Finite Group Databasebeta

← C31×#20864 #21477623

Subgroups — C31×#20864 #21477623

Order 7936.

1422 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#520C211yesmaximal, characteristic
2C31×#2911C211yesmaximal, characteristic, complemented
3C2×C31×#421C211yesmaximal, characteristic
4C31×#2896C211yesmaximal, characteristic
5C31×#2317C211yesmaximal, characteristic, complemented
6C31×#1860C214yesmaximal, complemented
7C31×#1859C214yesmaximal, complemented
8C2×C31×#343C211yesmaximal, characteristic, complemented
9C31×#1733C211yesmaximal, characteristic, complemented
10C22×#41×C31C2211yescharacteristic
11C22×M16×C31C2211yescharacteristic
12C22×M16×C31C2211yescharacteristic
13C31×#520—22no—
14C31×#520—22no—
15C31×#520—22no—
16C31×#520C2214yes—
17C2×C4×Q8×C31—21no—
18C2×C4×D4×C31—21no—
19C2×C4×D4×C31—22no—
20C2×C31×#117—21no—
21C2×C31×#117C2211yescharacteristic
22C2×C31×#117C2211yescharacteristic
23C2×C31×#117C2211yescharacteristic
24C22×C4⋊C4×C31—21no—
25C22×C4⋊C4×C31C2211yescharacteristic
26C31×#428—21no—
27C31×#428—21no—
28C31×#428C2212yescomplemented
29C31×#428C2212yescomplemented
30C31×#427—22no—
31C31×#421C418yescomplemented
32C2×C31×#104C2214yes—
33C31×#419—21no—
34C31×#418—21no—
35C31×#417C2212yes—
36C31×#417C2212yes—
37C31×#416—22no—
38C31×#389—22no—
39C31×#385—22no—
40C31×#384C2212yescomplemented
41C31×#382C2212yescomplemented
42C31×#343C2214yescomplemented
43C31×#339C2214yescomplemented
44C2×#41×C31—21no—
45C2×#41×C31—21no—
46C2×#41×C31—22no—
47C2×#41×C31—22no—
48C2×#41×C31C2×C414yes—
49C2×#41×C31D412yes—
50C2×#41×C31Q814yes—
51C2×#41×C31C2312yes—
52C22×Q8×C31D411yescharacteristic
53C22×D4×C31—21no—
54C22×D4×C31D411yescharacteristic
55C23×C4×C31—21no—
56C23×C4×C31—22no—
57C23×C4×C31D412yes—
58C23×C4×C31C2311yescharacteristic
59C2×M16×C31—21no—
60C2×M16×C31—22no—
61C2×M16×C31—21no—
62C2×M16×C31—22no—
63C2×M16×C31—22no—
64C2×M16×C31C2×C414yescomplemented
65C2×M16×C31C2312yes—
66C2×M16×C31C2312yes—
67C22×C8×C31—21no—
68C22×C8×C31—21no—
69C4×Q8×C31—24no—
70C4×Q8×C31—24no—
71C4×D4×C31—28no—
72C4×D4×C31—24no—
73C4×D4×C31—24no—
74C4×D4×C31—28no—
75C31×#117—24no—
76C31×#117—21no—
77C31×#117—21no—
78C31×#117—22no—
79C31×#117—24no—
80C31×#117—21no—
81C31×#117—21no—
82C31×#117—21no—
83C31×#117—21no—
84C31×#117C2×C414yescomplemented
85C31×#117D412yes—
86C31×#117C2312yes—
87C2×C4⋊C4×C31—22no—
88C2×C4⋊C4×C31—24no—
89C2×C4⋊C4×C31—21no—
90C2×C4⋊C4×C31—21no—
91C2×C4⋊C4×C31—21no—
92C2×C4⋊C4×C31—21no—
93C2×C4⋊C4×C31—21no—
94C2×C4⋊C4×C31—24no—
