Finite Group Databasebeta

← C31×#25625 #21482384

Subgroups — C31×#25625 #21482384

Order 7936.

1172 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×#3506C211yesmaximal, characteristic, complemented
2C31×#3153C211yesmaximal, characteristic, complemented
3C31×#3140C211yesmaximal, characteristic, complemented
4C31×#2544C211yesmaximal, characteristic, complemented
5C31×#2004C212yesmaximal, complemented
6C31×#2002C212yesmaximal, complemented
7C31×#1986C211yesmaximal, characteristic, complemented
8C31×#1980C211yesmaximal, characteristic, complemented
9C31×#1883C212yesmaximal, complemented
10C31×#1882C212yesmaximal, complemented
11C4×C31×#100C211yesmaximal, characteristic, complemented
12C2×C31×#137C2212yes—
13C2×C31×AGL(1,ℤ/8)C2212yes—
14C31×#559—22no—
15C31×#556—21no—
16C31×#555—21no—
17C31×#553—22no—
18C31×#550—21no—
19C31×#547—21no—
20C31×#538—22no—
21C31×#535C2212yes—
22C31×#533C2211yescharacteristic
23C31×#532C2211yescharacteristic
24C2×C31×#127C2211yescharacteristic
25C2×C31×#124C2211yescharacteristic
26C2×C31×#124C2211yescharacteristic
27C31×#525—22no—
28C31×#520C2212yes—
29C2×C4×D4×C31C2211yescharacteristic
30C31×#497—21no—
31C31×#496—21no—
32C31×#495—22no—
33C31×#489—21no—
34C31×#489—21no—
35C31×#488—22no—
36C2×C31×#104C2212yes—
37C2×C31×#104C2212yes—
38C31×#419C2212yes—
39C31×#418C2212yes—
40C2×C31×#100C2212yes—
41C4×M16×C31C2211yescharacteristic
42C4×M16×C31C2211yescharacteristic
43C31×#394—22no—
44C31×#391C2211yescharacteristic
45C31×#390C2211yescharacteristic
46C31×#386—22no—
47C4×C4⋊C4×C31C2211yescharacteristic
48C31×#343C2212yes—
49C31×#339C2212yes—
50C31×#339C2212yes—
51C2×#41×C31—22no—
52C2×#41×C31C2312yes—
53C22×Q8×C31C2311yescharacteristic
54C22×D4×C31—22no—
55C22×D4×C31C2311yescharacteristic
56C22×D4×C31C2311yescharacteristic
57C23×C4×C31D412yes—
58C31×#137—24no—
59C31×#137—42no—
60C31×AGL(1,ℤ/8)—24no—
61C31×AGL(1,ℤ/8)—42no—
62C2×Q16×C31—22no—
63C2×SD16×C31—22no—
64C2×SD16×C31—22no—
65C2×D8×C31—22no—
66C2×M16×C31C2312yes—
67C2×M16×C31C2312yes—
68C31×C4⋊Q8D411yescharacteristic, complemented
69C31×C4⋊Q8D411yescharacteristic, complemented
70C31×C4⋊Q8D411yescharacteristic, complemented
71C31×#127—22no—
72C31×#127—42no—
73C31×#127D411yescharacteristic, complemented
74C31×#126—24no—
75C31×#125D412yescomplemented
76C31×#124—24no—
77C31×#124—22no—
78C31×#124—21no—
79C31×#124—21no—
80C31×#124—21no—
81C31×#124—21no—
82C31×#124—42no—
83C31×#124D412yescomplemented
84C31×#123—24no—
85C31×#123—24no—
86C31×C22⋊Q8—22no—
87C31×C22⋊Q8—21no—
88C31×C22⋊Q8—21no—
89C31×#121—22no—
90C31×#121—24no—
91C31×#121—21no—
92C31×#121—21no—
93C31×#121—22no—
94C31×#121—22no—
95C31×#120—24no—
96C4×Q8×C31—22no—
97C4×Q8×C31—21no—
98C4×Q8×C31—21no—
99C4×D4×C31—24no—
100C4×D4×C31—22no—
101C4×D4×C31—24no—
102C4×D4×C31—21no—
103C4×D4×C31—21no—
104C4×D4×C31—21no—
105C4×D4×C31—21no—
106C4×D4×C31—21no—
107C4×D4×C31—21no—
108C31×#117—22no—
