Finite Group Databasebeta

← C31×#18193 #21474952

Subgroups — C31×#18193 #21474952

Order 7936.

1200 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3492C211yesmaximal, characteristic, complemented
2C31×#3002C212yesmaximal, complemented
3C31×#2406C211yesmaximal, characteristic, complemented
4C31×#1605C211yesmaximal, characteristic, complemented
5C31×#1604C211yesmaximal, characteristic, complemented
6C31×#1603C212yesmaximal, complemented
7C31×#1599C211yesmaximal, characteristic, complemented
8C31×#1598C211yesmaximal, characteristic, complemented
9C31×#1597C212yesmaximal, complemented
10C31×#1516C212yesmaximal, complemented
11C2×C31×#324C211yesmaximal, characteristic, complemented
12C22×#41×C31C2211yescharacteristic
13C2×C31×#135—22no—
14C31×#554—22no—
15C31×#553—22no—
16C31×#552—22no—
17C31×#547—22no—
18C31×#544C2211yescharacteristic
19C31×#539—21no—
20C31×#539C2212yes—
21C31×#537C2211yescharacteristic
22C31×#525—21no—
23C31×#525—21no—
24C31×#525C2212yes—
25C2×C31×#123—21no—
26C31×#520—21no—
27C22×C4⋊C4×C31C2211yescharacteristic
28C31×#452—22no—
29C31×#451C2212yes—
30C31×#450C2212yes—
31C31×#449—22no—
32C31×#448C2212yes—
33C31×#447C2212yes—
34C31×#416—22no—
35C2×C31×C22⋊C8C2212yes—
36C31×#395C2211yescharacteristic
37C31×#395C2211yescharacteristic
38C31×#394C2211yescharacteristic
39C31×#394C2211yescharacteristic
40C31×#386—21no—
41C31×#385—21no—
42C31×#385—21no—
43C31×#381C2211yescharacteristic
44C31×#328C2212yes—
45C31×#328C2212yes—
46C31×#327C2212yes—
47C31×#327C2212yes—
48C31×#324C2212yes—
49C31×#324C2211yescharacteristic
50C31×#324C2211yescharacteristic
51C2×#41×C31—21no—
52C2×#41×C31—21no—
53C2×#41×C31—22no—
54C2×#41×C31—22no—
55C2×#41×C31—44no—
56C2×#41×C31D412yes—
57C22×Q8×C31C2311yescharacteristic
58C22×D4×C31—21no—
59C22×D4×C31C2311yescharacteristic
60C23×C4×C31—21no—
61C23×C4×C31D411yescharacteristic
62C23×C4×C31D411yescharacteristic
63C23×C4×C31C2311yescharacteristic
64C31×#135—48no—
65C2×Q16×C31—22no—
66C2×SD16×C31—22no—
67C2×SD16×C31—22no—
68C2×D8×C31—22no—
69C22×C8×C31—22no—
70C31×C4⋊Q8—22no—
71C31×#126—22no—
72C31×#126—22no—
73C31×#125—22no—
74C31×#125—22no—
75C31×#125—22no—
76C31×#123—22no—
77C31×#123—22no—
78C31×#123—22no—
79C31×#123—22no—
80C31×#123—22no—
81C31×#123—44no—
82C31×C22⋊Q8—21no—
83C31×C22⋊Q8—21no—
84C31×C22⋊Q8—21no—
85C31×C22⋊Q8—22no—
86C31×C22⋊Q8—22no—
87C31×C22⋊Q8—21no—
88C31×C22⋊Q8—22no—
89C31×C22⋊Q8D412yes—
90C31×C22⋊Q8C2312yes—
91C31×#121—21no—
