Finite Group Databasebeta

← C31×#24415 #21481174

Subgroups — C31×#24415 #21481174

Order 7936.

1402 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3485C211yesmaximal, characteristic
2C31×#3056C212yesmaximal, complemented
3C2×C31×#455C211yesmaximal, characteristic, complemented
4C2×C31×#413C211yesmaximal, characteristic, complemented
5C31×#2432C211yesmaximal, characteristic, complemented
6C31×#2038C212yesmaximal, complemented
7C31×#2005C212yesmaximal, complemented
8C31×#1885C211yesmaximal, characteristic, complemented
9C31×#1871C212yesmaximal, complemented
10C31×#1856C211yesmaximal, characteristic, complemented
11C31×#1792C211yesmaximal, characteristic, complemented
12C2×C31×H(5,𝔽2)C2211yescharacteristic
13C22×#41×C31C2211yescharacteristic
14C2×C31×#137C2212yes—
15C22×M16×C31C2212yescomplemented
16C31×#547—22no—
17C31×#547—22no—
18C31×#546—22no—
19C31×#546—22no—
20C31×#536C2212yes—
21C31×#534—22no—
22C2×C31×C4⋊Q8C2211yescharacteristic, complemented
23C2×C31×#124C2211yescharacteristic
24C31×#525C2212yes—
25C2×C31×#123—22no—
26C2×C31×C22⋊Q8C2211yescharacteristic, complemented
27C31×#518C2211yescharacteristic
28C31×#470—22no—
29C31×#468—22no—
30C31×#467—22no—
31C31×#465—22no—
32C31×#455—24no—
33C31×#428C2212yescomplemented
34C2×C31×#104C2212yes—
35C2×C31×#104C2212yes—
36C31×#419C2212yescomplemented
37C31×#417C2212yes—
38C31×#413—24no—
39C2×C31×#101C2212yescomplemented
40C2×C31×#100C2211yescharacteristic
41C2×C31×#100C2211yescharacteristic
42C31×#394—22no—
43C31×#391C2211yescharacteristic, complemented
44C31×#390C2211yescharacteristic
45C31×#385—22no—
46C4×C22⋊C4×C31C2211yescharacteristic, complemented
47C31×#339C2212yescomplemented
48C31×#339C2212yescomplemented
49C31×H(5,𝔽2)—22no—
50C31×H(5,𝔽2)—22no—
51C31×H(5,𝔽2)—42no—
52C2×#41×C31—22no—
53C2×#41×C31—22no—
54C2×#41×C31—22no—
55C2×#41×C31—22no—
56C2×#41×C31—22no—
57C2×#41×C31—42no—
58C2×#41×C31C2312yes—
59C22×Q8×C31C2311yescharacteristic
60C22×Q8×C31C2311yescharacteristic
61C22×D4×C31—22no—
62C22×D4×C31—22no—
63C22×D4×C31D412yes—
64C22×D4×C31C2311yescharacteristic
65C23×C4×C31D412yes—
66C23×C4×C31C2311yescharacteristic
67C31×#137—24no—
68C31×#137—42no—
69C2×Q16×C31—22no—
70C2×SD16×C31—22no—
71C2×M16×C31—22no—
72C2×M16×C31—24no—
73C2×M16×C31—42no—
74C2×M16×C31C2312yes—
75C2×M16×C31C2312yes—
76C22×C8×C31—22no—
77C31×C4⋊Q8—22no—
78C31×C4⋊Q8—42no—
79C31×#124—22no—
80C31×#124—22no—
81C31×#124—22no—
82C31×#123—24no—
83C31×#123—24no—
84C31×#123—24no—
85C31×#123—22no—
86C31×#123—24no—
87C31×C22⋊Q8—22no—
