Finite Group Databasebeta

← C31×#24322 #21481081

Subgroups — C31×#24322 #21481081

Order 7936.

988 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3503C211yesmaximal, characteristic
2C31×#3078C211yesmaximal, characteristic, complemented
3C4×C31×#137C211yesmaximal, characteristic
4C31×#2568C211yesmaximal, characteristic, complemented
5C31×#2061C212yesmaximal, complemented
6C31×#2028C212yesmaximal, complemented
7C31×#1997C211yesmaximal, characteristic, complemented
8C31×#1986C211yesmaximal, characteristic, complemented
9C31×#1882C212yesmaximal
10C31×#1860C212yesmaximal
11C31×#1806C211yesmaximal, characteristic
12C2×C31×#137C2211yescharacteristic
13Q82×C31—21no—
14C31×#557—21no—
15C31×#554—21no—
16C31×#554—22no—
17D4×Q8×C31—22no—
18D4×Q8×C31—21no—
19C31×#544—22no—
20C31×#544C2212yes—
21C31×#533—21no—
22C31×#533C2211yescharacteristic
23C2×C31×C4⋊Q8—21no—
24C2×C31×C4⋊Q8C2211yescharacteristic
25C31×#520C2212yes—
26C4×#41×C31C2211yescharacteristic
27C2×C4×Q8×C31C2211yescharacteristic
28C31×#479—22no—
29C31×#477—21no—
30C31×#475—21no—
31C31×#441—21no—
32C31×#440—21no—
33C4×Q16×C31—21no—
34C4×SD16×C31—21no—
35C31×#428C2212yes—
36C31×#428C2211yescharacteristic
37C31×#423C2211yescharacteristic
38C2×C31×#104C2212yes—
39C31×#419C2212yes—
40C31×#419C2211yescharacteristic
41C31×#417C2211yescharacteristic
42C4×M16×C31C2211yescharacteristic
43C31×#400—22no—
44C31×#391—22no—
45C31×#391C2212yes—
46C31×#382C2212yes—
47C4×C4⋊C4×C31C2211yescharacteristic
48C31×#343C2212yes—
49C31×#343C2212yes—
50C31×#339C2212yes—
51C31×#339C2212yes—
52C31×#339C2212yes—
53C2×#41×C31C2311yescharacteristic
54C22×Q8×C31—22no—
55C22×Q8×C31D412yes—
56C22×Q8×C31C2311yescharacteristic
57C31×#137—22no—
58C31×#137Q814yescomplemented
59C2×Q16×C31—21no—
60C2×SD16×C31—21no—
61C2×M16×C31C2311yescharacteristic
62C2×M16×C31C2312yes—
63C2×M16×C31C2311yescharacteristic
64C31×C4⋊Q8—22no—
65C31×C4⋊Q8—22no—
66C31×C4⋊Q8—24no—
67C31×C4⋊Q8—22no—
68C31×C4⋊Q8—22no—
69C31×C4⋊Q8—21no—
70C31×C4⋊Q8—21no—
71C31×C4⋊Q8—21no—
72C31×C4⋊Q8—21no—
73C31×C4⋊Q8D411yescharacteristic
74C31×C4⋊Q8D411yescharacteristic
75C31×#125—24no—
76C31×#125—22no—
77C31×#125—22no—
78C31×#125D412yes—
79C31×C22⋊Q8—22no—
80C31×C22⋊Q8—22no—
81C31×C22⋊Q8—24no—
82C31×C22⋊Q8—24no—
83C31×C22⋊Q8—21no—
84C31×C22⋊Q8—21no—
85C31×C22⋊Q8—24no—
86C4×Q8×C31—22no—
87C4×Q8×C31—22no—
88C4×Q8×C31—22no—
89C4×Q8×C31—21no—
90C4×Q8×C31—21no—
91C4×Q8×C31—21no—
92C4×Q8×C31—21no—
93C4×Q8×C31—21no—
94C4×Q8×C31—21no—
95C4×Q8×C31D411yescharacteristic
96C4×Q8×C31D411yescharacteristic
97C4×D4×C31—22no—
98C4×D4×C31—21no—
99C4×D4×C31—21no—
100C4×D4×C31—24no—
101C4×D4×C31D411yescharacteristic
102C4×D4×C31D411yescharacteristic
103C31×#117—22no—
104C31×#117—21no—