92C31×#121—21no—
93C31×#121—21no—
94C31×#121—22no—
95C31×#121—21no—
96C31×#121D412yes—
97C31×#121C2312yes—
98C31×#120—22no—
99C4×Q8×C31—22no—
100C4×D4×C31—22no—
101C4×D4×C31—22no—
102C4×D4×C31—22no—
103C4×D4×C31—22no—
104C4×D4×C31—22no—
105C4×D4×C31—22no—
106C4×D4×C31—22no—
107C31×#117—22no—
108C31×#117—21no—
109C31×#117—21no—
110C31×#117D412yes—
111C2×C4⋊C4×C31—22no—
112C2×C4⋊C4×C31—22no—
113C2×C4⋊C4×C31—22no—
114C2×C4⋊C4×C31—21no—
115C2×C4⋊C4×C31—21no—
116C2×C4⋊C4×C31D411yescharacteristic
117C2×C4⋊C4×C31D411yescharacteristic
118C2×C4⋊C4×C31C2311yescharacteristic
119C2×C4⋊C4×C31C2311yescharacteristic
120C2×C22⋊C4×C31—21no—
121C2×C22⋊C4×C31—21no—
122C2×C22⋊C4×C31—21no—
123C2×C22⋊C4×C31—21no—
124C2×C22⋊C4×C31—21no—
125C31×#103—22no—
126C31×#103—22no—
127C31×#102—22no—
128C31×#102—22no—
129C31×C22⋊C8—22no—
130C31×C22⋊C8C2312yes—
131C31×C22⋊C8C2312yes—
132C31×#95—21no—
133C31×#95—21no—
134C31×#95—21no—
135C31×#95C2311yescharacteristic
136C31×#95C2311yescharacteristic
137C24×C31—21no—
138C24×C31C2×D411yescharacteristic
139#41×C31—28no—
140#41×C31—42no—
141#41×C31—44no—
142#41×C31—44no—
143#41×C31—44no—
144#41×C31—44no—
145#41×C31—42no—
146#41×C31—82no—
147C2×Q8×C31—21no—
148C2×Q8×C31—21no—
149C2×Q8×C31—22no—
150C2×Q8×C31—42no—
151C2×Q8×C31C2×D412yes—
152C2×D4×C31—21no—
153C2×D4×C31—24no—
154C2×D4×C31—22no—
155C2×D4×C31—22no—
156C2×D4×C31—22no—
157C2×D4×C31—21no—
158C2×D4×C31—22no—
159C2×D4×C31—42no—
160C2×D4×C31—42no—
161C2×D4×C31—41no—
162C2×D4×C31—41no—
163C2×D4×C31C2×D412yes—
164C22×C4×C31—22no—
165C22×C4×C31—21no—
166C22×C4×C31—21no—
167C22×C4×C31—21no—
168C22×C4×C31—21no—
169C22×C4×C31—22no—
170C22×C4×C31—21no—
171C22×C4×C31—22no—
172C22×C4×C31—22no—
173C22×C4×C31—21no—
174C22×C4×C31—21no—
175C22×C4×C31—41no—
176C22×C4×C31—42no—
177C22×C4×C31—42no—
178C22×C4×C31—42no—
179C22×C4×C31—42no—
180C22×C4×C31—42no—
181C22×C4×C31C2×D411yescharacteristic
182C22×C4×C31C2×D412yes—
183C22×C4×C31C2×D411yescharacteristic
184C22×C4×C31C2×D411yescharacteristic
185C22×C4×C31C2×D411yescharacteristic
186C22×C4×C31C2×D411yescharacteristic
187C22×C4×C31C2×D412yes—
188C22×C4×C31C2411yescharacteristic
189Q16×C31—44no—
190SD16×C31—44no—
191SD16×C31—44no—
192D8×C31—44no—
193C2×C8×C31—22no—
194C2×C8×C31—22no—
195C2×C8×C31—44no—
196C4⋊C4×C31—21no—
197C4⋊C4×C31—22no—
198C4⋊C4×C31—21no—
199C4⋊C4×C31—21no—
200C4⋊C4×C31—21no—
201C4⋊C4×C31—22no—
202C4⋊C4×C31—22no—
203C4⋊C4×C31—22no—
204C4⋊C4×C31—22no—
205C4⋊C4×C31—22no—
