Finite Group Databasebeta

← C31×#18426 #21475185

Subgroups — C31×#18426 #21475185

Order 7936.

1168 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3558C211yesmaximal, characteristic, complemented
2C31×#3217C211yesmaximal, characteristic, complemented
3C31×#3205C211yesmaximal, characteristic, complemented
4C31×#2657C211yesmaximal, characteristic, complemented
5C31×#1600C214yesmaximal, complemented
6C31×#1595C214yesmaximal, complemented
7C31×#1505C211yesmaximal, characteristic, complemented
8C31×#1500C211yesmaximal, characteristic, complemented
9C31×#1469C211yesmaximal, characteristic, complemented
10C31×#562—21no—
11C31×H(3,𝔽4)—21no—
12C31×#559—21no—
13C31×#548—21no—
14C31×#546—21no—
15C31×#546—22no—
16D42×C31—21no—
17C31×#542—21no—
18C31×#538—22no—
19C31×#538C2212yes—
20C31×#535—21no—
21C31×#534C2212yes—
22C31×#532C2211yescharacteristic
23C2×C31×#127—21no—
24C2×C31×#127C2211yescharacteristic, complemented
25C31×#529C2211yescharacteristic
26C2×C31×#124—21no—
27C2×C31×#124C2211yescharacteristic
28C31×#496—21no—
29C31×#493—21no—
30C31×#490—21no—
31C31×#486—21no—
32C31×#460—21no—
33C31×#459—22no—
34C31×#459—21no—
35C31×#450C2212yescomplemented
36C31×#447C2212yes—
37C31×#447C2214yescomplemented
38C31×#432C2211yescharacteristic, complemented
39C31×#431C2211yescharacteristic, complemented
40C31×#392—22no—
41C31×#392C2214yescomplemented
42C31×#376C2211yescharacteristic, complemented
43C31×#328C2212yescomplemented
44C31×#327C2212yes—
45C31×#327C2214yescomplemented
46C31×#323C2214yescomplemented
47C2×#41×C31—21no—
48C2×#41×C31—21no—
49C22×Q8×C31D411yescharacteristic
50C22×D4×C31—21no—
51C22×D4×C31—22no—
52C22×D4×C31—24no—
53C22×D4×C31—21no—
54C22×D4×C31—21no—
55C22×D4×C31D411yescharacteristic
56C22×D4×C31C2312yes—
57C2×SD16×C31—22no—
58C2×D8×C31—22no—
59C2×D8×C31—24no—
60C31×C4⋊Q8D411yescharacteristic, complemented
61C31×#127—21no—
62C31×#127—21no—
63C31×#127—21no—
64C31×#127—21no—
65C31×#127—21no—
66C31×#127—44no—
67C31×#127D411yescharacteristic, complemented
68C31×#126—22no—
69C31×#126—21no—
70C31×#125D411yescharacteristic, complemented
71C31×#124—21no—
72C31×#124—22no—
73C31×#124—22no—
74C31×#124—22no—
75C31×#124—21no—
76C31×#124—21no—
77C31×#124—21no—
78C31×#124—21no—
79C31×#124—21no—
80C31×#124—44no—
81C31×#124D411yescharacteristic, complemented
82C31×#123—22no—
83C31×#123—21no—
84C31×#123—22no—
85C31×#123—22no—
86C31×#123—22no—
87C31×C22⋊Q8—22no—
88C31×C22⋊Q8—21no—
89C31×C22⋊Q8—22no—
90C31×C22⋊Q8C2311yescharacteristic
91C31×#121—22no—
92C31×#121—22no—
93C31×#121—22no—
94C31×#121—21no—
95C31×#121—22no—
96C31×#121—21no—
97C31×#121—22no—
98C31×#121C2311yescharacteristic
99C31×#121C2312yes—
100C31×#120—22no—
101C31×#120—22no—
102C31×#120—22no—
103C31×#120—22no—
104C31×#120—22no—
105C4×Q8×C31—21no—
106C4×Q8×C31D411yescharacteristic, complemented
107C4×D4×C31—22no—
108C4×D4×C31—21no—
109C4×D4×C31—21no—
110C4×D4×C31—22no—
111C4×D4×C31—21no—
112C4×D4×C31D411yescharacteristic, complemented
113C4×D4×C31D412yescomplemented
