Finite Group Databasebeta

← C31×#18255 #21475014

Subgroups — C31×#18255 #21475014

Order 7936.

1260 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3537C211yesmaximal, characteristic
2C31×#3004C211yesmaximal, characteristic, complemented
3C2×C31×#448C211yesmaximal, characteristic, complemented
4C31×#2436C211yesmaximal, characteristic, complemented
5C31×#1606C214yesmaximal, complemented
6C31×#1601C214yesmaximal, complemented
7C31×#1507C211yesmaximal, characteristic, complemented
8C31×#1501C211yesmaximal, characteristic, complemented
9C31×#1471C211yesmaximal, characteristic, complemented
10C23×Q8×C31C2211yescharacteristic, complemented
11C2×C31×#137—21no—
12C22×SD16×C31—21no—
13C31×#560—22no—
14C31×#553—21no—
15C31×#553—21no—
16D42×C31—22no—
17D42×C31—21no—
18D42×C31—21no—
19C31×#540—21no—
20C31×#538—21no—
21C31×#538C2212yes—
22C31×#535—21no—
23C31×#534—21no—
24C31×#534C2212yes—
25C2×C31×#124—21no—
26C31×#525C2211yescharacteristic
27C2×C31×#121C2211yescharacteristic
28C2×C31×#121C2211yescharacteristic
29C31×#451—21no—
30C31×#451C2212yescomplemented
31C31×#448—22no—
32C31×#448—21no—
33C31×#448C2212yescomplemented
34C31×#448C2214yescomplemented
35C31×#419—21no—
36C2×C31×#103—21no—
37C31×#407C2211yescharacteristic, complemented
38C2×C31×C22⋊C8C2211yescharacteristic, complemented
39C31×#393C2214yescomplemented
40C31×#391—21no—
41C31×#386—22no—
42C4×C22⋊C4×C31C2211yescharacteristic, complemented
43C31×#328C2212yescomplemented
44C31×#327C2212yescomplemented
45C31×#327C2214yescomplemented
46C31×#323C2214yescomplemented
47C2×#41×C31D411yescharacteristic
48C22×Q8×C31—21no—
49C22×Q8×C31—21no—
50C22×Q8×C31—21no—
51C22×Q8×C31—21no—
52C22×Q8×C31—21no—
53C22×Q8×C31—44no—
54C22×Q8×C31C2312yes—
55C22×D4×C31—22no—
56C22×D4×C31—21no—
57C22×D4×C31—21no—
58C22×D4×C31—22no—
59C22×D4×C31D411yescharacteristic
60C22×D4×C31D412yes—
61C22×D4×C31D411yescharacteristic
62C23×C4×C31—21no—
63C23×C4×C31C2311yescharacteristic
64C31×#137—44no—
65C2×Q16×C31—22no—
66C2×SD16×C31—24no—
67C2×SD16×C31—22no—
68C2×SD16×C31—44no—
69C2×M16×C31—21no—
70C22×C8×C31—21no—
71C31×#127—22no—
72C31×#127—21no—
73C31×#126—22no—
74C31×#126—22no—
75C31×#125—21no—
76C31×#124—22no—
77C31×#124—44no—
78C31×#123—22no—
79C31×#123—21no—
80C31×#123—22no—
81C31×#123—22no—
82C31×#123—22no—
83C31×C22⋊Q8D411yescharacteristic
84C31×C22⋊Q8C2311yescharacteristic
85C31×#121—22no—
86C31×#121—21no—
87C31×#121—21no—
88C31×#121—22no—
89C31×#121—22no—
90C31×#121—22no—
91C31×#121—22no—
92C31×#121—21no—
93C31×#121—21no—
94C31×#121—21no—
95C31×#121—21no—
