Finite Group Databasebeta

← C31×#25566 #21482325

Subgroups — C31×#25566 #21482325

Order 7936.

1122 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3491C211yesmaximal, characteristic
2C31×#2484C211yesmaximal, characteristic, complemented
3C31×#2413C212yesmaximal, complemented
4C31×#2062C211yesmaximal, characteristic, complemented
5C31×#2044C211yesmaximal, characteristic
6C31×#2030C211yesmaximal, characteristic, complemented
7C31×#2022C211yesmaximal, characteristic
8C31×#1899C212yesmaximal, complemented
9C31×#1890C212yesmaximal, complemented
10C31×#1867C212yesmaximal, complemented
11C31×#1740C211yesmaximal, characteristic, complemented
12C31×#552—22no—
13C31×#552—22no—
14C31×#551—22no—
15C31×#551—22no—
16C31×#543C2211yescharacteristic
17C31×#539—21no—
18C31×#539—21no—
19C31×#534C2211yescharacteristic
20C31×#529—21no—
21C31×#529—21no—
22C2×C31×#123C2212yes—
23C2×C31×#123C2212yes—
24C31×#518C2212yes—
25C2×C31×#117C2212yes—
26C2×C31×#117C2211yescharacteristic
27C22×C4⋊C4×C31C2211yescharacteristic
28C2×C31×#99C2212yes—
29C2×C31×#99C2212yes—
30C31×#400—21no—
31C31×#398—22no—
32C31×#398—21no—
33C31×#396—22no—
34C31×#389—22no—
35C31×#388—22no—
36C31×#387—21no—
37C31×#387—21no—
38C31×#386C2212yes—
39C31×#385C2212yes—
40C31×#385C2211yescharacteristic
41C31×#385C2211yescharacteristic
42C31×#381C2211yescharacteristic
43C31×#381C2212yes—
44C31×#381C2211yescharacteristic
45C31×#381C2211yescharacteristic
46C31×#342C2212yes—
47C31×#342C2211yescharacteristic
48C31×#342C2212yes—
49C31×#342C2212yes—
50C31×#342C2211yescharacteristic
51C22×D4×C31C2312yes—
52C23×C4×C31C2311yescharacteristic
53C23×C4×C31C2311yescharacteristic
54C23×C4×C31C2311yescharacteristic
55C31×#126—22no—
56C31×#126—22no—
57C31×#126—22no—
58C31×#126—22no—
59C31×#125—21no—
60C31×#125—21no—
61C31×#125—22no—
62C31×#125—22no—
63C31×#124—21no—
64C31×#124—21no—
65C31×#123—22no—
66C31×#123—22no—
67C31×#123—22no—
68C31×#123—22no—
69C31×#123—22no—
70C31×#123—22no—
71C31×#123—22no—
72C31×#123—24no—
73C31×#123—42no—
74C31×C22⋊Q8—22no—
75C31×C22⋊Q8—22no—
76C31×C22⋊Q8—21no—
77C31×C22⋊Q8—21no—
78C31×C22⋊Q8—22no—
79C31×C22⋊Q8—22no—
80C31×#121—21no—
81C31×#121—21no—
82C31×#120—22no—
83C4×Q8×C31—21no—
84C4×Q8×C31—21no—
85C4×D4×C31—21no—
86C4×D4×C31—21no—
87C4×D4×C31—22no—
88C4×D4×C31—22no—
89C4×D4×C31—22no—
90C4×D4×C31—22no—
91C31×#117—22no—
92C31×#117—22no—
93C31×#117—22no—
94C31×#117—22no—
95C31×#117—22no—
96C31×#117—24no—
97C31×#117—42no—
98C2×C4⋊C4×C31—22no—
99C2×C4⋊C4×C31—22no—
100C2×C4⋊C4×C31—22no—
101C2×C4⋊C4×C31—21no—
102C2×C4⋊C4×C31—22no—
103C2×C4⋊C4×C31—21no—
104C2×C4⋊C4×C31—21no—
105C2×C4⋊C4×C31—21no—
106C2×C4⋊C4×C31—22no—
107C2×C4⋊C4×C31D411yescharacteristic
108C2×C4⋊C4×C31D412yes—
109C2×C4⋊C4×C31D411yescharacteristic
110C2×C4⋊C4×C31D412yes—
111C2×C22⋊C4×C31—22no—
112C2×C22⋊C4×C31—22no—
113C2×C22⋊C4×C31—22no—