206C4⋊C4×C31—22no—
207C4⋊C4×C31—22no—
208C4⋊C4×C31—22no—
209C4⋊C4×C31—22no—
210C4⋊C4×C31—42no—
211C4⋊C4×C31—42no—
212C4⋊C4×C31C2×D412yes—
213C4⋊C4×C31C2×D412yes—
214C22⋊C4×C31—22no—
215C22⋊C4×C31—22no—
216C22⋊C4×C31—22no—
217C22⋊C4×C31—22no—
218C22⋊C4×C31—22no—
219C22⋊C4×C31—22no—
220C22⋊C4×C31—22no—
221C22⋊C4×C31—22no—
222C22⋊C4×C31—22no—
223C22⋊C4×C31—42no—
224C22⋊C4×C31—42no—
225C22⋊C4×C31—42no—
226C22⋊C4×C31—42no—
227C22⋊C4×C31—42no—
228C42×C31—22no—
229C42×C31—22no—
230C42×C31—22no—
231#18193C3111yesmaximal, characteristic, direct product factor, Sylow
232C23×C31—21no—
233C23×C31—21no—
234C23×C31—22no—
235C23×C31—22no—
236C23×C31—22no—
237C23×C31—41no—
238C23×C31—41no—
239C23×C31—41no—
240C23×C31—41no—
241C23×C31—41no—
242C23×C31—41no—
243C23×C31—41no—
244C23×C31#12011yescharacteristic
245C23×C31#12011yescharacteristic
246C23×C31C22×D411yescharacteristic
247Q8×C31—24no—
248Q8×C31—42no—
249Q8×C31—42no—
250Q8×C31—42no—
251Q8×C31—81no—
252D4×C31—24no—
253D4×C31—42no—
254D4×C31—42no—
255D4×C31—42no—
256D4×C31—44no—
257D4×C31—44no—
258D4×C31—42no—
259D4×C31—42no—
260D4×C31—44no—
261D4×C31—44no—
262D4×C31—81no—
263C2×C4×C31—21no—
264C2×C4×C31—21no—
265C2×C4×C31—24no—
266C2×C4×C31—22no—
267C2×C4×C31—22no—
268C2×C4×C31—22no—
269C2×C4×C31—21no—
270C2×C4×C31—22no—
271C2×C4×C31—21no—
272C2×C4×C31—21no—
273C2×C4×C31—21no—
274C2×C4×C31—21no—
275C2×C4×C31—41no—
276C2×C4×C31—42no—
277C2×C4×C31—41no—
278C2×C4×C31—42no—
279C2×C4×C31—42no—
280C2×C4×C31—44no—
281C2×C4×C31—48no—
282C2×C4×C31—41no—
283C2×C4×C31—41no—
284C2×C4×C31—42no—
285C2×C4×C31—41no—
286C2×C4×C31—41no—
287C2×C4×C31—41no—
288C2×C4×C31—41no—
289C2×C4×C31—41no—
290C2×C4×C31—41no—
291C2×C4×C31—41no—
292C2×C4×C31—41no—
293C2×C4×C31—82no—
294C2×C4×C31—81no—
295C2×C4×C31—82no—
296C2×C4×C31#12012yes—
297C2×C4×C31#13512yescomplemented
298C2×C4×C31C22×D412yes—
299C8×C31—42no—
300C8×C31—42no—
301#3492C2×C3111yescharacteristic, complemented
302#3002C2×C3112yescomplemented
303#2406C2×C3111yescharacteristic, complemented
304#1605C2×C3111yescharacteristic, complemented
305#1604C2×C3111yescharacteristic, complemented
306#1603C2×C3112yescomplemented
307#1599C2×C3111yescharacteristic, complemented
308#1598C2×C3111yescharacteristic, complemented
309#1597C2×C3112yescomplemented
310#1516C2×C3112yescomplemented
311C2×#324C2×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—42no—
322C22×C31—42no—
323C22×C31—42no—
324C22×C31—42no—
325C22×C31—42no—
