Finite Group Databasebeta

← C31×#10954 #21467713

Subgroups — C31×#10954 #21467713

Order 7936.

966 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#421C211yesmaximal, characteristic
2C2×C31×#417C211yesmaximal, characteristic
3C2×C31×#410C211yesmaximal, characteristic
4C31×#2274C211yesmaximal, characteristic, complemented
5C2×C31×#344C211yesmaximal, characteristic, complemented
6C31×#1736C211yesmaximal, characteristic, complemented
7C2×C31×H(3,ℤ/4)C211yesmaximal, characteristic, complemented
8C22×#41×C31C2211yescharacteristic
9C22×M16×C31C2211yescharacteristic
10C22×M16×C31C2211yescharacteristic
11C2×C31×#117C2211yescharacteristic
12C2×C31×#117C2211yescharacteristic
13C2×C31×#117C2211yescharacteristic
14C22×C4⋊C4×C31C2211yescharacteristic
15C31×#421—22no—
16C31×#421C414yescomplemented
17C2×C31×#104—22no—
18C31×#417—22no—
19C31×#417—22no—
20C2×C31×#103C412yescomplemented
21C2×C31×#102C412yescomplemented
22C31×#410—22no—
23C31×#410C414yescomplemented
24C2×C31×#99—22no—
25C4×C4⋊C4×C31—22no—
26C4×C4⋊C4×C31—22no—
27C4×C22⋊C4×C31—22no—
28C31×#344—22no—
29C31×#340—22no—
30C31×H(3,ℤ/4)—22no—
31C2×#41×C31—21no—
32C2×#41×C31—21no—
33C2×#41×C31—21no—
34C2×#41×C31—21no—
35C2×#41×C31—21no—
36C2×#41×C31—21no—
37C2×#41×C31—21no—
38C2×#41×C31—21no—
39C2×#41×C31—21no—
40C2×#41×C31—21no—
41C2×#41×C31C2×C412yes—
42C2×#41×C31C2×C412yes—
43C22×Q8×C31C2×C411yescharacteristic
44C22×D4×C31—21no—
45C22×D4×C31C2×C411yescharacteristic
46C23×C4×C31—21no—
47C23×C4×C31—21no—
48C23×C4×C31C2311yescharacteristic
49C2×M16×C31—21no—
50C2×M16×C31—21no—
51C2×M16×C31—21no—
52C2×M16×C31—21no—
53C2×M16×C31—21no—
54C2×M16×C31—21no—
55C2×M16×C31—21no—
56C2×M16×C31—21no—
57C2×M16×C31—21no—
58C2×M16×C31—21no—
59C2×M16×C31—21no—
60C2×M16×C31C2×C412yes—
61C22×C8×C31—21no—
62C22×C8×C31C2×C412yes—
63C31×#117—21no—
64C31×#117—21no—
65C31×#117—21no—
66C31×#117—21no—
67C31×#117—21no—
68C31×#117—21no—
69C31×#117—21no—
70C31×#117C2×C412yes—
71C31×#117C2×C412yes—
72C31×#117C2×C412yes—
73C31×#117D412yes—
74C31×#117D412yes—
75C2×C4⋊C4×C31—21no—
76C2×C4⋊C4×C31—21no—
77C2×C4⋊C4×C31—21no—
78C2×C4⋊C4×C31—21no—
79C2×C4⋊C4×C31—21no—
80C2×C4⋊C4×C31—21no—
81C2×C4⋊C4×C31C2×C412yes—
82C2×C4⋊C4×C31D412yes—
83C2×C4⋊C4×C31D412yes—
84C2×C22⋊C4×C31—21no—
85C2×C22⋊C4×C31—21no—
86C2×C22⋊C4×C31D412yes—
87C2×C42×C31—21no—
88C2×C42×C31—21no—
89C2×C42×C31—24no—
90C2×C42×C31D412yes—
91C31×#104—28no—
92C31×#103—24no—
93C31×#102—24no—
94C31×#99—28no—
95C31×#95—24no—
96C24×C31—21no—
97C24×C31C2×D411yescharacteristic
98#41×C31—22no—
99#41×C31—22no—
100#41×C31—22no—
101#41×C31—22no—
102#41×C31—22no—
103#41×C31—22no—
104#41×C31—22no—
105#41×C31—22no—
106#41×C31—41no—
107#41×C31—41no—
108#41×C31—41no—
109#41×C31—41no—
110#41×C31—41no—
111#41×C31—41no—
112#41×C31—41no—
