Finite Group Databasebeta

← C31×#24117 #21480876

Subgroups — C31×#24117 #21480876

Order 7936.

1238 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#555C211yesmaximal, characteristic, complemented
2C2×C31×#476C211yesmaximal, characteristic, complemented
3C2×C4×D8×C31C211yesmaximal, characteristic, complemented
4C31×#2537C211yesmaximal, characteristic, complemented
5C31×#2034C212yesmaximal, complemented
6C31×#1999C212yesmaximal, complemented
7C31×#1990C211yesmaximal, characteristic, complemented
8C31×#1954C211yesmaximal, characteristic, complemented
9C31×#1875C212yesmaximal, complemented
10C31×#1854C212yesmaximal, complemented
11C31×#1802C211yesmaximal, characteristic, complemented
12C22×D8×C31C2211yescharacteristic
13C31×#555—24no—
14C31×#555—24no—
15C2×C31×C4⋊Q8C2211yescharacteristic
16C2×C31×#124—22no—
17C2×C31×#121—21no—
18C2×C31×#121—21no—
19C2×C31×#121C2212yes—
20C2×C31×#121C2212yes—
21C2×C4×D4×C31C2211yescharacteristic
22C2×C4×D4×C31C2211yescharacteristic
23C2×C4×D4×C31C2211yescharacteristic
24C2×C31×#117—21no—
25C2×C31×#117C2212yes—
26C31×#476—24no—
27C4×D8×C31—24no—
28C2×C31×#107C2211yescharacteristic
29C2×C31×#106C2212yes—
30C2×C31×C4⋊C8C2211yescharacteristic
31C2×C31×#102C2211yescharacteristic
32C2×C31×#102C2212yes—
33C2×C31×#102C2211yescharacteristic
34C2×C31×#102C2212yes—
35C2×C31×C22⋊C8C2212yes—
36C2×C31×C22⋊C8C2212yes—
37C2×C4×C8×C31C2211yescharacteristic
38C31×#394—22no—
39C31×#384C2211yescharacteristic
40C31×#380—21no—
41C31×#380C2212yes—
42C31×#340C2212yes—
43C31×#340C2212yes—
44C31×#340C2212yes—
45C22×Q8×C31—21no—
46C22×D4×C31—22no—
47C22×D4×C31—22no—
48C22×D4×C31D411yescharacteristic
49C22×D4×C31C2311yescharacteristic
50C22×D4×C31C2311yescharacteristic
51C23×C4×C31—21no—
52C23×C4×C31—21no—
53C23×C4×C31C2312yes—
54C2×D8×C31—22no—
55C2×D8×C31—22no—
56C2×D8×C31—22no—
57C22×C8×C31C2311yescharacteristic
58C22×C8×C31C2312yes—
59C22×C8×C31C2311yescharacteristic
60C31×C4⋊Q8—22no—
61C31×C4⋊Q8D414yescomplemented
62C31×#124—216no—
63C31×#121—28no—
64C31×#121—24no—
65C31×#121—24no—
66C31×#121—24no—
67C31×#121—42no—
68C31×#121—44no—
69C4×D4×C31—22no—
70C4×D4×C31—22no—
71C4×D4×C31—22no—
72C4×D4×C31—22no—
73C4×D4×C31—22no—
74C4×D4×C31D414yescomplemented
75C31×#117—28no—
76C31×#117—24no—
77C31×#117—24no—
78C2×C4⋊C4×C31—21no—
79C2×C4⋊C4×C31—21no—
80C2×C4⋊C4×C31—21no—
81C2×C4⋊C4×C31D411yescharacteristic
82C2×C4⋊C4×C31C2311yescharacteristic
83C2×C4⋊C4×C31C2312yes—
84C2×C4⋊C4×C31C2312yes—
85C2×C4⋊C4×C31C2311yescharacteristic
86C2×C22⋊C4×C31—22no—
87C2×C22⋊C4×C31—22no—
88C2×C22⋊C4×C31—21no—
89C2×C22⋊C4×C31—22no—
90C2×C22⋊C4×C31—21no—
91C2×C22⋊C4×C31—22no—
92C2×C22⋊C4×C31D412yes—
93C2×C42×C31—22no—
