Finite Group Databasebeta

← C31×#24172 #21480931

Subgroups — C31×#24172 #21480931

Order 7936.

1058 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×#3511C211yesmaximal, characteristic
2C31×#3037C211yesmaximal, characteristic, complemented
3C4×C31×#137C211yesmaximal, characteristic, complemented
4C31×#2568C211yesmaximal, characteristic, complemented
5C31×#2040C212yesmaximal, complemented
6C31×#2007C212yesmaximal, complemented
7C31×#1997C211yesmaximal, characteristic, complemented
8C31×#1980C211yesmaximal, characteristic, complemented
9C31×#1882C212yesmaximal, complemented
10C31×#1860C212yesmaximal, complemented
11C31×#1806C211yesmaximal, characteristic, complemented
12C2×C31×#137C2211yescharacteristic
13C2×C31×#137C2212yes—
14C31×#556—21no—
15C31×#555—21no—
16C31×#554—21no—
17C31×#553—22no—
18C31×#548—22no—
19C31×#547—21no—
20C31×#542—22no—
21C31×#537C2212yes—
22C31×#533—21no—
23C31×#532—21no—
24C31×#532C2211yescharacteristic
25C2×C31×C4⋊Q8C2211yescharacteristic
26C31×#520C2212yes—
27C4×#41×C31C2211yescharacteristic
28C2×C4×Q8×C31C2211yescharacteristic
29C31×#464—22no—
30C31×#462—21no—
31C31×#460—21no—
32C31×#441—21no—
33C31×#440—21no—
34C4×Q16×C31—21no—
35C4×SD16×C31—21no—
36C31×#428C2211yescharacteristic
37C31×#423C2211yescharacteristic
38C2×C31×#104C2212yes—
39C2×C31×#104C2212yes—
40C2×C31×#104C2212yes—
41C31×#419C2211yescharacteristic
42C31×#417C2212yes—
43C31×#417C2211yescharacteristic
44C4×M16×C31C2211yescharacteristic
45C31×#400—22no—
46C31×#391—22no—
47C31×#391C2212yes—
48C31×#382C2212yes—
49C4×C4⋊C4×C31C2211yescharacteristic
50C31×#343C2212yes—
51C31×#343C2212yes—
52C31×#339C2212yes—
53C2×#41×C31—22no—
54C2×#41×C31C2311yescharacteristic
55C2×#41×C31C2312yes—
56C22×Q8×C31D412yes—
57C22×Q8×C31C2311yescharacteristic
58C31×#137—22no—
59C31×#137—24no—
60C31×#137—22no—
61C31×#137—42no—
62C2×Q16×C31—21no—
63C2×Q16×C31—22no—
64C2×SD16×C31—21no—
65C2×SD16×C31—22no—
66C2×M16×C31C2311yescharacteristic
67C2×M16×C31C2312yes—
68C2×M16×C31C2311yescharacteristic
69C31×C4⋊Q8—21no—
70C31×C4⋊Q8—21no—
71C31×C4⋊Q8—21no—
72C31×C4⋊Q8—21no—
73C31×C4⋊Q8D411yescharacteristic
74C31×#127D411yescharacteristic
75C31×#126—24no—
76C31×#125—22no—
77C31×#124—22no—
78C31×#124—24no—
79C31×#124D412yes—
80C31×#123—24no—
81C31×C22⋊Q8—22no—
82C31×C22⋊Q8—24no—
83C31×C22⋊Q8—22no—
84C31×C22⋊Q8—22no—
85C31×#121—22no—
86C31×#121—22no—
87C31×#121—22no—
88C31×#121—21no—
89C31×#121—21no—
90C31×#121—24no—
91C4×Q8×C31—22no—
92C4×Q8×C31—22no—
93C4×Q8×C31—21no—
94C4×Q8×C31—21no—
95C4×Q8×C31—21no—
96C4×Q8×C31—21no—
97C4×Q8×C31D411yescharacteristic
98C4×Q8×C31D411yescharacteristic
99C4×D4×C31—22no—
100C4×D4×C31—22no—
101C4×D4×C31—21no—
102C4×D4×C31—21no—
103C4×D4×C31—24no—
104C4×D4×C31—21no—
105C4×D4×C31—21no—
106C4×D4×C31D411yescharacteristic
107C4×D4×C31D411yescharacteristic
108C31×#117—22no—
109C31×#117—22no—
110C31×#117—21no—
111C31×#117C2311yescharacteristic
