Finite Group Databasebeta

← C31×#18164 #21474923

Subgroups — C31×#18164 #21474923

Order 7936.

1386 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3492C211yesmaximal, characteristic, complemented
2C31×#3003C212yesmaximal, complemented
3C2×C31×#394C211yesmaximal, characteristic, complemented
4C31×#1599C212yesmaximal, complemented
5C31×#1598C212yesmaximal, complemented
6C31×#1597C214yesmaximal, complemented
7C31×#1504C212yesmaximal, complemented
8C31×#1463C211yesmaximal, characteristic, complemented
9C23×D4×C31C2211yescharacteristic
10C2×C31×AGL(1,ℤ/8)—22no—
11C31×#554—22no—
12C31×#553—22no—
13C31×#552—22no—
14C31×#547—22no—
15C31×#544C2211yescharacteristic, complemented
16C31×#539—21no—
17C31×#539C2212yescomplemented
18C31×#537C2211yescharacteristic, complemented
19C31×#525—21no—
20C31×#525—21no—
21C31×#525C2212yes—
22C31×#520—21no—
23C22×C4⋊C4×C31C2211yescharacteristic
24C22×C22⋊C4×C31—21no—
25C22×C22⋊C4×C31—21no—
26C31×#450—22no—
27C31×#450C2214yes—
28C31×#447—22no—
29C31×#447C2214yes—
30C31×#418—22no—
31C31×#407C2212yes—
32C31×#394—21no—
33C31×#394—21no—
34C31×#394C2212yes—
35C31×#394C2212yes—
36C2×C31×#95C2211yescharacteristic
37C31×#328C2214yescomplemented
38C31×#327C2214yescomplemented
39C31×#324C2212yescomplemented
40C31×#324C2212yescomplemented
41C25×C31—21no—
42C2×#41×C31—21no—
43C2×#41×C31—21no—
44C2×#41×C31D412yes—
45C22×D4×C31—22no—
46C22×D4×C31—22no—
47C22×D4×C31—21no—
48C22×D4×C31—44no—
49C22×D4×C31C2312yes—
50C23×C4×C31—21no—
51C23×C4×C31D411yescharacteristic
52C23×C4×C31D411yescharacteristic
53C23×C4×C31C2311yescharacteristic
54C31×AGL(1,ℤ/8)—48no—
55C2×SD16×C31—24no—
56C2×D8×C31—24no—
57C2×M16×C31—22no—
58C31×C4⋊Q8—22no—
59C31×#126—22no—
60C31×#126—22no—
61C31×#125—22no—
62C31×#125—22no—
63C31×#125—22no—
64C31×#123—22no—
65C31×#123—22no—
66C31×#123—22no—
67C31×#123—22no—
68C31×#123—22no—
69C31×C22⋊Q8—21no—
70C31×C22⋊Q8—21no—
71C31×C22⋊Q8—21no—
72C31×C22⋊Q8—22no—
73C31×C22⋊Q8—22no—
74C31×C22⋊Q8—21no—
75C31×C22⋊Q8—22no—
76C31×C22⋊Q8D412yes—
77C31×C22⋊Q8C2312yes—
78C31×#121—21no—
79C31×#121—21no—
80C31×#121—21no—
81C31×#121—22no—
82C31×#121—21no—
83C31×#121D412yes—
84C31×#121C2312yes—
85C31×#120—22no—
86C4×Q8×C31—22no—
87C4×D4×C31—22no—
88C4×D4×C31—22no—
89C4×D4×C31—22no—
90C4×D4×C31—22no—
91C4×D4×C31—22no—
92C4×D4×C31—22no—
93C4×D4×C31—22no—
94C31×#117—22no—
95C31×#117—21no—
96C31×#117—21no—
97C31×#117D412yes—
98C2×C4⋊C4×C31—22no—
99C2×C4⋊C4×C31—22no—
100C2×C4⋊C4×C31—22no—
101C2×C4⋊C4×C31D411yescharacteristic
102C2×C4⋊C4×C31D411yescharacteristic
103C2×C4⋊C4×C31C2311yescharacteristic
104C2×C4⋊C4×C31C2311yescharacteristic
105C2×C22⋊C4×C31—22no—
106C2×C22⋊C4×C31—22no—
107C2×C22⋊C4×C31—22no—
108C2×C22⋊C4×C31—22no—
