Finite Group Databasebeta

← C31×#18422 #21475181

Subgroups — C31×#18422 #21475181

Order 7936.

960 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3538C211yesmaximal, characteristic, complemented
2C31×#3193C211yesmaximal, characteristic, complemented
3C31×#3188C211yesmaximal, characteristic
4C31×#2639C211yesmaximal, characteristic, complemented
5C31×#1617C212yesmaximal, complemented
6C31×#1615C212yesmaximal, complemented
7C31×#1599C212yesmaximal, complemented
8C31×#1597C212yesmaximal, complemented
9C31×#1504C211yesmaximal, characteristic, complemented
10C2×C31×#328C211yesmaximal, characteristic, complemented
11C31×#1472C211yesmaximal, characteristic, complemented
12C31×#563—22no—
13C31×#552—21no—
14C31×#552—21no—
15C31×#551—22no—
16D4×Q8×C31—21no—
17D4×Q8×C31—21no—
18C31×#544—21no—
19C31×#544C2212yes—
20C31×#541—21no—
21C31×#539—21no—
22C31×#539C2212yes—
23C31×#536—21no—
24C31×#525C2211yescharacteristic
25C2×C31×C22⋊Q8C2211yescharacteristic
26C2×C31×C22⋊Q8C2211yescharacteristic
27C2×C31×#121C2211yescharacteristic
28C31×#484C2212yes—
29C31×#484C2211yescharacteristic
30C31×#482—21no—
31C31×#481C2211yescharacteristic
32C31×#480—21no—
33C31×#468—22no—
34C31×#466—21no—
35C31×#465—21no—
36C31×#450C2212yes—
37C31×#450C2211yescharacteristic
38C31×#449—21no—
39C31×#448—21no—
40C31×#447C2211yescharacteristic
41C31×#407C2211yescharacteristic
42C2×C31×C22⋊C8C2211yescharacteristic
43C31×#400C2212yes—
44C31×#397—21no—
45C31×#394C2212yes—
46C31×#393—21no—
47C31×#386—22no—
48C2×C31×#95C2211yescharacteristic
49C31×#328C2212yes—
50C31×#328C2212yes—
51C31×#328C2212yes—
52C31×#327C2212yes—
53C31×#324C2212yes—
54C31×#324C2212yes—
55C2×#41×C31D411yescharacteristic
56C22×Q8×C31—21no—
57C22×Q8×C31D411yescharacteristic
58C22×Q8×C31D411yescharacteristic
59C22×D4×C31—21no—
60C22×D4×C31C2311yescharacteristic
61C23×C4×C31—21no—
62C23×C4×C31C2311yescharacteristic
63C2×SD16×C31—22no—
64C2×SD16×C31—21no—
65C2×D8×C31—21no—
66C2×M16×C31—21no—
67C22×C8×C31—21no—
68C31×C4⋊Q8—21no—
69C31×C4⋊Q8—22no—
70C31×C4⋊Q8—22no—
71C31×C4⋊Q8—22no—
72C31×#126—22no—
73C31×#126—22no—
74C31×#126—22no—
75C31×#126—22no—
76C31×#125—21no—
77C31×#125—22no—
78C31×#125—22no—
79C31×#125—22no—
80C31×#125—22no—
81C31×#124—22no—
82C31×#123—22no—
83C31×#123—22no—
84C31×#123—22no—
85C31×#123—21no—
86C31×#123—22no—
87C31×C22⋊Q8—21no—
88C31×C22⋊Q8—21no—
89C31×C22⋊Q8—22no—
90C31×C22⋊Q8—22no—
91C31×C22⋊Q8—22no—
92C31×C22⋊Q8—21no—
93C31×C22⋊Q8—21no—
94C31×C22⋊Q8—21no—
95C31×C22⋊Q8—21no—