326C22×C31—42no—
327C22×C31—81no—
328C22×C31—81no—
329C22×C31—81no—
330C22×C31—81no—
331C22×C31B(4,𝔽2)12yes—
332C22×C31C2×#12011yescharacteristic
333C22×C31C2×#13511yescharacteristic
334C22×C31#57511yescharacteristic
335C4×C31—22no—
336C4×C31—22no—
337C4×C31—22no—
338C4×C31—41no—
339C4×C31—42no—
340C4×C31—42no—
341C4×C31—42no—
342C4×C31—41no—
343C4×C31—41no—
344C4×C31—42no—
345C4×C31—81no—
346C4×C31—81no—
347C22×#41C22×C3111yescharacteristic
348C2×#135—22no—
349#554—22no—
350#553—22no—
351#552—22no—
352#547—22no—
353#544C22×C3111yescharacteristic
354#539—21no—
355#539C22×C3112yes—
356#537C22×C3111yescharacteristic
357#525—21no—
358#525—21no—
359#525C22×C3112yes—
360C2×#123—21no—
361#520—21no—
362C22×C4⋊C4C22×C3111yescharacteristic
363#452—22no—
364#451C22×C3112yes—
365#450C22×C3112yes—
366#449—22no—
367#448C22×C3112yes—
368#447C22×C3112yes—
369#416—22no—
370C2×C22⋊C8C22×C3112yes—
371#395C22×C3111yescharacteristic
372#395C22×C3111yescharacteristic
373#394C22×C3111yescharacteristic
374#394C22×C3111yescharacteristic
375#386—21no—
376#385—21no—
377#385—21no—
378#381C22×C3111yescharacteristic
379#328C22×C3112yes—
380#328C22×C3112yes—
381#327C22×C3112yes—
382#327C22×C3112yes—
383#324C22×C3112yes—
384#324C22×C3111yescharacteristic
385#324C22×C3111yescharacteristic
386C2×C31—21no—
387C2×C31—21no—
388C2×C31—22no—
389C2×C31—41no—
390C2×C31—82no—
391C2×C31—82no—
392C2×C31#300211yescharacteristic
393C2×C31#302011yescharacteristic
394C2×C31C2×B(4,𝔽2)11yescharacteristic
395C2×#41—21no—
396C2×#41—21no—
397C2×#41—22no—
398C2×#41—22no—
399C2×#41—44no—
400C2×#41D4×C3112yes—
401C22×Q8C23×C3111yescharacteristic
402C22×D4—21no—
403C22×D4C23×C3111yescharacteristic
404C23×C4—21no—
405C23×C4D4×C3111yescharacteristic
406C23×C4D4×C3111yescharacteristic
407C23×C4C23×C3111yescharacteristic
408#135—48no—
409C2×Q16—22no—
410C2×SD16—22no—
411C2×SD16—22no—
412C2×D8—22no—
413C22×C8—22no—
414C4⋊Q8—22no—
415#126—22no—
416#126—22no—
417#125—22no—
418#125—22no—
419#125—22no—
420#123—22no—
421#123—22no—
422#123—22no—
423#123—22no—
424#123—22no—
425#123—44no—
426C22⋊Q8—21no—
427C22⋊Q8—21no—
428C22⋊Q8—21no—
429C22⋊Q8—22no—
430C22⋊Q8—22no—
431C22⋊Q8—21no—
432C22⋊Q8—22no—
433C22⋊Q8D4×C3112yes—
434C22⋊Q8C23×C3112yes—
435#121—21no—
436#121—21no—
437#121—21no—
438#121—22no—
439#121—21no—
440#121D4×C3112yes—
441#121C23×C3112yes—
442#120—22no—
443C4×Q8—22no—
444C4×D4—22no—
445C4×D4—22no—
446C4×D4—22no—
447C4×D4—22no—
448C4×D4—22no—
