Finite Group Databasebeta

← C31×#24086 #21480845

Subgroups — C31×#24086 #21480845

Order 7936.

1014 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3495C211yesmaximal, characteristic
2C31×#3092C211yesmaximal, characteristic
3C2×C31×#428C211yesmaximal, characteristic, complemented
4C31×#2502C211yesmaximal, characteristic, complemented
5C31×#2060C212yesmaximal, complemented
6C31×#2038C212yesmaximal, complemented
7C31×#1901C211yesmaximal, characteristic, complemented
8C31×#1871C211yesmaximal, characteristic
9C31×#1859C212yesmaximal, complemented
10C31×#1857C212yesmaximal, complemented
11C31×#1826C211yesmaximal, characteristic, complemented
12C22×M16×C31C2211yescharacteristic
13C31×#556—21no—
14C31×#554—21no—
15C31×#552—21no—
16C31×#552—22no—
17C31×#552—21no—
18C31×#551—22no—
19C31×#544C2211yescharacteristic
20C31×#542—21no—
21C31×#541—21no—
22C31×#536C2211yescharacteristic
23C31×#533—21no—
24C31×#529—21no—
25C31×#525C2211yescharacteristic
26C2×C31×C22⋊Q8C2211yescharacteristic
27C31×#520C2211yescharacteristic
28C31×#519C2211yescharacteristic
29C2×C31×#117C2211yescharacteristic
30C22×C4⋊C4×C31C2211yescharacteristic
31C31×#484—21no—
32C31×#483—21no—
33C31×#481—21no—
34C31×#480—21no—
35C31×#428—21no—
36C31×#428—21no—
37C31×#428C2212yes—
38C31×#428C2212yes—
39C31×#428C2212yes—
40C2×C31×#107—21no—
41C2×C31×#106—21no—
42C2×C31×#104C2212yes—
43C31×#417C2212yes—
44C2×C31×#101C2212yes—
45C31×#407C2211yescharacteristic
46C31×#398—22no—
47C31×#391C2212yes—
48C31×#385—22no—
49C31×#384C2212yes—
50C4×C22⋊C4×C31C2211yescharacteristic
51C31×#343C2212yes—
52C31×#339C2212yes—
53C31×#339C2212yes—
54C31×#339C2212yes—
55C2×#41×C31C2311yescharacteristic
56C22×Q8×C31C2311yescharacteristic
57C23×C4×C31—21no—
58C23×C4×C31C2311yescharacteristic
59C2×M16×C31—21no—
60C2×M16×C31—24no—
61C2×M16×C31C2312yes—
62C2×M16×C31C2312yes—
63C22×C8×C31—21no—
64C31×C4⋊Q8—21no—
65C31×C4⋊Q8—21no—
66C31×C4⋊Q8—21no—
67C31×#126—22no—
68C31×#126—22no—
69C31×#126—22no—
70C31×#126—22no—
71C31×#125—22no—
72C31×#125—22no—
73C31×#125—22no—
74C31×#125—21no—
75C31×#125—21no—
76C31×#124—22no—
77C31×#124—21no—
78C31×#123—22no—
79C31×#123—22no—
80C31×#123—21no—
81C31×#123—22no—
82C31×C22⋊Q8—22no—
83C31×C22⋊Q8—22no—
84C31×C22⋊Q8—21no—
85C31×C22⋊Q8—21no—
86C31×C22⋊Q8—21no—
87C31×C22⋊Q8—21no—
88C31×C22⋊Q8—22no—
89C31×C22⋊Q8—21no—
90C31×C22⋊Q8—21no—
91C31×C22⋊Q8D411yescharacteristic
92C31×C22⋊Q8D411yescharacteristic
93C31×#121D411yescharacteristic
94C31×#121D411yescharacteristic
95C31×#120—21no—
96C4×Q8×C31—22no—
97C4×Q8×C31—21no—
98C4×Q8×C31—21no—
99C4×Q8×C31—21no—
100C4×Q8×C31—21no—
101C4×Q8×C31—21no—
102C4×D4×C31—22no—
103C4×D4×C31—21no—
104C4×D4×C31—22no—
105C4×D4×C31—21no—
106C4×D4×C31—21no—
107C31×#117—22no—
108C31×#117—22no—
109C31×#117—21no—
110C31×#117—21no—
111C31×#117—22no—
112C31×#117—21no—
113C31×#117D411yescharacteristic
114C31×#117D411yescharacteristic