96C31×C22⋊Q8—21no—
97C31×C22⋊Q8—21no—
98C31×C22⋊Q8—22no—
99C31×C22⋊Q8D411yescharacteristic
100C31×C22⋊Q8D412yes—
101C31×C22⋊Q8C2312yes—
102C31×C22⋊Q8C2311yescharacteristic
103C31×#121—21no—
104C31×#121—21no—
105C31×#121—21no—
106C31×#121—21no—
107C31×#121D411yescharacteristic
108C31×#121C2311yescharacteristic
109C31×#120—21no—
110C4×Q8×C31—21no—
111C4×Q8×C31—21no—
112C4×D4×C31—22no—
113C4×D4×C31—21no—
114C4×D4×C31—21no—
115C4×D4×C31—22no—
116C31×#117—21no—
117C31×#117—22no—
118C31×#117—22no—
119C31×#117D411yescharacteristic
120C2×C4⋊C4×C31—21no—
121C2×C4⋊C4×C31D411yescharacteristic
122C2×C4⋊C4×C31D411yescharacteristic
123C2×C4⋊C4×C31C2311yescharacteristic
124C2×C4⋊C4×C31C2312yes—
125C2×C22⋊C4×C31—21no—
126C2×C22⋊C4×C31—21no—
127C2×C22⋊C4×C31—22no—
128C2×C22⋊C4×C31—22no—
129C2×C22⋊C4×C31D412yes—
130C31×#107—21no—
131C31×#107—22no—
132C31×#106—21no—
133C31×#103—22no—
134C31×#103—21no—
135C31×#102—21no—
136C31×#102—21no—
137C31×#102—21no—
138C31×#102—22no—
139C31×C22⋊C8—21no—
140C31×C22⋊C8—21no—
141C31×C22⋊C8C2311yescharacteristic
142C31×C22⋊C8C2311yescharacteristic
143C31×C22⋊C8C2312yes—
144C31×#95—22no—
145C31×#95—21no—
146C31×#95—22no—
147C31×#95C2312yes—
148C24×C31—21no—
149C24×C31C2×D411yescharacteristic
150#41×C31—81no—
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×C31—81no—
158C2×Q8×C31C2×D412yes—
159C2×Q8×C31C2×D411yescharacteristic
160C2×D4×C31—21no—
161C2×D4×C31—21no—
162C2×D4×C31—21no—
163C2×D4×C31—21no—
164C2×D4×C31—21no—
165C2×D4×C31—44no—
166C2×D4×C31—41no—
167C2×D4×C31—41no—
168C2×D4×C31—81no—
169C2×D4×C31C2×D411yescharacteristic
170C22×C4×C31—21no—
171C22×C4×C31—22no—
172C22×C4×C31—21no—
173C22×C4×C31—21no—
174C22×C4×C31—21no—
175C22×C4×C31—22no—
176C22×C4×C31—42no—
177C22×C4×C31—81no—
178C22×C4×C31—81no—
179C22×C4×C31—81no—
180C22×C4×C31C2×D412yes—
181C22×C4×C31C2×D412yes—
182C22×C4×C31C2×D411yescharacteristic
183C22×C4×C31C2×D411yescharacteristic
184C22×C4×C31C2×D411yescharacteristic
185C22×C4×C31C2×D411yescharacteristic
186C22×C4×C31C2×D411yescharacteristic
187C22×C4×C31C2411yescharacteristic
188SD16×C31—81no—
189SD16×C31—82no—
190D8×C31—81no—
191M16×C31—81no—
192C2×C8×C31—21no—
193C2×C8×C31—21no—
194C2×C8×C31—22no—
195C2×C8×C31—81no—
196C4⋊C4×C31—22no—
197C4⋊C4×C31—22no—
198C4⋊C4×C31—22no—
199C4⋊C4×C31—22no—
200C4⋊C4×C31—22no—
201C4⋊C4×C31—22no—
202C4⋊C4×C31—22no—
203C4⋊C4×C31—22no—
204C4⋊C4×C31—22no—
205C4⋊C4×C31—21no—
206C4⋊C4×C31—21no—
207C4⋊C4×C31—21no—
208C4⋊C4×C31—21no—
209C4⋊C4×C31—22no—