95C2×C4⋊C4×C31—24no—
96C2×C4⋊C4×C31D412yes—
97C2×C4⋊C4×C31D411yescharacteristic
98C2×C4⋊C4×C31D411yescharacteristic
99C2×C4⋊C4×C31C2311yescharacteristic
100C2×C4⋊C4×C31C2311yescharacteristic
101C2×C4⋊C4×C31C2312yes—
102C2×C22⋊C4×C31—22no—
103C2×C22⋊C4×C31—21no—
104C2×C22⋊C4×C31—21no—
105C2×C22⋊C4×C31—21no—
106C2×C42×C31—22no—
107C2×C42×C31—21no—
108C2×C42×C31—21no—
109C2×C42×C31C2312yes—
110C31×#107—22no—
111C31×#107—22no—
112C31×#106—22no—
113C31×#106—22no—
114C31×#104C2×C4116yescomplemented
115C31×#103—22no—
116C31×#103—22no—
117C31×#102—22no—
118C31×#102—22no—
119C31×#95—24no—
120C24×C31—21no—
121C24×C31C2×D411yescharacteristic
122#41×C31—24no—
123#41×C31—24no—
124#41×C31—24no—
125#41×C31—24no—
126#41×C31—24no—
127#41×C31—22no—
128#41×C31—22no—
129#41×C31C4⋊C418yescomplemented
130#41×C31C22×C414yes—
131#41×C31C2×Q814yes—
132C2×Q8×C31—21no—
133C2×Q8×C31—22no—
134C2×Q8×C31C4⋊C414yescomplemented
135C2×Q8×C31C2×D412yes—
136C2×D4×C31—24no—
137C2×D4×C31—22no—
138C2×D4×C31—24no—
139C2×D4×C31—22no—
140C2×D4×C31—21no—
141C2×D4×C31—22no—
142C2×D4×C31C4⋊C414yescomplemented
143C2×D4×C31C2×D412yes—
144C22×C4×C31—21no—
145C22×C4×C31—24no—
146C22×C4×C31—22no—
147C22×C4×C31—24no—
148C22×C4×C31—22no—
149C22×C4×C31—21no—
150C22×C4×C31—22no—
151C22×C4×C31—21no—
152C22×C4×C31—22no—
153C22×C4×C31—28no—
154C22×C4×C31—22no—
155C22×C4×C31—21no—
156C22×C4×C31—21no—
157C22×C4×C31—82no—
158C22×C4×C31—84no—
159C22×C4×C31C22×C412yes—
160C22×C4×C31C2×D414yes—
161C22×C4×C31C2×D411yescharacteristic
162C22×C4×C31C2×D411yescharacteristic
163C22×C4×C31C2×D411yescharacteristic
164C22×C4×C31C2×D411yescharacteristic
165C22×C4×C31C2×D411yescharacteristic
166C22×C4×C31C2×Q812yes—
167C22×C4×C31#4112yes—
168C22×C4×C31#4112yes—
169C22×C4×C31C2411yescharacteristic
170M16×C31—24no—
171M16×C31—22no—
172M16×C31—22no—
173M16×C31—22no—
174M16×C31—42no—
175M16×C31C22×C414yescomplemented
176C2×C8×C31—24no—
177C2×C8×C31—22no—
178C2×C8×C31—22no—
179C2×C8×C31—42no—
180C4⋊C4×C31—28no—
181C4⋊C4×C31—24no—
182C4⋊C4×C31—22no—
183C4⋊C4×C31—21no—
184C4⋊C4×C31—22no—
185C4⋊C4×C31—21no—
186C4⋊C4×C31—22no—
187C4⋊C4×C31—22no—
188C4⋊C4×C31—21no—
189C4⋊C4×C31—21no—
190C4⋊C4×C31—21no—
191C4⋊C4×C31—21no—
192C4⋊C4×C31—84no—
193C4⋊C4×C31C2×D412yes—
194C4⋊C4×C31C2×D412yes—
195C22⋊C4×C31—28no—
196C22⋊C4×C31—24no—
197C22⋊C4×C31—22no—
198C22⋊C4×C31—22no—
199C22⋊C4×C31—41no—
200C22⋊C4×C31—41no—
201C42×C31—28no—
202C42×C31—22no—
203C42×C31—22no—
204C42×C31—22no—
205C42×C31—41no—
206C42×C31—41no—