88C31×C22⋊Q8—22no—
89C31×C22⋊Q8—22no—
90C31×C22⋊Q8—22no—
91C31×C22⋊Q8D412yes—
92C31×C22⋊Q8D412yes—
93C31×#121—24no—
94C31×#121—24no—
95C31×#121D412yes—
96C31×#121D412yes—
97C31×#120—24no—
98C31×#120—22no—
99C4×D4×C31—22no—
100C4×D4×C31—22no—
101C4×D4×C31—22no—
102C4×D4×C31—22no—
103C31×#117C2312yes—
104C2×C4⋊C4×C31—22no—
105C2×C4⋊C4×C31—22no—
106C2×C4⋊C4×C31—21no—
107C2×C4⋊C4×C31—22no—
108C2×C4⋊C4×C31C2311yescharacteristic
109C2×C22⋊C4×C31—22no—
110C2×C22⋊C4×C31—22no—
111C2×C22⋊C4×C31—22no—
112C2×C22⋊C4×C31—21no—
113C2×C22⋊C4×C31—22no—
114C2×C22⋊C4×C31—21no—
115C2×C22⋊C4×C31—22no—
116C2×C42×C31C2311yescharacteristic
117C2×C42×C31C2311yescharacteristic
118C31×#107—22no—
119C31×#106—22no—
120C31×#104—24no—
121C31×#104—42no—
122C31×#103—22no—
123C31×#103—22no—
124C31×#103—22no—
125C31×#102—22no—
126C31×#101—24no—
127C31×#100—22no—
128C31×#100D414yescomplemented
129C31×#95—21no—
130C31×#95—22no—
131C24×C31—22no—
132C24×C31C2×D411yescharacteristic
133C24×C31C2×D412yes—
134#41×C31—22no—
135#41×C31—22no—
136#41×C31—22no—
137#41×C31—22no—
138#41×C31—44no—
139#41×C31—44no—
140#41×C31—44no—
141#41×C31—44no—
142#41×C31—44no—
143#41×C31—44no—
144C2×Q8×C31—21no—
145C2×Q8×C31—22no—
146C2×Q8×C31—21no—
147C2×Q8×C31—22no—
148C2×Q8×C31—22no—
149C2×Q8×C31—21no—
150C2×Q8×C31—21no—
151C2×Q8×C31—22no—
152C2×Q8×C31C2×D412yes—
153C2×D4×C31—24no—
154C2×D4×C31—24no—
155C2×D4×C31—24no—
156C2×D4×C31—22no—
157C2×D4×C31—22no—
158C2×D4×C31—22no—
159C2×D4×C31—22no—
160C2×D4×C31—22no—
161C2×D4×C31—42no—
162C2×D4×C31—44no—
163C2×D4×C31—44no—
164C2×D4×C31—44no—
165C2×D4×C31—44no—
166C2×D4×C31C2×D412yes—
167C2×D4×C31C2×D412yes—
168C2×D4×C31#4112yes—
169C22×C4×C31—21no—
170C22×C4×C31—21no—
171C22×C4×C31—22no—
172C22×C4×C31—22no—
173C22×C4×C31—24no—
174C22×C4×C31—21no—
175C22×C4×C31—22no—
176C22×C4×C31—22no—
177C22×C4×C31—22no—
178C22×C4×C31—22no—
179C22×C4×C31—22no—
180C22×C4×C31—21no—
181C22×C4×C31—21no—
182C22×C4×C31—21no—
183C22×C4×C31—42no—
184C22×C4×C31C2×D412yes—
185C22×C4×C31C2×D411yescharacteristic
186C22×C4×C31C2×D412yes—
187C22×C4×C31C2×D412yes—
188C22×C4×C31C2411yescharacteristic
189Q16×C31—44no—
190SD16×C31—44no—
191M16×C31—22no—
192M16×C31—22no—
193M16×C31—22no—
194M16×C31—44no—
195M16×C31C2×D414yes—
196C2×C8×C31—22no—
197C2×C8×C31—22no—
198C2×C8×C31—44no—
199C4⋊C4×C31—24no—
200C4⋊C4×C31—24no—
201C4⋊C4×C31—22no—
202C4⋊C4×C31—22no—
203C4⋊C4×C31—22no—
204C4⋊C4×C31—44no—
205C4⋊C4×C31—41no—
206C4⋊C4×C31—41no—
207C4⋊C4×C31C2×D412yes—
208C4⋊C4×C31C2×D412yes—