210C4⋊C4×C31—22no—
211C4⋊C4×C31—22no—
212C4⋊C4×C31—21no—
213C4⋊C4×C31—21no—
214C4⋊C4×C31—21no—
215C4⋊C4×C31C2×D411yescharacteristic
216C4⋊C4×C31C2×D412yes—
217C4⋊C4×C31C2×D411yescharacteristic
218C22⋊C4×C31—22no—
219C22⋊C4×C31—22no—
220C22⋊C4×C31—22no—
221C22⋊C4×C31—22no—
222C22⋊C4×C31—22no—
223C22⋊C4×C31—22no—
224C22⋊C4×C31—21no—
225C22⋊C4×C31—21no—
226C22⋊C4×C31—22no—
227C22⋊C4×C31—21no—
228C22⋊C4×C31—21no—
229C22⋊C4×C31—21no—
230C22⋊C4×C31—41no—
231C22⋊C4×C31—41no—
232C22⋊C4×C31—82no—
233C22⋊C4×C31—82no—
234C42×C31—22no—
235C42×C31—22no—
236C42×C31—22no—
237C42×C31—21no—
238#18422C3111yesmaximal, characteristic, direct product factor, Sylow
239C23×C31—21no—
240C23×C31—21no—
241C23×C31—22no—
242C23×C31—21no—
243C23×C31—41no—
244C23×C31—42no—
245C23×C31—42no—
246C23×C31—41no—
247C23×C31—41no—
248C23×C31—41no—
249C23×C31#12011yescharacteristic
250C23×C31#12011yescharacteristic
251C23×C31C22×D411yescharacteristic
252Q8×C31—42no—
253Q8×C31—41no—
254Q8×C31—42no—
255Q8×C31—42no—
256Q8×C31—81no—
257Q8×C31—82no—
258D4×C31—41no—
259D4×C31—41no—
260D4×C31—41no—
261D4×C31—42no—
262D4×C31—41no—
263D4×C31—41no—
264D4×C31—42no—
265D4×C31—81no—
266D4×C31—82no—
267D4×C31—82no—
268C2×C4×C31—21no—
269C2×C4×C31—21no—
270C2×C4×C31—22no—
271C2×C4×C31—22no—
272C2×C4×C31—21no—
273C2×C4×C31—22no—
274C2×C4×C31—22no—
275C2×C4×C31—22no—
276C2×C4×C31—22no—
277C2×C4×C31—41no—
278C2×C4×C31—41no—
279C2×C4×C31—41no—
280C2×C4×C31—41no—
281C2×C4×C31—41no—
282C2×C4×C31—41no—
283C2×C4×C31—42no—
284C2×C4×C31—42no—
285C2×C4×C31—42no—
286C2×C4×C31—42no—
287C2×C4×C31—42no—
288C2×C4×C31—42no—
289C2×C4×C31—41no—
290C2×C4×C31—42no—
291C2×C4×C31—81no—
292C2×C4×C31—82no—
293C2×C4×C31—84no—
294C2×C4×C31#12011yescharacteristic
295C2×C4×C31#12011yescharacteristic
296C2×C4×C31C22×D411yescharacteristic
297C2×C4×C31C22×D411yescharacteristic
298C8×C31—41no—
299C8×C31—41no—
300C8×C31—42no—
301#3538C2×C3111yescharacteristic, complemented
302#3193C2×C3111yescharacteristic, complemented
303#3188C2×C3111yescharacteristic
304#2639C2×C3111yescharacteristic, complemented
305#1617C2×C3112yescomplemented
306#1615C2×C3112yescomplemented
307#1599C2×C3112yescomplemented
308#1597C2×C3112yescomplemented
309#1504C2×C3111yescharacteristic, complemented
310C2×#328C2×C3111yescharacteristic, complemented
311#1472C2×C3111yescharacteristic, complemented
312C22×C31—21no—
313C22×C31—21no—
314C22×C31—21no—
315C22×C31—21no—
316C22×C31—21no—
317C22×C31—21no—
318C22×C31—21no—
319C22×C31—42no—