115C31×#117C2311yescharacteristic
116C31×#117C2311yescharacteristic
117C31×#117C2311yescharacteristic
118C2×C4⋊C4×C31—21no—
119C2×C4⋊C4×C31—21no—
120C2×C4⋊C4×C31—21no—
121C2×C4⋊C4×C31—21no—
122C2×C4⋊C4×C31—21no—
123C2×C4⋊C4×C31—21no—
124C2×C4⋊C4×C31—24no—
125C2×C4⋊C4×C31D412yes—
126C2×C4⋊C4×C31D411yescharacteristic
127C2×C4⋊C4×C31D411yescharacteristic
128C2×C4⋊C4×C31C2312yes—
129C2×C4⋊C4×C31C2311yescharacteristic
130C2×C22⋊C4×C31—21no—
131C2×C22⋊C4×C31—21no—
132C2×C22⋊C4×C31D412yes—
133C2×C42×C31—21no—
134C2×C42×C31C2312yes—
135C31×#107—22no—
136C31×#107—22no—
137C31×#107—22no—
138C31×#106—22no—
139C31×#106—22no—
140C31×#106—22no—
141C31×#104—24no—
142C31×#103—22no—
143C31×#102—22no—
144C31×#101—24no—
145C31×C22⋊C8—21no—
146C31×C22⋊C8—21no—
147C31×#95—21no—
148C31×#95—24no—
149C24×C31C2×D411yescharacteristic
150#41×C31—22no—
151#41×C31—22no—
152C2×Q8×C31—22no—
153C2×Q8×C31—21no—
154C2×Q8×C31—21no—
155C2×Q8×C31—22no—
156C2×Q8×C31—22no—
157C2×Q8×C31C2×D411yescharacteristic
158C2×D4×C31—21no—
159C2×D4×C31—21no—
160C2×D4×C31—21no—
161C2×D4×C31C2×D411yescharacteristic
162C22×C4×C31—22no—
163C22×C4×C31—22no—
164C22×C4×C31—21no—
165C22×C4×C31—22no—
166C22×C4×C31—22no—
167C22×C4×C31—21no—
168C22×C4×C31—21no—
169C22×C4×C31—22no—
170C22×C4×C31—41no—
171C22×C4×C31—41no—
172C22×C4×C31—82no—
173C22×C4×C31C2×D412yes—
174C22×C4×C31C2×D411yescharacteristic
175C22×C4×C31C2×D411yescharacteristic
176C22×C4×C31C2×D411yescharacteristic
177C22×C4×C31C2×D411yescharacteristic
178C22×C4×C31C2×D411yescharacteristic
179C22×C4×C31C2×D412yes—
180C22×C4×C31C2411yescharacteristic
181M16×C31—24no—
182M16×C31—22no—
183M16×C31—22no—
184M16×C31—22no—
185M16×C31#4114yes—
186C2×C8×C31—24no—
187C2×C8×C31—22no—
188C2×C8×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—22no—
195C4⋊C4×C31—22no—
196C4⋊C4×C31—22no—
197C4⋊C4×C31—21no—
198C4⋊C4×C31—21no—
199C4⋊C4×C31—22no—
200C4⋊C4×C31—22no—
201C4⋊C4×C31—21no—
202C4⋊C4×C31—21no—
203C4⋊C4×C31—21no—
204C4⋊C4×C31—21no—
205C4⋊C4×C31—22no—
206C4⋊C4×C31—22no—
207C4⋊C4×C31—41no—
208C4⋊C4×C31—41no—
209C4⋊C4×C31—84no—
210C4⋊C4×C31C2×D411yescharacteristic
211C4⋊C4×C31C2×D411yescharacteristic
212C4⋊C4×C31C2×D411yescharacteristic
213C4⋊C4×C31C2×D411yescharacteristic
214C4⋊C4×C31C2×D411yescharacteristic
215C4⋊C4×C31C2×D411yescharacteristic
216C22⋊C4×C31—24no—
217C22⋊C4×C31—22no—
218C22⋊C4×C31—22no—
219C22⋊C4×C31—22no—
220C22⋊C4×C31—21no—
221C22⋊C4×C31—21no—
222C22⋊C4×C31—22no—
223C22⋊C4×C31—22no—
224C22⋊C4×C31—21no—
225C22⋊C4×C31—21no—
226C22⋊C4×C31—21no—
227C22⋊C4×C31—21no—
228C22⋊C4×C31—21no—
229C22⋊C4×C31—21no—
230C22⋊C4×C31—21no—
231C42×C31—22no—
232C42×C31—22no—
233C42×C31—22no—
234C42×C31—21no—
235C42×C31—21no—
236C42×C31—21no—
237C42×C31—21no—
238C42×C31—21no—
239C42×C31—22no—
240C42×C31—22no—
241#24086C3111yesmaximal, characteristic, direct product factor, Sylow
242C23×C31—22no—
243C23×C31—22no—
244C23×C31—21no—
