76C31×#126—22no—
77C31×#126—22no—
78C31×#126—22no—
79C31×#126—21no—
80C31×#125C2311yescharacteristic
81C31×#125C2312yes—
82C31×#124—21no—
83C31×#124—22no—
84C31×#124—22no—
85C31×#124—21no—
86C31×#124—21no—
87C31×#124—21no—
88C31×#124D412yes—
89C31×#124C2311yescharacteristic
90C31×#124C2312yes—
91C31×#123—22no—
92C31×#123—22no—
93C31×#123—21no—
94C31×#123—22no—
95C31×#123—22no—
96C31×C22⋊Q8—22no—
97C31×C22⋊Q8—22no—
98C31×C22⋊Q8—21no—
99C31×C22⋊Q8—22no—
100C31×C22⋊Q8—22no—
101C31×C22⋊Q8—22no—
102C31×C22⋊Q8—21no—
103C31×C22⋊Q8—21no—
104C31×C22⋊Q8—22no—
105C31×C22⋊Q8—22no—
106C31×C22⋊Q8D411yescharacteristic
107C31×#121—21no—
108C31×#121—22no—
109C31×#121—22no—
110C31×#121—21no—
111C31×#121—21no—
112C31×#121D411yescharacteristic
113C31×#120—22no—
114C4×Q8×C31—21no—
115C4×Q8×C31—21no—
116C4×Q8×C31—21no—
117C4×Q8×C31D411yescharacteristic
118C4×D4×C31—22no—
119C4×D4×C31—21no—
120C4×D4×C31—22no—
121C4×D4×C31—21no—
122C4×D4×C31—21no—
123C4×D4×C31—21no—
124C4×D4×C31D411yescharacteristic
125C31×#117—22no—
126C31×#117—21no—
127C31×#117D412yes—
128C2×C22⋊C4×C31—22no—
129C2×C22⋊C4×C31—21no—
130C2×C22⋊C4×C31—21no—
131C2×C42×C31C2311yescharacteristic
132C31×#107—21no—
133C31×#107—22no—
134C31×#106—21no—
135C31×#106—22no—
136C31×C4⋊C8—21no—
137C31×#103—21no—
138C31×#102—21no—
139C31×#102—21no—
140C31×#102—21no—
141C31×#102—22no—
142C31×#102—22no—
143C31×C22⋊C8—21no—
144C31×#97C2311yescharacteristic
145C31×#97C2312yes—
146C31×#97C2311yescharacteristic
147C31×#97C2312yes—
148C4×C8×C31—21no—
149C4×C8×C31—21no—
150C24×C31—21no—
151#41×C31—42no—
152#41×C31—42no—
153#41×C31—82no—
154C2×Q8×C31—22no—
155C2×Q8×C31—21no—
156C2×Q8×C31—21no—
157C2×Q8×C31—21no—
158C2×Q8×C31—21no—
159C2×Q8×C31—42no—
160C2×Q8×C31—82no—
161C2×Q8×C31C2×D411yescharacteristic
162C2×Q8×C31C2×D412yes—
163C2×D4×C31—21no—
164C2×D4×C31—22no—
165C2×D4×C31—22no—
166C2×D4×C31—21no—
167C2×D4×C31—21no—
168C2×D4×C31—21no—
169C2×D4×C31—21no—
170C2×D4×C31—42no—
171C2×D4×C31—41no—
172C2×D4×C31C2×D411yescharacteristic
173C2×D4×C31C2×D412yes—
174C22×C4×C31—22no—
175C22×C4×C31—22no—
176C22×C4×C31—22no—
177C22×C4×C31—21no—
178C22×C4×C31—21no—
179C22×C4×C31—41no—
180C22×C4×C31C2×D411yescharacteristic
181C22×C4×C31C2×D411yescharacteristic
182C22×C4×C31C2×D411yescharacteristic
183Q16×C31—44no—
184SD16×C31—42no—
185SD16×C31—44no—
186SD16×C31—44no—
187D8×C31—42no—
188D8×C31—44no—
189M16×C31—24no—
190M16×C31—42no—
191C2×C8×C31—21no—
192C2×C8×C31—22no—
193C2×C8×C31—21no—
194C2×C8×C31—22no—
195C4⋊C4×C31—22no—
196C4⋊C4×C31—22no—