320C22×C31—42no—
321C22×C31—41no—
322C22×C31—41no—
323C22×C31—41no—
324C22×C31—82no—
325C22×C31—82no—
326C22×C31—81no—
327C22×C31—81no—
328C22×C31—81no—
329C22×C31#45412yes—
330C22×C31C2×#12011yescharacteristic
331C22×C31#57711yescharacteristic
332C22×C31#57811yescharacteristic
333C4×C31—21no—
334C4×C31—21no—
335C4×C31—21no—
336C4×C31—21no—
337C4×C31—42no—
338C4×C31—41no—
339C4×C31—42no—
340C4×C31—41no—
341C4×C31—41no—
342C4×C31—42no—
343C4×C31—42no—
344C4×C31—82no—
345C4×C31—81no—
346#563—22no—
347#552—21no—
348#552—21no—
349#551—22no—
350D4×Q8—21no—
351D4×Q8—21no—
352#544—21no—
353#544C22×C3112yes—
354#541—21no—
355#539—21no—
356#539C22×C3112yes—
357#536—21no—
358#525C22×C3111yescharacteristic
359C2×C22⋊Q8C22×C3111yescharacteristic
360C2×C22⋊Q8C22×C3111yescharacteristic
361C2×#121C22×C3111yescharacteristic
362#484C22×C3112yes—
363#484C22×C3111yescharacteristic
364#482—21no—
365#481C22×C3111yescharacteristic
366#480—21no—
367#468—22no—
368#466—21no—
369#465—21no—
370#450C22×C3112yes—
371#450C22×C3111yescharacteristic
372#449—21no—
373#448—21no—
374#447C22×C3111yescharacteristic
375#407C22×C3111yescharacteristic
376C2×C22⋊C8C22×C3111yescharacteristic
377#400C22×C3112yes—
378#397—21no—
379#394C22×C3112yes—
380#393—21no—
381#386—22no—
382C2×#95C22×C3111yescharacteristic
383#328C22×C3112yes—
384#328C22×C3112yes—
385#328C22×C3112yes—
386#327C22×C3112yes—
387#324C22×C3112yes—
388#324C22×C3112yes—
389C2×C31—21no—
390C2×C31—21no—
391C2×C31—22no—
392C2×C31—41no—
393C2×C31—81no—
394C2×C31—161no—
395C2×C31#301311yescharacteristic
396C2×C31C2×#45411yescharacteristic
397C2×C31#302611yescharacteristic
398C2×#41D4×C3111yescharacteristic
399C22×Q8—21no—
400C22×Q8D4×C3111yescharacteristic
401C22×Q8D4×C3111yescharacteristic
402C22×D4—21no—
403C22×D4C23×C3111yescharacteristic
404C23×C4—21no—
405C23×C4C23×C3111yescharacteristic
406C2×SD16—22no—
407C2×SD16—21no—
408C2×D8—21no—
409C2×M16—21no—
410C22×C8—21no—
411C4⋊Q8—21no—
412C4⋊Q8—22no—
413C4⋊Q8—22no—
414C4⋊Q8—22no—
415#126—22no—
416#126—22no—
417#126—22no—
418#126—22no—
419#125—21no—
420#125—22no—
421#125—22no—
422#125—22no—
423#125—22no—
424#124—22no—
425#123—22no—
426#123—22no—
427#123—22no—
428#123—21no—
429#123—22no—
430C22⋊Q8—21no—
431C22⋊Q8—21no—
432C22⋊Q8—22no—
433C22⋊Q8—22no—
434C22⋊Q8—22no—
435C22⋊Q8—21no—
436C22⋊Q8—21no—
437C22⋊Q8—21no—
438C22⋊Q8—21no—
439C22⋊Q8—21no—
440C22⋊Q8—21no—
441C22⋊Q8—22no—
442C22⋊Q8D4×C3112yes—
443C22⋊Q8D4×C3111yescharacteristic
444C22⋊Q8C23×C3112yes—