96C31×#121—21no—
97C31×#121—21no—
98C31×#121—22no—
99C31×#121D412yes—
100C31×#121D411yescharacteristic
101C31×#121C2312yes—
102C31×#121C2311yescharacteristic
103C31×#120—22no—
104C31×#120—22no—
105C31×#120—22no—
106C31×#120—21no—
107C31×#120—22no—
108C31×#120—22no—
109C31×#120—22no—
110C4×D4×C31—22no—
111C4×D4×C31—21no—
112C4×D4×C31—21no—
113C4×D4×C31—21no—
114C4×D4×C31—21no—
115C4×D4×C31—22no—
116C31×#117—21no—
117C31×#117D411yescharacteristic
118C2×C4⋊C4×C31D411yescharacteristic
119C2×C4⋊C4×C31D411yescharacteristic
120C2×C22⋊C4×C31—21no—
121C2×C22⋊C4×C31—24no—
122C2×C22⋊C4×C31—21no—
123C2×C22⋊C4×C31C2312yes—
124C2×C42×C31—21no—
125C31×#103—24no—
126C31×#103—22no—
127C31×#103—22no—
128C31×C22⋊C8—21no—
129C31×C22⋊C8—21no—
130C31×C22⋊C8C2312yes—
131C31×C22⋊C8C2312yes—
132C31×#95—24no—
133C31×#95C2312yes—
134C24×C31—22no—
135C24×C31—21no—
136C24×C31C2×D411yescharacteristic
137C24×C31C2×D412yes—
138#41×C31—42no—
139C2×Q8×C31—24no—
140C2×Q8×C31—24no—
141C2×Q8×C31—24no—
142C2×Q8×C31—22no—
143C2×Q8×C31—22no—
144C2×Q8×C31—44no—
145C2×Q8×C31—42no—
146C2×Q8×C31—42no—
147C2×Q8×C31—42no—
148C2×Q8×C31—44no—
149C2×Q8×C31—44no—
150C2×Q8×C31—41no—
151C2×Q8×C31—41no—
152C2×Q8×C31C2×D411yescharacteristic
153C2×D4×C31—22no—
154C2×D4×C31—21no—
155C2×D4×C31—21no—
156C2×D4×C31—22no—
157C2×D4×C31—22no—
158C2×D4×C31—22no—
159C2×D4×C31—22no—
160C2×D4×C31—22no—
161C2×D4×C31—22no—
162C2×D4×C31—21no—
163C2×D4×C31—22no—
164C2×D4×C31—22no—
165C2×D4×C31—21no—
166C2×D4×C31—21no—
167C2×D4×C31—21no—
168C2×D4×C31—42no—
169C2×D4×C31—42no—
170C2×D4×C31—44no—
171C2×D4×C31—82no—
172C2×D4×C31—82no—
173C2×D4×C31—84no—
174C2×D4×C31—84no—
175C2×D4×C31C2×D411yescharacteristic
176C2×D4×C31C2×D412yes—
177C22×C4×C31—22no—
178C22×C4×C31—22no—
179C22×C4×C31—22no—
180C22×C4×C31—21no—
181C22×C4×C31—21no—
182C22×C4×C31—22no—
183C22×C4×C31—42no—
184C22×C4×C31—44no—
185C22×C4×C31—41no—
186C22×C4×C31—41no—
187C22×C4×C31C2×D411yescharacteristic
188C22×C4×C31C2×D411yescharacteristic
189C22×C4×C31C2×D411yescharacteristic
190C22×C4×C31C2×D412yes—
191C22×C4×C31C2×D411yescharacteristic
192C22×C4×C31C2×D411yescharacteristic
193C22×C4×C31C2411yescharacteristic
194Q16×C31—44no—
195SD16×C31—48no—
196SD16×C31—44no—
197M16×C31—42no—
198C2×C8×C31—22no—
199C2×C8×C31—22no—
200C2×C8×C31—42no—
201C4⋊C4×C31—22no—
202C4⋊C4×C31—22no—
203C4⋊C4×C31—21no—
204C4⋊C4×C31—21no—
205C4⋊C4×C31—21no—
206C4⋊C4×C31—21no—
207C4⋊C4×C31—21no—
208C4⋊C4×C31C2×D412yes—
209C4⋊C4×C31C2×D411yescharacteristic
210C4⋊C4×C31C2×D411yescharacteristic
211C22⋊C4×C31—22no—
212C22⋊C4×C31—22no—