209C22⋊C4×C31—24no—
210C22⋊C4×C31—24no—
211C22⋊C4×C31—24no—
212C22⋊C4×C31—22no—
213C22⋊C4×C31—22no—
214C22⋊C4×C31—22no—
215C22⋊C4×C31—22no—
216C22⋊C4×C31—22no—
217C22⋊C4×C31—42no—
218C22⋊C4×C31—44no—
219C22⋊C4×C31—42no—
220C22⋊C4×C31—44no—
221C42×C31—22no—
222C42×C31—21no—
223C42×C31—21no—
224C42×C31—22no—
225#24415C3111yesmaximal, characteristic, direct product factor, Sylow
226C23×C31—22no—
227C23×C31—21no—
228C23×C31—21no—
229C23×C31—22no—
230C23×C31—22no—
231C23×C31—41no—
232C23×C31—41no—
233C23×C31—42no—
234C23×C31—42no—
235C23×C31—44no—
236C23×C31—82no—
237C23×C31—82no—
238C23×C31#12012yes—
239C23×C31#12114yes—
240C23×C31C22×D411yescharacteristic
241Q8×C31—22no—
242Q8×C31—22no—
243Q8×C31—42no—
244Q8×C31—42no—
245Q8×C31—42no—
246Q8×C31—42no—
247D4×C31—24no—
248D4×C31—22no—
249D4×C31—24no—
250D4×C31—22no—
251D4×C31—48no—
252D4×C31—48no—
253D4×C31—42no—
254D4×C31—42no—
255D4×C31—82no—
256D4×C31—82no—
257C2×C4×C31—21no—
258C2×C4×C31—21no—
259C2×C4×C31—22no—
260C2×C4×C31—22no—
261C2×C4×C31—22no—
262C2×C4×C31—22no—
263C2×C4×C31—22no—
264C2×C4×C31—22no—
265C2×C4×C31—22no—
266C2×C4×C31—22no—
267C2×C4×C31—22no—
268C2×C4×C31—22no—
269C2×C4×C31—22no—
270C2×C4×C31—44no—
271C2×C4×C31—44no—
272C2×C4×C31—41no—
273C2×C4×C31—44no—
274C2×C4×C31—41no—
275C2×C4×C31—42no—
276C2×C4×C31—42no—
277C2×C4×C31—41no—
278C2×C4×C31—42no—
279C2×C4×C31—41no—
280C2×C4×C31—41no—
281C2×C4×C31—41no—
282C2×C4×C31—41no—
283C2×C4×C31—41no—
284C2×C4×C31—41no—
285C2×C4×C31—82no—
286C2×C4×C31—82no—
287C2×C4×C31#12011yescharacteristic
288C2×C4×C31#12011yescharacteristic
289C2×C4×C31C22×D411yescharacteristic
290C2×C4×C31C22×D412yes—
291C2×C4×C31C2×#4111yescharacteristic
292C2×C4×C31H(5,𝔽2)12yes—
293C8×C31—24no—
294C8×C31—42no—
295#3485C2×C3111yescharacteristic
296#3056C2×C3112yescomplemented
297C2×#455C2×C3111yescharacteristic, complemented
298C2×#413C2×C3111yescharacteristic, complemented
299#2432C2×C3111yescharacteristic, complemented
300#2038C2×C3112yescomplemented
301#2005C2×C3112yescomplemented
302#1885C2×C3111yescharacteristic, complemented
303#1871C2×C3112yescomplemented
304#1856C2×C3111yescharacteristic, complemented
305#1792C2×C3111yescharacteristic, complemented
306C22×C31—21no—
307C22×C31—21no—
308C22×C31—22no—
309C22×C31—24no—
310C22×C31—22no—
311C22×C31—22no—
312C22×C31—44no—
313C22×C31—42no—
314C22×C31—42no—
315C22×C31—42no—
316C22×C31—42no—
317C22×C31—42no—
318C22×C31—81no—
319C22×C31—82no—
320C22×C31—81no—
321C22×C31—164no—
322C22×C31C2×#12011yescharacteristic