112C2×C4⋊C4×C31—22no—
113C2×C4⋊C4×C31—21no—
114C2×C4⋊C4×C31—21no—
115C2×C4⋊C4×C31D412yes—
116C2×C4⋊C4×C31C2311yescharacteristic
117C2×C4⋊C4×C31C2312yes—
118C2×C42×C31—21no—
119C2×C42×C31—21no—
120C2×C42×C31C2312yes—
121C2×C42×C31C2311yescharacteristic
122C31×#107—21no—
123C31×#106—21no—
124C31×C4⋊C8—21no—
125C31×C4⋊C8—21no—
126C31×#104—24no—
127C31×#104—42no—
128C31×#104—42no—
129C31×#103—21no—
130C31×#103—22no—
131C31×#103—21no—
132C31×#103—21no—
133C31×#102—21no—
134C31×#102—22no—
135C31×#97—21no—
136C4×C8×C31—21no—
137C31×#95—21no—
138C31×#95—24no—
139C31×#95—24no—
140#41×C31—22no—
141#41×C31—22no—
142#41×C31—21no—
143#41×C31—21no—
144#41×C31—21no—
145#41×C31—22no—
146#41×C31—22no—
147#41×C31—21no—
148#41×C31—84no—
149C2×Q8×C31—22no—
150C2×Q8×C31—22no—
151C2×Q8×C31—21no—
152C2×Q8×C31—21no—
153C2×Q8×C31—21no—
154C2×Q8×C31—21no—
155C2×Q8×C31—21no—
156C2×Q8×C31—44no—
157C2×Q8×C31C2×D412yes—
158C2×Q8×C31C2×D411yescharacteristic
159C2×Q8×C31#4112yescomplemented
160C2×D4×C31—21no—
161C2×D4×C31—22no—
162C2×D4×C31—24no—
163C2×D4×C31—21no—
164C2×D4×C31—21no—
165C2×D4×C31C2×D412yes—
166C2×D4×C31C2×D411yescharacteristic
167C22×C4×C31—21no—
168C22×C4×C31—24no—
169C22×C4×C31—21no—
170C22×C4×C31—21no—
171C22×C4×C31—21no—
172C22×C4×C31—22no—
173C22×C4×C31—22no—
174C22×C4×C31—21no—
175C22×C4×C31C2×D411yescharacteristic
176C22×C4×C31C2×D414yes—
177C22×C4×C31C2×D411yescharacteristic
178C22×C4×C31C2411yescharacteristic
179Q16×C31—24no—
180Q16×C31—44no—
181SD16×C31—24no—
182SD16×C31—44no—
183M16×C31—21no—
184M16×C31—22no—
185M16×C31—21no—
186M16×C31—22no—
187M16×C31—21no—
188M16×C31#4112yescomplemented
189C2×C8×C31—21no—
190C2×C8×C31—21no—
191C2×C8×C31—22no—
192C4⋊C4×C31—24no—
193C4⋊C4×C31—24no—
194C4⋊C4×C31—22no—
195C4⋊C4×C31—22no—
196C4⋊C4×C31—21no—
197C4⋊C4×C31—21no—
198C4⋊C4×C31—21no—
199C4⋊C4×C31—21no—
200C4⋊C4×C31—22no—
201C4⋊C4×C31—22no—
202C4⋊C4×C31—21no—
203C4⋊C4×C31—21no—
204C4⋊C4×C31—42no—
205C4⋊C4×C31—41no—
206C4⋊C4×C31—41no—
207C4⋊C4×C31C2×D411yescharacteristic
208C4⋊C4×C31C2×D411yescharacteristic
209C22⋊C4×C31—24no—
210C22⋊C4×C31—24no—
211C22⋊C4×C31—24no—
212C22⋊C4×C31—21no—
213C22⋊C4×C31—21no—
214C22⋊C4×C31—21no—
215C22⋊C4×C31—22no—
216C22⋊C4×C31—22no—
217C42×C31—24no—
218C42×C31—21no—
219C42×C31—21no—
220C42×C31—21no—
221C42×C31—22no—
222C42×C31—22no—
223C42×C31—41no—
224C42×C31—41no—
225C42×C31C2×D411yescharacteristic
226C42×C31C2×D411yescharacteristic
227C42×C31C2×D411yescharacteristic
228C42×C31C2×D411yescharacteristic
229#24172C3111yesmaximal, characteristic, direct product factor, Sylow
230C23×C31—21no—
231C23×C31—22no—
232C23×C31C22×D411yescharacteristic
233Q8×C31—22no—
234Q8×C31—21no—
235Q8×C31—21no—
236Q8×C31—22no—
237Q8×C31—22no—
238Q8×C31—22no—
239Q8×C31—41no—
240Q8×C31—41no—
241Q8×C31—44no—
242Q8×C31—82no—
243D4×C31—21no—
244D4×C31—21no—
245D4×C31—22no—
246D4×C31—22no—
247D4×C31—42no—