213C22⋊C4×C31—22no—
214C22⋊C4×C31—22no—
215C22⋊C4×C31—22no—
216C22⋊C4×C31—21no—
217C22⋊C4×C31—21no—
218C22⋊C4×C31—21no—
219C22⋊C4×C31—22no—
220C22⋊C4×C31—21no—
221C22⋊C4×C31—22no—
222C22⋊C4×C31—22no—
223C22⋊C4×C31—22no—
224C22⋊C4×C31—21no—
225C22⋊C4×C31—48no—
226C22⋊C4×C31—42no—
227C42×C31—22no—
228C42×C31—21no—
229C42×C31—42no—
230#18255C3111yesmaximal, characteristic, direct product factor, Sylow
231C23×C31—22no—
232C23×C31—22no—
233C23×C31—22no—
234C23×C31—22no—
235C23×C31—21no—
236C23×C31—24no—
237C23×C31—41no—
238C23×C31—42no—
239C23×C31—42no—
240C23×C31—42no—
241C23×C31—81no—
242C23×C31—81no—
243C23×C31—82no—
244C23×C31—82no—
245C23×C31—82no—
246C23×C31#12011yescharacteristic
247C23×C31#12011yescharacteristic
248C23×C31C22×D411yescharacteristic
249Q8×C31—24no—
250Q8×C31—24no—
251Q8×C31—41no—
252Q8×C31—42no—
253Q8×C31—42no—
254Q8×C31—44no—
255Q8×C31—44no—
256D4×C31—42no—
257D4×C31—41no—
258D4×C31—44no—
259D4×C31—42no—
260D4×C31—44no—
261D4×C31—42no—
262D4×C31—42no—
263D4×C31—42no—
264D4×C31—81no—
265D4×C31—82no—
266D4×C31—82no—
267D4×C31—81no—
268D4×C31—82no—
269D4×C31—82no—
270D4×C31—82no—
271D4×C31—82no—
272D4×C31—82no—
273C2×C4×C31—22no—
274C2×C4×C31—22no—
275C2×C4×C31—22no—
276C2×C4×C31—24no—
277C2×C4×C31—21no—
278C2×C4×C31—22no—
279C2×C4×C31—22no—
280C2×C4×C31—21no—
281C2×C4×C31—21no—
282C2×C4×C31—41no—
283C2×C4×C31—41no—
284C2×C4×C31—41no—
285C2×C4×C31—42no—
286C2×C4×C31—44no—
287C2×C4×C31—44no—
288C2×C4×C31—42no—
289C2×C4×C31—42no—
290C2×C4×C31—42no—
291C2×C4×C31—42no—
292C2×C4×C31—42no—
293C2×C4×C31—42no—
294C2×C4×C31—81no—
295C2×C4×C31—82no—
296C2×C4×C31—82no—
297C2×C4×C31#12011yescharacteristic
298C2×C4×C31#12011yescharacteristic
299C2×C4×C31C22×D411yescharacteristic
300C2×C4×C31C22×D411yescharacteristic
301C8×C31—42no—
302C8×C31—42no—
303#3537C2×C3111yescharacteristic
304#3004C2×C3111yescharacteristic, complemented
305C2×#448C2×C3111yescharacteristic, complemented
306#2436C2×C3111yescharacteristic, complemented
307#1606C2×C3114yescomplemented
308#1601C2×C3114yescomplemented
309#1507C2×C3111yescharacteristic, complemented
310#1501C2×C3111yescharacteristic, complemented
311#1471C2×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—44no—
320C22×C31—42no—
321C22×C31—42no—
322C22×C31—41no—
323C22×C31—42no—
324C22×C31—42no—
325C22×C31—41no—
326C22×C31—41no—
327C22×C31—42no—
328C22×C31—42no—
329C22×C31—84no—
330C22×C31—162no—
331C22×C31—164no—
332C22×C31#45512yes—
333C22×C31C2×#12011yescharacteristic
334C22×C31#57712yes—
335C4×C31—21no—
336C4×C31—21no—
337C4×C31—21no—