114C2×C22⋊C4×C31D412yes—
115C2×C22⋊C4×C31D412yes—
116C2×C22⋊C4×C31C2311yescharacteristic
117C2×C22⋊C4×C31C2311yescharacteristic
118C2×C22⋊C4×C31C2312yes—
119C2×C22⋊C4×C31C2311yescharacteristic
120C2×C22⋊C4×C31C2311yescharacteristic
121C2×C22⋊C4×C31C2312yes—
122C2×C22⋊C4×C31C2312yes—
123C2×C42×C31—22no—
124C2×C42×C31D411yescharacteristic
125C2×C42×C31D411yescharacteristic
126C31×#99—24no—
127C31×#99—42no—
128C31×#95—22no—
129C31×#95—21no—
130C31×#95—21no—
131C31×#95—21no—
132C31×#95—22no—
133C31×#95—22no—
134C31×#95—22no—
135C31×#95—21no—
136C31×#95—21no—
137C31×#95—21no—
138C31×#95—21no—
139C31×#95—21no—
140C24×C31C2×D412yes—
141C24×C31C2411yescharacteristic
142C2×Q8×C31—21no—
143C2×Q8×C31—21no—
144C2×D4×C31—22no—
145C2×D4×C31—22no—
146C2×D4×C31—22no—
147C2×D4×C31—22no—
148C2×D4×C31—22no—
149C2×D4×C31—22no—
150C2×D4×C31—21no—
151C2×D4×C31—21no—
152C22×C4×C31—22no—
153C22×C4×C31—21no—
154C22×C4×C31—21no—
155C22×C4×C31—22no—
156C22×C4×C31—21no—
157C22×C4×C31—22no—
158C22×C4×C31—22no—
159C22×C4×C31—22no—
160C22×C4×C31—21no—
161C22×C4×C31—21no—
162C22×C4×C31—21no—
163C22×C4×C31—22no—
164C22×C4×C31—22no—
165C22×C4×C31—22no—
166C22×C4×C31—21no—
167C22×C4×C31—21no—
168C22×C4×C31—42no—
169C22×C4×C31C2×D412yes—
170C22×C4×C31C2×D411yescharacteristic
171C22×C4×C31C2×D411yescharacteristic
172C22×C4×C31C2×D412yes—
173C22×C4×C31C2×D412yes—
174C22×C4×C31C2×D411yescharacteristic
175C22×C4×C31C2×D411yescharacteristic
176C22×C4×C31C2×D412yes—
177C22×C4×C31C2×D411yescharacteristic
178C22×C4×C31C2×D411yescharacteristic
179C22×C4×C31C2×D412yes—
180C4⋊C4×C31—22no—
181C4⋊C4×C31—22no—
182C4⋊C4×C31—22no—
183C4⋊C4×C31—22no—
184C4⋊C4×C31—21no—
185C4⋊C4×C31—21no—
186C4⋊C4×C31—21no—
187C4⋊C4×C31—21no—
188C4⋊C4×C31—22no—
189C4⋊C4×C31—22no—
190C4⋊C4×C31—22no—
191C4⋊C4×C31—22no—
192C4⋊C4×C31—22no—
193C4⋊C4×C31—22no—
194C4⋊C4×C31—44no—
195C4⋊C4×C31—44no—
196C4⋊C4×C31—44no—
197C4⋊C4×C31—42no—
198C4⋊C4×C31—42no—
199C4⋊C4×C31—41no—
200C4⋊C4×C31—41no—
201C4⋊C4×C31—42no—
202C22⋊C4×C31—22no—
203C22⋊C4×C31—22no—
204C22⋊C4×C31—22no—
205C22⋊C4×C31—22no—
206C22⋊C4×C31—22no—
207C22⋊C4×C31—22no—
208C22⋊C4×C31—22no—
209C22⋊C4×C31—22no—
210C22⋊C4×C31—22no—
211C22⋊C4×C31—22no—
212C22⋊C4×C31—22no—
213C22⋊C4×C31—22no—
214C22⋊C4×C31—22no—
215C22⋊C4×C31—22no—
216C22⋊C4×C31—22no—
217C22⋊C4×C31—21no—
218C22⋊C4×C31—22no—
219C22⋊C4×C31—21no—
220C22⋊C4×C31—44no—
221C22⋊C4×C31—44no—
222C22⋊C4×C31#4114yes—
223C42×C31—22no—
224C42×C31—22no—
225C42×C31—21no—
226C42×C31—21no—
227C42×C31—21no—
228C42×C31—21no—
229C42×C31—44no—
230#25566C3111yesmaximal, characteristic, direct product factor, Sylow
231C23×C31—22no—
232C23×C31—22no—
233C23×C31—22no—
234C23×C31—41no—
235C23×C31—41no—
236C23×C31—42no—
237C23×C31—42no—
238C23×C31C22×D411yescharacteristic
239C23×C31C22×D412yes—
240C23×C31C2×#4112yes—
241C23×C31H(5,𝔽2)11yescharacteristic