114C31×#117—21no—
115C2×C22⋊C4×C31—22no—
116C2×C22⋊C4×C31—24no—
117C2×C22⋊C4×C31D412yes—
118C2×C22⋊C4×C31C2312yes—
119C2×C42×C31—21no—
120C2×C42×C31C2311yescharacteristic
121C31×C4⋊C8—21no—
122C31×C4⋊C8—21no—
123C31×#102—22no—
124C31×#102—24no—
125C31×#102—22no—
126C31×C22⋊C8C2312yes—
127C31×C22⋊C8C2312yes—
128C31×#97—21no—
129C4×C8×C31—21no—
130C31×#95—21no—
131C31×#95C2312yes—
132C24×C31—22no—
133C24×C31—22no—
134C24×C31C2×D412yes—
135#41×C31—82no—
136#41×C31—82no—
137C2×Q8×C31—21no—
138C2×Q8×C31—21no—
139C2×Q8×C31—21no—
140C2×Q8×C31—42no—
141C2×Q8×C31C2×D411yescharacteristic
142C2×D4×C31—22no—
143C2×D4×C31—21no—
144C2×D4×C31—22no—
145C2×D4×C31—22no—
146C2×D4×C31—22no—
147C2×D4×C31—22no—
148C2×D4×C31—22no—
149C2×D4×C31—21no—
150C2×D4×C31—21no—
151C2×D4×C31—21no—
152C2×D4×C31—21no—
153C2×D4×C31—22no—
154C2×D4×C31—22no—
155C2×D4×C31—42no—
156C2×D4×C31—42no—
157C2×D4×C31—42no—
158C2×D4×C31—48no—
159C2×D4×C31—41no—
160C2×D4×C31—41no—
161C2×D4×C31—41no—
162C2×D4×C31—41no—
163C2×D4×C31—82no—
164C2×D4×C31—82no—
165C2×D4×C31—82no—
166C2×D4×C31—84no—
167C2×D4×C31—84no—
168C2×D4×C31—84no—
169C2×D4×C31C2×D411yescharacteristic
170C2×D4×C31C2×D412yes—
171C22×C4×C31—22no—
172C22×C4×C31—22no—
173C22×C4×C31—22no—
174C22×C4×C31—21no—
175C22×C4×C31—22no—
176C22×C4×C31C2×D411yescharacteristic
177C22×C4×C31C2×D411yescharacteristic
178C22×C4×C31C2×D412yes—
179C22×C4×C31C2411yescharacteristic
180SD16×C31—82no—
181D8×C31—84no—
182D8×C31—82no—
183C2×C8×C31—22no—
184C2×C8×C31—22no—
185C4⋊C4×C31—22no—
186C4⋊C4×C31—22no—
187C4⋊C4×C31—21no—
188C4⋊C4×C31—21no—
189C4⋊C4×C31—21no—
190C4⋊C4×C31—21no—
191C4⋊C4×C31—21no—
192C4⋊C4×C31C2×D412yes—
193C4⋊C4×C31C2×D411yescharacteristic
194C4⋊C4×C31C2×D411yescharacteristic
195C22⋊C4×C31—22no—
196C22⋊C4×C31—22no—
197C22⋊C4×C31—22no—
198C22⋊C4×C31—22no—
199C22⋊C4×C31—22no—
200C22⋊C4×C31—21no—
201C22⋊C4×C31—21no—
202C22⋊C4×C31—22no—
203C22⋊C4×C31—22no—
204C22⋊C4×C31—21no—
205C22⋊C4×C31—21no—
206C22⋊C4×C31—22no—
207C22⋊C4×C31—22no—
208C22⋊C4×C31—22no—
209C22⋊C4×C31—21no—
210C22⋊C4×C31—22no—
211C22⋊C4×C31—48no—
212C42×C31—24no—
213C42×C31—22no—
214C42×C31—21no—
215C42×C31C2×D411yescharacteristic
216C42×C31C2×D411yescharacteristic
217C42×C31C2×D411yescharacteristic
218C42×C31C2×D411yescharacteristic
219#18426C3111yesmaximal, characteristic, direct product factor, Sylow
220C23×C31—22no—
221C23×C31—21no—
222C23×C31—22no—
223C23×C31—24no—
224C23×C31—42no—
225C23×C31—42no—
226C23×C31—42no—
227C23×C31—42no—
228C23×C31—44no—
229C23×C31—42no—
230C23×C31—44no—
231C23×C31—82no—
232C23×C31—82no—
233C23×C31—82no—
234C23×C31C22×D411yescharacteristic
235Q8×C31—41no—
236Q8×C31—42no—
237Q8×C31—81no—
238Q8×C31—81no—
239D4×C31—42no—
240D4×C31—41no—
241D4×C31—44no—
242D4×C31—42no—
243D4×C31—42no—
244D4×C31—42no—
245D4×C31—44no—
246D4×C31—42no—
247D4×C31—42no—
248D4×C31—81no—
249D4×C31—81no—
250D4×C31—82no—
251D4×C31—82no—
252D4×C31—82no—
253D4×C31—82no—
254D4×C31—82no—
255D4×C31—82no—
256D4×C31—82no—