109C2×C22⋊C4×C31—44no—
110C2×C22⋊C4×C31—44no—
111C31×#102—24no—
112C31×#102—24no—
113C31×C22⋊C8—22no—
114C31×C22⋊C8C2314yes—
115C31×#95—21no—
116C31×#95C2312yes—
117C24×C31—24no—
118C24×C31—22no—
119C24×C31—42no—
120C24×C31—42no—
121C24×C31—42no—
122C24×C31—42no—
123C24×C31—42no—
124C24×C31—41no—
125C24×C31—41no—
126C24×C31C2×D411yescharacteristic
127#41×C31—42no—
128#41×C31—44no—
129#41×C31—42no—
130#41×C31—82no—
131C2×Q8×C31—21no—
132C2×Q8×C31—21no—
133C2×Q8×C31C2×D412yes—
134C2×D4×C31—28no—
135C2×D4×C31—21no—
136C2×D4×C31—22no—
137C2×D4×C31—22no—
138C2×D4×C31—22no—
139C2×D4×C31—21no—
140C2×D4×C31—24no—
141C2×D4×C31—44no—
142C2×D4×C31—44no—
143C2×D4×C31—48no—
144C2×D4×C31—44no—
145C2×D4×C31—42no—
146C2×D4×C31C2×D412yes—
147C22×C4×C31—22no—
148C22×C4×C31—21no—
149C22×C4×C31—21no—
150C22×C4×C31—21no—
151C22×C4×C31—22no—
152C22×C4×C31—21no—
153C22×C4×C31—22no—
154C22×C4×C31—22no—
155C22×C4×C31—22no—
156C22×C4×C31—41no—
157C22×C4×C31—42no—
158C22×C4×C31—42no—
159C22×C4×C31—42no—
160C22×C4×C31—42no—
161C22×C4×C31—42no—
162C22×C4×C31C2×D411yescharacteristic
163C22×C4×C31C2×D412yes—
164C22×C4×C31C2×D411yescharacteristic
165C22×C4×C31C2×D411yescharacteristic
166C22×C4×C31C2×D411yescharacteristic
167C22×C4×C31C2×D411yescharacteristic
168C22×C4×C31C2×D412yes—
169C22×C4×C31C2411yescharacteristic
170SD16×C31—48no—
171D8×C31—48no—
172M16×C31—44no—
173C2×C8×C31—24no—
174C4⋊C4×C31—21no—
175C4⋊C4×C31—22no—
176C4⋊C4×C31—21no—
177C4⋊C4×C31—21no—
178C4⋊C4×C31—21no—
179C4⋊C4×C31—22no—
180C4⋊C4×C31—22no—
181C4⋊C4×C31—22no—
182C4⋊C4×C31—22no—
183C4⋊C4×C31—22no—
184C4⋊C4×C31—22no—
185C4⋊C4×C31—22no—
186C4⋊C4×C31—22no—
187C4⋊C4×C31—22no—
188C4⋊C4×C31C2×D412yes—
189C4⋊C4×C31C2×D412yes—
190C22⋊C4×C31—22no—
191C22⋊C4×C31—22no—
192C22⋊C4×C31—22no—
193C22⋊C4×C31—22no—
194C22⋊C4×C31—22no—
195C22⋊C4×C31—22no—
196C22⋊C4×C31—22no—
197C22⋊C4×C31—22no—
198C22⋊C4×C31—22no—
199C22⋊C4×C31—44no—
200C22⋊C4×C31—44no—
201C22⋊C4×C31—44no—
202C22⋊C4×C31—44no—
203C42×C31—22no—
204C42×C31—22no—
205C42×C31—22no—
206#18164C3111yesmaximal, characteristic, direct product factor, Sylow
207C23×C31—21no—
208C23×C31—21no—
209C23×C31—22no—
210C23×C31—22no—
211C23×C31—24no—
212C23×C31—22no—
213C23×C31—42no—
214C23×C31—44no—
215C23×C31—48no—
216C23×C31—48no—
217C23×C31—42no—
218C23×C31—42no—
219C23×C31—42no—
220C23×C31—42no—
221C23×C31—42no—
222C23×C31—44no—
223C23×C31—84no—
224C23×C31—84no—
225C23×C31—82no—
226C23×C31—82no—
227C23×C31—82no—
228C23×C31—82no—
229C23×C31#12011yescharacteristic
230C23×C31#12011yescharacteristic
231C23×C31C22×D411yescharacteristic
232Q8×C31—42no—
233Q8×C31—42no—
234Q8×C31—81no—
235D4×C31—28no—
236D4×C31—42no—
237D4×C31—44no—
238D4×C31—48no—
239D4×C31—42no—