338C4×C31—42no—
339C4×C31—41no—
340C4×C31—42no—
341C4×C31—41no—
342C4×C31—44no—
343C4×C31—41no—
344C4×C31—82no—
345C4×C31B(4,𝔽2)12yes—
346C23×Q8C22×C3111yescharacteristic, complemented
347C2×#137—21no—
348C22×SD16—21no—
349#560—22no—
350#553—21no—
351#553—21no—
352D42—22no—
353D42—21no—
354D42—21no—
355#540—21no—
356#538—21no—
357#538C22×C3112yes—
358#535—21no—
359#534—21no—
360#534C22×C3112yes—
361C2×#124—21no—
362#525C22×C3111yescharacteristic
363C2×#121C22×C3111yescharacteristic
364C2×#121C22×C3111yescharacteristic
365#451—21no—
366#451C22×C3112yescomplemented
367#448—22no—
368#448—21no—
369#448C22×C3112yescomplemented
370#448C22×C3114yescomplemented
371#419—21no—
372C2×#103—21no—
373#407C22×C3111yescharacteristic, complemented
374C2×C22⋊C8C22×C3111yescharacteristic, complemented
375#393C22×C3114yescomplemented
376#391—21no—
377#386—22no—
378C4×C22⋊C4C22×C3111yescharacteristic, complemented
379#328C22×C3112yescomplemented
380#327C22×C3112yescomplemented
381#327C22×C3114yescomplemented
382#323C22×C3114yescomplemented
383C2×C31—22no—
384C2×C31—22no—
385C2×C31—41no—
386C2×C31—82no—
387C2×C31—82no—
388C2×C31—81no—
389C2×C31#300911yescharacteristic
390C2×C31C2×#45511yescharacteristic
391C2×C31C2×B(4,𝔽2)11yescharacteristic
392C2×#41D4×C3111yescharacteristic
393C22×Q8—21no—
394C22×Q8—21no—
395C22×Q8—21no—
396C22×Q8—21no—
397C22×Q8—21no—
398C22×Q8—44no—
399C22×Q8C23×C3112yes—
400C22×D4—22no—
401C22×D4—21no—
402C22×D4—21no—
403C22×D4—22no—
404C22×D4D4×C3111yescharacteristic
405C22×D4D4×C3112yes—
406C22×D4D4×C3111yescharacteristic
407C23×C4—21no—
408C23×C4C23×C3111yescharacteristic
409#137—44no—
410C2×Q16—22no—
411C2×SD16—24no—
412C2×SD16—22no—
413C2×SD16—44no—
414C2×M16—21no—
415C22×C8—21no—
416#127—22no—
417#127—21no—
418#126—22no—
419#126—22no—
420#125—21no—
421#124—22no—
422#124—44no—
423#123—22no—
424#123—21no—
425#123—22no—
426#123—22no—
427#123—22no—
428C22⋊Q8D4×C3111yescharacteristic
429C22⋊Q8C23×C3111yescharacteristic
430#121—22no—
431#121—21no—
432#121—21no—
433#121—22no—
434#121—22no—
435#121—22no—
436#121—22no—
437#121—21no—
438#121—21no—
439#121—21no—
440#121—21no—
441#121—21no—
442#121—21no—
443#121—22no—
444#121D4×C3112yes—
445#121D4×C3111yescharacteristic
446#121C23×C3112yes—
447#121C23×C3111yescharacteristic
448#120—22no—
449#120—22no—
450#120—22no—
451#120—21no—
452#120—22no—
453#120—22no—
454#120—22no—
455C4×D4—22no—
456C4×D4—21no—
457C4×D4—21no—
458C4×D4—21no—
459C4×D4—21no—
460C4×D4—22no—
461#117—21no—
462#117D4×C3111yescharacteristic
463C2×C4⋊C4D4×C3111yescharacteristic