257D4×C31—82no—
258D4×C31—816no—
259D4×C31—82no—
260C2×C4×C31—22no—
261C2×C4×C31—22no—
262C2×C4×C31—24no—
263C2×C4×C31—21no—
264C2×C4×C31—22no—
265C2×C4×C31—22no—
266C2×C4×C31—21no—
267C2×C4×C31—21no—
268C2×C4×C31—41no—
269C2×C4×C31—42no—
270C2×C4×C31—42no—
271C2×C4×C31—42no—
272C2×C4×C31—42no—
273C2×C4×C31—82no—
274C2×C4×C31—82no—
275C2×C4×C31—82no—
276C2×C4×C31#12011yescharacteristic
277C2×C4×C31#12011yescharacteristic
278C2×C4×C31#12011yescharacteristic
279C2×C4×C31#12011yescharacteristic
280C2×C4×C31C22×D411yescharacteristic
281C2×C4×C31C22×D411yescharacteristic
282C8×C31—42no—
283C8×C31—42no—
284#3558C2×C3111yescharacteristic, complemented
285#3217C2×C3111yescharacteristic, complemented
286#3205C2×C3111yescharacteristic, complemented
287#2657C2×C3111yescharacteristic, complemented
288#1600C2×C3114yescomplemented
289#1595C2×C3114yescomplemented
290#1505C2×C3111yescharacteristic, complemented
291#1500C2×C3111yescharacteristic, complemented
292#1469C2×C3111yescharacteristic, complemented
293C22×C31—21no—
294C22×C31—21no—
295C22×C31—42no—
296C22×C31—42no—
297C22×C31—41no—
298C22×C31—41no—
299C22×C31—42no—
300C22×C31—42no—
301C22×C31—42no—
302C22×C31—41no—
303C22×C31—42no—
304C22×C31—84no—
305C22×C31—84no—
306C22×C31—84no—
307C22×C31—82no—
308C22×C31—82no—
309C22×C31—82no—
310C22×C31—164no—
311C22×C31C2×#12011yescharacteristic
312C22×C31#57612yes—
313C4×C31—21no—
314C4×C31—21no—
315C4×C31—21no—
316C4×C31—21no—
317C4×C31—21no—
318C4×C31—44no—
319C4×C31—41no—
320C4×C31—42no—
321C4×C31—41no—
322C4×C31—42no—
323C4×C31—41no—
324C4×C31#45312yes—
325#562—21no—
326H(3,𝔽4)—21no—
327#559—21no—
328#548—21no—
329#546—21no—
330#546—22no—
331D42—21no—
332#542—21no—
333#538—22no—
334#538C22×C3112yes—
335#535—21no—
336#534C22×C3112yes—
337#532C22×C3111yescharacteristic
338C2×#127—21no—
339C2×#127C22×C3111yescharacteristic, complemented
340#529C22×C3111yescharacteristic
341C2×#124—21no—
342C2×#124C22×C3111yescharacteristic
343#496—21no—
344#493—21no—
345#490—21no—
346#486—21no—
347#460—21no—
348#459—22no—
349#459—21no—
350#450C22×C3112yescomplemented
351#447C22×C3112yes—
352#447C22×C3114yescomplemented
353#432C22×C3111yescharacteristic, complemented
354#431C22×C3111yescharacteristic, complemented
355#392—22no—
356#392C22×C3114yescomplemented
357#376C22×C3111yescharacteristic, complemented
358#328C22×C3112yescomplemented
359#327C22×C3112yes—
360#327C22×C3114yescomplemented
361#323C22×C3114yescomplemented
362C2×C31—22no—
363C2×C31—82no—
364C2×C31—82no—
365C2×C31—81no—
366C2×C31—162no—
367C2×C31#300811yescharacteristic
368C2×C31C2×#45311yescharacteristic
369C2×C31#302711yescharacteristic
370C2×#41—21no—
371C2×#41—21no—
372C22×Q8D4×C3111yescharacteristic
373C22×D4—21no—
374C22×D4—22no—
375C22×D4—24no—
376C22×D4—21no—
377C22×D4—21no—
378C22×D4D4×C3111yescharacteristic
379C22×D4C23×C3112yes—
380C2×SD16—22no—
381C2×D8—22no—
382C2×D8—24no—
383C4⋊Q8D4×C3111yescharacteristic, complemented
384#127—21no—
385#127—21no—
386#127—21no—
387#127—21no—