240D4×C31—42no—
241D4×C31—42no—
242D4×C31—44no—
243D4×C31—44no—
244D4×C31—81no—
245C2×C4×C31—21no—
246C2×C4×C31—21no—
247C2×C4×C31—24no—
248C2×C4×C31—22no—
249C2×C4×C31—22no—
250C2×C4×C31—21no—
251C2×C4×C31—22no—
252C2×C4×C31—21no—
253C2×C4×C31—21no—
254C2×C4×C31—21no—
255C2×C4×C31—21no—
256C2×C4×C31—42no—
257C2×C4×C31—41no—
258C2×C4×C31—41no—
259C2×C4×C31—42no—
260C2×C4×C31—42no—
261C2×C4×C31—44no—
262C2×C4×C31—42no—
263C2×C4×C31—41no—
264C2×C4×C31—41no—
265C2×C4×C31—42no—
266C2×C4×C31—42no—
267C2×C4×C31—42no—
268C2×C4×C31—82no—
269C2×C4×C31—81no—
270C2×C4×C31—82no—
271C2×C4×C31—84no—
272C2×C4×C31#12012yes—
273C2×C4×C31AGL(1,ℤ/8)12yescomplemented
274C2×C4×C31C22×D412yes—
275C8×C31—44no—
276#3492C2×C3111yescharacteristic, complemented
277#3003C2×C3112yescomplemented
278C2×#394C2×C3111yescharacteristic, complemented
279#1599C2×C3112yescomplemented
280#1598C2×C3112yescomplemented
281#1597C2×C3114yescomplemented
282#1504C2×C3112yescomplemented
283#1463C2×C3111yescharacteristic, complemented
284C22×C31—24no—
285C22×C31—24no—
286C22×C31—21no—
287C22×C31—21no—
288C22×C31—21no—
289C22×C31—21no—
290C22×C31—21no—
291C22×C31—21no—
292C22×C31—44no—
293C22×C31—44no—
294C22×C31—42no—
295C22×C31—42no—
296C22×C31—42no—
297C22×C31—44no—
298C22×C31—82no—
299C22×C31—82no—
300C22×C31—88no—
301C22×C31—88no—
302C22×C31—82no—
303C22×C31—82no—
304C22×C31#45412yes—
305C22×C31B(4,𝔽2)12yes—
306C22×C31C2×#12011yescharacteristic
307C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
308C22×C31#57511yescharacteristic
309C4×C31—22no—
310C4×C31—22no—
311C4×C31—22no—
312C4×C31—41no—
313C4×C31—42no—
314C4×C31—42no—
315C4×C31—41no—
316C4×C31—41no—
317C4×C31—42no—
318C4×C31—82no—
319C23×D4C22×C3111yescharacteristic
320C2×AGL(1,ℤ/8)—22no—
321#554—22no—
322#553—22no—
323#552—22no—
324#547—22no—
325#544C22×C3111yescharacteristic, complemented
326#539—21no—
327#539C22×C3112yescomplemented
328#537C22×C3111yescharacteristic, complemented
329#525—21no—
330#525—21no—
331#525C22×C3112yes—
332#520—21no—
333C22×C4⋊C4C22×C3111yescharacteristic
334C22×C22⋊C4—21no—
335C22×C22⋊C4—21no—
336#450—22no—
337#450C22×C3114yes—
338#447—22no—
339#447C22×C3114yes—
340#418—22no—
341#407C22×C3112yes—
342#394—21no—
343#394—21no—
344#394C22×C3112yes—
345#394C22×C3112yes—
346C2×#95C22×C3111yescharacteristic
347#328C22×C3114yescomplemented
348#327C22×C3114yescomplemented
349#324C22×C3112yescomplemented
350#324C22×C3112yescomplemented
351C2×C31—21no—
352C2×C31—21no—
353C2×C31—22no—
354C2×C31—22no—
355C2×C31—82no—
356C2×C31—84no—
357C2×C31#300611yescharacteristic
358C2×C31C2×#45411yescharacteristic
359C2×C31C2×B(4,𝔽2)11yescharacteristic
360C25—21no—
361C2×#41—21no—
362C2×#41—21no—
363C2×#41D4×C3112yes—