323C22×C31C2×#12111yescharacteristic
324C22×C31#53411yescharacteristic
325C4×C31—21no—
326C4×C31—21no—
327C4×C31—42no—
328C4×C31—42no—
329C4×C31—42no—
330C4×C31—42no—
331C4×C31—42no—
332C4×C31—81no—
333C4×C31—81no—
334C4×C31—81no—
335C4×C31D4212yes—
336C4×C31#54612yes—
337C2×H(5,𝔽2)C22×C3111yescharacteristic
338C22×#41C22×C3111yescharacteristic
339C2×#137C22×C3112yes—
340C22×M16C22×C3112yescomplemented
341#547—22no—
342#547—22no—
343#546—22no—
344#546—22no—
345#536C22×C3112yes—
346#534—22no—
347C2×C4⋊Q8C22×C3111yescharacteristic, complemented
348C2×#124C22×C3111yescharacteristic
349#525C22×C3112yes—
350C2×#123—22no—
351C2×C22⋊Q8C22×C3111yescharacteristic, complemented
352#518C22×C3111yescharacteristic
353#470—22no—
354#468—22no—
355#467—22no—
356#465—22no—
357#455—24no—
358#428C22×C3112yescomplemented
359C2×#104C22×C3112yes—
360C2×#104C22×C3112yes—
361#419C22×C3112yescomplemented
362#417C22×C3112yes—
363#413—24no—
364C2×#101C22×C3112yescomplemented
365C2×#100C22×C3111yescharacteristic
366C2×#100C22×C3111yescharacteristic
367#394—22no—
368#391C22×C3111yescharacteristic, complemented
369#390C22×C3111yescharacteristic
370#385—22no—
371C4×C22⋊C4C22×C3111yescharacteristic, complemented
372#339C22×C3112yescomplemented
373#339C22×C3112yescomplemented
374C2×C31—21no—
375C2×C31—21no—
376C2×C31—42no—
377C2×C31—44no—
378C2×C31—82no—
379C2×C31#241011yescharacteristic
380C2×C31#301811yescharacteristic
381C2×C31#302211yescharacteristic
382H(5,𝔽2)—22no—
383H(5,𝔽2)—22no—
384H(5,𝔽2)—42no—
385C2×#41—22no—
386C2×#41—22no—
387C2×#41—22no—
388C2×#41—22no—
389C2×#41—22no—
390C2×#41—42no—
391C2×#41C23×C3112yes—
392C22×Q8C23×C3111yescharacteristic
393C22×Q8C23×C3111yescharacteristic
394C22×D4—22no—
395C22×D4—22no—
396C22×D4D4×C3112yes—
397C22×D4C23×C3111yescharacteristic
398C23×C4D4×C3112yes—
399C23×C4C23×C3111yescharacteristic
400#137—24no—
401#137—42no—
402C2×Q16—22no—
403C2×SD16—22no—
404C2×M16—22no—
405C2×M16—24no—
406C2×M16—42no—
407C2×M16C23×C3112yes—
408C2×M16C23×C3112yes—
409C22×C8—22no—
410C4⋊Q8—22no—
411C4⋊Q8—42no—
412#124—22no—
413#124—22no—
414#124—22no—
415#123—24no—
416#123—24no—
417#123—24no—
418#123—22no—
419#123—24no—
420C22⋊Q8—22no—
421C22⋊Q8—22no—
422C22⋊Q8—22no—
423C22⋊Q8—22no—
424C22⋊Q8D4×C3112yes—
425C22⋊Q8D4×C3112yes—
426#121—24no—
427#121—24no—
428#121D4×C3112yes—
429#121D4×C3112yes—
430#120—24no—
431#120—22no—
432C4×D4—22no—
433C4×D4—22no—
434C4×D4—22no—
435C4×D4—22no—
436#117C23×C3112yes—
437C2×C4⋊C4—22no—
438C2×C4⋊C4—22no—
439C2×C4⋊C4—21no—
440C2×C4⋊C4—22no—