242C23×C312−1+411yescharacteristic
243Q8×C31—81no—
244Q8×C31—81no—
245D4×C31—42no—
246D4×C31—42no—
247D4×C31—42no—
248D4×C31—42no—
249D4×C31—81no—
250D4×C31—81no—
251C2×C4×C31—21no—
252C2×C4×C31—21no—
253C2×C4×C31—21no—
254C2×C4×C31—21no—
255C2×C4×C31—22no—
256C2×C4×C31—22no—
257C2×C4×C31—22no—
258C2×C4×C31—22no—
259C2×C4×C31—24no—
260C2×C4×C31—22no—
261C2×C4×C31—21no—
262C2×C4×C31—21no—
263C2×C4×C31—21no—
264C2×C4×C31—21no—
265C2×C4×C31—22no—
266C2×C4×C31—22no—
267C2×C4×C31—22no—
268C2×C4×C31—24no—
269C2×C4×C31—22no—
270C2×C4×C31—22no—
271C2×C4×C31—42no—
272C2×C4×C31—42no—
273C2×C4×C31—41no—
274C2×C4×C31—42no—
275C2×C4×C31—41no—
276C2×C4×C31—41no—
277C2×C4×C31—42no—
278C2×C4×C31—41no—
279C2×C4×C31—41no—
280C2×C4×C31—42no—
281C2×C4×C31—41no—
282C2×C4×C31—42no—
283C2×C4×C31—41no—
284C2×C4×C31—41no—
285C2×C4×C31—41no—
286C2×C4×C31—41no—
287C2×C4×C31—42no—
288C2×C4×C31—41no—
289C2×C4×C31—41no—
290C2×C4×C31—41no—
291C2×C4×C31—41no—
292C2×C4×C31—41no—
293C2×C4×C31—42no—
294C2×C4×C31—42no—
295C2×C4×C31—42no—
296C2×C4×C31—82no—
297C2×C4×C31—82no—
298C2×C4×C31#12011yescharacteristic
299C2×C4×C31#12011yescharacteristic
300C2×C4×C31#12012yes—
301#3491C2×C3111yescharacteristic
302#2484C2×C3111yescharacteristic, complemented
303#2413C2×C3112yescomplemented
304#2062C2×C3111yescharacteristic, complemented
305#2044C2×C3111yescharacteristic
306#2030C2×C3111yescharacteristic, complemented
307#2022C2×C3111yescharacteristic
308#1899C2×C3112yescomplemented
309#1890C2×C3112yescomplemented
310#1867C2×C3112yescomplemented
311#1740C2×C3111yescharacteristic, complemented
312C22×C31—21no—
313C22×C31—22no—
314C22×C31—21no—
315C22×C31—22no—
316C22×C31—42no—
317C22×C31—41no—
318C22×C31—41no—
319C22×C31—42no—
320C22×C31—42no—
321C22×C31—42no—
322C22×C31—82no—
323C22×C31—82no—
324C22×C31C2×#12011yescharacteristic
325C22×C31#53211yescharacteristic
326C22×C31#53711yescharacteristic
327C22×C31#54612yes—
328C22×C31#54712yes—
329C4×C31—21no—
330C4×C31—22no—
331C4×C31—21no—
332C4×C31—44no—
333C4×C31—42no—
334C4×C31—42no—
335C4×C31—41no—
336C4×C31—42no—
337C4×C31—41no—
338C4×C31—41no—
339C4×C31—42no—
340C4×C31—41no—
341C4×C31—82no—
342C4×C31—81no—
343C4×C31—81no—
344C4×C31—81no—
345#552—22no—
346#552—22no—
347#551—22no—
348#551—22no—
349#543C22×C3111yescharacteristic
350#539—21no—
351#539—21no—
352#534C22×C3111yescharacteristic
353#529—21no—
354#529—21no—
355C2×#123C22×C3112yes—
356C2×#123C22×C3112yes—
357#518C22×C3112yes—
358C2×#117C22×C3112yes—
359C2×#117C22×C3111yescharacteristic
360C22×C4⋊C4C22×C3111yescharacteristic
361C2×#99C22×C3112yes—
362C2×#99C22×C3112yes—
363#400—21no—
364#398—22no—
365#398—21no—
366#396—22no—
367#389—22no—
368#388—22no—
369#387—21no—
370#387—21no—
371#386C22×C3112yes—
372#385C22×C3112yes—
373#385C22×C3111yescharacteristic
374#385C22×C3111yescharacteristic
375#381C22×C3111yescharacteristic