388#127—21no—
389#127—44no—
390#127D4×C3111yescharacteristic, complemented
391#126—22no—
392#126—21no—
393#125D4×C3111yescharacteristic, complemented
394#124—21no—
395#124—22no—
396#124—22no—
397#124—22no—
398#124—21no—
399#124—21no—
400#124—21no—
401#124—21no—
402#124—21no—
403#124—44no—
404#124D4×C3111yescharacteristic, complemented
405#123—22no—
406#123—21no—
407#123—22no—
408#123—22no—
409#123—22no—
410C22⋊Q8—22no—
411C22⋊Q8—21no—
412C22⋊Q8—22no—
413C22⋊Q8C23×C3111yescharacteristic
414#121—22no—
415#121—22no—
416#121—22no—
417#121—21no—
418#121—22no—
419#121—21no—
420#121—22no—
421#121C23×C3111yescharacteristic
422#121C23×C3112yes—
423#120—22no—
424#120—22no—
425#120—22no—
426#120—22no—
427#120—22no—
428C4×Q8—21no—
429C4×Q8D4×C3111yescharacteristic, complemented
430C4×D4—22no—
431C4×D4—21no—
432C4×D4—21no—
433C4×D4—22no—
434C4×D4—21no—
435C4×D4D4×C3111yescharacteristic, complemented
436C4×D4D4×C3112yescomplemented
437#117—21no—
438C2×C22⋊C4—22no—
439C2×C22⋊C4—24no—
440C2×C22⋊C4D4×C3112yes—
441C2×C22⋊C4C23×C3112yes—
442C2×C42—21no—
443C2×C42C23×C3111yescharacteristic
444C4⋊C8—21no—
445C4⋊C8—21no—
446#102—22no—
447#102—24no—
448#102—22no—
449C22⋊C8C23×C3112yes—
450C22⋊C8C23×C3112yes—
451#97—21no—
452C4×C8—21no—
453#95—21no—
454#95C23×C3112yes—
455C31#1842611yescharacteristic, direct product factor, Sylow
456C24—22no—
457C24—22no—
458C24C2×D4×C3112yes—
459#41—82no—
460#41—82no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—42no—
465C2×Q8C2×D4×C3111yescharacteristic
466C2×D4—22no—
467C2×D4—21no—
468C2×D4—22no—
469C2×D4—22no—
470C2×D4—22no—
471C2×D4—22no—
472C2×D4—22no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—22no—
478C2×D4—22no—
479C2×D4—42no—
480C2×D4—42no—
481C2×D4—42no—
482C2×D4—48no—
483C2×D4—41no—
484C2×D4—41no—
485C2×D4—41no—
486C2×D4—41no—
487C2×D4—82no—
488C2×D4—82no—
489C2×D4—82no—
490C2×D4—84no—
491C2×D4—84no—
492C2×D4—84no—
493C2×D4C2×D4×C3111yescharacteristic
494C2×D4C2×D4×C3112yes—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—21no—
499C22×C4—22no—
500C22×C4C2×D4×C3111yescharacteristic
501C22×C4C2×D4×C3111yescharacteristic
502C22×C4C2×D4×C3112yes—
503C22×C4C24×C3111yescharacteristic
504SD16—82no—
505D8—84no—
506D8—82no—
507C2×C8—22no—
508C2×C8—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4C2×D4×C3112yes—
517C4⋊C4C2×D4×C3111yescharacteristic
518C4⋊C4C2×D4×C3111yescharacteristic
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—22no—
522C22⋊C4—22no—
523C22⋊C4—22no—
524C22⋊C4—21no—
525C22⋊C4—21no—
526C22⋊C4—22no—
527C22⋊C4—22no—
528C22⋊C4—21no—
529C22⋊C4—21no—
530C22⋊C4—22no—
531C22⋊C4—22no—
532C22⋊C4—22no—
533C22⋊C4—21no—
534C22⋊C4—22no—
535C22⋊C4—48no—
536C42—24no—
537C42—22no—
538C42—21no—
539C42C2×D4×C3111yescharacteristic
540C42C2×D4×C3111yescharacteristic
541C42C2×D4×C3111yescharacteristic
542C42C2×D4×C3111yescharacteristic
543C23—22no—
544C23—21no—
545C23—22no—
546C23—24no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—44no—
552C23—42no—