441C2×C4⋊C4C23×C3111yescharacteristic
442C2×C22⋊C4—22no—
443C2×C22⋊C4—22no—
444C2×C22⋊C4—22no—
445C2×C22⋊C4—21no—
446C2×C22⋊C4—22no—
447C2×C22⋊C4—21no—
448C2×C22⋊C4—22no—
449C2×C42C23×C3111yescharacteristic
450C2×C42C23×C3111yescharacteristic
451#107—22no—
452#106—22no—
453#104—24no—
454#104—42no—
455#103—22no—
456#103—22no—
457#103—22no—
458#102—22no—
459#101—24no—
460#100—22no—
461#100D4×C3114yescomplemented
462#95—21no—
463#95—22no—
464C31#2441511yescharacteristic, direct product factor, Sylow
465C24—22no—
466C24C2×D4×C3111yescharacteristic
467C24C2×D4×C3112yes—
468#41—22no—
469#41—22no—
470#41—22no—
471#41—22no—
472#41—44no—
473#41—44no—
474#41—44no—
475#41—44no—
476#41—44no—
477#41—44no—
478C2×Q8—21no—
479C2×Q8—22no—
480C2×Q8—21no—
481C2×Q8—22no—
482C2×Q8—22no—
483C2×Q8—21no—
484C2×Q8—21no—
485C2×Q8—22no—
486C2×Q8C2×D4×C3112yes—
487C2×D4—24no—
488C2×D4—24no—
489C2×D4—24no—
490C2×D4—22no—
491C2×D4—22no—
492C2×D4—22no—
493C2×D4—22no—
494C2×D4—22no—
495C2×D4—42no—
496C2×D4—44no—
497C2×D4—44no—
498C2×D4—44no—
499C2×D4—44no—
500C2×D4C2×D4×C3112yes—
501C2×D4C2×D4×C3112yes—
502C2×D4#41×C3112yes—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—22no—
506C22×C4—22no—
507C22×C4—24no—
508C22×C4—21no—
509C22×C4—22no—
510C22×C4—22no—
511C22×C4—22no—
512C22×C4—22no—
513C22×C4—22no—
514C22×C4—21no—
515C22×C4—21no—
516C22×C4—21no—
517C22×C4—42no—
518C22×C4C2×D4×C3112yes—
519C22×C4C2×D4×C3111yescharacteristic
520C22×C4C2×D4×C3112yes—
521C22×C4C2×D4×C3112yes—
522C22×C4C24×C3111yescharacteristic
523Q16—44no—
524SD16—44no—
525M16—22no—
526M16—22no—
527M16—22no—
528M16—44no—
529M16C2×D4×C3114yes—
530C2×C8—22no—
531C2×C8—22no—
532C2×C8—44no—
533C4⋊C4—24no—
534C4⋊C4—24no—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—44no—
539C4⋊C4—41no—
540C4⋊C4—41no—
541C4⋊C4C2×D4×C3112yes—
542C4⋊C4C2×D4×C3112yes—
543C22⋊C4—24no—
544C22⋊C4—24no—
545C22⋊C4—24no—
546C22⋊C4—22no—
547C22⋊C4—22no—
548C22⋊C4—22no—
549C22⋊C4—22no—
550C22⋊C4—22no—
551C22⋊C4—42no—
552C22⋊C4—44no—
553C22⋊C4—42no—
554C22⋊C4—44no—
555C42—22no—
556C42—21no—
557C42—21no—
558C42—22no—
559C23—22no—
560C23—21no—
561C23—21no—
562C23—22no—
563C23—22no—
564C23—41no—
565C23—41no—
566C23—42no—
567C23—42no—
568C23—44no—
569C23—82no—
570C23—82no—
571C23C31×#12012yes—
572C23C31×#12114yes—
573C23C22×D4×C3111yescharacteristic
574Q8—22no—
575Q8—22no—
576Q8—42no—
577Q8—42no—
578Q8—42no—
579Q8—42no—
580D4—24no—
581D4—22no—
582D4—24no—
583D4—22no—
584D4—48no—
585D4—48no—
586D4—42no—
587D4—42no—
588D4—82no—