105C31×#117C2311yescharacteristic
106C31×#117C2312yes—
107C2×C4⋊C4×C31—22no—
108C2×C4⋊C4×C31—22no—
109C2×C4⋊C4×C31—21no—
110C2×C4⋊C4×C31—21no—
111C2×C4⋊C4×C31D412yes—
112C2×C4⋊C4×C31C2311yescharacteristic
113C2×C4⋊C4×C31C2312yes—
114C2×C42×C31—21no—
115C2×C42×C31—21no—
116C2×C42×C31C2312yes—
117C2×C42×C31C2311yescharacteristic
118C31×#107—21no—
119C31×#107—22no—
120C31×#106—21no—
121C31×#106—22no—
122C31×C4⋊C8—21no—
123C31×C4⋊C8—21no—
124C31×#104—24no—
125C31×#103—21no—
126C31×#103—21no—
127C31×#103—21no—
128C31×#103—22no—
129C31×#103—22no—
130C31×#102—21no—
131C31×#97—21no—
132C4×C8×C31—21no—
133C31×#95—21no—
134C31×#95—24no—
135C31×#95—24no—
136#41×C31—21no—
137#41×C31—21no—
138#41×C31C2×Q812yes—
139#41×C31#4112yes—
140C2×Q8×C31—24no—
141C2×Q8×C31—21no—
142C2×Q8×C31—22no—
143C2×Q8×C31—21no—
144C2×Q8×C31—22no—
145C2×Q8×C31—21no—
146C2×Q8×C31—21no—
147C2×Q8×C31—21no—
148C2×Q8×C31—21no—
149C2×Q8×C31—21no—
150C2×Q8×C31—44no—
151C2×Q8×C31—84no—
152C2×Q8×C31C2×D411yescharacteristic
153C2×Q8×C31C2×Q812yescomplemented
154C2×D4×C31—21no—
155C2×D4×C31C2×D411yescharacteristic
156C22×C4×C31—21no—
157C22×C4×C31—21no—
158C22×C4×C31—24no—
159C22×C4×C31—21no—
160C22×C4×C31—21no—
161C22×C4×C31—22no—
162C22×C4×C31—21no—
163C22×C4×C31C2×D411yescharacteristic
164C22×C4×C31C2×D414yes—
165C22×C4×C31C2×D411yescharacteristic
166C22×C4×C31C2411yescharacteristic
167Q16×C31—24no—
168SD16×C31—24no—
169M16×C31—21no—
170M16×C31—22no—
171M16×C31—21no—
172M16×C31—21no—
173M16×C31—22no—
174M16×C31C2×Q812yescomplemented
175C2×C8×C31—21no—
176C2×C8×C31—21no—
177C2×C8×C31—22no—
178C4⋊C4×C31—24no—
179C4⋊C4×C31—24no—
180C4⋊C4×C31—24no—
181C4⋊C4×C31—24no—
182C4⋊C4×C31—21no—
183C4⋊C4×C31—22no—
184C4⋊C4×C31—22no—
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—21no—
193C4⋊C4×C31—21no—
194C4⋊C4×C31—42no—
195C4⋊C4×C31—41no—
196C4⋊C4×C31—41no—
197C4⋊C4×C31C2×D412yes—
198C4⋊C4×C31C2×D411yescharacteristic
199C4⋊C4×C31C2×D412yes—
200C4⋊C4×C31C2×D411yescharacteristic
201C22⋊C4×C31—24no—
202C22⋊C4×C31—22no—
203C22⋊C4×C31—21no—
204C22⋊C4×C31—21no—
205C22⋊C4×C31—21no—
206C42×C31—24no—
207C42×C31—21no—
208C42×C31—21no—
209C42×C31—21no—
210C42×C31—22no—
211C42×C31—22no—
212C42×C31—22no—
213C42×C31—41no—
214C42×C31—41no—
215C42×C31C2×D411yescharacteristic
216C42×C31C2×D411yescharacteristic
217C42×C31C2×D411yescharacteristic
218C42×C31C2×D411yescharacteristic
219#24322C3111yesmaximal, characteristic, direct product factor, Sylow
220C23×C31—21no—
221C23×C31C22×D411yescharacteristic
222Q8×C31—21no—
223Q8×C31—22no—
224Q8×C31—22no—
225Q8×C31—42no—
226Q8×C31—41no—
227Q8×C31—41no—
228Q8×C31—44no—
229Q8×C31—42no—
230Q8×C31—82no—
231Q8×C31—84no—
232Q8×C31C22⋊Q812yes—
233D4×C31—22no—
234D4×C31—22no—