109C31×#117—22no—
110C31×#117C2312yes—
111C2×C4⋊C4×C31—22no—
112C2×C4⋊C4×C31—21no—
113C2×C4⋊C4×C31C2311yescharacteristic
114C2×C22⋊C4×C31—22no—
115C2×C22⋊C4×C31—21no—
116C2×C22⋊C4×C31—21no—
117C2×C22⋊C4×C31—22no—
118C2×C22⋊C4×C31D412yes—
119C2×C42×C31C2311yescharacteristic
120C2×C42×C31C2311yescharacteristic
121C2×C42×C31C2311yescharacteristic
122C31×#104—42no—
123C31×#104—42no—
124C31×#103—22no—
125C31×#103—22no—
126C31×#102—22no—
127C31×#102—22no—
128C31×#100—24no—
129C31×#97—21no—
130C31×#97—21no—
131C4×C8×C31—21no—
132C4×C8×C31—21no—
133C31×#95—21no—
134C31×#95—22no—
135C24×C31—21no—
136C24×C31C2×D412yes—
137#41×C31—22no—
138#41×C31—22no—
139#41×C31—22no—
140#41×C31—22no—
141#41×C31—84no—
142C2×Q8×C31—21no—
143C2×Q8×C31—21no—
144C2×Q8×C31—21no—
145C2×Q8×C31—21no—
146C2×Q8×C31—22no—
147C2×Q8×C31—22no—
148C2×Q8×C31C2×D412yes—
149C2×D4×C31—22no—
150C2×D4×C31—24no—
151C2×D4×C31—24no—
152C2×D4×C31—21no—
153C2×D4×C31—21no—
154C2×D4×C31—21no—
155C2×D4×C31—21no—
156C2×D4×C31—21no—
157C2×D4×C31—21no—
158C2×D4×C31—22no—
159C2×D4×C31—22no—
160C2×D4×C31—22no—
161C2×D4×C31—22no—
162C2×D4×C31—44no—
163C2×D4×C31—44no—
164C2×D4×C31—82no—
165C2×D4×C31C2×D412yes—
166C22×C4×C31—21no—
167C22×C4×C31—24no—
168C22×C4×C31—22no—
169C22×C4×C31—22no—
170C22×C4×C31—22no—
171C22×C4×C31—21no—
172C22×C4×C31—21no—
173C22×C4×C31—82no—
174C22×C4×C31C2×D412yes—
175C22×C4×C31C2×D411yescharacteristic
176C22×C4×C31C2×D411yescharacteristic
177C22×C4×C31C2411yescharacteristic
178Q16×C31—44no—
179SD16×C31—44no—
180SD16×C31—44no—
181D8×C31—44no—
182M16×C31—22no—
183M16×C31—22no—
184M16×C31—22no—
185M16×C31#4114yescomplemented
186C2×C8×C31—22no—
187C2×C8×C31—22no—
188C4⋊C4×C31—24no—
189C4⋊C4×C31—21no—
190C4⋊C4×C31—21no—
191C4⋊C4×C31—22no—
192C4⋊C4×C31—21no—
193C4⋊C4×C31—22no—
194C4⋊C4×C31—21no—
195C4⋊C4×C31—21no—
196C4⋊C4×C31—21no—
197C4⋊C4×C31—42no—
198C4⋊C4×C31C2×D412yes—
199C4⋊C4×C31C2×D412yes—
200C22⋊C4×C31—24no—
201C22⋊C4×C31—24no—
202C22⋊C4×C31—24no—
203C22⋊C4×C31—22no—
204C22⋊C4×C31—22no—
205C22⋊C4×C31—22no—
206C22⋊C4×C31—22no—
207C22⋊C4×C31—22no—
208C22⋊C4×C31—44no—
209C22⋊C4×C31—41no—
210C22⋊C4×C31—41no—
211C22⋊C4×C31—41no—
212C22⋊C4×C31—41no—
213C22⋊C4×C31—82no—
214C42×C31—24no—
215C42×C31—22no—
216C42×C31—22no—
217C42×C31—22no—
218C42×C31C2×D411yescharacteristic
219C42×C31C2×D411yescharacteristic
220C42×C31C2×D411yescharacteristic
221C42×C31C2×D411yescharacteristic
222#25625C3111yesmaximal, characteristic, direct product factor, Sylow
223C23×C31—22no—
224C23×C31—22no—
225C23×C31—22no—
226C23×C31—22no—
227C23×C31—42no—
228C23×C31—42no—
229C23×C31—42no—
230C23×C31—42no—
231C23×C31—41no—
232C23×C31—41no—
233C23×C31—41no—
234C23×C31C22×D411yescharacteristic
235Q8×C31—22no—
236Q8×C31—22no—
237Q8×C31—42no—
238Q8×C31—42no—
239Q8×C31—42no—
240Q8×C31—81no—
241D4×C31—22no—
242D4×C31—22no—
243D4×C31—42no—
244D4×C31—42no—
245D4×C31—42no—
246D4×C31—42no—
247D4×C31—42no—
