Finite Group Databasebeta

← C31×#18242 #21475001

Subgroups — C31×#18242 #21475001

Order 7936.

992 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3561C211yesmaximal, characteristic, complemented
2C31×#3114C211yesmaximal, characteristic, complemented
3C31×#3101C211yesmaximal, characteristic, complemented
4C31×#2631C211yesmaximal, characteristic, complemented
5C31×#1612C212yesmaximal, complemented
6C31×#1608C212yesmaximal, complemented
7C31×#1600C212yesmaximal, complemented
8C31×#1596C212yesmaximal, complemented
9C31×#1521C211yesmaximal, characteristic, complemented
10C31×#1515C211yesmaximal, characteristic, complemented
11C31×#1469C211yesmaximal, characteristic, complemented
12C31×#563—21no—
13C31×#562—21no—
14C31×#556—21no—
15C31×#551—22no—
16D4×Q8×C31—21no—
17C31×#548—21no—
18C31×#546—21no—
19C31×#543C2212yes—
20C31×#542—21no—
21C31×#542—22no—
22C31×#538C2212yes—
23C31×#536—21no—
24C31×#533C2211yescharacteristic
25C31×#529C2211yescharacteristic
26C2×C31×#124C2211yescharacteristic
27C2×C4×D4×C31C2211yescharacteristic
28C31×#483C2212yes—
29C31×#483C2211yescharacteristic
30C31×#480C2211yescharacteristic
31C31×#478—21no—
32C31×#476—21no—
33C31×#463—21no—
34C31×#461—21no—
35C31×#450C2212yes—
36C31×#450C2211yescharacteristic
37C31×#447C2211yescharacteristic
38C31×#442—21no—
39C31×#440—21no—
40C4×SD16×C31—22no—
41C31×#432C2211yescharacteristic
42C31×#431C2211yescharacteristic
43C31×#397C2212yes—
44C31×#393—21no—
45C31×#392—21no—
46C31×#392C2212yes—
47C31×#386—22no—
48C31×#383C2211yescharacteristic
49C31×#328C2212yes—
50C31×#328C2212yes—
51C31×#328C2212yes—
52C31×#327C2212yes—
53C31×#323C2212yes—
54C31×#323C2212yes—
55C2×#41×C31—21no—
56C22×Q8×C31—21no—
57C22×Q8×C31D411yescharacteristic
58C22×D4×C31—22no—
59C22×D4×C31D411yescharacteristic
60C22×D4×C31C2311yescharacteristic
61C23×C4×C31—21no—
62C2×SD16×C31—22no—
63C2×SD16×C31—21no—
64C2×D8×C31—21no—
65C31×C4⋊Q8—21no—
66C31×C4⋊Q8—22no—
67C31×C4⋊Q8D411yescharacteristic, complemented
68C31×C4⋊Q8D411yescharacteristic, complemented
69C31×#126—22no—
70C31×#126—22no—
71C31×#126—22no—
72C31×#126—21no—
73C31×#125—21no—
74C31×#125D411yescharacteristic, complemented
75C31×#124—21no—
76C31×#124—22no—
77C31×#124—22no—
78C31×#124—21no—
79C31×#124—21no—
80C31×#124—21no—
81C31×#124—21no—
82C31×#124D411yescharacteristic, complemented
83C31×#123—22no—
84C31×#123—22no—
85C31×#123—22no—
86C31×#123—21no—
87C31×#123—22no—
88C31×C22⋊Q8—22no—
89C31×C22⋊Q8—21no—
90C31×C22⋊Q8—22no—
91C31×C22⋊Q8—22no—
92C31×C22⋊Q8—22no—
93C31×C22⋊Q8—21no—
94C31×C22⋊Q8—22no—
95C31×C22⋊Q8—22no—
96C31×C22⋊Q8—22no—
97C31×C22⋊Q8C2311yescharacteristic
98C31×C22⋊Q8C2312yes—
99C31×#121—21no—
100C31×#121—22no—
101C31×#121—22no—
102C31×#121C2311yescharacteristic
103C31×#120—22no—
104C4×Q8×C31—21no—
105C4×Q8×C31—21no—
106C4×Q8×C31D412yescomplemented
107C4×Q8×C31D411yescharacteristic, complemented