364C22×D4—22no—
365C22×D4—22no—
366C22×D4—21no—
367C22×D4—44no—
368C22×D4C23×C3112yes—
369C23×C4—21no—
370C23×C4D4×C3111yescharacteristic
371C23×C4D4×C3111yescharacteristic
372C23×C4C23×C3111yescharacteristic
373AGL(1,ℤ/8)—48no—
374C2×SD16—24no—
375C2×D8—24no—
376C2×M16—22no—
377C4⋊Q8—22no—
378#126—22no—
379#126—22no—
380#125—22no—
381#125—22no—
382#125—22no—
383#123—22no—
384#123—22no—
385#123—22no—
386#123—22no—
387#123—22no—
388C22⋊Q8—21no—
389C22⋊Q8—21no—
390C22⋊Q8—21no—
391C22⋊Q8—22no—
392C22⋊Q8—22no—
393C22⋊Q8—21no—
394C22⋊Q8—22no—
395C22⋊Q8D4×C3112yes—
396C22⋊Q8C23×C3112yes—
397#121—21no—
398#121—21no—
399#121—21no—
400#121—22no—
401#121—21no—
402#121D4×C3112yes—
403#121C23×C3112yes—
404#120—22no—
405C4×Q8—22no—
406C4×D4—22no—
407C4×D4—22no—
408C4×D4—22no—
409C4×D4—22no—
410C4×D4—22no—
411C4×D4—22no—
412C4×D4—22no—
413#117—22no—
414#117—21no—
415#117—21no—
416#117D4×C3112yes—
417C2×C4⋊C4—22no—
418C2×C4⋊C4—22no—
419C2×C4⋊C4—22no—
420C2×C4⋊C4D4×C3111yescharacteristic
421C2×C4⋊C4D4×C3111yescharacteristic
422C2×C4⋊C4C23×C3111yescharacteristic
423C2×C4⋊C4C23×C3111yescharacteristic
424C2×C22⋊C4—22no—
425C2×C22⋊C4—22no—
426C2×C22⋊C4—22no—
427C2×C22⋊C4—22no—
428C2×C22⋊C4—44no—
429C2×C22⋊C4—44no—
430#102—24no—
431#102—24no—
432C22⋊C8—22no—
433C22⋊C8C23×C3114yes—
434#95—21no—
435#95C23×C3112yes—
436C31#1816411yescharacteristic, direct product factor, Sylow
437C24—24no—
438C24—22no—
439C24—42no—
440C24—42no—
441C24—42no—
442C24—42no—
443C24—42no—
444C24—41no—
445C24—41no—
446C24C2×D4×C3111yescharacteristic
447#41—42no—
448#41—44no—
449#41—42no—
450#41—82no—
451C2×Q8—21no—
452C2×Q8—21no—
453C2×Q8C2×D4×C3112yes—
454C2×D4—28no—
455C2×D4—21no—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—21no—
460C2×D4—24no—
461C2×D4—44no—
462C2×D4—44no—
463C2×D4—48no—
464C2×D4—44no—
465C2×D4—42no—
466C2×D4C2×D4×C3112yes—
467C22×C4—22no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—22no—
472C22×C4—21no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—41no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4C2×D4×C3111yescharacteristic
483C22×C4C2×D4×C3112yes—
484C22×C4C2×D4×C3111yescharacteristic
485C22×C4C2×D4×C3111yescharacteristic
486C22×C4C2×D4×C3111yescharacteristic
487C22×C4C2×D4×C3111yescharacteristic
488C22×C4C2×D4×C3112yes—
489C22×C4C24×C3111yescharacteristic
490SD16—48no—
491D8—48no—
492M16—44no—
493C2×C8—24no—
494C4⋊C4—21no—
495C4⋊C4—22no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4C2×D4×C3112yes—
509C4⋊C4C2×D4×C3112yes—
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—22no—
513C22⋊C4—22no—
514C22⋊C4—22no—
515C22⋊C4—22no—
516C22⋊C4—22no—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—44no—
520C22⋊C4—44no—