248D4×C31—42no—
249D4×C31—41no—
250D4×C31—42no—
251D4×C31—41no—
252D4×C31—42no—
253D4×C31—84no—
254C2×C4×C31—22no—
255C2×C4×C31—21no—
256C2×C4×C31—22no—
257C2×C4×C31—21no—
258C2×C4×C31—21no—
259C2×C4×C31—24no—
260C2×C4×C31—22no—
261C2×C4×C31—41no—
262C2×C4×C31—42no—
263C2×C4×C31—41no—
264C2×C4×C31—41no—
265C2×C4×C31—42no—
266C2×C4×C31—42no—
267C2×C4×C31—41no—
268C2×C4×C31—41no—
269C2×C4×C31—42no—
270C2×C4×C31—44no—
271C2×C4×C31—42no—
272C2×C4×C31—42no—
273C2×C4×C31—42no—
274C2×C4×C31—41no—
275C2×C4×C31—42no—
276C2×C4×C31—84no—
277C2×C4×C31#12011yescharacteristic
278C2×C4×C31#12011yescharacteristic
279C2×C4×C31#12011yescharacteristic
280C2×C4×C31#12011yescharacteristic
281C2×C4×C31C22×D411yescharacteristic
282C2×C4×C31C22×D411yescharacteristic
283C2×C4×C31C2×#4111yescharacteristic
284C2×C4×C31C2×#4111yescharacteristic
285C2×C4×C31H(5,𝔽2)11yescharacteristic
286C2×C4×C312−1+411yescharacteristic
287C8×C31—22no—
288C8×C31—41no—
289C8×C31—42no—
290#3511C2×C3111yescharacteristic
291#3037C2×C3111yescharacteristic, complemented
292C4×#137C2×C3111yescharacteristic, complemented
293#2568C2×C3111yescharacteristic, complemented
294#2040C2×C3112yescomplemented
295#2007C2×C3112yescomplemented
296#1997C2×C3111yescharacteristic, complemented
297#1980C2×C3111yescharacteristic, complemented
298#1882C2×C3112yescomplemented
299#1860C2×C3112yescomplemented
300#1806C2×C3111yescharacteristic, complemented
301C22×C31—21no—
302C22×C31—21no—
303C22×C31—42no—
304C22×C31—41no—
305C22×C31—41no—
306C22×C31—42no—
307C22×C31—41no—
308C22×C31—42no—
309C22×C31C2×#12011yescharacteristic
310C22×C31#54811yescharacteristic
311C22×C31#55011yescharacteristic
312C4×C31—21no—
313C4×C31—21no—
314C4×C31—21no—
315C4×C31—42no—
316C4×C31—41no—
317C4×C31—42no—
318C4×C31—41no—
319C4×C31—44no—
320C4×C31—41no—
321C4×C31—82no—
322C4×C31—81no—
323C4×C31#45512yes—
324C4×C31#52511yescharacteristic
325C4×C31#53711yescharacteristic
326C4×C31#54611yescharacteristic
327C4×C31#54711yescharacteristic
328C2×#137C22×C3111yescharacteristic
329C2×#137C22×C3112yes—
330#556—21no—
331#555—21no—
332#554—21no—
333#553—22no—
334#548—22no—
335#547—21no—
336#542—22no—
337#537C22×C3112yes—
338#533—21no—
339#532—21no—
340#532C22×C3111yescharacteristic
341C2×C4⋊Q8C22×C3111yescharacteristic
342#520C22×C3112yes—
343C4×#41C22×C3111yescharacteristic
344C2×C4×Q8C22×C3111yescharacteristic
345#464—22no—
346#462—21no—
347#460—21no—
348#441—21no—
349#440—21no—
350C4×Q16—21no—
351C4×SD16—21no—
352#428C22×C3111yescharacteristic
353#423C22×C3111yescharacteristic
354C2×#104C22×C3112yes—
355C2×#104C22×C3112yes—
356C2×#104C22×C3112yes—
357#419C22×C3111yescharacteristic
358#417C22×C3112yes—
359#417C22×C3111yescharacteristic
360C4×M16C22×C3111yescharacteristic
361#400—22no—
362#391—22no—
363#391C22×C3112yes—
364#382C22×C3112yes—
365C4×C4⋊C4C22×C3111yescharacteristic
366#343C22×C3112yes—
367#343C22×C3112yes—