464C2×C4⋊C4D4×C3111yescharacteristic
465C2×C22⋊C4—21no—
466C2×C22⋊C4—24no—
467C2×C22⋊C4—21no—
468C2×C22⋊C4C23×C3112yes—
469C2×C42—21no—
470#103—24no—
471#103—22no—
472#103—22no—
473C22⋊C8—21no—
474C22⋊C8—21no—
475C22⋊C8C23×C3112yes—
476C22⋊C8C23×C3112yes—
477#95—24no—
478#95C23×C3112yes—
479C31#1825511yescharacteristic, direct product factor, Sylow
480C24—22no—
481C24—21no—
482C24C2×D4×C3111yescharacteristic
483C24C2×D4×C3112yes—
484#41—42no—
485C2×Q8—24no—
486C2×Q8—24no—
487C2×Q8—24no—
488C2×Q8—22no—
489C2×Q8—22no—
490C2×Q8—44no—
491C2×Q8—42no—
492C2×Q8—42no—
493C2×Q8—42no—
494C2×Q8—44no—
495C2×Q8—44no—
496C2×Q8—41no—
497C2×Q8—41no—
498C2×Q8C2×D4×C3111yescharacteristic
499C2×D4—22no—
500C2×D4—21no—
501C2×D4—21no—
502C2×D4—22no—
503C2×D4—22no—
504C2×D4—22no—
505C2×D4—22no—
506C2×D4—22no—
507C2×D4—22no—
508C2×D4—21no—
509C2×D4—22no—
510C2×D4—22no—
511C2×D4—21no—
512C2×D4—21no—
513C2×D4—21no—
514C2×D4—42no—
515C2×D4—42no—
516C2×D4—44no—
517C2×D4—82no—
518C2×D4—82no—
519C2×D4—84no—
520C2×D4—84no—
521C2×D4C2×D4×C3111yescharacteristic
522C2×D4C2×D4×C3112yes—
523C22×C4—22no—
524C22×C4—22no—
525C22×C4—22no—
526C22×C4—21no—
527C22×C4—21no—
528C22×C4—22no—
529C22×C4—42no—
530C22×C4—44no—
531C22×C4—41no—
532C22×C4—41no—
533C22×C4C2×D4×C3111yescharacteristic
534C22×C4C2×D4×C3111yescharacteristic
535C22×C4C2×D4×C3111yescharacteristic
536C22×C4C2×D4×C3112yes—
537C22×C4C2×D4×C3111yescharacteristic
538C22×C4C2×D4×C3111yescharacteristic
539C22×C4C24×C3111yescharacteristic
540Q16—44no—
541SD16—48no—
542SD16—44no—
543M16—42no—
544C2×C8—22no—
545C2×C8—22no—
546C2×C8—42no—
547C4⋊C4—22no—
548C4⋊C4—22no—
549C4⋊C4—21no—
550C4⋊C4—21no—
551C4⋊C4—21no—
552C4⋊C4—21no—
553C4⋊C4—21no—
554C4⋊C4C2×D4×C3112yes—
555C4⋊C4C2×D4×C3111yescharacteristic
556C4⋊C4C2×D4×C3111yescharacteristic
557C22⋊C4—22no—
558C22⋊C4—22no—
559C22⋊C4—22no—
560C22⋊C4—22no—
561C22⋊C4—22no—
562C22⋊C4—21no—
563C22⋊C4—21no—
564C22⋊C4—21no—
565C22⋊C4—22no—
566C22⋊C4—21no—
567C22⋊C4—22no—
568C22⋊C4—22no—
569C22⋊C4—22no—
570C22⋊C4—21no—
571C22⋊C4—48no—
572C22⋊C4—42no—
573C42—22no—
574C42—21no—
575C42—42no—
576C23—22no—
577C23—22no—
578C23—22no—
579C23—22no—
580C23—21no—
581C23—24no—
582C23—41no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23—81no—
587C23—81no—
588C23—82no—
589C23—82no—
590C23—82no—
591C23C31×#12011yescharacteristic
592C23C31×#12011yescharacteristic
593C23C22×D4×C3111yescharacteristic
594Q8—24no—
595Q8—24no—
596Q8—41no—
597Q8—42no—
598Q8—42no—
599Q8—44no—
600Q8—44no—
601D4—42no—
602D4—41no—
603D4—44no—
604D4—42no—