245C23×C31—41no—
246C23×C31#12011yescharacteristic
247C23×C31#12011yescharacteristic
248C23×C31C22×D411yescharacteristic
249Q8×C31—22no—
250Q8×C31—41no—
251Q8×C31—41no—
252Q8×C31—42no—
253Q8×C31—82no—
254D4×C31—22no—
255D4×C31—42no—
256D4×C31—42no—
257D4×C31—81no—
258C2×C4×C31—22no—
259C2×C4×C31—21no—
260C2×C4×C31—21no—
261C2×C4×C31—21no—
262C2×C4×C31—21no—
263C2×C4×C31—22no—
264C2×C4×C31—22no—
265C2×C4×C31—24no—
266C2×C4×C31—21no—
267C2×C4×C31—21no—
268C2×C4×C31—21no—
269C2×C4×C31—21no—
270C2×C4×C31—21no—
271C2×C4×C31—22no—
272C2×C4×C31—21no—
273C2×C4×C31—21no—
274C2×C4×C31—22no—
275C2×C4×C31—24no—
276C2×C4×C31—42no—
277C2×C4×C31—42no—
278C2×C4×C31—42no—
279C2×C4×C31—41no—
280C2×C4×C31—41no—
281C2×C4×C31—42no—
282C2×C4×C31—44no—
283C2×C4×C31—42no—
284C2×C4×C31—81no—
285C2×C4×C31—82no—
286C2×C4×C31—82no—
287C2×C4×C31—82no—
288C2×C4×C31#12011yescharacteristic
289C2×C4×C31#12011yescharacteristic
290C2×C4×C31C22×D411yescharacteristic
291C2×C4×C31C22×D411yescharacteristic
292C2×C4×C31C2×#4112yes—
293C2×C4×C312−1+412yes—
294C8×C31—24no—
295C8×C31—42no—
296#3495C2×C3111yescharacteristic
297#3092C2×C3111yescharacteristic
298C2×#428C2×C3111yescharacteristic, complemented
299#2502C2×C3111yescharacteristic, complemented
300#2060C2×C3112yescomplemented
301#2038C2×C3112yescomplemented
302#1901C2×C3111yescharacteristic, complemented
303#1871C2×C3111yescharacteristic
304#1859C2×C3112yescomplemented
305#1857C2×C3112yescomplemented
306#1826C2×C3111yescharacteristic, complemented
307C22×C31—21no—
308C22×C31—21no—
309C22×C31—21no—
310C22×C31—21no—
311C22×C31—21no—
312C22×C31—21no—
313C22×C31—21no—
314C22×C31—44no—
315C22×C31—41no—
316C22×C31—41no—
317C22×C31—41no—
318C22×C31#45612yes—
319C22×C31C2×#12011yescharacteristic
320C22×C31#54712yes—
321C4×C31—21no—
322C4×C31—21no—
323C4×C31—44no—
324C4×C31—42no—
325C4×C31—41no—
326C4×C31—41no—
327C4×C31—41no—
328C4×C31—41no—
329C4×C31—42no—
330C4×C31—41no—
331C4×C31—41no—
332C4×C31—41no—
333C4×C31—82no—
334C4×C31#52511yescharacteristic
335C4×C31#53611yescharacteristic
336C4×C31#54712yes—
337C22×M16C22×C3111yescharacteristic
338#556—21no—
339#554—21no—
340#552—21no—
341#552—22no—
342#552—21no—
343#551—22no—
344#544C22×C3111yescharacteristic
345#542—21no—
346#541—21no—
347#536C22×C3111yescharacteristic
348#533—21no—
349#529—21no—
350#525C22×C3111yescharacteristic
351C2×C22⋊Q8C22×C3111yescharacteristic
352#520C22×C3111yescharacteristic
353#519C22×C3111yescharacteristic
354C2×#117C22×C3111yescharacteristic
355C22×C4⋊C4C22×C3111yescharacteristic
356#484—21no—
357#483—21no—
358#481—21no—
359#480—21no—
360#428—21no—
361#428—21no—
362#428C22×C3112yes—
363#428C22×C3112yes—
364#428C22×C3112yes—
365C2×#107—21no—
366C2×#106—21no—
367C2×#104C22×C3112yes—
368#417C22×C3112yes—
369C2×#101C22×C3112yes—
370#407C22×C3111yescharacteristic
371#398—22no—
372#391C22×C3112yes—
373#385—22no—
374#384C22×C3112yes—
