Finite Group Databasebeta

← C31×#25896 #21482655

Subgroups — C31×#25896 #21482655

Order 7936.

1322 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#548C211yesmaximal, characteristic
2C2×C31×#489C211yesmaximal, characteristic, complemented
3C2×C31×#477C211yesmaximal, characteristic, complemented
4C31×#2547C211yesmaximal, characteristic, complemented
5C31×#2036C212yesmaximal, complemented
6C31×#2033C212yesmaximal, complemented
7C31×#1983C211yesmaximal, characteristic, complemented
8C31×#1955C211yesmaximal, characteristic, complemented
9C31×#1788C212yesmaximal, complemented
10C31×#1787C212yesmaximal, complemented
11C31×#1763C211yesmaximal, characteristic, complemented
12C22×#41×C31C2211yescharacteristic
13C23×Q8×C31C2211yescharacteristic
14C31×#548—24no—
15C31×#548—24no—
16C2×C31×C4⋊Q8C2211yescharacteristic
17C2×C31×#124—21no—
18C2×C31×#124—21no—
19C2×C31×#124C2211yescharacteristic
20C2×C31×C22⋊Q8—21no—
21C2×C31×C22⋊Q8C2211yescharacteristic
22C2×C31×#121—21no—
23C2×C31×#121C2211yescharacteristic
24C2×C4×Q8×C31C2211yescharacteristic
25C2×C4×D4×C31—21no—
26C2×C4×D4×C31C2211yescharacteristic
27C31×#489—24no—
28C31×#477—24no—
29C2×C31×#107C2212yes—
30C2×C31×C4⋊C8C2211yescharacteristic
31C2×C31×#103C2212yes—
32C2×C31×#103C2212yes—
33C2×C31×#103C2212yes—
34C2×C31×#103C2212yes—
35C2×C31×#102C2212yes—
36C2×C31×#102C2212yes—
37C2×C31×C22⋊C8C2212yes—
38C2×C31×C22⋊C8C2212yes—
39C2×C31×#97C2211yescharacteristic
40C31×#397—21no—
41C31×#397—21no—
42C31×#391—21no—
43C31×#380C2212yes—
44C31×#380C2212yes—
45C4×C4⋊C4×C31C2211yescharacteristic
46C31×#340C2212yes—
47C2×#41×C31—22no—
48C2×#41×C31—22no—
49C2×#41×C31—24no—
50C2×#41×C31—42no—
51C22×Q8×C31—22no—
52C22×Q8×C31—22no—
53C22×Q8×C31—21no—
54C22×Q8×C31—21no—
55C22×Q8×C31—24no—
56C22×Q8×C31—42no—
57C22×Q8×C31C2311yescharacteristic
58C22×Q8×C31C2311yescharacteristic
59C22×D4×C31—21no—
60C22×D4×C31—21no—
61C22×D4×C31D411yescharacteristic
62C22×D4×C31C2311yescharacteristic
63C23×C4×C31—21no—
64C23×C4×C31—21no—
65C23×C4×C31C2311yescharacteristic
66C23×C4×C31C2311yescharacteristic
67C22×C8×C31C2312yes—
68C22×C8×C31C2312yes—
69C31×C4⋊Q8—22no—
70C31×C4⋊Q8—22no—
71C31×#124—28no—
72C31×#124—22no—
73C31×#124—44no—
74C31×#124D414yes—
75C31×C22⋊Q8—28no—
76C31×C22⋊Q8—22no—
77C31×C22⋊Q8—22no—
78C31×#121—28no—
79C31×#121—22no—
80C31×#121—22no—
81C4×Q8×C31—22no—
82C4×Q8×C31D414yes—
83C4×D4×C31—28no—
84C4×D4×C31—22no—
85C4×D4×C31—22no—
86C2×C4⋊C4×C31—21no—
87C2×C4⋊C4×C31—21no—
88C2×C4⋊C4×C31—21no—
89C2×C4⋊C4×C31D411yescharacteristic
90C2×C4⋊C4×C31C2312yes—
91C2×C4⋊C4×C31C2312yes—
92C2×C4⋊C4×C31C2311yescharacteristic
93C2×C22⋊C4×C31—21no—
94C2×C22⋊C4×C31—21no—
95C2×C22⋊C4×C31—21no—
96C2×C22⋊C4×C31—21no—
97C2×C22⋊C4×C31—22no—