376#381C22×C3112yes—
377#381C22×C3111yescharacteristic
378#381C22×C3111yescharacteristic
379#342C22×C3112yes—
380#342C22×C3111yescharacteristic
381#342C22×C3112yes—
382#342C22×C3112yes—
383#342C22×C3111yescharacteristic
384C2×C31—22no—
385C2×C31—21no—
386C2×C31—21no—
387C2×C31—22no—
388C2×C31—82no—
389C2×C31#243011yescharacteristic
390C2×C31#302511yescharacteristic
391C2×C31#302811yescharacteristic
392C22×D4C23×C3112yes—
393C23×C4C23×C3111yescharacteristic
394C23×C4C23×C3111yescharacteristic
395C23×C4C23×C3111yescharacteristic
396#126—22no—
397#126—22no—
398#126—22no—
399#126—22no—
400#125—21no—
401#125—21no—
402#125—22no—
403#125—22no—
404#124—21no—
405#124—21no—
406#123—22no—
407#123—22no—
408#123—22no—
409#123—22no—
410#123—22no—
411#123—22no—
412#123—22no—
413#123—24no—
414#123—42no—
415C22⋊Q8—22no—
416C22⋊Q8—22no—
417C22⋊Q8—21no—
418C22⋊Q8—21no—
419C22⋊Q8—22no—
420C22⋊Q8—22no—
421#121—21no—
422#121—21no—
423#120—22no—
424C4×Q8—21no—
425C4×Q8—21no—
426C4×D4—21no—
427C4×D4—21no—
428C4×D4—22no—
429C4×D4—22no—
430C4×D4—22no—
431C4×D4—22no—
432#117—22no—
433#117—22no—
434#117—22no—
435#117—22no—
436#117—22no—
437#117—24no—
438#117—42no—
439C2×C4⋊C4—21no—
440C2×C4⋊C4—22no—
441C2×C4⋊C4—22no—
442C2×C4⋊C4—21no—
443C2×C4⋊C4—22no—
444C2×C4⋊C4—22no—
445C2×C4⋊C4—21no—
446C2×C4⋊C4—21no—
447C2×C4⋊C4—22no—
448C2×C4⋊C4D4×C3111yescharacteristic
449C2×C4⋊C4D4×C3111yescharacteristic
450C2×C4⋊C4D4×C3112yes—
451C2×C4⋊C4D4×C3112yes—
452C2×C22⋊C4—22no—
453C2×C22⋊C4—22no—
454C2×C22⋊C4—22no—
455C2×C22⋊C4D4×C3112yes—
456C2×C22⋊C4D4×C3112yes—
457C2×C22⋊C4C23×C3112yes—
458C2×C22⋊C4C23×C3111yescharacteristic
459C2×C22⋊C4C23×C3111yescharacteristic
460C2×C22⋊C4C23×C3111yescharacteristic
461C2×C22⋊C4C23×C3111yescharacteristic
462C2×C22⋊C4C23×C3112yes—
463C2×C22⋊C4C23×C3112yes—
464C2×C42—22no—
465C2×C42D4×C3111yescharacteristic
466C2×C42D4×C3111yescharacteristic
467#99—24no—
468#99—42no—
469#95—21no—
470#95—21no—
471#95—21no—
472#95—21no—
473#95—21no—
474#95—22no—
475#95—22no—
476#95—21no—
477#95—21no—
478#95—21no—
479#95—22no—
480#95—22no—
481C31#2556611yescharacteristic, direct product factor, Sylow
482C24C2×D4×C3112yes—
483C24C24×C3111yescharacteristic
484C2×Q8—21no—
485C2×Q8—21no—
486C2×D4—22no—
487C2×D4—22no—
488C2×D4—22no—
489C2×D4—22no—
490C2×D4—22no—
491C2×D4—22no—
492C2×D4—21no—
493C2×D4—21no—
494C22×C4—22no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—22no—
498C22×C4—21no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—22no—
506C22×C4—22no—
507C22×C4—22no—
508C22×C4—21no—
509C22×C4—21no—
510C22×C4—42no—
511C22×C4C2×D4×C3112yes—
512C22×C4C2×D4×C3111yescharacteristic
513C22×C4C2×D4×C3111yescharacteristic
514C22×C4C2×D4×C3112yes—
515C22×C4C2×D4×C3112yes—
516C22×C4C2×D4×C3111yescharacteristic
517C22×C4C2×D4×C3111yescharacteristic
518C22×C4C2×D4×C3112yes—