449C4×D4—22no—
450C4×D4—22no—
451#117—22no—
452#117—21no—
453#117—21no—
454#117D4×C3112yes—
455C2×C4⋊C4—22no—
456C2×C4⋊C4—22no—
457C2×C4⋊C4—22no—
458C2×C4⋊C4—21no—
459C2×C4⋊C4—21no—
460C2×C4⋊C4D4×C3111yescharacteristic
461C2×C4⋊C4D4×C3111yescharacteristic
462C2×C4⋊C4C23×C3111yescharacteristic
463C2×C4⋊C4C23×C3111yescharacteristic
464C2×C22⋊C4—21no—
465C2×C22⋊C4—21no—
466C2×C22⋊C4—21no—
467C2×C22⋊C4—21no—
468C2×C22⋊C4—21no—
469#103—22no—
470#103—22no—
471#102—22no—
472#102—22no—
473C22⋊C8—22no—
474C22⋊C8C23×C3112yes—
475C22⋊C8C23×C3112yes—
476#95—21no—
477#95—21no—
478#95—21no—
479#95C23×C3111yescharacteristic
480#95C23×C3111yescharacteristic
481C31#1819311yescharacteristic, direct product factor, Sylow
482C24—21no—
483C24C2×D4×C3111yescharacteristic
484#41—28no—
485#41—42no—
486#41—44no—
487#41—44no—
488#41—44no—
489#41—44no—
490#41—42no—
491#41—82no—
492C2×Q8—21no—
493C2×Q8—21no—
494C2×Q8—22no—
495C2×Q8—42no—
496C2×Q8C2×D4×C3112yes—
497C2×D4—21no—
498C2×D4—24no—
499C2×D4—22no—
500C2×D4—22no—
501C2×D4—22no—
502C2×D4—21no—
503C2×D4—22no—
504C2×D4—42no—
505C2×D4—42no—
506C2×D4—41no—
507C2×D4—41no—
508C2×D4C2×D4×C3112yes—
509C22×C4—22no—
510C22×C4—21no—
511C22×C4—21no—
512C22×C4—21no—
513C22×C4—21no—
514C22×C4—22no—
515C22×C4—21no—
516C22×C4—22no—
517C22×C4—22no—
518C22×C4—21no—
519C22×C4—21no—
520C22×C4—41no—
521C22×C4—42no—
522C22×C4—42no—
523C22×C4—42no—
524C22×C4—42no—
525C22×C4—42no—
526C22×C4C2×D4×C3111yescharacteristic
527C22×C4C2×D4×C3112yes—
528C22×C4C2×D4×C3111yescharacteristic
529C22×C4C2×D4×C3111yescharacteristic
530C22×C4C2×D4×C3111yescharacteristic
531C22×C4C2×D4×C3111yescharacteristic
532C22×C4C2×D4×C3112yes—
533C22×C4C24×C3111yescharacteristic
534Q16—44no—
535SD16—44no—
536SD16—44no—
537D8—44no—
538C2×C8—22no—
539C2×C8—22no—
540C2×C8—44no—
541C4⋊C4—21no—
542C4⋊C4—22no—
543C4⋊C4—21no—
544C4⋊C4—21no—
545C4⋊C4—21no—
546C4⋊C4—22no—
547C4⋊C4—22no—
548C4⋊C4—22no—
549C4⋊C4—22no—
550C4⋊C4—22no—
551C4⋊C4—22no—
552C4⋊C4—22no—
553C4⋊C4—22no—
554C4⋊C4—22no—
555C4⋊C4—42no—
556C4⋊C4—42no—
557C4⋊C4C2×D4×C3112yes—
558C4⋊C4C2×D4×C3112yes—
559C22⋊C4—22no—
560C22⋊C4—22no—
561C22⋊C4—22no—
562C22⋊C4—22no—
563C22⋊C4—22no—
564C22⋊C4—22no—
565C22⋊C4—22no—
566C22⋊C4—22no—
567C22⋊C4—22no—
568C22⋊C4—42no—
569C22⋊C4—42no—
570C22⋊C4—42no—
571C22⋊C4—42no—
572C22⋊C4—42no—
573C42—22no—
574C42—22no—
575C42—22no—
576C23—22no—
577C23—21no—
578C23—21no—
579C23—22no—
580C23—22no—
581C23—41no—
582C23—41no—