235D4×C31—21no—
236D4×C31C22⋊Q812yes—
237C2×C4×C31—21no—
238C2×C4×C31—22no—
239C2×C4×C31—22no—
240C2×C4×C31—22no—
241C2×C4×C31—22no—
242C2×C4×C31—21no—
243C2×C4×C31—21no—
244C2×C4×C31—24no—
245C2×C4×C31—22no—
246C2×C4×C31—22no—
247C2×C4×C31—41no—
248C2×C4×C31—42no—
249C2×C4×C31—41no—
250C2×C4×C31—41no—
251C2×C4×C31—42no—
252C2×C4×C31—44no—
253C2×C4×C31—42no—
254C2×C4×C31—42no—
255C2×C4×C31—42no—
256C2×C4×C31—41no—
257C2×C4×C31—42no—
258C2×C4×C31—84no—
259C2×C4×C31#12011yescharacteristic
260C2×C4×C31#12011yescharacteristic
261C2×C4×C31#12011yescharacteristic
262C2×C4×C31#12011yescharacteristic
263C2×C4×C31C22×D411yescharacteristic
264C2×C4×C31C22×D411yescharacteristic
265C2×C4×C31C22×Q811yescharacteristic
266C2×C4×C31C2×#4111yescharacteristic
267C2×C4×C312−1+411yescharacteristic
268C2×C4×C312−1+411yescharacteristic
269C8×C31—22no—
270C8×C31—41no—
271C8×C31—42no—
272#3503C2×C3111yescharacteristic
273#3078C2×C3111yescharacteristic, complemented
274C4×#137C2×C3111yescharacteristic
275#2568C2×C3111yescharacteristic, complemented
276#2061C2×C3112yescomplemented
277#2028C2×C3112yescomplemented
278#1997C2×C3111yescharacteristic, complemented
279#1986C2×C3111yescharacteristic, complemented
280#1882C2×C3112yes—
281#1860C2×C3112yes—
282#1806C2×C3111yescharacteristic
283C22×C31—21no—
284C22×C31—21no—
285C22×C31—22no—
286C22×C31—41no—
287C22×C31—41no—
288C22×C31C2×#12011yescharacteristic
289C22×C31#54811yescharacteristic
290C22×C31D4×Q811yescharacteristic
291C4×C31—22no—
292C4×C31—21no—
293C4×C31—21no—
294C4×C31—21no—
295C4×C31—42no—
296C4×C31—41no—
297C4×C31—44no—
298C4×C31—42no—
299C4×C31—41no—
300C4×C31—42no—
301C4×C31—82no—
302C4×C31—81no—
303C4×C31#45512yes—
304C4×C31C2×C22⋊Q811yescharacteristic
305C4×C31#53611yescharacteristic
306C4×C31#54811yescharacteristic
307C4×C31D4×Q811yescharacteristic
308C2×#137C22×C3111yescharacteristic
309Q82—21no—
310#557—21no—
311#554—21no—
312#554—22no—
313D4×Q8—22no—
314D4×Q8—21no—
315#544—22no—
316#544C22×C3112yes—
317#533—21no—
318#533C22×C3111yescharacteristic
319C2×C4⋊Q8—21no—
320C2×C4⋊Q8C22×C3111yescharacteristic
321#520C22×C3112yes—
322C4×#41C22×C3111yescharacteristic
323C2×C4×Q8C22×C3111yescharacteristic
324#479—22no—
325#477—21no—
326#475—21no—
327#441—21no—
328#440—21no—
329C4×Q16—21no—
330C4×SD16—21no—
331#428C22×C3112yes—
332#428C22×C3111yescharacteristic
333#423C22×C3111yescharacteristic
334C2×#104C22×C3112yes—
335#419C22×C3112yes—
336#419C22×C3111yescharacteristic
337#417C22×C3111yescharacteristic
338C4×M16C22×C3111yescharacteristic
339#400—22no—
340#391—22no—
341#391C22×C3112yes—
342#382C22×C3112yes—
343C4×C4⋊C4C22×C3111yescharacteristic
344#343C22×C3112yes—
345#343C22×C3112yes—
346#339C22×C3112yes—