248D4×C31—42no—
249D4×C31—42no—
250D4×C31—81no—
251D4×C31—84no—
252D4×C31—84no—
253D4×C31—88no—
254C2×C4×C31—22no—
255C2×C4×C31—22no—
256C2×C4×C31—22no—
257C2×C4×C31—22no—
258C2×C4×C31—22no—
259C2×C4×C31—41no—
260C2×C4×C31—41no—
261C2×C4×C31—44no—
262C2×C4×C31—41no—
263C2×C4×C31—41no—
264C2×C4×C31—42no—
265C2×C4×C31—42no—
266C2×C4×C31—42no—
267C2×C4×C31—41no—
268C2×C4×C31—41no—
269C2×C4×C31—41no—
270C2×C4×C31—41no—
271C2×C4×C31—41no—
272C2×C4×C31—41no—
273C2×C4×C31—82no—
274C2×C4×C31—84no—
275C2×C4×C31#12011yescharacteristic
276C2×C4×C31#12011yescharacteristic
277C2×C4×C31#12011yescharacteristic
278C2×C4×C31#12011yescharacteristic
279C2×C4×C31C22×D411yescharacteristic
280C2×C4×C31C22×D411yescharacteristic
281C2×C4×C31C2×#4112yes—
282C2×C4×C31H(5,𝔽2)12yes—
283C8×C31—24no—
284C8×C31—42no—
285#3506C2×C3111yescharacteristic, complemented
286#3153C2×C3111yescharacteristic, complemented
287#3140C2×C3111yescharacteristic, complemented
288#2544C2×C3111yescharacteristic, complemented
289#2004C2×C3112yescomplemented
290#2002C2×C3112yescomplemented
291#1986C2×C3111yescharacteristic, complemented
292#1980C2×C3111yescharacteristic, complemented
293#1883C2×C3112yescomplemented
294#1882C2×C3112yescomplemented
295C4×#100C2×C3111yescharacteristic, complemented
296C22×C31—21no—
297C22×C31—21no—
298C22×C31—44no—
299C22×C31—42no—
300C22×C31—42no—
301C22×C31—42no—
302C22×C31—42no—
303C22×C31—42no—
304C22×C31—42no—
305C22×C31—82no—
306C22×C31—82no—
307C22×C31—82no—
308C22×C31—81no—
309C22×C31—81no—
310C22×C31—81no—
311C22×C31C2×#12011yescharacteristic
312C22×C31#53211yescharacteristic
313C22×C31#53511yescharacteristic
314C4×C31—21no—
315C4×C31—21no—
316C4×C31—21no—
317C4×C31—21no—
318C4×C31—42no—
319C4×C31—42no—
320C4×C31—42no—
321C4×C31—42no—
322C4×C31—42no—
323C4×C31—81no—
324C4×C31—81no—
325C4×C31#54612yes—
326C4×C31#55012yes—
327C2×#137C22×C3112yes—
328C2×AGL(1,ℤ/8)C22×C3112yes—
329#559—22no—
330#556—21no—
331#555—21no—
332#553—22no—
333#550—21no—
334#547—21no—
335#538—22no—
336#535C22×C3112yes—
337#533C22×C3111yescharacteristic
338#532C22×C3111yescharacteristic
339C2×#127C22×C3111yescharacteristic
340C2×#124C22×C3111yescharacteristic
341C2×#124C22×C3111yescharacteristic
342#525—22no—
343#520C22×C3112yes—
344C2×C4×D4C22×C3111yescharacteristic
345#497—21no—
346#496—21no—
347#495—22no—
348#489—21no—
349#489—21no—
350#488—22no—
351C2×#104C22×C3112yes—
352C2×#104C22×C3112yes—
353#419C22×C3112yes—
354#418C22×C3112yes—
355C2×#100C22×C3112yes—
356C4×M16C22×C3111yescharacteristic
357C4×M16C22×C3111yescharacteristic
358#394—22no—
359#391C22×C3111yescharacteristic
360#390C22×C3111yescharacteristic
361#386—22no—
362C4×C4⋊C4C22×C3111yescharacteristic
363#343C22×C3112yes—
364#339C22×C3112yes—
365#339C22×C3112yes—
366C2×C31—21no—
367C2×C31—21no—
368C2×C31—82no—
369C2×C31—82no—
370C2×C31—161no—
371C2×C31#243111yescharacteristic
372C2×C31#301811yescharacteristic