375C4×C22⋊C4C22×C3111yescharacteristic
376#343C22×C3112yes—
377#339C22×C3112yes—
378#339C22×C3112yes—
379#339C22×C3112yes—
380C2×C31—22no—
381C2×C31—22no—
382C2×C31—41no—
383C2×C31—81no—
384C2×C31#240611yescharacteristic
385C2×C31C2×#45611yescharacteristic
386C2×C31#302111yescharacteristic
387C2×#41C23×C3111yescharacteristic
388C22×Q8C23×C3111yescharacteristic
389C23×C4—21no—
390C23×C4C23×C3111yescharacteristic
391C2×M16—21no—
392C2×M16—24no—
393C2×M16C23×C3112yes—
394C2×M16C23×C3112yes—
395C22×C8—21no—
396C4⋊Q8—21no—
397C4⋊Q8—21no—
398C4⋊Q8—21no—
399#126—22no—
400#126—22no—
401#126—22no—
402#126—22no—
403#125—22no—
404#125—22no—
405#125—22no—
406#125—21no—
407#125—21no—
408#124—22no—
409#124—21no—
410#123—22no—
411#123—22no—
412#123—21no—
413#123—22no—
414C22⋊Q8—22no—
415C22⋊Q8—22no—
416C22⋊Q8—21no—
417C22⋊Q8—21no—
418C22⋊Q8—21no—
419C22⋊Q8—21no—
420C22⋊Q8—22no—
421C22⋊Q8—21no—
422C22⋊Q8—21no—
423C22⋊Q8D4×C3111yescharacteristic
424C22⋊Q8D4×C3111yescharacteristic
425#121D4×C3111yescharacteristic
426#121D4×C3111yescharacteristic
427#120—21no—
428C4×Q8—22no—
429C4×Q8—21no—
430C4×Q8—21no—
431C4×Q8—21no—
432C4×Q8—21no—
433C4×Q8—21no—
434C4×D4—22no—
435C4×D4—21no—
436C4×D4—22no—
437C4×D4—21no—
438C4×D4—21no—
439#117—22no—
440#117—22no—
441#117—21no—
442#117—21no—
443#117—22no—
444#117—21no—
445#117D4×C3111yescharacteristic
446#117D4×C3111yescharacteristic
447#117C23×C3111yescharacteristic
448#117C23×C3111yescharacteristic
449#117C23×C3111yescharacteristic
450C2×C4⋊C4—21no—
451C2×C4⋊C4—21no—
452C2×C4⋊C4—21no—
453C2×C4⋊C4—21no—
454C2×C4⋊C4—21no—
455C2×C4⋊C4—21no—
456C2×C4⋊C4—24no—
457C2×C4⋊C4D4×C3112yes—
458C2×C4⋊C4D4×C3111yescharacteristic
459C2×C4⋊C4D4×C3111yescharacteristic
460C2×C4⋊C4C23×C3112yes—
461C2×C4⋊C4C23×C3111yescharacteristic
462C2×C22⋊C4—21no—
463C2×C22⋊C4—21no—
464C2×C22⋊C4D4×C3112yes—
465C2×C42—21no—
466C2×C42C23×C3112yes—
467#107—22no—
468#107—22no—
469#107—22no—
470#106—22no—
471#106—22no—
472#106—22no—
473#104—24no—
474#103—22no—
475#102—22no—
476#101—24no—
477C22⋊C8—21no—
478C22⋊C8—21no—
479#95—21no—
480#95—24no—
481C31#2408611yescharacteristic, direct product factor, Sylow
482C24C2×D4×C3111yescharacteristic
483#41—22no—
484#41—22no—
485C2×Q8—22no—
486C2×Q8—21no—
487C2×Q8—21no—
488C2×Q8—22no—
489C2×Q8—22no—
490C2×Q8C2×D4×C3111yescharacteristic
491C2×D4—21no—
492C2×D4—21no—
493C2×D4—21no—
494C2×D4C2×D4×C3111yescharacteristic
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—21no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—22no—
503C22×C4—41no—
504C22×C4—41no—
505C22×C4—82no—
506C22×C4C2×D4×C3112yes—
507C22×C4C2×D4×C3111yescharacteristic
508C22×C4C2×D4×C3111yescharacteristic
509C22×C4C2×D4×C3111yescharacteristic
510C22×C4C2×D4×C3111yescharacteristic
511C22×C4C2×D4×C3111yescharacteristic
512C22×C4C2×D4×C3112yes—
513C22×C4C24×C3111yescharacteristic
514M16—24no—
515M16—22no—
516M16—22no—
517M16—22no—
518M16#41×C3114yes—