519C22×C4C2×D4×C3111yescharacteristic
520C22×C4C2×D4×C3111yescharacteristic
521C22×C4C2×D4×C3112yes—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—44no—
537C4⋊C4—44no—
538C4⋊C4—44no—
539C4⋊C4—42no—
540C4⋊C4—42no—
541C4⋊C4—41no—
542C4⋊C4—41no—
543C4⋊C4—42no—
544C22⋊C4—22no—
545C22⋊C4—22no—
546C22⋊C4—22no—
547C22⋊C4—22no—
548C22⋊C4—22no—
549C22⋊C4—22no—
550C22⋊C4—22no—
551C22⋊C4—22no—
552C22⋊C4—22no—
553C22⋊C4—22no—
554C22⋊C4—22no—
555C22⋊C4—22no—
556C22⋊C4—22no—
557C22⋊C4—22no—
558C22⋊C4—22no—
559C22⋊C4—21no—
560C22⋊C4—22no—
561C22⋊C4—21no—
562C22⋊C4—44no—
563C22⋊C4—44no—
564C22⋊C4#41×C3114yes—
565C42—22no—
566C42—22no—
567C42—21no—
568C42—21no—
569C42—21no—
570C42—21no—
571C42—44no—
572C23—22no—
573C23—22no—
574C23—22no—
575C23—41no—
576C23—41no—
577C23—42no—
578C23—42no—
579C23C22×D4×C3111yescharacteristic
580C23C22×D4×C3112yes—
581C23C2×#41×C3112yes—
582C23C31×H(5,𝔽2)11yescharacteristic
583C23C31×2−1+411yescharacteristic
584Q8—81no—
585Q8—81no—
586D4—42no—
587D4—42no—
588D4—42no—
589D4—42no—
590D4—81no—
591D4—81no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—22no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—24no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—42no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—82no—
638C2×C4—82no—
639C2×C4C31×#12011yescharacteristic
640C2×C4C31×#12011yescharacteristic
641C2×C4C31×#12012yes—
642C22—21no—
643C22—22no—
644C22—21no—
645C22—22no—
646C22—42no—
647C22—41no—
648C22—41no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—82no—
653C22—82no—
654C22C2×C31×#12011yescharacteristic
655C22C31×#53211yescharacteristic
656C22C31×#53711yescharacteristic
657C22C31×#54612yes—
658C22C31×#54712yes—
659C4—21no—
660C4—22no—
661C4—21no—
662C4—44no—
663C4—42no—
664C4—42no—
665C4—41no—
666C4—42no—
667C4—41no—
668C4—41no—
669C4—42no—
670C4—41no—
671C4—82no—
672C4—81no—
673C4—81no—
674C4—81no—
675C2—22no—
676C2—21no—
677C2—21no—
678C2—22no—
679C2—82no—
680C2C31×#243011yescharacteristic
681C2C31×#302511yescharacteristic
682C2C31×#302811yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC24C24×C31
prime radicalC31×#534C22
LCSG ▷ C24 ▷ 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×#2484C2semidirect
3. C31×#2413C2semidirect
4. C31×#2062C2semidirect
6. C31×#2030C2semidirect
8. C31×#1899C2semidirect
9. C31×#1890C2semidirect
10. C31×#1867C2semidirect
11. C31×#1740C2semidirect
230. #25566C31direct (N × H)
302. #2484C2×C31semidirect
303. #2413C2×C31semidirect
304. #2062C2×C31semidirect
306. #2030C2×C31semidirect
308. #1899C2×C31semidirect
309. #1890C2×C31semidirect
310. #1867C2×C31semidirect
311. #1740C2×C31semidirect
481. C31#25566direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#34912
C31×#24842
C31×#24132
C31×#20622
C31×#20442
C31×#20302
C31×#20222
C31×#18992
C31×#18902
C31×#18672
C31×#17402
#2556631