373C2×C31#301911yescharacteristic
374C2×#41—22no—
375C2×#41C23×C3112yes—
376C22×Q8C23×C3111yescharacteristic
377C22×D4—22no—
378C22×D4C23×C3111yescharacteristic
379C22×D4C23×C3111yescharacteristic
380C23×C4D4×C3112yes—
381#137—24no—
382#137—42no—
383AGL(1,ℤ/8)—24no—
384AGL(1,ℤ/8)—42no—
385C2×Q16—22no—
386C2×SD16—22no—
387C2×SD16—22no—
388C2×D8—22no—
389C2×M16C23×C3112yes—
390C2×M16C23×C3112yes—
391C4⋊Q8D4×C3111yescharacteristic, complemented
392C4⋊Q8D4×C3111yescharacteristic, complemented
393C4⋊Q8D4×C3111yescharacteristic, complemented
394#127—22no—
395#127—42no—
396#127D4×C3111yescharacteristic, complemented
397#126—24no—
398#125D4×C3112yescomplemented
399#124—24no—
400#124—22no—
401#124—21no—
402#124—21no—
403#124—21no—
404#124—21no—
405#124—42no—
406#124D4×C3112yescomplemented
407#123—24no—
408#123—24no—
409C22⋊Q8—22no—
410C22⋊Q8—21no—
411C22⋊Q8—21no—
412#121—22no—
413#121—24no—
414#121—21no—
415#121—21no—
416#121—22no—
417#121—22no—
418#120—24no—
419C4×Q8—22no—
420C4×Q8—21no—
421C4×Q8—21no—
422C4×D4—24no—
423C4×D4—22no—
424C4×D4—24no—
425C4×D4—21no—
426C4×D4—21no—
427C4×D4—21no—
428C4×D4—21no—
429C4×D4—21no—
430C4×D4—21no—
431#117—22no—
432#117—22no—
433#117C23×C3112yes—
434C2×C4⋊C4—22no—
435C2×C4⋊C4—21no—
436C2×C4⋊C4C23×C3111yescharacteristic
437C2×C22⋊C4—22no—
438C2×C22⋊C4—21no—
439C2×C22⋊C4—21no—
440C2×C22⋊C4—22no—
441C2×C22⋊C4D4×C3112yes—
442C2×C42C23×C3111yescharacteristic
443C2×C42C23×C3111yescharacteristic
444C2×C42C23×C3111yescharacteristic
445#104—42no—
446#104—42no—
447#103—22no—
448#103—22no—
449#102—22no—
450#102—22no—
451#100—24no—
452#97—21no—
453#97—21no—
454C4×C8—21no—
455C4×C8—21no—
456#95—21no—
457#95—22no—
458C31#2562511yescharacteristic, direct product factor, Sylow
459C24—21no—
460C24C2×D4×C3112yes—
461#41—22no—
462#41—22no—
463#41—22no—
464#41—22no—
465#41—84no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—22no—
471C2×Q8—22no—
472C2×Q8C2×D4×C3112yes—
473C2×D4—22no—
474C2×D4—24no—
475C2×D4—24no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4—21no—
480C2×D4—21no—
481C2×D4—21no—
482C2×D4—22no—
483C2×D4—22no—
484C2×D4—22no—
485C2×D4—22no—
486C2×D4—44no—
487C2×D4—44no—
488C2×D4—82no—
489C2×D4C2×D4×C3112yes—
490C22×C4—21no—
491C22×C4—24no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—82no—
498C22×C4C2×D4×C3112yes—
499C22×C4C2×D4×C3111yescharacteristic
500C22×C4C2×D4×C3111yescharacteristic
501C22×C4C24×C3111yescharacteristic
502Q16—44no—
503SD16—44no—
504SD16—44no—
505D8—44no—
506M16—22no—
507M16—22no—
508M16—22no—
509M16#41×C3114yescomplemented
510C2×C8—22no—
511C2×C8—22no—
512C4⋊C4—24no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—22no—
516C4⋊C4—21no—
517C4⋊C4—22no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—42no—
522C4⋊C4C2×D4×C3112yes—
523C4⋊C4C2×D4×C3112yes—
524C22⋊C4—24no—
525C22⋊C4—24no—
526C22⋊C4—24no—
527C22⋊C4—22no—
528C22⋊C4—22no—
529C22⋊C4—22no—