98C2×C22⋊C4×C31—22no—
99C2×C22⋊C4×C31D411yescharacteristic
100C2×C22⋊C4×C31D411yescharacteristic
101C2×C42×C31—21no—
102C2×C42×C31—21no—
103C2×C42×C31C2311yescharacteristic
104C31×#107—24no—
105C31×C4⋊C8—22no—
106C31×#103—24no—
107C31×#103—24no—
108C31×#103—24no—
109C31×#103—42no—
110C31×#102—24no—
111C31×#102—42no—
112C31×C22⋊C8—42no—
113C31×C22⋊C8—42no—
114C31×#97—22no—
115C31×#95—21no—
116C31×#95—22no—
117C31×#95—22no—
118C24×C31—21no—
119C24×C31C2×D411yescharacteristic
120C24×C31C2×D411yescharacteristic
121#41×C31—48no—
122#41×C31—44no—
123#41×C31—44no—
124C2×Q8×C31—24no—
125C2×Q8×C31—21no—
126C2×Q8×C31—21no—
127C2×Q8×C31—22no—
128C2×Q8×C31—22no—
129C2×Q8×C31—22no—
130C2×Q8×C31—44no—
131C2×Q8×C31—42no—
132C2×Q8×C31—42no—
133C2×Q8×C31—48no—
134C2×Q8×C31—44no—
135C2×Q8×C31—44no—
136C2×Q8×C31—42no—
137C2×Q8×C31C2×D412yes—
138C2×Q8×C31C2×D412yes—
139C2×Q8×C31#4114yes—
140C2×D4×C31—24no—
141C2×D4×C31—24no—
142C2×D4×C31—22no—
143C2×D4×C31—22no—
144C2×D4×C31—21no—
145C2×D4×C31—22no—
146C2×D4×C31—22no—
147C2×D4×C31—41no—
148C2×D4×C31—44no—
149C2×D4×C31—42no—
150C2×D4×C31—42no—
151C2×D4×C31C2×D412yes—
152C22×C4×C31—21no—
153C22×C4×C31—24no—
154C22×C4×C31—24no—
155C22×C4×C31—21no—
156C22×C4×C31—21no—
157C22×C4×C31—21no—
158C22×C4×C31—21no—
159C22×C4×C31—21no—
160C22×C4×C31—21no—
161C22×C4×C31—22no—
162C22×C4×C31—22no—
163C22×C4×C31—22no—
164C22×C4×C31—22no—
165C22×C4×C31—22no—
166C22×C4×C31—22no—
167C22×C4×C31—44no—
168C22×C4×C31—44no—
169C22×C4×C31C2×D411yescharacteristic
170C22×C4×C31C2×D411yescharacteristic
171C22×C4×C31C2×D411yescharacteristic
172C22×C4×C31C2×D411yescharacteristic
173C22×C4×C31C2411yescharacteristic
174C2×C8×C31—22no—
175C2×C8×C31—22no—
176C2×C8×C31—22no—
177C2×C8×C31—22no—
178C2×C8×C31—22no—
179C2×C8×C31—22no—
180C4⋊C4×C31—24no—
181C4⋊C4×C31—24no—
182C4⋊C4×C31—22no—
183C4⋊C4×C31—21no—
184C4⋊C4×C31—22no—
185C4⋊C4×C31—22no—
186C4⋊C4×C31—22no—
187C4⋊C4×C31—22no—
188C4⋊C4×C31—42no—
189C4⋊C4×C31C2×D412yes—
190C22⋊C4×C31—24no—
191C22⋊C4×C31—24no—
192C22⋊C4×C31—24no—
193C22⋊C4×C31—24no—
194C22⋊C4×C31—22no—
195C22⋊C4×C31—22no—
196C22⋊C4×C31—44no—
197C22⋊C4×C31—44no—
198C42×C31—24no—
199C42×C31—42no—
200C42×C31C2×D412yes—
201C42×C31C2×D412yes—
202#25896C3111yesmaximal, characteristic, direct product factor, Sylow
203C23×C31—21no—
204C23×C31—22no—
205C23×C31—21no—
206C23×C31—22no—
207C23×C31—21no—
208C23×C31—22no—
209C23×C31—22no—
210C23×C31—21no—
211C23×C31—21no—
212C23×C31—22no—
213C23×C31—42no—
214C23×C31—41no—
215C23×C31—41no—
216C23×C31—42no—
217C23×C31AGL(1,ℤ/8)12yes—
218C23×C31C22×D411yescharacteristic
219Q8×C31—22no—
220Q8×C31—24no—
221Q8×C31—24no—
222Q8×C31—48no—
223Q8×C31—41no—
224Q8×C31—41no—
225Q8×C31—84no—
226Q8×C31#12414yescomplemented
227D4×C31—24no—