197C4⋊C4×C31—22no—
198C4⋊C4×C31—22no—
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—22no—
206C4⋊C4×C31—22no—
207C4⋊C4×C31—21no—
208C4⋊C4×C31—21no—
209C4⋊C4×C31C2×D412yes—
210C4⋊C4×C31C2×D411yescharacteristic
211C4⋊C4×C31C2×D412yes—
212C4⋊C4×C31C2×D411yescharacteristic
213C22⋊C4×C31—22no—
214C22⋊C4×C31—22no—
215C22⋊C4×C31—22no—
216C22⋊C4×C31—22no—
217C22⋊C4×C31—22no—
218C22⋊C4×C31—21no—
219C22⋊C4×C31—22no—
220C22⋊C4×C31—22no—
221C22⋊C4×C31—22no—
222C22⋊C4×C31—21no—
223C22⋊C4×C31—21no—
224C22⋊C4×C31—22no—
225C22⋊C4×C31—22no—
226C22⋊C4×C31—21no—
227C22⋊C4×C31—21no—
228C22⋊C4×C31—41no—
229C22⋊C4×C31—41no—
230C42×C31—22no—
231C42×C31—22no—
232C42×C31—21no—
233C42×C31C2×D411yescharacteristic
234C42×C31C2×D411yescharacteristic
235C42×C31C2×D411yescharacteristic
236C42×C31C2411yescharacteristic
237#19019C3111yesmaximal, characteristic, direct product factor, Sylow
238C23×C31—22no—
239C23×C31—21no—
240C23×C31—42no—
241C23×C31—41no—
242C23×C31—81no—
243C23×C31—81no—
244C23×C31#12011yescharacteristic
245Q8×C31—42no—
246Q8×C31—41no—
247Q8×C31—41no—
248Q8×C31—41no—
249Q8×C31—42no—
250Q8×C31—42no—
251Q8×C31—81no—
252Q8×C31—82no—
253D4×C31—42no—
254D4×C31—41no—
255D4×C31—41no—
256D4×C31—41no—
257D4×C31—41no—
258D4×C31—41no—
259D4×C31—42no—
260D4×C31—42no—
261D4×C31—42no—
262D4×C31—82no—
263D4×C31—82no—
264D4×C31—82no—
265C2×C4×C31—22no—
266C2×C4×C31—21no—
267C2×C4×C31—22no—
268C2×C4×C31—21no—
269C2×C4×C31—22no—
270C2×C4×C31—22no—
271C2×C4×C31—21no—
272C2×C4×C31—41no—
273C2×C4×C31—41no—
274C2×C4×C31—42no—
275C2×C4×C31—41no—
276C2×C4×C31—81no—
277C2×C4×C31—82no—
278C2×C4×C31—82no—
279C2×C4×C31—82no—
280C2×C4×C31—82no—
281C2×C4×C31#12011yescharacteristic
282C2×C4×C31#12011yescharacteristic
283C2×C4×C31#12011yescharacteristic
284C2×C4×C31C22×D411yescharacteristic
285C2×C4×C31C22×D411yescharacteristic
286C2×C4×C31C22×D411yescharacteristic
287C8×C31—24no—
288C8×C31—22no—
289C8×C31—42no—
290C8×C31—41no—
291#3561C2×C3111yescharacteristic, complemented
292#3235C2×C3111yescharacteristic, complemented
293#3171C2×C3111yescharacteristic, complemented
294#3153C2×C3111yescharacteristic, complemented
295#1661C2×C3112yescomplemented
296#1660C2×C3112yescomplemented
297#1657C2×C3112yescomplemented
298#1656C2×C3112yescomplemented
299#1529C2×C3111yescharacteristic, complemented
300#1527C2×C3111yescharacteristic, complemented
301#1526C2×C3111yescharacteristic, complemented
302C22×C31—21no—
303C22×C31—21no—
304C22×C31—21no—
305C22×C31—41no—
306C22×C31—42no—
307C22×C31—42no—