553C23—44no—
554C23—82no—
555C23—82no—
556C23—82no—
557C23C22×D4×C3111yescharacteristic
558Q8—41no—
559Q8—42no—
560Q8—81no—
561Q8—81no—
562D4—42no—
563D4—41no—
564D4—44no—
565D4—42no—
566D4—42no—
567D4—42no—
568D4—44no—
569D4—42no—
570D4—42no—
571D4—81no—
572D4—81no—
573D4—82no—
574D4—82no—
575D4—82no—
576D4—82no—
577D4—82no—
578D4—82no—
579D4—82no—
580D4—82no—
581D4—816no—
582D4—82no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—82no—
597C2×C4—82no—
598C2×C4—82no—
599C2×C4C31×#12011yescharacteristic
600C2×C4C31×#12011yescharacteristic
601C2×C4C31×#12011yescharacteristic
602C2×C4C31×#12011yescharacteristic
603C2×C4C22×D4×C3111yescharacteristic
604C2×C4C22×D4×C3111yescharacteristic
605C8—42no—
606C8—42no—
607C22—21no—
608C22—21no—
609C22—42no—
610C22—42no—
611C22—41no—
612C22—41no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—41no—
617C22—42no—
618C22—84no—
619C22—84no—
620C22—84no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22—164no—
625C22C2×C31×#12011yescharacteristic
626C22C31×#57612yes—
627C4—21no—
628C4—21no—
629C4—21no—
630C4—21no—
631C4—21no—
632C4—44no—
633C4—41no—
634C4—42no—
635C4—41no—
636C4—42no—
637C4—41no—
638C4C31×#45312yes—
639C2—22no—
640C2—82no—
641C2—81no—
642C2—82no—
643C2—162no—
644C2C31×#300811yescharacteristic
645C2C2×C31×#45311yescharacteristic
646C2C31×#302711yescharacteristic
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×#3558C2semidirect
2. C31×#3217C2semidirect
3. C31×#3205C2semidirect
4. C31×#2657C2semidirect
5. C31×#1600C2semidirect
6. C31×#1595C2semidirect
7. C31×#1505C2semidirect
8. C31×#1500C2semidirect
9. C31×#1469C2semidirect
24. C2×C31×#127C22semidirect
35. C31×#450C22semidirect
37. C31×#447C22semidirect
38. C31×#432C22semidirect
39. C31×#431C22semidirect
41. C31×#392C22semidirect
42. C31×#376C22semidirect
43. C31×#328C22semidirect
45. C31×#327C22semidirect
46. C31×#323C22semidirect
60. C31×C4⋊Q8D4semidirect
67. C31×#127D4semidirect
70. C31×#125D4semidirect
81. C31×#124D4semidirect
106. C4×Q8×C31D4semidirect
112. C4×D4×C31D4semidirect
113. C4×D4×C31D4semidirect
219. #18426C31direct (N × H)
284. #3558C2×C31semidirect
285. #3217C2×C31semidirect
286. #3205C2×C31semidirect
287. #2657C2×C31semidirect
288. #1600C2×C31semidirect
289. #1595C2×C31semidirect
290. #1505C2×C31semidirect
291. #1500C2×C31semidirect
292. #1469C2×C31semidirect
339. C2×#127C22×C31semidirect
350. #450C22×C31semidirect
352. #447C22×C31semidirect
353. #432C22×C31semidirect
354. #431C22×C31semidirect
356. #392C22×C31semidirect
357. #376C22×C31semidirect
358. #328C22×C31semidirect
360. #327C22×C31semidirect
361. #323C22×C31semidirect
383. C4⋊Q8D4×C31semidirect
390. #127D4×C31semidirect
393. #125D4×C31semidirect
404. #124D4×C31semidirect
429. C4×Q8D4×C31semidirect
435. C4×D4D4×C31semidirect
436. C4×D4D4×C31semidirect
455. C31#18426direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35582
C31×#32172
C31×#32052
C31×#26572
C31×#16002
C31×#15952
C31×#15052
C31×#15002
C31×#14692
#1842631