519C2×C8—24no—
520C2×C8—22no—
521C2×C8—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—41no—
541C4⋊C4—41no—
542C4⋊C4—84no—
543C4⋊C4C2×D4×C3111yescharacteristic
544C4⋊C4C2×D4×C3111yescharacteristic
545C4⋊C4C2×D4×C3111yescharacteristic
546C4⋊C4C2×D4×C3111yescharacteristic
547C4⋊C4C2×D4×C3111yescharacteristic
548C4⋊C4C2×D4×C3111yescharacteristic
549C22⋊C4—24no—
550C22⋊C4—22no—
551C22⋊C4—22no—
552C22⋊C4—22no—
553C22⋊C4—21no—
554C22⋊C4—21no—
555C22⋊C4—22no—
556C22⋊C4—22no—
557C22⋊C4—21no—
558C22⋊C4—21no—
559C22⋊C4—21no—
560C22⋊C4—21no—
561C22⋊C4—21no—
562C22⋊C4—21no—
563C22⋊C4—21no—
564C42—22no—
565C42—22no—
566C42—22no—
567C42—21no—
568C42—21no—
569C42—21no—
570C42—21no—
571C42—21no—
572C42—22no—
573C42—22no—
574C23—22no—
575C23—22no—
576C23—21no—
577C23—41no—
578C23C31×#12011yescharacteristic
579C23C31×#12011yescharacteristic
580C23C22×D4×C3111yescharacteristic
581Q8—22no—
582Q8—41no—
583Q8—41no—
584Q8—42no—
585Q8—82no—
586D4—22no—
587D4—42no—
588D4—42no—
589D4—81no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—24no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—22no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—22no—
607C2×C4—24no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—81no—
617C2×C4—82no—
618C2×C4—82no—
619C2×C4—82no—
620C2×C4C31×#12011yescharacteristic
621C2×C4C31×#12011yescharacteristic
622C2×C4C22×D4×C3111yescharacteristic
623C2×C4C22×D4×C3111yescharacteristic
624C2×C4C2×#41×C3112yes—
625C2×C4C31×2−1+412yes—
626C8—24no—
627C8—42no—
628C22—21no—
629C22—21no—
630C22—21no—
631C22—21no—
632C22—21no—
633C22—21no—
634C22—21no—
635C22—44no—
636C22—41no—
637C22—41no—
638C22—41no—
639C22C31×#45612yes—
640C22C2×C31×#12011yescharacteristic
641C22C31×#54712yes—
642C4—21no—
643C4—21no—
644C4—44no—
645C4—42no—
646C4—41no—
647C4—41no—
648C4—41no—
649C4—41no—
650C4—42no—
651C4—41no—
652C4—41no—
653C4—41no—
654C4—82no—
655C4C31×#52511yescharacteristic
656C4C31×#53611yescharacteristic
657C4C31×#54712yes—
658C2—22no—
659C2—22no—
660C2—41no—
661C2—81no—
662C2C31×#240611yescharacteristic
663C2C2×C31×#45611yescharacteristic
664C2C31×#302111yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#525C22
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
3. C2×C31×#428C2semidirect
4. C31×#2502C2semidirect
5. C31×#2060C2semidirect
6. C31×#2038C2semidirect
7. C31×#1901C2semidirect
9. C31×#1859C2semidirect
10. C31×#1857C2semidirect
11. C31×#1826C2semidirect
241. #24086C31direct (N × H)
298. C2×#428C2×C31semidirect
299. #2502C2×C31semidirect
300. #2060C2×C31semidirect
301. #2038C2×C31semidirect
302. #1901C2×C31semidirect
304. #1859C2×C31semidirect
305. #1857C2×C31semidirect
306. #1826C2×C31semidirect
481. C31#24086direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#34952
C31×#30922
C2×C31×#4282
C31×#25022
C31×#20602
C31×#20382
C31×#19012
C31×#18712
C31×#18592
C31×#18572
C31×#18262
#2408631