108C4×D4×C31—21no—
109C4×D4×C31—22no—
110C4×D4×C31—21no—
111C4×D4×C31—22no—
112C4×D4×C31—21no—
113C4×D4×C31—21no—
114C4×D4×C31—21no—
115C4×D4×C31—21no—
116C4×D4×C31D411yescharacteristic, complemented
117C31×#117—22no—
118C31×#117—21no—
119C31×#117—21no—
120C2×C4⋊C4×C31C2311yescharacteristic
121C2×C22⋊C4×C31—22no—
122C2×C22⋊C4×C31—21no—
123C2×C22⋊C4×C31—22no—
124C2×C22⋊C4×C31—22no—
125C2×C22⋊C4×C31D412yes—
126C2×C22⋊C4×C31C2312yes—
127C2×C42×C31C2311yescharacteristic
128C31×#107—21no—
129C31×#106—22no—
130C31×#106—21no—
131C31×C4⋊C8—21no—
132C31×C4⋊C8—21no—
133C31×#103—22no—
134C31×#103—21no—
135C31×#102—21no—
136C31×#102—22no—
137C31×#102—21no—
138C31×#102—21no—
139C31×C22⋊C8C2312yes—
140C31×C22⋊C8C2311yescharacteristic
141C31×C22⋊C8C2311yescharacteristic
142C31×#97—21no—
143C4×C8×C31—21no—
144C31×#95—22no—
145C31×#95—21no—
146C31×#95—21no—
147C31×#95C2312yes—
148C24×C31—21no—
149C24×C31C2×D412yes—
150#41×C31—82no—
151C2×Q8×C31—22no—
152C2×Q8×C31—21no—
153C2×Q8×C31—21no—
154C2×Q8×C31—21no—
155C2×Q8×C31—81no—
156C2×Q8×C31—82no—
157C2×Q8×C31C2×D411yescharacteristic
158C2×Q8×C31C2×D412yes—
159C2×D4×C31—21no—
160C2×D4×C31—22no—
161C2×D4×C31—22no—
162C2×D4×C31—21no—
163C2×D4×C31—21no—
164C2×D4×C31—21no—
165C2×D4×C31—21no—
166C2×D4×C31—42no—
167C2×D4×C31—44no—
168C2×D4×C31—81no—
169C2×D4×C31—82no—
170C2×D4×C31—82no—
171C2×D4×C31C2×D411yescharacteristic
172C22×C4×C31—22no—
173C22×C4×C31—22no—
174C22×C4×C31—22no—
175C22×C4×C31—21no—
176C22×C4×C31—22no—
177C22×C4×C31—21no—
178C22×C4×C31—41no—
179C22×C4×C31—41no—
180C22×C4×C31—82no—
181C22×C4×C31C2×D411yescharacteristic
182C22×C4×C31C2×D412yes—
183C22×C4×C31C2×D411yescharacteristic
184C22×C4×C31C2411yescharacteristic
185SD16×C31—42no—
186SD16×C31—44no—
187D8×C31—42no—
188C2×C8×C31—22no—
189C2×C8×C31—21no—
190C2×C8×C31—21no—
191C4⋊C4×C31—22no—
192C4⋊C4×C31—22no—
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—22no—
202C4⋊C4×C31—21no—
203C4⋊C4×C31—22no—
204C4⋊C4×C31—21no—
205C4⋊C4×C31—21no—
206C4⋊C4×C31C2×D411yescharacteristic
207C4⋊C4×C31C2×D412yes—
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—22no—
217C22⋊C4×C31—22no—
218C22⋊C4×C31—21no—
219C22⋊C4×C31—21no—
220C22⋊C4×C31—21no—
221C22⋊C4×C31—22no—
222C22⋊C4×C31—22no—
223C22⋊C4×C31—21no—
224C22⋊C4×C31—41no—
225C22⋊C4×C31—41no—
226C22⋊C4×C31—82no—
227C22⋊C4×C31—82no—
228C42×C31—22no—
229C42×C31—22no—
230C42×C31—21no—
231C42×C31C2×D411yescharacteristic
232C42×C31C2×D411yescharacteristic
233C42×C31C2×D411yescharacteristic
234C42×C31C2×D411yescharacteristic
235#18242C3111yesmaximal, characteristic, direct product factor, Sylow
236C23×C31—22no—
237C23×C31—22no—
238C23×C31—21no—
239C23×C31—22no—
240C23×C31—22no—
241C23×C31—42no—
242C23×C31—41no—
243C23×C31—42no—
244C23×C31—42no—
245C23×C31—41no—
246C23×C31—42no—
247C23×C31C22×D411yescharacteristic
248Q8×C31—41no—