368#339C22×C3112yes—
369C2×C31—21no—
370C2×C31—21no—
371C2×C31—81no—
372C2×C31—82no—
373C2×C31#240711yescharacteristic
374C2×C31C2×#45511yescharacteristic
375C2×C31#302011yescharacteristic
376C2×#41—22no—
377C2×#41C23×C3111yescharacteristic
378C2×#41C23×C3112yes—
379C22×Q8D4×C3112yes—
380C22×Q8C23×C3111yescharacteristic
381#137—22no—
382#137—24no—
383#137—22no—
384#137—42no—
385C2×Q16—21no—
386C2×Q16—22no—
387C2×SD16—21no—
388C2×SD16—22no—
389C2×M16C23×C3112yes—
390C2×M16C23×C3111yescharacteristic
391C2×M16C23×C3111yescharacteristic
392C4⋊Q8—21no—
393C4⋊Q8—21no—
394C4⋊Q8—21no—
395C4⋊Q8—21no—
396C4⋊Q8D4×C3111yescharacteristic
397#127D4×C3111yescharacteristic
398#126—24no—
399#125—22no—
400#124—22no—
401#124—24no—
402#124D4×C3112yes—
403#123—24no—
404C22⋊Q8—22no—
405C22⋊Q8—24no—
406C22⋊Q8—22no—
407C22⋊Q8—22no—
408#121—22no—
409#121—22no—
410#121—22no—
411#121—21no—
412#121—21no—
413#121—24no—
414C4×Q8—22no—
415C4×Q8—22no—
416C4×Q8—21no—
417C4×Q8—21no—
418C4×Q8—21no—
419C4×Q8—21no—
420C4×Q8D4×C3111yescharacteristic
421C4×Q8D4×C3111yescharacteristic
422C4×D4—22no—
423C4×D4—22no—
424C4×D4—21no—
425C4×D4—21no—
426C4×D4—24no—
427C4×D4—21no—
428C4×D4—21no—
429C4×D4D4×C3111yescharacteristic
430C4×D4D4×C3111yescharacteristic
431#117—22no—
432#117—22no—
433#117—21no—
434#117C23×C3111yescharacteristic
435C2×C4⋊C4—22no—
436C2×C4⋊C4—21no—
437C2×C4⋊C4—21no—
438C2×C4⋊C4D4×C3112yes—
439C2×C4⋊C4C23×C3111yescharacteristic
440C2×C4⋊C4C23×C3112yes—
441C2×C42—21no—
442C2×C42—21no—
443C2×C42C23×C3112yes—
444C2×C42C23×C3111yescharacteristic
445#107—21no—
446#106—21no—
447C4⋊C8—21no—
448C4⋊C8—21no—
449#104—24no—
450#104—42no—
451#104—42no—
452#103—21no—
453#103—22no—
454#103—21no—
455#103—21no—
456#102—21no—
457#102—22no—
458#97—21no—
459C4×C8—21no—
460#95—21no—
461#95—24no—
462#95—24no—
463C31#2417211yescharacteristic, direct product factor, Sylow
464#41—22no—
465#41—22no—
466#41—21no—
467#41—21no—
468#41—21no—
469#41—22no—
470#41—22no—
471#41—21no—
472#41—84no—
473C2×Q8—22no—
474C2×Q8—22no—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—21no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8—44no—
481C2×Q8C2×D4×C3112yes—
482C2×Q8C2×D4×C3111yescharacteristic
483C2×Q8#41×C3112yescomplemented
484C2×D4—21no—
485C2×D4—22no—
486C2×D4—24no—
487C2×D4—21no—
488C2×D4—21no—
489C2×D4C2×D4×C3112yes—
490C2×D4C2×D4×C3111yescharacteristic
491C22×C4—21no—
492C22×C4—24no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—21no—
499C22×C4C2×D4×C3111yescharacteristic
500C22×C4C2×D4×C3114yes—
501C22×C4C2×D4×C3111yescharacteristic
502C22×C4C24×C3111yescharacteristic
503Q16—24no—
504Q16—44no—
505SD16—24no—
506SD16—44no—
507M16—21no—
508M16—22no—
509M16—21no—
510M16—22no—
511M16—21no—
512M16#41×C3112yescomplemented
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—22no—
516C4⋊C4—24no—
517C4⋊C4—24no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—42no—