347#339C22×C3112yes—
348#339C22×C3112yes—
349C2×C31—21no—
350C2×C31—21no—
351C2×C31—42no—
352C2×C31#240911yescharacteristic
353C2×C31C2×#45511yescharacteristic
354C2×C31#302011yescharacteristic
355C2×#41C23×C3111yescharacteristic
356C22×Q8—22no—
357C22×Q8D4×C3112yes—
358C22×Q8C23×C3111yescharacteristic
359#137—22no—
360#137Q8×C3114yescomplemented
361C2×Q16—21no—
362C2×SD16—21no—
363C2×M16C23×C3111yescharacteristic
364C2×M16C23×C3112yes—
365C2×M16C23×C3111yescharacteristic
366C4⋊Q8—22no—
367C4⋊Q8—22no—
368C4⋊Q8—24no—
369C4⋊Q8—22no—
370C4⋊Q8—22no—
371C4⋊Q8—21no—
372C4⋊Q8—21no—
373C4⋊Q8—21no—
374C4⋊Q8—21no—
375C4⋊Q8D4×C3111yescharacteristic
376C4⋊Q8D4×C3111yescharacteristic
377#125—24no—
378#125—22no—
379#125—22no—
380#125D4×C3112yes—
381C22⋊Q8—22no—
382C22⋊Q8—22no—
383C22⋊Q8—24no—
384C22⋊Q8—24no—
385C22⋊Q8—21no—
386C22⋊Q8—21no—
387C22⋊Q8—24no—
388C4×Q8—22no—
389C4×Q8—22no—
390C4×Q8—22no—
391C4×Q8—21no—
392C4×Q8—21no—
393C4×Q8—21no—
394C4×Q8—21no—
395C4×Q8—21no—
396C4×Q8—21no—
397C4×Q8D4×C3111yescharacteristic
398C4×Q8D4×C3111yescharacteristic
399C4×D4—22no—
400C4×D4—21no—
401C4×D4—21no—
402C4×D4—24no—
403C4×D4D4×C3111yescharacteristic
404C4×D4D4×C3111yescharacteristic
405#117—22no—
406#117—21no—
407#117C23×C3111yescharacteristic
408#117C23×C3112yes—
409C2×C4⋊C4—22no—
410C2×C4⋊C4—22no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4D4×C3112yes—
414C2×C4⋊C4C23×C3111yescharacteristic
415C2×C4⋊C4C23×C3112yes—
416C2×C42—21no—
417C2×C42—21no—
418C2×C42C23×C3112yes—
419C2×C42C23×C3111yescharacteristic
420#107—21no—
421#107—22no—
422#106—21no—
423#106—22no—
424C4⋊C8—21no—
425C4⋊C8—21no—
426#104—24no—
427#103—21no—
428#103—21no—
429#103—21no—
430#103—22no—
431#103—22no—
432#102—21no—
433#97—21no—
434C4×C8—21no—
435#95—21no—
436#95—24no—
437#95—24no—
438C31#2432211yescharacteristic, direct product factor, Sylow
439#41—21no—
440#41—21no—
441#41C2×Q8×C3112yes—
442#41#41×C3112yes—
443C2×Q8—24no—
444C2×Q8—21no—
445C2×Q8—22no—
446C2×Q8—21no—
447C2×Q8—22no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—21no—
453C2×Q8—44no—
454C2×Q8—84no—
455C2×Q8C2×D4×C3111yescharacteristic
456C2×Q8C2×Q8×C3112yescomplemented
457C2×D4—21no—
458C2×D4C2×D4×C3111yescharacteristic
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—24no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4C2×D4×C3111yescharacteristic
467C22×C4C2×D4×C3114yes—
468C22×C4C2×D4×C3111yescharacteristic
469C22×C4C24×C3111yescharacteristic
470Q16—24no—
471SD16—24no—
472M16—21no—
473M16—22no—
474M16—21no—
475M16—21no—
476M16—22no—
477M16C2×Q8×C3112yescomplemented
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—22no—
481C4⋊C4—24no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—21no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—42no—