249Q8×C31—42no—
250Q8×C31—42no—
251Q8×C31—42no—
252Q8×C31—81no—
253Q8×C31—82no—
254D4×C31—22no—
255D4×C31—22no—
256D4×C31—41no—
257D4×C31—41no—
258D4×C31—41no—
259D4×C31—42no—
260D4×C31—42no—
261D4×C31—81no—
262D4×C31—82no—
263D4×C31—82no—
264D4×C31—88no—
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—42no—
276C2×C4×C31—42no—
277C2×C4×C31—42no—
278C2×C4×C31—41no—
279C2×C4×C31—42no—
280C2×C4×C31—42no—
281C2×C4×C31—82no—
282C2×C4×C31—82no—
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—41no—
293C8×C31—41no—
294C8×C31—42no—
295#3561C2×C3111yescharacteristic, complemented
296#3114C2×C3111yescharacteristic, complemented
297#3101C2×C3111yescharacteristic, complemented
298#2631C2×C3111yescharacteristic, complemented
299#1612C2×C3112yescomplemented
300#1608C2×C3112yescomplemented
301#1600C2×C3112yescomplemented
302#1596C2×C3112yescomplemented
303#1521C2×C3111yescharacteristic, complemented
304#1515C2×C3111yescharacteristic, complemented
305#1469C2×C3111yescharacteristic, complemented
306C22×C31—21no—
307C22×C31—21no—
308C22×C31—44no—
309C22×C31—41no—
310C22×C31—42no—
311C22×C31—41no—
312C22×C31—42no—
313C22×C31—42no—
314C22×C31—42no—
315C22×C31—41no—
316C22×C31—82no—
317C22×C31—82no—
318C22×C31—82no—
319C22×C31—81no—
320C22×C31—81no—
321C22×C31C2×#12011yescharacteristic
322C22×C31#57712yes—
323C4×C31—21no—
324C4×C31—21no—
325C4×C31—21no—
326C4×C31—21no—
327C4×C31—21no—
328C4×C31—21no—
329C4×C31—41no—
330C4×C31—41no—
331C4×C31—42no—
332C4×C31—42no—
333C4×C31—41no—
334C4×C31—42no—
335C4×C31—82no—
336C4×C31—81no—
337#563—21no—
338#562—21no—
339#556—21no—
340#551—22no—
341D4×Q8—21no—
342#548—21no—
343#546—21no—
344#543C22×C3112yes—
345#542—21no—
346#542—22no—
347#538C22×C3112yes—
348#536—21no—
349#533C22×C3111yescharacteristic
350#529C22×C3111yescharacteristic
351C2×#124C22×C3111yescharacteristic
352C2×C4×D4C22×C3111yescharacteristic
353#483C22×C3112yes—
354#483C22×C3111yescharacteristic
355#480C22×C3111yescharacteristic
356#478—21no—
357#476—21no—
358#463—21no—
359#461—21no—
360#450C22×C3112yes—
361#450C22×C3111yescharacteristic
362#447C22×C3111yescharacteristic
363#442—21no—
364#440—21no—
365C4×SD16—22no—
366#432C22×C3111yescharacteristic
367#431C22×C3111yescharacteristic
368#397C22×C3112yes—
369#393—21no—
370#392—21no—
371#392C22×C3112yes—
372#386—22no—
373#383C22×C3111yescharacteristic
374#328C22×C3112yes—
375#328C22×C3112yes—
376#328C22×C3112yes—
377#327C22×C3112yes—
378#323C22×C3112yes—
379#323C22×C3112yes—
380C2×C31—22no—
381C2×C31—81no—
382C2×C31—82no—
383C2×C31—82no—
384C2×C31#301011yescharacteristic
385C2×C31#301911yescharacteristic
386C2×C31#302511yescharacteristic
387C2×#41—21no—
388C22×Q8—21no—
389C22×Q8D4×C3111yescharacteristic
390C22×D4—22no—
391C22×D4D4×C3111yescharacteristic
392C22×D4C23×C3111yescharacteristic
393C23×C4—21no—
394C2×SD16—22no—
395C2×SD16—21no—