308C22×C31—41no—
309C22×C31—41no—
310C22×C31—42no—
311C22×C31—81no—
312C22×C31—81no—
313C22×C31—81no—
314C22×C31—162no—
315C22×C31C2×#12011yescharacteristic
316C4×C31—21no—
317C4×C31—21no—
318C4×C31—21no—
319C4×C31—21no—
320C4×C31—21no—
321C4×C31—41no—
322C4×C31—42no—
323C4×C31—42no—
324C4×C31—41no—
325C4×C31—42no—
326C4×C31—41no—
327C4×C31—81no—
328C4×C31#45312yes—
329C2×#137—21no—
330C2×AGL(1,ℤ/8)—21no—
331#563C22×C3112yes—
332#562C22×C3112yes—
333#556—21no—
334#551—22no—
335D4×Q8—21no—
336#548—21no—
337#546—21no—
338#543—21no—
339#542—21no—
340#542—22no—
341#538—21no—
342#536—21no—
343#533C22×C3111yescharacteristic
344#532C22×C3111yescharacteristic
345#529C22×C3111yescharacteristic
346C2×#124C22×C3111yescharacteristic
347#501C22×C3111yescharacteristic, complemented
348#501C22×C3112yescomplemented
349#497C22×C3112yes—
350#496C22×C3112yes—
351#496C22×C3111yescharacteristic, complemented
352#495—22no—
353#490C22×C3111yescharacteristic, complemented
354#490C22×C3112yescomplemented
355#489C22×C3111yescharacteristic, complemented
356#482—22no—
357#476—21no—
358#474—21no—
359#469—21no—
360#468—21no—
361C4×SD16—21no—
362C4×D8—21no—
363#432C22×C3111yescharacteristic
364C4×M16C22×C3111yescharacteristic
365C4×M16C22×C3111yescharacteristic
366#330C22×C3112yescomplemented
367#330C22×C3112yescomplemented
368#330C22×C3112yescomplemented
369#329C22×C3112yes—
370#329C22×C3112yes—
371#329C22×C3112yescomplemented
372C2×C31—41no—
373C2×C31—81no—
374C2×C31—82no—
375C2×C31—161no—
376C2×C31C2×#45311yescharacteristic
377C2×C31#301811yescharacteristic
378C2×C31#302011yescharacteristic
379C2×#41—21no—
380C2×#41—21no—
381C22×Q8—21no—
382C22×Q8D4×C3111yescharacteristic
383C22×D4D4×C3111yescharacteristic
384#137—44no—
385AGL(1,ℤ/8)—44no—
386C2×Q16—22no—
387C2×SD16—22no—
388C2×SD16—21no—
389C2×SD16—22no—
390C2×D8—22no—
391C2×D8—21no—
392C2×M16—21no—
393C2×M16—21no—
394C4⋊Q8—21no—
395C4⋊Q8—22no—
396C4⋊Q8D4×C3111yescharacteristic
397C4⋊Q8D4×C3111yescharacteristic
398C4⋊Q8C23×C3111yescharacteristic
399#127C23×C3111yescharacteristic
400#126—22no—
401#126—22no—
402#126—22no—
403#126—21no—
404#125C23×C3111yescharacteristic
405#125C23×C3112yes—
406#124—21no—
407#124—22no—
408#124—22no—
409#124—21no—
410#124—21no—
411#124—21no—
412#124D4×C3112yes—
413#124C23×C3112yes—
414#124C23×C3111yescharacteristic
415#123—22no—
416#123—22no—
417#123—21no—
418#123—22no—
419#123—22no—
420C22⋊Q8—22no—
421C22⋊Q8—22no—
422C22⋊Q8—21no—
423C22⋊Q8—22no—
424C22⋊Q8—22no—
425C22⋊Q8—22no—