94C2×C42×C31C2311yescharacteristic
95C31×#107—22no—
96C31×#106—24no—
97C31×C4⋊C8—22no—
98C31×#102—22no—
99C31×#102—22no—
100C31×#102—24no—
101C31×#102—42no—
102C31×C22⋊C8—24no—
103C31×C22⋊C8—42no—
104C4×C8×C31—22no—
105C31×#95—22no—
106C24×C31—21no—
107C24×C31—21no—
108C24×C31C2×D412yes—
109C2×Q8×C31—24no—
110C2×Q8×C31—81no—
111C2×Q8×C31—81no—
112C2×D4×C31—28no—
113C2×D4×C31—22no—
114C2×D4×C31—22no—
115C2×D4×C31—22no—
116C2×D4×C31—21no—
117C2×D4×C31—21no—
118C2×D4×C31—21no—
119C2×D4×C31—21no—
120C2×D4×C31—21no—
121C2×D4×C31—21no—
122C2×D4×C31—21no—
123C2×D4×C31—21no—
124C2×D4×C31—21no—
125C2×D4×C31—21no—
126C2×D4×C31—44no—
127C2×D4×C31—44no—
128C2×D4×C31—82no—
129C2×D4×C31—82no—
130C2×D4×C31—82no—
131C2×D4×C31C2×D412yes—
132C2×D4×C31#4112yes—
133C22×C4×C31—21no—
134C22×C4×C31—22no—
135C22×C4×C31—24no—
136C22×C4×C31—22no—
137C22×C4×C31—22no—
138C22×C4×C31—22no—
139C22×C4×C31—22no—
140C22×C4×C31—22no—
141C22×C4×C31—22no—
142C22×C4×C31—21no—
143C22×C4×C31—22no—
144C22×C4×C31—42no—
145C22×C4×C31—42no—
146C22×C4×C31—42no—
147C22×C4×C31—42no—
148C22×C4×C31—42no—
149C22×C4×C31C2×D412yes—
150C22×C4×C31C2×D411yescharacteristic
151C22×C4×C31C2×D411yescharacteristic
152C22×C4×C31C2411yescharacteristic
153D8×C31—28no—
154C2×C8×C31—21no—
155C2×C8×C31—21no—
156C2×C8×C31—21no—
157C2×C8×C31—22no—
158C2×C8×C31—22no—
159C2×C8×C31—21no—
160C2×C8×C31—22no—
161C2×C8×C31—21no—
162C2×C8×C31#4112yes—
163C4⋊C4×C31—24no—
164C4⋊C4×C31—22no—
165C4⋊C4×C31—21no—
166C4⋊C4×C31—22no—
167C4⋊C4×C31—21no—
168C4⋊C4×C31—21no—
169C4⋊C4×C31—22no—
170C4⋊C4×C31—22no—
171C4⋊C4×C31—42no—
172C4⋊C4×C31—81no—
173C4⋊C4×C31C2×D414yes—
174C4⋊C4×C31C2×D412yes—
175C22⋊C4×C31—28no—
176C22⋊C4×C31—28no—
177C22⋊C4×C31—24no—
178C22⋊C4×C31—24no—
179C22⋊C4×C31—44no—
180C22⋊C4×C31—44no—
181C22⋊C4×C31—42no—
182C42×C31—28no—
183C42×C31C2×D412yes—
184C42×C31C2×D412yes—
185#24117C3111yesmaximal, characteristic, direct product factor, Sylow
186C23×C31—22no—
187C23×C31—22no—
188C23×C31—22no—
189C23×C31—22no—
190C23×C31—22no—
191C23×C31—22no—
192C23×C31—22no—
193C23×C31—22no—
194C23×C31—22no—
195C23×C31—41no—
196C23×C31—41no—
197C23×C31—41no—
198C23×C31—41no—
199C23×C31—41no—
200C23×C31—41no—
201C23×C31—41no—
202C23×C31—41no—
203C23×C31—41no—
204C23×C31—41no—
205C23×C31—41no—
206C23×C31—41no—
207C23×C31C22×D411yescharacteristic
208Q8×C31—84no—
209D4×C31—22no—
210D4×C31—22no—
211D4×C31—22no—
212D4×C31—22no—
213D4×C31—41no—
214D4×C31—41no—
215D4×C31—41no—
216D4×C31—41no—
217D4×C31—44no—
218D4×C31—88no—
219D4×C31—88no—
220C2×C4×C31—21no—
221C2×C4×C31—21no—
222C2×C4×C31—22no—
223C2×C4×C31—22no—
224C2×C4×C31—24no—
225C2×C4×C31—22no—
226C2×C4×C31—22no—
227C2×C4×C31—41no—
228C2×C4×C31—41no—