396C2×D8—21no—
397C4⋊Q8—21no—
398C4⋊Q8—22no—
399C4⋊Q8D4×C3111yescharacteristic, complemented
400C4⋊Q8D4×C3111yescharacteristic, complemented
401#126—22no—
402#126—22no—
403#126—22no—
404#126—21no—
405#125—21no—
406#125D4×C3111yescharacteristic, complemented
407#124—21no—
408#124—22no—
409#124—22no—
410#124—21no—
411#124—21no—
412#124—21no—
413#124—21no—
414#124D4×C3111yescharacteristic, complemented
415#123—22no—
416#123—22no—
417#123—22no—
418#123—21no—
419#123—22no—
420C22⋊Q8—22no—
421C22⋊Q8—21no—
422C22⋊Q8—22no—
423C22⋊Q8—22no—
424C22⋊Q8—22no—
425C22⋊Q8—21no—
426C22⋊Q8—22no—
427C22⋊Q8—22no—
428C22⋊Q8—22no—
429C22⋊Q8C23×C3111yescharacteristic
430C22⋊Q8C23×C3112yes—
431#121—21no—
432#121—22no—
433#121—22no—
434#121C23×C3111yescharacteristic
435#120—22no—
436C4×Q8—21no—
437C4×Q8—21no—
438C4×Q8D4×C3111yescharacteristic, complemented
439C4×Q8D4×C3112yescomplemented
440C4×D4—21no—
441C4×D4—22no—
442C4×D4—21no—
443C4×D4—22no—
444C4×D4—21no—
445C4×D4—21no—
446C4×D4—21no—
447C4×D4—21no—
448C4×D4D4×C3111yescharacteristic, complemented
449#117—22no—
450#117—21no—
451#117—21no—
452C2×C4⋊C4C23×C3111yescharacteristic
453C2×C22⋊C4—22no—
454C2×C22⋊C4—21no—
455C2×C22⋊C4—22no—
456C2×C22⋊C4—22no—
457C2×C22⋊C4D4×C3112yes—
458C2×C22⋊C4C23×C3112yes—
459C2×C42C23×C3111yescharacteristic
460#107—21no—
461#106—22no—
462#106—21no—
463C4⋊C8—21no—
464C4⋊C8—21no—
465#103—22no—
466#103—21no—
467#102—21no—
468#102—22no—
469#102—21no—
470#102—21no—
471C22⋊C8C23×C3112yes—
472C22⋊C8C23×C3111yescharacteristic
473C22⋊C8C23×C3111yescharacteristic
474#97—21no—
475C4×C8—21no—
476#95—21no—
477#95—21no—
478#95—22no—
479#95C23×C3112yes—
480C31#1824211yescharacteristic, direct product factor, Sylow
481C24—21no—
482C24C2×D4×C3112yes—
483#41—82no—
484C2×Q8—22no—
485C2×Q8—21no—
486C2×Q8—21no—
487C2×Q8—21no—
488C2×Q8—81no—
489C2×Q8—82no—
490C2×Q8C2×D4×C3111yescharacteristic
491C2×Q8C2×D4×C3112yes—
492C2×D4—21no—
493C2×D4—22no—
494C2×D4—22no—
495C2×D4—21no—
496C2×D4—21no—
497C2×D4—21no—
498C2×D4—21no—
499C2×D4—42no—
500C2×D4—44no—
501C2×D4—81no—
502C2×D4—82no—
503C2×D4—82no—
504C2×D4C2×D4×C3111yescharacteristic
505C22×C4—22no—
506C22×C4—22no—
507C22×C4—22no—
508C22×C4—21no—
509C22×C4—22no—
510C22×C4—21no—
511C22×C4—41no—
512C22×C4—41no—
513C22×C4—82no—
514C22×C4C2×D4×C3111yescharacteristic
515C22×C4C2×D4×C3112yes—
516C22×C4C2×D4×C3111yescharacteristic
517C22×C4C24×C3111yescharacteristic
518SD16—42no—
519SD16—44no—
520D8—42no—
521C2×C8—22no—
522C2×C8—21no—
523C2×C8—21no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—22no—
535C4⋊C4—21no—
536C4⋊C4—22no—
537C4⋊C4—21no—
538C4⋊C4—21no—
539C4⋊C4C2×D4×C3111yescharacteristic
540C4⋊C4C2×D4×C3112yes—
541C4⋊C4C2×D4×C3111yescharacteristic
542C22⋊C4—22no—
543C22⋊C4—22no—
544C22⋊C4—22no—
545C22⋊C4—22no—
546C22⋊C4—22no—
547C22⋊C4—21no—
548C22⋊C4—22no—
549C22⋊C4—22no—
550C22⋊C4—22no—