445C22⋊Q8C23×C3111yescharacteristic
446#121—21no—
447#121—21no—
448#121—21no—
449#121—21no—
450#121D4×C3111yescharacteristic
451#121C23×C3111yescharacteristic
452#120—21no—
453C4×Q8—21no—
454C4×Q8—21no—
455C4×D4—22no—
456C4×D4—21no—
457C4×D4—21no—
458C4×D4—22no—
459#117—21no—
460#117—22no—
461#117—22no—
462#117D4×C3111yescharacteristic
463C2×C4⋊C4—21no—
464C2×C4⋊C4D4×C3111yescharacteristic
465C2×C4⋊C4D4×C3111yescharacteristic
466C2×C4⋊C4C23×C3111yescharacteristic
467C2×C4⋊C4C23×C3112yes—
468C2×C22⋊C4—21no—
469C2×C22⋊C4—21no—
470C2×C22⋊C4—22no—
471C2×C22⋊C4—22no—
472C2×C22⋊C4D4×C3112yes—
473#107—21no—
474#107—22no—
475#106—21no—
476#103—22no—
477#103—21no—
478#102—21no—
479#102—21no—
480#102—21no—
481#102—22no—
482C22⋊C8—21no—
483C22⋊C8—21no—
484C22⋊C8C23×C3112yes—
485C22⋊C8C23×C3111yescharacteristic
486C22⋊C8C23×C3111yescharacteristic
487#95—21no—
488#95—22no—
489#95—22no—
490#95C23×C3112yes—
491C31#1842211yescharacteristic, direct product factor, Sylow
492C24—21no—
493C24C2×D4×C3111yescharacteristic
494#41—81no—
495C2×Q8—22no—
496C2×Q8—21no—
497C2×Q8—21no—
498C2×Q8—21no—
499C2×Q8—81no—
500C2×Q8—82no—
501C2×Q8—81no—
502C2×Q8C2×D4×C3112yes—
503C2×Q8C2×D4×C3111yescharacteristic
504C2×D4—21no—
505C2×D4—21no—
506C2×D4—21no—
507C2×D4—21no—
508C2×D4—21no—
509C2×D4—44no—
510C2×D4—41no—
511C2×D4—41no—
512C2×D4—81no—
513C2×D4C2×D4×C3111yescharacteristic
514C22×C4—21no—
515C22×C4—22no—
516C22×C4—21no—
517C22×C4—21no—
518C22×C4—21no—
519C22×C4—22no—
520C22×C4—42no—
521C22×C4—81no—
522C22×C4—81no—
523C22×C4—81no—
524C22×C4C2×D4×C3112yes—
525C22×C4C2×D4×C3112yes—
526C22×C4C2×D4×C3111yescharacteristic
527C22×C4C2×D4×C3111yescharacteristic
528C22×C4C2×D4×C3111yescharacteristic
529C22×C4C2×D4×C3111yescharacteristic
530C22×C4C2×D4×C3111yescharacteristic
531C22×C4C24×C3111yescharacteristic
532SD16—81no—
533SD16—82no—
534D8—81no—
535M16—81no—
536C2×C8—21no—
537C2×C8—21no—
538C2×C8—22no—
539C2×C8—81no—
540C4⋊C4—22no—
541C4⋊C4—22no—
542C4⋊C4—22no—
543C4⋊C4—22no—
544C4⋊C4—22no—
545C4⋊C4—22no—
546C4⋊C4—22no—
547C4⋊C4—22no—
548C4⋊C4—22no—
549C4⋊C4—21no—
550C4⋊C4—21no—
551C4⋊C4—21no—
552C4⋊C4—21no—
553C4⋊C4—22no—
554C4⋊C4—22no—
555C4⋊C4—22no—
556C4⋊C4—21no—
557C4⋊C4—21no—
558C4⋊C4—21no—
559C4⋊C4C2×D4×C3111yescharacteristic
560C4⋊C4C2×D4×C3112yes—
561C4⋊C4C2×D4×C3111yescharacteristic
562C22⋊C4—22no—
563C22⋊C4—22no—
564C22⋊C4—22no—
565C22⋊C4—22no—
566C22⋊C4—22no—
567C22⋊C4—22no—
568C22⋊C4—21no—
569C22⋊C4—21no—
570C22⋊C4—22no—
571C22⋊C4—21no—