229C2×C4×C31—42no—
230C2×C4×C31—42no—
231C2×C4×C31—42no—
232C2×C4×C31—42no—
233C2×C4×C31—42no—
234C2×C4×C31—42no—
235C2×C4×C31—48no—
236C2×C4×C31—41no—
237C2×C4×C31—42no—
238C2×C4×C31—41no—
239C2×C4×C31—41no—
240C2×C4×C31—42no—
241C2×C4×C31—42no—
242C2×C4×C31—84no—
243C2×C4×C31#12012yes—
244C2×C4×C31#12012yes—
245C2×C4×C31#13512yes—
246C2×C4×C31AGL(1,ℤ/8)12yes—
247C2×C4×C31C22×D411yescharacteristic
248C2×C4×C31C22×D411yescharacteristic
249C2×C4×C31C2×#4111yescharacteristic
250C2×C4×C31C2×#4111yescharacteristic
251C2×C4×C31H(5,𝔽2)11yescharacteristic
252C2×C4×C312−1+411yescharacteristic
253C8×C31—22no—
254C8×C31—41no—
255C8×C31—42no—
256C2×#555C2×C3111yescharacteristic, complemented
257C2×#476C2×C3111yescharacteristic, complemented
258C2×C4×D8C2×C3111yescharacteristic, complemented
259#2537C2×C3111yescharacteristic, complemented
260#2034C2×C3112yescomplemented
261#1999C2×C3112yescomplemented
262#1990C2×C3111yescharacteristic, complemented
263#1954C2×C3111yescharacteristic, complemented
264#1875C2×C3112yescomplemented
265#1854C2×C3112yescomplemented
266#1802C2×C3111yescharacteristic, complemented
267C22×C31—42no—
268C22×C31—42no—
269C22×C31—42no—
270C22×C31—42no—
271C22×C31—42no—
272C22×C31—42no—
273C22×C31—42no—
274C22×C31—42no—
275C22×C31—42no—
276C22×C31—42no—
277C22×C31—42no—
278C22×C31—42no—
279C22×C31—42no—
280C22×C31—81no—
281C22×C31—81no—
282C22×C31—81no—
283C22×C31—81no—
284C22×C31—81no—
285C22×C31—81no—
286C22×C31—81no—
287C22×C31—81no—
288C22×C31C2×#12011yescharacteristic
289C22×C31#54611yescharacteristic
290C22×C31#54711yescharacteristic
291C22×C31C2×#13511yescharacteristic
292C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
293C22×C31#57611yescharacteristic
294C22×C31#57711yescharacteristic
295C4×C31—22no—
296C4×C31—42no—
297C4×C31—42no—
298C4×C31—44no—
299C4×C31—81no—
300C4×C31—82no—
301C4×C31#44914yes—
302C4×C31#52511yescharacteristic
303C4×C31#53711yescharacteristic
304C4×C31#54811yescharacteristic
305C4×C31#55011yescharacteristic
306C22×D8C22×C3111yescharacteristic
307#555—24no—
308#555—24no—
309C2×C4⋊Q8C22×C3111yescharacteristic
310C2×#124—22no—
311C2×#121—21no—
312C2×#121—21no—
313C2×#121C22×C3112yes—
314C2×#121C22×C3112yes—
315C2×C4×D4C22×C3111yescharacteristic
316C2×C4×D4C22×C3111yescharacteristic
317C2×C4×D4C22×C3111yescharacteristic
318C2×#117—21no—
319C2×#117C22×C3112yes—
320#476—24no—
321C4×D8—24no—
322C2×#107C22×C3111yescharacteristic
323C2×#106C22×C3112yes—
324C2×C4⋊C8C22×C3111yescharacteristic
325C2×#102C22×C3111yescharacteristic
326C2×#102C22×C3112yes—
327C2×#102C22×C3111yescharacteristic
328C2×#102C22×C3112yes—
329C2×C22⋊C8C22×C3112yes—
330C2×C22⋊C8C22×C3112yes—
331C2×C4×C8C22×C3111yescharacteristic