583C23—41no—
584C23—41no—
585C23—41no—
586C23—41no—
587C23—41no—
588C23C31×#12011yescharacteristic
589C23C31×#12011yescharacteristic
590C23C22×D4×C3111yescharacteristic
591Q8—24no—
592Q8—42no—
593Q8—42no—
594Q8—42no—
595Q8—81no—
596D4—24no—
597D4—42no—
598D4—42no—
599D4—42no—
600D4—44no—
601D4—44no—
602D4—44no—
603D4—44no—
604D4—42no—
605D4—42no—
606D4—81no—
607C2×C4—24no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—21no—
614C2×C4—22no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—48no—
626C2×C4—41no—
627C2×C4—42no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—82no—
638C2×C4—81no—
639C2×C4—82no—
640C2×C4C31×#12012yes—
641C2×C4C31×#13512yescomplemented
642C2×C4C22×D4×C3112yes—
643C8—42no—
644C8—42no—
645C22—21no—
646C22—21no—
647C22—21no—
648C22—21no—
649C22—21no—
650C22—21no—
651C22—21no—
652C22—42no—
653C22—42no—
654C22—42no—
655C22—42no—
656C22—42no—
657C22—42no—
658C22—42no—
659C22—42no—
660C22—81no—
661C22—81no—
662C22—81no—
663C22—81no—
664C22C31×B(4,𝔽2)12yes—
665C22C2×C31×#12011yescharacteristic
666C22C2×C31×#13511yescharacteristic
667C22C31×#57511yescharacteristic
668C4—22no—
669C4—22no—
670C4—22no—
671C4—41no—
672C4—42no—
673C4—42no—
674C4—42no—
675C4—41no—
676C4—41no—
677C4—42no—
678C4—81no—
679C4—81no—
680C2—21no—
681C2—21no—
682C2—22no—
683C2—41no—
684C2—82no—
685C2—82no—
686C2C31×#300211yescharacteristic
687C2C31×#302011yescharacteristic
688C2C2×C31×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
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×#3492C2semidirect
2. C31×#3002C2semidirect
3. C31×#2406C2semidirect
4. C31×#1605C2semidirect
5. C31×#1604C2semidirect
6. C31×#1603C2semidirect
7. C31×#1599C2semidirect
8. C31×#1598C2semidirect
9. C31×#1597C2semidirect
10. C31×#1516C2semidirect
11. C2×C31×#324C2semidirect
231. #18193C31direct (N × H)
297. C2×C4×C31#135semidirect
301. #3492C2×C31semidirect
302. #3002C2×C31semidirect
303. #2406C2×C31semidirect
304. #1605C2×C31semidirect
305. #1604C2×C31semidirect
306. #1603C2×C31semidirect
307. #1599C2×C31semidirect
308. #1598C2×C31semidirect
309. #1597C2×C31semidirect
310. #1516C2×C31semidirect
311. C2×#324C2×C31semidirect
481. C31#18193direct (N × H)
641. C2×C4C31×#135semidirect

Maximal subgroups

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

subgroupindex
C31×#34922
C31×#30022
C31×#24062
C31×#16052
C31×#16042
C31×#16032
C31×#15992
C31×#15982
C31×#15972
C31×#15162
C2×C31×#3242
#1819331