551C22⋊C4—21no—
552C22⋊C4—21no—
553C22⋊C4—21no—
554C22⋊C4—22no—
555C22⋊C4—22no—
556C22⋊C4—21no—
557C22⋊C4—41no—
558C22⋊C4—41no—
559C22⋊C4—82no—
560C22⋊C4—82no—
561C42—22no—
562C42—22no—
563C42—21no—
564C42C2×D4×C3111yescharacteristic
565C42C2×D4×C3111yescharacteristic
566C42C2×D4×C3111yescharacteristic
567C42C2×D4×C3111yescharacteristic
568C23—22no—
569C23—22no—
570C23—21no—
571C23—22no—
572C23—22no—
573C23—42no—
574C23—41no—
575C23—42no—
576C23—42no—
577C23—41no—
578C23—42no—
579C23C22×D4×C3111yescharacteristic
580Q8—41no—
581Q8—42no—
582Q8—42no—
583Q8—42no—
584Q8—81no—
585Q8—82no—
586D4—22no—
587D4—22no—
588D4—41no—
589D4—41no—
590D4—41no—
591D4—42no—
592D4—42no—
593D4—81no—
594D4—82no—
595D4—82no—
596D4—88no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—21no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—21no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—82no—
614C2×C4—82no—
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—41no—
625C8—41no—
626C8—42no—
627C22—21no—
628C22—21no—
629C22—44no—
630C22—41no—
631C22—42no—
632C22—41no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—41no—
637C22—82no—
638C22—82no—
639C22—82no—
640C22—81no—
641C22—81no—
642C22C2×C31×#12011yescharacteristic
643C22C31×#57712yes—
644C4—21no—
645C4—21no—
646C4—21no—
647C4—21no—
648C4—21no—
649C4—21no—
650C4—41no—
651C4—41no—
652C4—42no—
653C4—42no—
654C4—41no—
655C4—42no—
656C4—82no—
657C4—81no—
658C2—22no—
659C2—82no—
660C2—82no—
661C2—81no—
662C2C31×#301011yescharacteristic
663C2C31×#301911yescharacteristic
664C2C31×#302511yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
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
1. C31×#3561C2semidirect
2. C31×#3114C2semidirect
3. C31×#3101C2semidirect
4. C31×#2631C2semidirect
5. C31×#1612C2semidirect
6. C31×#1608C2semidirect
7. C31×#1600C2semidirect
8. C31×#1596C2semidirect
9. C31×#1521C2semidirect
10. C31×#1515C2semidirect
11. C31×#1469C2semidirect
67. C31×C4⋊Q8D4semidirect
68. C31×C4⋊Q8D4semidirect
74. C31×#125D4semidirect
82. C31×#124D4semidirect
106. C4×Q8×C31D4semidirect
107. C4×Q8×C31D4semidirect
116. C4×D4×C31D4semidirect
235. #18242C31direct (N × H)
295. #3561C2×C31semidirect
296. #3114C2×C31semidirect
297. #3101C2×C31semidirect
298. #2631C2×C31semidirect
299. #1612C2×C31semidirect
300. #1608C2×C31semidirect
301. #1600C2×C31semidirect
302. #1596C2×C31semidirect
303. #1521C2×C31semidirect
304. #1515C2×C31semidirect
305. #1469C2×C31semidirect
399. C4⋊Q8D4×C31semidirect
400. C4⋊Q8D4×C31semidirect
406. #125D4×C31semidirect
414. #124D4×C31semidirect
438. C4×Q8D4×C31semidirect
439. C4×Q8D4×C31semidirect
448. C4×D4D4×C31semidirect
480. C31#18242direct (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×#31142
C31×#31012
C31×#26312
C31×#16122
C31×#16082
C31×#16002
C31×#15962
C31×#15212
C31×#15152
C31×#14692
#1824231