332#394—22no—
333#384C22×C3111yescharacteristic
334#380—21no—
335#380C22×C3112yes—
336#340C22×C3112yes—
337#340C22×C3112yes—
338#340C22×C3112yes—
339C2×C31—82no—
340C2×C31—82no—
341C2×C31—82no—
342C2×C31#240711yescharacteristic
343C2×C31C2×#44911yescharacteristic
344C2×C31#301211yescharacteristic
345C2×C31#330311yescharacteristic
346C2×C31#332511yescharacteristic
347C2×C31#334111yescharacteristic
348C2×C31#335411yescharacteristic
349C22×Q8—21no—
350C22×D4—22no—
351C22×D4—22no—
352C22×D4D4×C3111yescharacteristic
353C22×D4C23×C3111yescharacteristic
354C22×D4C23×C3111yescharacteristic
355C23×C4—21no—
356C23×C4—21no—
357C23×C4C23×C3112yes—
358C2×D8—22no—
359C2×D8—22no—
360C2×D8—22no—
361C22×C8C23×C3111yescharacteristic
362C22×C8C23×C3112yes—
363C22×C8C23×C3111yescharacteristic
364C4⋊Q8—22no—
365C4⋊Q8D4×C3114yescomplemented
366#124—216no—
367#121—28no—
368#121—24no—
369#121—24no—
370#121—24no—
371#121—42no—
372#121—44no—
373C4×D4—22no—
374C4×D4—22no—
375C4×D4—22no—
376C4×D4—22no—
377C4×D4—22no—
378C4×D4D4×C3114yescomplemented
379#117—28no—
380#117—24no—
381#117—24no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4D4×C3111yescharacteristic
386C2×C4⋊C4C23×C3111yescharacteristic
387C2×C4⋊C4C23×C3112yes—
388C2×C4⋊C4C23×C3111yescharacteristic
389C2×C4⋊C4C23×C3112yes—
390C2×C22⋊C4—22no—
391C2×C22⋊C4—22no—
392C2×C22⋊C4—21no—
393C2×C22⋊C4—22no—
394C2×C22⋊C4—21no—
395C2×C22⋊C4—22no—
396C2×C22⋊C4D4×C3112yes—
397C2×C42—22no—
398C2×C42C23×C3111yescharacteristic
399#107—22no—
400#106—24no—
401C4⋊C8—22no—
402#102—22no—
403#102—22no—
404#102—24no—
405#102—42no—
406C22⋊C8—24no—
407C22⋊C8—42no—
408C4×C8—22no—
409#95—22no—
410C31#2411711yescharacteristic, direct product factor, Sylow
411C24—21no—
412C24—21no—
413C24C2×D4×C3112yes—
414C2×Q8—24no—
415C2×Q8—81no—
416C2×Q8—81no—
417C2×D4—28no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—44no—
432C2×D4—44no—
433C2×D4—82no—
434C2×D4—82no—
435C2×D4—82no—
436C2×D4C2×D4×C3112yes—
437C2×D4#41×C3112yes—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4C2×D4×C3112yes—
455C22×C4C2×D4×C3111yescharacteristic
456C22×C4C2×D4×C3111yescharacteristic
457C22×C4C24×C3111yescharacteristic
458D8—28no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—21no—
465C2×C8—22no—
466C2×C8—21no—
467C2×C8#41×C3112yes—
468C4⋊C4—24no—
469C4⋊C4—22no—
470C4⋊C4—21no—
471C4⋊C4—22no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—42no—
477C4⋊C4—81no—
478C4⋊C4C2×D4×C3114yes—
479C4⋊C4C2×D4×C3112yes—
480C22⋊C4—28no—
481C22⋊C4—28no—
482C22⋊C4—24no—
483C22⋊C4—24no—
484C22⋊C4—44no—
485C22⋊C4—44no—
486C22⋊C4—42no—
487C42—28no—
488C42C2×D4×C3112yes—