228D4×C31—44no—
229D4×C31—48no—
230D4×C31—48no—
231D4×C31—41no—
232D4×C31—41no—
233C2×C4×C31—22no—
234C2×C4×C31—24no—
235C2×C4×C31—21no—
236C2×C4×C31—21no—
237C2×C4×C31—22no—
238C2×C4×C31—22no—
239C2×C4×C31—22no—
240C2×C4×C31—21no—
241C2×C4×C31—21no—
242C2×C4×C31—22no—
243C2×C4×C31—22no—
244C2×C4×C31—48no—
245C2×C4×C31—48no—
246C2×C4×C31—42no—
247C2×C4×C31—41no—
248C2×C4×C31—41no—
249C2×C4×C31—41no—
250C2×C4×C31—41no—
251C2×C4×C31—41no—
252C2×C4×C31—41no—
253C2×C4×C31—42no—
254C2×C4×C31—42no—
255C2×C4×C31—42no—
256C2×C4×C31—42no—
257C2×C4×C31—42no—
258C2×C4×C31—42no—
259C2×C4×C31—42no—
260C2×C4×C31—42no—
261C2×C4×C31#12012yes—
262C2×C4×C31#12012yes—
263C2×C4×C31C22×D411yescharacteristic
264C2×C4×C31C22×D411yescharacteristic
265C2×C4×C31C2×#4112yes—
266C2×C4×C31H(5,𝔽2)12yes—
267C8×C31—42no—
268C8×C31—42no—
269C2×#548C2×C3111yescharacteristic
270C2×#489C2×C3111yescharacteristic, complemented
271C2×#477C2×C3111yescharacteristic, complemented
272#2547C2×C3111yescharacteristic, complemented
273#2036C2×C3112yescomplemented
274#2033C2×C3112yescomplemented
275#1983C2×C3111yescharacteristic, complemented
276#1955C2×C3111yescharacteristic, complemented
277#1788C2×C3112yescomplemented
278#1787C2×C3112yescomplemented
279#1763C2×C3111yescharacteristic, complemented
280C22×C31—22no—
281C22×C31—24no—
282C22×C31—41no—
283C22×C31—41no—
284C22×C31—42no—
285C22×C31—42no—
286C22×C31—41no—
287C22×C31—41no—
288C22×C31—42no—
289C22×C31—41no—
290C22×C31—42no—
291C22×C31—42no—
292C22×C31—42no—
293C22×C31—82no—
294C22×C31—82no—
295C22×C31C2×#12011yescharacteristic
296C22×C31#54612yes—
297C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
298C22×C31#57511yescharacteristic
299C22×C31#57812yes—
300C4×C31—24no—
301C4×C31—22no—
302C4×C31—22no—
303C4×C31—41no—
304C4×C31—41no—
305C4×C31—42no—
306C4×C31—42no—
307C4×C31—82no—
308C4×C31—82no—
309C4×C31C2×#12411yescharacteristic
310C4×C31#53511yescharacteristic
311C4×C31#54612yes—
312C22×#41C22×C3111yescharacteristic
313C23×Q8C22×C3111yescharacteristic
314#548—24no—
315#548—24no—
316C2×C4⋊Q8C22×C3111yescharacteristic
317C2×#124—21no—
318C2×#124—21no—
319C2×#124C22×C3111yescharacteristic
320C2×C22⋊Q8—21no—
321C2×C22⋊Q8C22×C3111yescharacteristic
322C2×#121—21no—
323C2×#121C22×C3111yescharacteristic
324C2×C4×Q8C22×C3111yescharacteristic
325C2×C4×D4—21no—
326C2×C4×D4C22×C3111yescharacteristic
327#489—24no—
328#477—24no—
329C2×#107C22×C3112yes—
330C2×C4⋊C8C22×C3111yescharacteristic
331C2×#103C22×C3112yes—
332C2×#103C22×C3112yes—
333C2×#103C22×C3112yes—
334C2×#103C22×C3112yes—
335C2×#102C22×C3112yes—
336C2×#102C22×C3112yes—
337C2×C22⋊C8C22×C3112yes—
338C2×C22⋊C8C22×C3112yes—
339C2×#97C22×C3111yescharacteristic
340#397—21no—
341#397—21no—