572C22⋊C4—21no—
573C22⋊C4—21no—
574C22⋊C4—41no—
575C22⋊C4—41no—
576C22⋊C4—82no—
577C22⋊C4—82no—
578C42—22no—
579C42—22no—
580C42—22no—
581C42—21no—
582C23—22no—
583C23—21no—
584C23—21no—
585C23—21no—
586C23—41no—
587C23—42no—
588C23—42no—
589C23—41no—
590C23—41no—
591C23—41no—
592C23C31×#12011yescharacteristic
593C23C31×#12011yescharacteristic
594C23C22×D4×C3111yescharacteristic
595Q8—42no—
596Q8—41no—
597Q8—42no—
598Q8—42no—
599Q8—81no—
600Q8—82no—
601D4—41no—
602D4—41no—
603D4—41no—
604D4—42no—
605D4—42no—
606D4—41no—
607D4—41no—
608D4—81no—
609D4—82no—
610D4—82no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—21no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—41no—
633C2×C4—42no—
634C2×C4—81no—
635C2×C4—82no—
636C2×C4—84no—
637C2×C4C31×#12011yescharacteristic
638C2×C4C31×#12011yescharacteristic
639C2×C4C22×D4×C3111yescharacteristic
640C2×C4C22×D4×C3111yescharacteristic
641C8—41no—
642C8—41no—
643C8—42no—
644C22—21no—
645C22—21no—
646C22—21no—
647C22—21no—
648C22—21no—
649C22—21no—
650C22—21no—
651C22—42no—
652C22—42no—
653C22—41no—
654C22—41no—
655C22—41no—
656C22—82no—
657C22—82no—
658C22—81no—
659C22—81no—
660C22—81no—
661C22C31×#45412yes—
662C22C2×C31×#12011yescharacteristic
663C22C31×#57711yescharacteristic
664C22C31×#57811yescharacteristic
665C4—21no—
666C4—21no—
667C4—21no—
668C4—21no—
669C4—42no—
670C4—41no—
671C4—42no—
672C4—41no—
673C4—41no—
674C4—42no—
675C4—42no—
676C4—82no—
677C4—81no—
678C2—21no—
679C2—21no—
680C2—22no—
681C2—41no—
682C2—81no—
683C2—161no—
684C2C31×#301311yescharacteristic
685C2C2×C31×#45411yescharacteristic
686C2C31×#302611yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#3188C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C31×#3538C2semidirect
2. C31×#3193C2semidirect
4. C31×#2639C2semidirect
5. C31×#1617C2semidirect
6. C31×#1615C2semidirect
7. C31×#1599C2semidirect
8. C31×#1597C2semidirect
9. C31×#1504C2semidirect
10. C2×C31×#328C2semidirect
11. C31×#1472C2semidirect
238. #18422C31direct (N × H)
301. #3538C2×C31semidirect
302. #3193C2×C31semidirect
304. #2639C2×C31semidirect
305. #1617C2×C31semidirect
306. #1615C2×C31semidirect
307. #1599C2×C31semidirect
308. #1597C2×C31semidirect
309. #1504C2×C31semidirect
310. C2×#328C2×C31semidirect
311. #1472C2×C31semidirect
491. C31#18422direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35382
C31×#31932
C31×#31882
C31×#26392
C31×#16172
C31×#16152
C31×#15992
C31×#15972
C31×#15042
C2×C31×#3282
C31×#14722
#1842231