489C42C2×D4×C3112yes—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23C22×D4×C3111yescharacteristic
512Q8—84no—
513D4—22no—
514D4—22no—
515D4—22no—
516D4—22no—
517D4—44no—
518D4—41no—
519D4—41no—
520D4—41no—
521D4—41no—
522D4—88no—
523D4—88no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—48no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—84no—
547C2×C4C31×#12012yes—
548C2×C4C31×#12012yes—
549C2×C4C31×#13512yes—
550C2×C4C31×AGL(1,ℤ/8)12yes—
551C2×C4C22×D4×C3111yescharacteristic
552C2×C4C22×D4×C3111yescharacteristic
553C2×C4C2×#41×C3111yescharacteristic
554C2×C4C2×#41×C3111yescharacteristic
555C2×C4C31×H(5,𝔽2)11yescharacteristic
556C2×C4C31×2−1+411yescharacteristic
557C8—22no—
558C8—41no—
559C8—42no—
560C22—42no—
561C22—42no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22—81no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22—81no—
578C22—81no—
579C22—81no—
580C22—81no—
581C22C2×C31×#12011yescharacteristic
582C22C31×#54611yescharacteristic
583C22C31×#54711yescharacteristic
584C22C2×C31×#13511yescharacteristic
585C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
586C22C31×#57611yescharacteristic
587C22C31×#57711yescharacteristic
588C4—22no—
589C4—42no—
590C4—42no—
591C4—44no—
592C4—81no—
593C4—82no—
594C4C31×#44914yes—
595C4C31×#52511yescharacteristic
596C4C31×#53711yescharacteristic
597C4C31×#54811yescharacteristic
598C4C31×#55011yescharacteristic
599C2—82no—
600C2—82no—
601C2—82no—
602C2C31×#240711yescharacteristic
603C2C2×C31×#44911yescharacteristic
604C2C31×#301211yescharacteristic
605C2C31×#330311yescharacteristic
606C2C31×#332511yescharacteristic
607C2C31×#334111yescharacteristic
608C2C31×#335411yescharacteristic
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C2×C4×D4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C31×#555C2semidirect
2. C2×C31×#476C2semidirect
3. C2×C4×D8×C31C2semidirect
4. C31×#2537C2semidirect
5. C31×#2034C2semidirect
6. C31×#1999C2semidirect
7. C31×#1990C2semidirect
8. C31×#1954C2semidirect
9. C31×#1875C2semidirect
10. C31×#1854C2semidirect
11. C31×#1802C2semidirect
61. C31×C4⋊Q8D4semidirect
74. C4×D4×C31D4semidirect
185. #24117C31direct (N × H)
256. C2×#555C2×C31semidirect
257. C2×#476C2×C31semidirect
258. C2×C4×D8C2×C31semidirect
259. #2537C2×C31semidirect
260. #2034C2×C31semidirect
261. #1999C2×C31semidirect
262. #1990C2×C31semidirect
263. #1954C2×C31semidirect
264. #1875C2×C31semidirect
265. #1854C2×C31semidirect
266. #1802C2×C31semidirect
365. C4⋊Q8D4×C31semidirect
378. C4×D4D4×C31semidirect
410. C31#24117direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5552
C2×C31×#4762
C2×C4×D8×C312
C31×#25372
C31×#20342
C31×#19992
C31×#19902
C31×#19542
C31×#18752
C31×#18542
C31×#18022
#2411731