529C4⋊C4—41no—
530C4⋊C4—41no—
531C4⋊C4C2×D4×C3111yescharacteristic
532C4⋊C4C2×D4×C3111yescharacteristic
533C22⋊C4—24no—
534C22⋊C4—24no—
535C22⋊C4—24no—
536C22⋊C4—21no—
537C22⋊C4—21no—
538C22⋊C4—21no—
539C22⋊C4—22no—
540C22⋊C4—22no—
541C42—24no—
542C42—21no—
543C42—21no—
544C42—21no—
545C42—22no—
546C42—22no—
547C42—41no—
548C42—41no—
549C42C2×D4×C3111yescharacteristic
550C42C2×D4×C3111yescharacteristic
551C42C2×D4×C3111yescharacteristic
552C42C2×D4×C3111yescharacteristic
553C23—21no—
554C23—22no—
555C23C22×D4×C3111yescharacteristic
556Q8—22no—
557Q8—21no—
558Q8—21no—
559Q8—22no—
560Q8—22no—
561Q8—22no—
562Q8—44no—
563Q8—41no—
564Q8—41no—
565Q8—82no—
566D4—21no—
567D4—21no—
568D4—22no—
569D4—22no—
570D4—42no—
571D4—42no—
572D4—41no—
573D4—42no—
574D4—41no—
575D4—42no—
576D4—84no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—84no—
600C2×C4C31×#12011yescharacteristic
601C2×C4C31×#12011yescharacteristic
602C2×C4C31×#12011yescharacteristic
603C2×C4C31×#12011yescharacteristic
604C2×C4C22×D4×C3111yescharacteristic
605C2×C4C22×D4×C3111yescharacteristic
606C2×C4C2×#41×C3111yescharacteristic
607C2×C4C2×#41×C3111yescharacteristic
608C2×C4C31×H(5,𝔽2)11yescharacteristic
609C2×C4C31×2−1+411yescharacteristic
610C8—22no—
611C8—41no—
612C8—42no—
613C22—21no—
614C22—21no—
615C22—42no—
616C22—41no—
617C22—41no—
618C22—42no—
619C22—41no—
620C22—42no—
621C22C2×C31×#12011yescharacteristic
622C22C31×#54811yescharacteristic
623C22C31×#55011yescharacteristic
624C4—21no—
625C4—21no—
626C4—21no—
627C4—42no—
628C4—41no—
629C4—42no—
630C4—41no—
631C4—44no—
632C4—41no—
633C4—82no—
634C4—81no—
635C4C31×#45512yes—
636C4C31×#52511yescharacteristic
637C4C31×#53711yescharacteristic
638C4C31×#54611yescharacteristic
639C4C31×#54711yescharacteristic
640C2—21no—
641C2—21no—
642C2—81no—
643C2—82no—
644C2C31×#240711yescharacteristic
645C2C2×C31×#45511yescharacteristic
646C2C31×#302011yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#3511C2
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
2. C31×#3037C2semidirect
3. C4×C31×#137C2semidirect
4. C31×#2568C2semidirect
5. C31×#2040C2semidirect
6. C31×#2007C2semidirect
7. C31×#1997C2semidirect
8. C31×#1980C2semidirect
9. C31×#1882C2semidirect
10. C31×#1860C2semidirect
11. C31×#1806C2semidirect
159. C2×Q8×C31#41semidirect
188. M16×C31#41semidirect
229. #24172C31direct (N × H)
291. #3037C2×C31semidirect
292. C4×#137C2×C31semidirect
293. #2568C2×C31semidirect
294. #2040C2×C31semidirect
295. #2007C2×C31semidirect
296. #1997C2×C31semidirect
297. #1980C2×C31semidirect
298. #1882C2×C31semidirect
299. #1860C2×C31semidirect
300. #1806C2×C31semidirect
463. C31#24172direct (N × H)
483. C2×Q8#41×C31semidirect
512. M16#41×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#35112
C31×#30372
C4×C31×#1372
C31×#25682
C31×#20402
C31×#20072
C31×#19972
C31×#19802
C31×#18822
C31×#18602
C31×#18062
#2417231