342#391—21no—
343#380C22×C3112yes—
344#380C22×C3112yes—
345C4×C4⋊C4C22×C3111yescharacteristic
346#340C22×C3112yes—
347C2×C31—42no—
348C2×C31—81no—
349C2×C31—82no—
350C2×C31#243511yescharacteristic
351C2×C31#300611yescharacteristic
352C2×C31#301111yescharacteristic
353C2×C31#330812yes—
354C2×C31#331312yes—
355C2×#41—22no—
356C2×#41—22no—
357C2×#41—24no—
358C2×#41—42no—
359C22×Q8—22no—
360C22×Q8—22no—
361C22×Q8—21no—
362C22×Q8—21no—
363C22×Q8—24no—
364C22×Q8—42no—
365C22×Q8C23×C3111yescharacteristic
366C22×Q8C23×C3111yescharacteristic
367C22×D4—21no—
368C22×D4—21no—
369C22×D4D4×C3111yescharacteristic
370C22×D4C23×C3111yescharacteristic
371C23×C4—21no—
372C23×C4—21no—
373C23×C4C23×C3111yescharacteristic
374C23×C4C23×C3111yescharacteristic
375C22×C8C23×C3112yes—
376C22×C8C23×C3112yes—
377C4⋊Q8—22no—
378C4⋊Q8—22no—
379#124—28no—
380#124—22no—
381#124—44no—
382#124D4×C3114yes—
383C22⋊Q8—28no—
384C22⋊Q8—22no—
385C22⋊Q8—22no—
386#121—28no—
387#121—22no—
388#121—22no—
389C4×Q8—22no—
390C4×Q8D4×C3114yes—
391C4×D4—28no—
392C4×D4—22no—
393C4×D4—22no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4D4×C3111yescharacteristic
398C2×C4⋊C4C23×C3112yes—
399C2×C4⋊C4C23×C3112yes—
400C2×C4⋊C4C23×C3111yescharacteristic
401C2×C22⋊C4—21no—
402C2×C22⋊C4—21no—
403C2×C22⋊C4—21no—
404C2×C22⋊C4—21no—
405C2×C22⋊C4—22no—
406C2×C22⋊C4—22no—
407C2×C22⋊C4D4×C3111yescharacteristic
408C2×C22⋊C4D4×C3111yescharacteristic
409C2×C42—21no—
410C2×C42—21no—
411C2×C42C23×C3111yescharacteristic
412#107—24no—
413C4⋊C8—22no—
414#103—24no—
415#103—24no—
416#103—24no—
417#103—42no—
418#102—24no—
419#102—42no—
420C22⋊C8—42no—
421C22⋊C8—42no—
422#97—22no—
423#95—21no—
424#95—22no—
425#95—22no—
426C31#2589611yescharacteristic, direct product factor, Sylow
427C24—21no—
428C24C2×D4×C3111yescharacteristic
429C24C2×D4×C3111yescharacteristic
430#41—48no—
431#41—44no—
432#41—44no—
433C2×Q8—24no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—44no—
440C2×Q8—42no—
441C2×Q8—42no—
442C2×Q8—48no—
443C2×Q8—44no—
444C2×Q8—44no—
445C2×Q8—42no—
446C2×Q8C2×D4×C3112yes—
447C2×Q8C2×D4×C3112yes—
448C2×Q8#41×C3114yes—
449C2×D4—24no—
450C2×D4—24no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—21no—
454C2×D4—22no—
455C2×D4—22no—
456C2×D4—41no—
457C2×D4—44no—
458C2×D4—42no—
459C2×D4—42no—
460C2×D4C2×D4×C3112yes—
461C22×C4—21no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—44no—
477C22×C4—44no—
478C22×C4C2×D4×C3111yescharacteristic
479C22×C4C2×D4×C3111yescharacteristic
480C22×C4C2×D4×C3111yescharacteristic
481C22×C4C2×D4×C3111yescharacteristic
482C22×C4C24×C3111yescharacteristic
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—22no—
492C4⋊C4—21no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—42no—