207C42×C31C22×C414yescomplemented
208#20864C3111yesmaximal, characteristic, direct product factor, Sylow
209C23×C31—22no—
210C23×C31—24no—
211C23×C31—41no—
212C23×C31—41no—
213C23×C31—42no—
214C23×C31—42no—
215C23×C31C4×D412yes—
216C23×C31#12011yescharacteristic
217C23×C31#12011yescharacteristic
218C23×C31C22⋊Q812yes—
219C23×C31C22×D411yescharacteristic
220Q8×C31—22no—
221Q8×C31—24no—
222Q8×C31C2×C4⋊C414yes—
223D4×C31—24no—
224D4×C31—24no—
225D4×C31—28no—
226D4×C31—22no—
227D4×C31—24no—
228D4×C31C2×C4⋊C414yes—
229C2×C4×C31—22no—
230C2×C4×C31—24no—
231C2×C4×C31—24no—
232C2×C4×C31—24no—
233C2×C4×C31—28no—
234C2×C4×C31—24no—
235C2×C4×C31—22no—
236C2×C4×C31—24no—
237C2×C4×C31—22no—
238C2×C4×C31—42no—
239C2×C4×C31—42no—
240C2×C4×C31—42no—
241C2×C4×C31—44no—
242C2×C4×C31—42no—
243C2×C4×C31—44no—
244C2×C4×C31—42no—
245C2×C4×C31—42no—
246C2×C4×C31—41no—
247C2×C4×C31—41no—
248C2×C4×C31—82no—
249C2×C4×C31—88no—
250C2×C4×C31C2×C4⋊C412yes—
251C2×C4×C31C2×C4⋊C412yes—
252C2×C4×C31C4×D412yes—
253C2×C4×C31C4×D412yes—
254C2×C4×C31C4×D412yes—
255C2×C4×C31#12011yescharacteristic
256C2×C4×C31#12011yescharacteristic
257C2×C4×C31C22⋊Q812yes—
258C2×C4×C31C22⋊Q812yes—
259C2×C4×C31C22⋊Q812yes—
260C2×C4×C31#12312yes—
261C2×C4×C31#12312yes—
262C2×C4×C31C23×C411yescharacteristic
263C2×C4×C31C22×D411yescharacteristic
264C2×C4×C31C22×D411yescharacteristic
265C2×C4×C31C22×Q811yescharacteristic
266C2×C4×C31C2×#4111yescharacteristic
267C2×C4×C31C2×#4111yescharacteristic
268C8×C31—24no—
269C8×C31—42no—
270C2×#520C2×C3111yescharacteristic
271#2911C2×C3111yescharacteristic, complemented
272C2×#421C2×C3111yescharacteristic
273#2896C2×C3111yescharacteristic
274#2317C2×C3111yescharacteristic, complemented
275#1860C2×C3114yescomplemented
276#1859C2×C3114yescomplemented
277C2×#343C2×C3111yescharacteristic, complemented
278#1733C2×C3111yescharacteristic, complemented
279C22×C31—21no—
280C22×C31—21no—
281C22×C31—21no—
282C22×C31—21no—
283C22×C31—21no—
284C22×C31—21no—
285C22×C31—24no—
286C22×C31—42no—
287C22×C31—42no—
288C22×C31—48no—
289C22×C31—42no—
290C22×C31—42no—
291C22×C31#38512yes—
292C22×C31#38512yes—
293C22×C31C2×C4×D411yescharacteristic
294C22×C31C2×#12011yescharacteristic
295C22×C31C2×C22⋊Q811yescharacteristic
296C4×C31—24no—
297C4×C31—44no—
298C4×C31—44no—
299C4×C31—42no—
300C4×C31—42no—
301C4×C31—81no—
302C4×C31—81no—
303C4×C31#38512yes—
304C4×C31#38512yes—
305C4×C31C22×C4⋊C411yescharacteristic
306C4×C31C2×C4×D411yescharacteristic
307C4×C31C2×C22⋊Q811yescharacteristic