426C22⋊Q8—21no—
427C22⋊Q8—21no—
428C22⋊Q8—22no—
429C22⋊Q8—22no—
430C22⋊Q8D4×C3111yescharacteristic
431#121—21no—
432#121—22no—
433#121—22no—
434#121—21no—
435#121—21no—
436#121D4×C3111yescharacteristic
437#120—22no—
438C4×Q8—21no—
439C4×Q8—21no—
440C4×Q8—21no—
441C4×Q8D4×C3111yescharacteristic
442C4×D4—22no—
443C4×D4—21no—
444C4×D4—22no—
445C4×D4—21no—
446C4×D4—21no—
447C4×D4—21no—
448C4×D4D4×C3111yescharacteristic
449#117—22no—
450#117—21no—
451#117D4×C3112yes—
452C2×C22⋊C4—22no—
453C2×C22⋊C4—21no—
454C2×C22⋊C4—21no—
455C2×C42C23×C3111yescharacteristic
456#107—21no—
457#107—22no—
458#106—21no—
459#106—22no—
460C4⋊C8—21no—
461#103—21no—
462#102—21no—
463#102—21no—
464#102—21no—
465#102—22no—
466#102—22no—
467C22⋊C8—21no—
468#97C23×C3112yes—
469#97C23×C3111yescharacteristic
470#97C23×C3111yescharacteristic
471#97C23×C3112yes—
472C4×C8—21no—
473C4×C8—21no—
474C31#1901911yescharacteristic, direct product factor, Sylow
475C24—21no—
476#41—42no—
477#41—42no—
478#41—82no—
479C2×Q8—22no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—21no—
484C2×Q8—42no—
485C2×Q8—82no—
486C2×Q8C2×D4×C3111yescharacteristic
487C2×Q8C2×D4×C3112yes—
488C2×D4—21no—
489C2×D4—22no—
490C2×D4—22no—
491C2×D4—21no—
492C2×D4—21no—
493C2×D4—21no—
494C2×D4—21no—
495C2×D4—42no—
496C2×D4—41no—
497C2×D4C2×D4×C3111yescharacteristic
498C2×D4C2×D4×C3112yes—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—41no—
505C22×C4C2×D4×C3111yescharacteristic
506C22×C4C2×D4×C3111yescharacteristic
507C22×C4C2×D4×C3111yescharacteristic
508Q16—44no—
509SD16—42no—
510SD16—44no—
511SD16—44no—
512D8—42no—
513D8—44no—
514M16—24no—
515M16—42no—
516C2×C8—21no—
517C2×C8—22no—
518C2×C8—21no—
519C2×C8—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4C2×D4×C3112yes—
535C4⋊C4C2×D4×C3111yescharacteristic
536C4⋊C4C2×D4×C3112yes—
537C4⋊C4C2×D4×C3111yescharacteristic
538C22⋊C4—22no—
539C22⋊C4—22no—
540C22⋊C4—22no—
541C22⋊C4—22no—
542C22⋊C4—22no—
543C22⋊C4—21no—
544C22⋊C4—22no—
545C22⋊C4—22no—
546C22⋊C4—22no—
547C22⋊C4—21no—
548C22⋊C4—21no—
549C22⋊C4—22no—
550C22⋊C4—22no—
551C22⋊C4—21no—
552C22⋊C4—21no—
553C22⋊C4—41no—
554C22⋊C4—41no—
555C42—22no—
556C42—22no—
557C42—21no—
558C42C2×D4×C3111yescharacteristic
559C42C2×D4×C3111yescharacteristic
560C42C2×D4×C3111yescharacteristic
561C42C24×C3111yescharacteristic
562C23—22no—
563C23—21no—
564C23—42no—
565C23—41no—
566C23—81no—
567C23—81no—
568C23C31×#12011yescharacteristic
569Q8—42no—
570Q8—41no—
571Q8—41no—
572Q8—41no—
573Q8—42no—
574Q8—42no—
575Q8—81no—
576Q8—82no—