530C22⋊C4—22no—
531C22⋊C4—22no—
532C22⋊C4—44no—
533C22⋊C4—41no—
534C22⋊C4—41no—
535C22⋊C4—41no—
536C22⋊C4—41no—
537C22⋊C4—82no—
538C42—24no—
539C42—22no—
540C42—22no—
541C42—22no—
542C42C2×D4×C3111yescharacteristic
543C42C2×D4×C3111yescharacteristic
544C42C2×D4×C3111yescharacteristic
545C42C2×D4×C3111yescharacteristic
546C23—22no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23C22×D4×C3111yescharacteristic
558Q8—22no—
559Q8—22no—
560Q8—42no—
561Q8—42no—
562Q8—42no—
563Q8—81no—
564D4—22no—
565D4—22no—
566D4—42no—
567D4—42no—
568D4—42no—
569D4—42no—
570D4—42no—
571D4—42no—
572D4—42no—
573D4—81no—
574D4—84no—
575D4—84no—
576D4—88no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—44no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—82no—
597C2×C4—84no—
598C2×C4C31×#12011yescharacteristic
599C2×C4C31×#12011yescharacteristic
600C2×C4C31×#12011yescharacteristic
601C2×C4C31×#12011yescharacteristic
602C2×C4C22×D4×C3111yescharacteristic
603C2×C4C22×D4×C3111yescharacteristic
604C2×C4C2×#41×C3112yes—
605C2×C4C31×H(5,𝔽2)12yes—
606C8—24no—
607C8—42no—
608C22—21no—
609C22—21no—
610C22—44no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22—82no—
618C22—82no—
619C22—82no—
620C22—81no—
621C22—81no—
622C22—81no—
623C22C2×C31×#12011yescharacteristic
624C22C31×#53211yescharacteristic
625C22C31×#53511yescharacteristic
626C4—21no—
627C4—21no—
628C4—21no—
629C4—21no—
630C4—42no—
631C4—42no—
632C4—42no—
633C4—42no—
634C4—42no—
635C4—81no—
636C4—81no—
637C4C31×#54612yes—
638C4C31×#55012yes—
639C2—21no—
640C2—21no—
641C2—82no—
642C2—82no—
643C2—161no—
644C2C31×#243111yescharacteristic
645C2C31×#301811yescharacteristic
646C2C31×#301911yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#3506C2semidirect
2. C31×#3153C2semidirect
3. C31×#3140C2semidirect
4. C31×#2544C2semidirect
5. C31×#2004C2semidirect
6. C31×#2002C2semidirect
7. C31×#1986C2semidirect
8. C31×#1980C2semidirect
9. C31×#1883C2semidirect
10. C31×#1882C2semidirect
11. C4×C31×#100C2semidirect
68. C31×C4⋊Q8D4semidirect
69. C31×C4⋊Q8D4semidirect
70. C31×C4⋊Q8D4semidirect
73. C31×#127D4semidirect
75. C31×#125D4semidirect
83. C31×#124D4semidirect
185. M16×C31#41semidirect
222. #25625C31direct (N × H)
285. #3506C2×C31semidirect
286. #3153C2×C31semidirect
287. #3140C2×C31semidirect
288. #2544C2×C31semidirect
289. #2004C2×C31semidirect
290. #2002C2×C31semidirect
291. #1986C2×C31semidirect
292. #1980C2×C31semidirect
293. #1883C2×C31semidirect
294. #1882C2×C31semidirect
295. C4×#100C2×C31semidirect
391. C4⋊Q8D4×C31semidirect
392. C4⋊Q8D4×C31semidirect
393. C4⋊Q8D4×C31semidirect
396. #127D4×C31semidirect
398. #125D4×C31semidirect
406. #124D4×C31semidirect
458. C31#25625direct (N × H)
509. M16#41×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#35062
C31×#31532
C31×#31402
C31×#25442
C31×#20042
C31×#20022
C31×#19862
C31×#19802
C31×#18832
C31×#18822
C4×C31×#1002
#2562531