589D4—82no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—41no—
606C2×C4—44no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—82no—
619C2×C4—82no—
620C2×C4C31×#12011yescharacteristic
621C2×C4C31×#12011yescharacteristic
622C2×C4C22×D4×C3111yescharacteristic
623C2×C4C22×D4×C3112yes—
624C2×C4C2×#41×C3111yescharacteristic
625C2×C4C31×H(5,𝔽2)12yes—
626C8—24no—
627C8—42no—
628C22—21no—
629C22—21no—
630C22—22no—
631C22—24no—
632C22—22no—
633C22—22no—
634C22—44no—
635C22—42no—
636C22—42no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—81no—
641C22—82no—
642C22—81no—
643C22—164no—
644C22C2×C31×#12011yescharacteristic
645C22C2×C31×#12111yescharacteristic
646C22C31×#53411yescharacteristic
647C4—21no—
648C4—21no—
649C4—42no—
650C4—42no—
651C4—42no—
652C4—42no—
653C4—42no—
654C4—81no—
655C4—81no—
656C4—81no—
657C4D42×C3112yes—
658C4C31×#54612yes—
659C2—21no—
660C2—21no—
661C2—42no—
662C2—44no—
663C2—82no—
664C2C31×#241011yescharacteristic
665C2C31×#301811yescharacteristic
666C2C31×#302211yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#3485C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#3056C2semidirect
3. C2×C31×#455C2semidirect
4. C2×C31×#413C2semidirect
5. C31×#2432C2semidirect
6. C31×#2038C2semidirect
7. C31×#2005C2semidirect
8. C31×#1885C2semidirect
9. C31×#1871C2semidirect
10. C31×#1856C2semidirect
11. C31×#1792C2semidirect
15. C22×M16×C31C22semidirect
22. C2×C31×C4⋊Q8C22semidirect
26. C2×C31×C22⋊Q8C22semidirect
33. C31×#428C22semidirect
36. C31×#419C22semidirect
39. C2×C31×#101C22semidirect
43. C31×#391C22semidirect
46. C4×C22⋊C4×C31C22semidirect
47. C31×#339C22semidirect
48. C31×#339C22semidirect
128. C31×#100D4semidirect
225. #24415C31direct (N × H)
296. #3056C2×C31semidirect
297. C2×#455C2×C31semidirect
298. C2×#413C2×C31semidirect
299. #2432C2×C31semidirect
300. #2038C2×C31semidirect
301. #2005C2×C31semidirect
302. #1885C2×C31semidirect
303. #1871C2×C31semidirect
304. #1856C2×C31semidirect
305. #1792C2×C31semidirect
340. C22×M16C22×C31semidirect
347. C2×C4⋊Q8C22×C31semidirect
351. C2×C22⋊Q8C22×C31semidirect
358. #428C22×C31semidirect
361. #419C22×C31semidirect
364. C2×#101C22×C31semidirect
368. #391C22×C31semidirect
371. C4×C22⋊C4C22×C31semidirect
372. #339C22×C31semidirect
373. #339C22×C31semidirect
461. #100D4×C31semidirect
464. C31#24415direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#34852
C31×#30562
C2×C31×#4552
C2×C31×#4132
C31×#24322
C31×#20382
C31×#20052
C31×#18852
C31×#18712
C31×#18562
C31×#17922
#2441531