308C4×C31C2×#12311yescharacteristic
309C22×#41C22×C3111yescharacteristic
310C22×M16C22×C3111yescharacteristic
311C22×M16C22×C3111yescharacteristic
312#520—22no—
313#520—22no—
314#520—22no—
315#520C22×C3114yes—
316C2×C4×Q8—21no—
317C2×C4×D4—21no—
318C2×C4×D4—22no—
319C2×#117—21no—
320C2×#117C22×C3111yescharacteristic
321C2×#117C22×C3111yescharacteristic
322C2×#117C22×C3111yescharacteristic
323C22×C4⋊C4—21no—
324C22×C4⋊C4C22×C3111yescharacteristic
325#428—21no—
326#428—21no—
327#428C22×C3112yescomplemented
328#428C22×C3112yescomplemented
329#427—22no—
330#421C4×C3118yescomplemented
331C2×#104C22×C3114yes—
332#419—21no—
333#418—21no—
334#417C22×C3112yes—
335#417C22×C3112yes—
336#416—22no—
337#389—22no—
338#385—22no—
339#384C22×C3112yescomplemented
340#382C22×C3112yescomplemented
341#343C22×C3114yescomplemented
342#339C22×C3114yescomplemented
343C2×C31—21no—
344C2×C31—22no—
345C2×C31—21no—
346C2×C31—22no—
347C2×C31—44no—
348C2×C31C2×#38511yescharacteristic
349C2×C31#301911yescharacteristic
350C2×C31#302011yescharacteristic
351C2×#41—21no—
352C2×#41—21no—
353C2×#41—22no—
354C2×#41—22no—
355C2×#41C2×C4×C3114yes—
356C2×#41D4×C3112yes—
357C2×#41Q8×C3114yes—
358C2×#41C23×C3112yes—
359C22×Q8D4×C3111yescharacteristic
360C22×D4—21no—
361C22×D4D4×C3111yescharacteristic
362C23×C4—21no—
363C23×C4—22no—
364C23×C4D4×C3112yes—
365C23×C4C23×C3111yescharacteristic
366C2×M16—21no—
367C2×M16—22no—
368C2×M16—21no—
369C2×M16—22no—
370C2×M16—22no—
371C2×M16C2×C4×C3114yescomplemented
372C2×M16C23×C3112yes—
373C2×M16C23×C3112yes—
374C22×C8—21no—
375C22×C8—21no—
376C4×Q8—24no—
377C4×Q8—24no—
378C4×D4—28no—
379C4×D4—24no—
380C4×D4—24no—
381C4×D4—28no—
382#117—24no—
383#117—21no—
384#117—21no—
385#117—22no—
386#117—24no—
387#117—21no—
388#117—21no—
389#117—21no—
390#117—21no—
391#117C2×C4×C3114yescomplemented
392#117D4×C3112yes—
393#117C23×C3112yes—
394C2×C4⋊C4—22no—
395C2×C4⋊C4—24no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—24no—
402C2×C4⋊C4—24no—
403C2×C4⋊C4D4×C3112yes—
404C2×C4⋊C4D4×C3111yescharacteristic
405C2×C4⋊C4D4×C3111yescharacteristic
406C2×C4⋊C4C23×C3111yescharacteristic
407C2×C4⋊C4C23×C3111yescharacteristic
408C2×C4⋊C4C23×C3112yes—
409C2×C22⋊C4—22no—
410C2×C22⋊C4—21no—
411C2×C22⋊C4—21no—
412C2×C22⋊C4—21no—
413C2×C42—22no—
414C2×C42—21no—
415C2×C42—21no—
416C2×C42C23×C3112yes—
417#107—22no—
418#107—22no—
419#106—22no—
420#106—22no—
421#104C2×C4×C31116yescomplemented
422#103—22no—
423#103—22no—
424#102—22no—
425#102—22no—
426#95—24no—