498C4⋊C4C2×D4×C3112yes—
499C22⋊C4—24no—
500C22⋊C4—24no—
501C22⋊C4—24no—
502C22⋊C4—24no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—44no—
506C22⋊C4—44no—
507C42—24no—
508C42—42no—
509C42C2×D4×C3112yes—
510C42C2×D4×C3112yes—
511C23—21no—
512C23—22no—
513C23—21no—
514C23—22no—
515C23—21no—
516C23—22no—
517C23—22no—
518C23—21no—
519C23—21no—
520C23—22no—
521C23—42no—
522C23—41no—
523C23—41no—
524C23—42no—
525C23C31×AGL(1,ℤ/8)12yes—
526C23C22×D4×C3111yescharacteristic
527Q8—22no—
528Q8—24no—
529Q8—24no—
530Q8—48no—
531Q8—41no—
532Q8—41no—
533Q8—84no—
534Q8C31×#12414yescomplemented
535D4—24no—
536D4—44no—
537D4—48no—
538D4—48no—
539D4—41no—
540D4—41no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—48no—
553C2×C4—48no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4C31×#12012yes—
570C2×C4C31×#12012yes—
571C2×C4C22×D4×C3111yescharacteristic
572C2×C4C22×D4×C3111yescharacteristic
573C2×C4C2×#41×C3112yes—
574C2×C4C31×H(5,𝔽2)12yes—
575C8—42no—
576C8—42no—
577C22—22no—
578C22—24no—
579C22—41no—
580C22—41no—
581C22—42no—
582C22—42no—
583C22—41no—
584C22—41no—
585C22—42no—
586C22—41no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—82no—
591C22—82no—
592C22C2×C31×#12011yescharacteristic
593C22C31×#54612yes—
594C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
595C22C31×#57511yescharacteristic
596C22C31×#57812yes—
597C4—24no—
598C4—22no—
599C4—22no—
600C4—41no—
601C4—41no—
602C4—42no—
603C4—42no—
604C4—82no—
605C4—82no—
606C4C2×C31×#12411yescharacteristic
607C4C31×#53511yescharacteristic
608C4C31×#54612yes—
609C2—42no—
610C2—81no—
611C2—82no—
612C2C31×#243511yescharacteristic
613C2C31×#300611yescharacteristic
614C2C31×#301111yescharacteristic
615C2C31×#330812yes—
616C2C31×#331312yes—
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC2×C31×#548C2
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
2. C2×C31×#489C2semidirect
3. C2×C31×#477C2semidirect
4. C31×#2547C2semidirect
5. C31×#2036C2semidirect
6. C31×#2033C2semidirect
7. C31×#1983C2semidirect
8. C31×#1955C2semidirect
9. C31×#1788C2semidirect
10. C31×#1787C2semidirect
11. C31×#1763C2semidirect
202. #25896C31direct (N × H)
226. Q8×C31#124semidirect
270. C2×#489C2×C31semidirect
271. C2×#477C2×C31semidirect
272. #2547C2×C31semidirect
273. #2036C2×C31semidirect
274. #2033C2×C31semidirect
275. #1983C2×C31semidirect
276. #1955C2×C31semidirect
277. #1788C2×C31semidirect
278. #1787C2×C31semidirect
279. #1763C2×C31semidirect
426. C31#25896direct (N × H)
534. Q8C31×#124semidirect

Maximal subgroups

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

subgroupindex
C2×C31×#5482
C2×C31×#4892
C2×C31×#4772
C31×#25472
C31×#20362
C31×#20332
C31×#19832
C31×#19552
C31×#17882
C31×#17872
C31×#17632
#2589631