577D4—42no—
578D4—41no—
579D4—41no—
580D4—41no—
581D4—41no—
582D4—41no—
583D4—42no—
584D4—42no—
585D4—42no—
586D4—82no—
587D4—82no—
588D4—82no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—81no—
601C2×C4—82no—
602C2×C4—82no—
603C2×C4—82no—
604C2×C4—82no—
605C2×C4C31×#12011yescharacteristic
606C2×C4C31×#12011yescharacteristic
607C2×C4C31×#12011yescharacteristic
608C2×C4C22×D4×C3111yescharacteristic
609C2×C4C22×D4×C3111yescharacteristic
610C2×C4C22×D4×C3111yescharacteristic
611C8—24no—
612C8—22no—
613C8—42no—
614C8—41no—
615C22—21no—
616C22—21no—
617C22—21no—
618C22—41no—
619C22—42no—
620C22—42no—
621C22—41no—
622C22—41no—
623C22—42no—
624C22—81no—
625C22—81no—
626C22—81no—
627C22—162no—
628C22C2×C31×#12011yescharacteristic
629C4—21no—
630C4—21no—
631C4—21no—
632C4—21no—
633C4—21no—
634C4—41no—
635C4—42no—
636C4—42no—
637C4—41no—
638C4—42no—
639C4—41no—
640C4—81no—
641C4C31×#45312yes—
642C2—41no—
643C2—82no—
644C2—81no—
645C2—161no—
646C2C2×C31×#45311yescharacteristic
647C2C31×#301811yescharacteristic
648C2C31×#302011yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC42C24×C31
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
1. C31×#3561C2semidirect
2. C31×#3235C2semidirect
3. C31×#3171C2semidirect
4. C31×#3153C2semidirect
5. C31×#1661C2semidirect
6. C31×#1660C2semidirect
7. C31×#1657C2semidirect
8. C31×#1656C2semidirect
9. C31×#1529C2semidirect
10. C31×#1527C2semidirect
11. C31×#1526C2semidirect
30. C31×#501C22semidirect
31. C31×#501C22semidirect
34. C31×#496C22semidirect
36. C31×#490C22semidirect
37. C31×#490C22semidirect
38. C31×#489C22semidirect
49. C31×#330C22semidirect
50. C31×#330C22semidirect
51. C31×#330C22semidirect
54. C31×#329C22semidirect
237. #19019C31direct (N × H)
291. #3561C2×C31semidirect
292. #3235C2×C31semidirect
293. #3171C2×C31semidirect
294. #3153C2×C31semidirect
295. #1661C2×C31semidirect
296. #1660C2×C31semidirect
297. #1657C2×C31semidirect
298. #1656C2×C31semidirect
299. #1529C2×C31semidirect
300. #1527C2×C31semidirect
301. #1526C2×C31semidirect
347. #501C22×C31semidirect
348. #501C22×C31semidirect
351. #496C22×C31semidirect
353. #490C22×C31semidirect
354. #490C22×C31semidirect
355. #489C22×C31semidirect
366. #330C22×C31semidirect
367. #330C22×C31semidirect
368. #330C22×C31semidirect
371. #329C22×C31semidirect
474. C31#19019direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35612
C31×#32352
C31×#31712
C31×#31532
C31×#16612
C31×#16602
C31×#16572
C31×#16562
C31×#15292
C31×#15272
C31×#15262
#1901931