427C31#2086411yescharacteristic, direct product factor, Sylow
428C24—21no—
429C24C2×D4×C3111yescharacteristic
430#41—24no—
431#41—24no—
432#41—24no—
433#41—24no—
434#41—24no—
435#41—22no—
436#41—22no—
437#41C4⋊C4×C3118yescomplemented
438#41C22×C4×C3114yes—
439#41C2×Q8×C3114yes—
440C2×Q8—21no—
441C2×Q8—22no—
442C2×Q8C4⋊C4×C3114yescomplemented
443C2×Q8C2×D4×C3112yes—
444C2×D4—24no—
445C2×D4—22no—
446C2×D4—24no—
447C2×D4—22no—
448C2×D4—21no—
449C2×D4—22no—
450C2×D4C4⋊C4×C3114yescomplemented
451C2×D4C2×D4×C3112yes—
452C22×C4—21no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—28no—
462C22×C4—22no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—82no—
466C22×C4—84no—
467C22×C4C22×C4×C3112yes—
468C22×C4C2×D4×C3114yes—
469C22×C4C2×D4×C3111yescharacteristic
470C22×C4C2×D4×C3111yescharacteristic
471C22×C4C2×D4×C3111yescharacteristic
472C22×C4C2×D4×C3111yescharacteristic
473C22×C4C2×D4×C3111yescharacteristic
474C22×C4C2×Q8×C3112yes—
475C22×C4#41×C3112yes—
476C22×C4#41×C3112yes—
477C22×C4C24×C3111yescharacteristic
478M16—24no—
479M16—22no—
480M16—22no—
481M16—22no—
482M16—42no—
483M16C22×C4×C3114yescomplemented
484C2×C8—24no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—42no—
488C4⋊C4—28no—
489C4⋊C4—24no—
490C4⋊C4—22no—
491C4⋊C4—21no—
492C4⋊C4—22no—
493C4⋊C4—21no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—84no—
501C4⋊C4C2×D4×C3112yes—
502C4⋊C4C2×D4×C3112yes—
503C22⋊C4—28no—
504C22⋊C4—24no—
505C22⋊C4—22no—
506C22⋊C4—22no—
507C22⋊C4—41no—
508C22⋊C4—41no—
509C42—28no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—41no—
514C42—41no—
515C42C22×C4×C3114yescomplemented
516C23—22no—
517C23—24no—
518C23—41no—
519C23—41no—
520C23—42no—
521C23—42no—
522C23C4×D4×C3112yes—
523C23C31×#12011yescharacteristic
524C23C31×#12011yescharacteristic
525C23C31×C22⋊Q812yes—
526C23C22×D4×C3111yescharacteristic
527Q8—22no—
528Q8—24no—
529Q8C2×C4⋊C4×C3114yes—
530D4—24no—
531D4—24no—
532D4—28no—
533D4—22no—
534D4—24no—
535D4C2×C4⋊C4×C3114yes—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—28no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—82no—
556C2×C4—88no—
557C2×C4C2×C4⋊C4×C3112yes—
558C2×C4C2×C4⋊C4×C3112yes—
559C2×C4C4×D4×C3112yes—
560C2×C4C4×D4×C3112yes—
561C2×C4C4×D4×C3112yes—
562C2×C4C31×#12011yescharacteristic
563C2×C4C31×#12011yescharacteristic
564C2×C4C31×C22⋊Q812yes—
565C2×C4C31×C22⋊Q812yes—
566C2×C4C31×C22⋊Q812yes—
567C2×C4C31×#12312yes—
568C2×C4C31×#12312yes—
569C2×C4C23×C4×C3111yescharacteristic
570C2×C4C22×D4×C3111yescharacteristic