498C4⋊C4—41no—
499C4⋊C4—41no—
500C4⋊C4C2×D4×C3112yes—
501C4⋊C4C2×D4×C3111yescharacteristic
502C4⋊C4C2×D4×C3112yes—
503C4⋊C4C2×D4×C3111yescharacteristic
504C22⋊C4—24no—
505C22⋊C4—22no—
506C22⋊C4—21no—
507C22⋊C4—21no—
508C22⋊C4—21no—
509C42—24no—
510C42—21no—
511C42—21no—
512C42—21no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—41no—
517C42—41no—
518C42C2×D4×C3111yescharacteristic
519C42C2×D4×C3111yescharacteristic
520C42C2×D4×C3111yescharacteristic
521C42C2×D4×C3111yescharacteristic
522C23—21no—
523C23C22×D4×C3111yescharacteristic
524Q8—21no—
525Q8—22no—
526Q8—22no—
527Q8—42no—
528Q8—44no—
529Q8—42no—
530Q8—41no—
531Q8—41no—
532Q8—82no—
533Q8—84no—
534Q8C31×C22⋊Q812yes—
535D4—22no—
536D4—22no—
537D4—21no—
538D4C31×C22⋊Q812yes—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—84no—
561C2×C4C31×#12011yescharacteristic
562C2×C4C31×#12011yescharacteristic
563C2×C4C31×#12011yescharacteristic
564C2×C4C31×#12011yescharacteristic
565C2×C4C22×D4×C3111yescharacteristic
566C2×C4C22×D4×C3111yescharacteristic
567C2×C4C22×Q8×C3111yescharacteristic
568C2×C4C2×#41×C3111yescharacteristic
569C2×C4C31×2−1+411yescharacteristic
570C2×C4C31×2−1+411yescharacteristic
571C8—22no—
572C8—41no—
573C8—42no—
574C22—21no—
575C22—21no—
576C22—22no—
577C22—41no—
578C22—41no—
579C22C2×C31×#12011yescharacteristic
580C22C31×#54811yescharacteristic
581C22D4×Q8×C3111yescharacteristic
582C4—22no—
583C4—21no—
584C4—21no—
585C4—21no—
586C4—42no—
587C4—41no—
588C4—44no—
589C4—42no—
590C4—41no—
591C4—42no—
592C4—82no—
593C4—81no—
594C4C31×#45512yes—
595C4C2×C31×C22⋊Q811yescharacteristic
596C4C31×#53611yescharacteristic
597C4C31×#54811yescharacteristic
598C4D4×Q8×C3111yescharacteristic
599C2—21no—
600C2—21no—
601C2—42no—
602C2C31×#240911yescharacteristic
603C2C2×C31×#45511yescharacteristic
604C2C31×#302011yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC2×#41×C31C23
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×#3078C2semidirect
4. C31×#2568C2semidirect
5. C31×#2061C2semidirect
6. C31×#2028C2semidirect
7. C31×#1997C2semidirect
8. C31×#1986C2semidirect
58. C31×#137Q8semidirect
153. C2×Q8×C31C2×Q8semidirect
174. M16×C31C2×Q8semidirect
219. #24322C31direct (N × H)
273. #3078C2×C31semidirect
275. #2568C2×C31semidirect
276. #2061C2×C31semidirect
277. #2028C2×C31semidirect
278. #1997C2×C31semidirect
279. #1986C2×C31semidirect
360. #137Q8×C31semidirect
438. C31#24322direct (N × H)
456. C2×Q8C2×Q8×C31semidirect
477. M16C2×Q8×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#35032
C31×#30782
C4×C31×#1372
C31×#25682
C31×#20612
C31×#20282
C31×#19972
C31×#19862
C31×#18822
C31×#18602
C31×#18062
#2432231