605D4—44no—
606D4—42no—
607D4—42no—
608D4—42no—
609D4—81no—
610D4—82no—
611D4—82no—
612D4—81no—
613D4—82no—
614D4—82no—
615D4—82no—
616D4—82no—
617D4—82no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—24no—
622C2×C4—21no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—81no—
640C2×C4—82no—
641C2×C4—82no—
642C2×C4C31×#12011yescharacteristic
643C2×C4C31×#12011yescharacteristic
644C2×C4C22×D4×C3111yescharacteristic
645C2×C4C22×D4×C3111yescharacteristic
646C8—42no—
647C8—42no—
648C22—21no—
649C22—21no—
650C22—21no—
651C22—21no—
652C22—21no—
653C22—21no—
654C22—21no—
655C22—44no—
656C22—42no—
657C22—42no—
658C22—41no—
659C22—42no—
660C22—42no—
661C22—41no—
662C22—41no—
663C22—42no—
664C22—42no—
665C22—84no—
666C22—162no—
667C22—164no—
668C22C31×#45512yes—
669C22C2×C31×#12011yescharacteristic
670C22C31×#57712yes—
671C4—21no—
672C4—21no—
673C4—21no—
674C4—42no—
675C4—41no—
676C4—42no—
677C4—41no—
678C4—44no—
679C4—41no—
680C4—82no—
681C4C31×B(4,𝔽2)12yes—
682C2—22no—
683C2—22no—
684C2—41no—
685C2—82no—
686C2—82no—
687C2—81no—
688C2C31×#300911yescharacteristic
689C2C2×C31×#45511yescharacteristic
690C2C2×C31×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#3537C2
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
2. C31×#3004C2semidirect
3. C2×C31×#448C2semidirect
4. C31×#2436C2semidirect
5. C31×#1606C2semidirect
6. C31×#1601C2semidirect
7. C31×#1507C2semidirect
8. C31×#1501C2semidirect
9. C31×#1471C2semidirect
10. C23×Q8×C31C22semidirect
30. C31×#451C22semidirect
33. C31×#448C22semidirect
34. C31×#448C22semidirect
37. C31×#407C22semidirect
38. C2×C31×C22⋊C8C22semidirect
39. C31×#393C22semidirect
42. C4×C22⋊C4×C31C22semidirect
43. C31×#328C22semidirect
44. C31×#327C22semidirect
45. C31×#327C22semidirect
46. C31×#323C22semidirect
230. #18255C31direct (N × H)
304. #3004C2×C31semidirect
305. C2×#448C2×C31semidirect
306. #2436C2×C31semidirect
307. #1606C2×C31semidirect
308. #1601C2×C31semidirect
309. #1507C2×C31semidirect
310. #1501C2×C31semidirect
311. #1471C2×C31semidirect
346. C23×Q8C22×C31semidirect
366. #451C22×C31semidirect
369. #448C22×C31semidirect
370. #448C22×C31semidirect
373. #407C22×C31semidirect
374. C2×C22⋊C8C22×C31semidirect
375. #393C22×C31semidirect
378. C4×C22⋊C4C22×C31semidirect
379. #328C22×C31semidirect
380. #327C22×C31semidirect
381. #327C22×C31semidirect
382. #323C22×C31semidirect
479. C31#18255direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35372
C31×#30042
C2×C31×#4482
C31×#24362
C31×#16062
C31×#16012
C31×#15072
C31×#15012
C31×#14712
#1825531