571C2×C4C22×D4×C3111yescharacteristic
572C2×C4C22×Q8×C3111yescharacteristic
573C2×C4C2×#41×C3111yescharacteristic
574C2×C4C2×#41×C3111yescharacteristic
575C8—24no—
576C8—42no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—24no—
584C22—42no—
585C22—42no—
586C22—48no—
587C22—42no—
588C22—42no—
589C22C31×#38512yes—
590C22C31×#38512yes—
591C22C2×C4×D4×C3111yescharacteristic
592C22C2×C31×#12011yescharacteristic
593C22C2×C31×C22⋊Q811yescharacteristic
594C4—24no—
595C4—44no—
596C4—44no—
597C4—42no—
598C4—42no—
599C4—81no—
600C4—81no—
601C4C31×#38512yes—
602C4C31×#38512yes—
603C4C22×C4⋊C4×C3111yescharacteristic
604C4C2×C4×D4×C3111yescharacteristic
605C4C2×C31×C22⋊Q811yescharacteristic
606C4C2×C31×#12311yescharacteristic
607C2—21no—
608C2—22no—
609C2—21no—
610C2—22no—
611C2—44no—
612C2C2×C31×#38511yescharacteristic
613C2C31×#301911yescharacteristic
614C2C31×#302011yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutatorC2×C4C23×C4×C31
Frattini, cosocleC22×C4C24×C31
prime radicalC22×#41×C31C22
LCSG ▷ C2×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×#2911C2semidirect
5. C31×#2317C2semidirect
6. C31×#1860C2semidirect
7. C31×#1859C2semidirect
8. C2×C31×#343C2semidirect
9. C31×#1733C2semidirect
28. C31×#428C22semidirect
29. C31×#428C22semidirect
31. C31×#421C4semidirect
40. C31×#384C22semidirect
41. C31×#382C22semidirect
42. C31×#343C22semidirect
43. C31×#339C22semidirect
64. C2×M16×C31C2×C4semidirect
84. C31×#117C2×C4semidirect
114. C31×#104C2×C4semidirect
129. #41×C31C4⋊C4semidirect
134. C2×Q8×C31C4⋊C4semidirect
142. C2×D4×C31C4⋊C4semidirect
175. M16×C31C22×C4semidirect
207. C42×C31C22×C4semidirect
208. #20864C31direct (N × H)
271. #2911C2×C31semidirect
274. #2317C2×C31semidirect
275. #1860C2×C31semidirect
276. #1859C2×C31semidirect
277. C2×#343C2×C31semidirect
278. #1733C2×C31semidirect
327. #428C22×C31semidirect
328. #428C22×C31semidirect
330. #421C4×C31semidirect
339. #384C22×C31semidirect
340. #382C22×C31semidirect
341. #343C22×C31semidirect
342. #339C22×C31semidirect
371. C2×M16C2×C4×C31semidirect
391. #117C2×C4×C31semidirect
421. #104C2×C4×C31semidirect
427. C31#20864direct (N × H)
437. #41C4⋊C4×C31semidirect
442. C2×Q8C4⋊C4×C31semidirect
450. C2×D4C4⋊C4×C31semidirect
483. M16C22×C4×C31semidirect
515. C42C22×C4×C31semidirect

Maximal subgroups

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

subgroupindex
C2×C31×#5202
C31×#29112
C2×C31×#4212
C31×#28962
C31×#23172
C31×#18602
C31×#18592
C2×C31×#3432
C31×#17332
#2086431