521C22⋊C4—44no—
522C22⋊C4—44no—
523C42—22no—
524C42—22no—
525C42—22no—
526C23—22no—
527C23—22no—
528C23—21no—
529C23—21no—
530C23—24no—
531C23—22no—
532C23—42no—
533C23—44no—
534C23—48no—
535C23—48no—
536C23—42no—
537C23—42no—
538C23—44no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—84no—
543C23—84no—
544C23—82no—
545C23—82no—
546C23—82no—
547C23—82no—
548C23C31×#12011yescharacteristic
549C23C31×#12011yescharacteristic
550C23C22×D4×C3111yescharacteristic
551Q8—42no—
552Q8—42no—
553Q8—81no—
554D4—28no—
555D4—42no—
556D4—44no—
557D4—48no—
558D4—42no—
559D4—44no—
560D4—44no—
561D4—42no—
562D4—42no—
563D4—81no—
564C2×C4—24no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—82no—
588C2×C4—81no—
589C2×C4—82no—
590C2×C4—84no—
591C2×C4C31×#12012yes—
592C2×C4C31×AGL(1,ℤ/8)12yescomplemented
593C2×C4C22×D4×C3112yes—
594C8—44no—
595C22—24no—
596C22—24no—
597C22—21no—
598C22—21no—
599C22—21no—
600C22—21no—
601C22—21no—
602C22—21no—
603C22—44no—
604C22—44no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—44no—
609C22—82no—
610C22—82no—
611C22—88no—
612C22—88no—
613C22—82no—
614C22—82no—
615C22C31×#45412yes—
616C22C31×B(4,𝔽2)12yes—
617C22C2×C31×#12011yescharacteristic
618C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
619C22C31×#57511yescharacteristic
620C4—22no—
621C4—22no—
622C4—22no—
623C4—41no—
624C4—42no—
625C4—42no—
626C4—41no—
627C4—41no—
628C4—42no—
629C4—82no—
630C2—21no—
631C2—21no—
632C2—22no—
633C2—22no—
634C2—84no—
635C2—82no—
636C2C31×#300611yescharacteristic
637C2C2×C31×#45411yescharacteristic
638C2C2×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×#3003C2semidirect
3. C2×C31×#394C2semidirect
4. C31×#1599C2semidirect
5. C31×#1598C2semidirect
6. C31×#1597C2semidirect
7. C31×#1504C2semidirect
8. C31×#1463C2semidirect
15. C31×#544C22semidirect
17. C31×#539C22semidirect
18. C31×#537C22semidirect
37. C31×#328C22semidirect
38. C31×#327C22semidirect
39. C31×#324C22semidirect
40. C31×#324C22semidirect
206. #18164C31direct (N × H)
273. C2×C4×C31AGL(1,ℤ/8)semidirect
276. #3492C2×C31semidirect
277. #3003C2×C31semidirect
278. C2×#394C2×C31semidirect
279. #1599C2×C31semidirect
280. #1598C2×C31semidirect
281. #1597C2×C31semidirect
282. #1504C2×C31semidirect
283. #1463C2×C31semidirect
325. #544C22×C31semidirect
327. #539C22×C31semidirect
328. #537C22×C31semidirect
347. #328C22×C31semidirect
348. #327C22×C31semidirect
349. #324C22×C31semidirect
350. #324C22×C31semidirect
436. C31#18164direct (N × H)
592. C2×C4C31×AGL(1,ℤ/8)semidirect

Maximal subgroups

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

subgroupindex
C31×#34922
C31×#30032
C2×C31×#3942
C31×#15992
C31×#15982
C31×#15972
C31×#15042
C31×#14632
#1816431