113#41×C31—41no—
114C2×Q8×C31—21no—
115C2×Q8×C31—21no—
116C2×Q8×C31—21no—
117C2×Q8×C31—21no—
118C2×Q8×C31—21no—
119C2×Q8×C31C4212yescomplemented
120C2×D4×C31—22no—
121C2×D4×C31—22no—
122C2×D4×C31—21no—
123C2×D4×C31—21no—
124C2×D4×C31—21no—
125C2×D4×C31—21no—
126C2×D4×C31—21no—
127C2×D4×C31—41no—
128C2×D4×C31—41no—
129C2×D4×C31—41no—
130C2×D4×C31—41no—
131C2×D4×C31C4212yescomplemented
132C22×C4×C31—21no—
133C22×C4×C31—22no—
134C22×C4×C31—22no—
135C22×C4×C31—22no—
136C22×C4×C31—22no—
137C22×C4×C31—22no—
138C22×C4×C31—22no—
139C22×C4×C31—22no—
140C22×C4×C31—41no—
141C22×C4×C31—41no—
142C22×C4×C31—41no—
143C22×C4×C31—41no—
144C22×C4×C31—41no—
145C22×C4×C31—82no—
146C22×C4×C31C22×C411yescharacteristic
147C22×C4×C31C22×C411yescharacteristic
148C22×C4×C31C22×C411yescharacteristic
149C22×C4×C31C2×D411yescharacteristic
150C22×C4×C31C2×D411yescharacteristic
151C22×C4×C31C2×D411yescharacteristic
152C22×C4×C31C2×D411yescharacteristic
153C22×C4×C31C2×D411yescharacteristic
154C22×C4×C31#4111yescharacteristic
155C22×C4×C31#4111yescharacteristic
156C22×C4×C31#4111yescharacteristic
157C22×C4×C31#4111yescharacteristic
158C22×C4×C31#4111yescharacteristic
159C22×C4×C31#4111yescharacteristic
160M16×C31—22no—
161M16×C31—22no—
162M16×C31—41no—
163M16×C31—41no—
164M16×C31—41no—
165M16×C31—41no—
166M16×C31—41no—
167M16×C31—41no—
168C2×C8×C31—22no—
169C2×C8×C31—22no—
170C2×C8×C31—22no—
171C2×C8×C31—22no—
172C2×C8×C31—41no—
173C2×C8×C31—41no—
174C4⋊C4×C31—22no—
175C4⋊C4×C31—22no—
176C4⋊C4×C31—22no—
177C4⋊C4×C31—22no—
178C4⋊C4×C31—22no—
179C4⋊C4×C31—22no—
180C4⋊C4×C31—22no—
181C4⋊C4×C31—22no—
182C4⋊C4×C31—41no—
183C4⋊C4×C31C22⋊C414yes—
184C22⋊C4×C31—22no—
185C22⋊C4×C31—22no—
186C22⋊C4×C31—22no—
187C22⋊C4×C31—22no—
188C22⋊C4×C31—41no—
189C22⋊C4×C31C22⋊C414yes—
190C42×C31—22no—
191C42×C31—22no—
192C42×C31—22no—
193C42×C31—22no—
194C42×C31—41no—
195C42×C31—84no—
196C42×C31C22⋊C414yes—
197#10954C3111yesmaximal, characteristic, direct product factor, Sylow
198C23×C31—22no—
199C23×C31—41no—
200C23×C31—41no—
201C23×C31—41no—
202C23×C31—41no—
203C23×C31—81no—
204C23×C31—81no—
205C23×C31C2×C22⋊C411yescharacteristic
206C23×C31C4×D411yescharacteristic
207C23×C31C4×D411yescharacteristic
208C23×C31#12011yescharacteristic
209C23×C31#12111yescharacteristic
210C23×C31#12111yescharacteristic
211C23×C31#12311yescharacteristic
212Q8×C31—22no—
213Q8×C31—22no—
214Q8×C31—41no—
215Q8×C31—41no—
216D4×C31—22no—
217D4×C31—22no—
218D4×C31—42no—
219D4×C31—42no—
220D4×C31—42no—
221D4×C31—42no—
222D4×C31—41no—
223D4×C31—41no—
224C2×C4×C31—22no—
225C2×C4×C31—24no—
226C2×C4×C31—24no—
227C2×C4×C31—24no—
228C2×C4×C31—42no—
229C2×C4×C31—42no—
230C2×C4×C31—44no—
231C2×C4×C31—42no—
232C2×C4×C31—42no—
233C2×C4×C31—41no—
234C2×C4×C31—41no—
235C2×C4×C31—42no—
236C2×C4×C31—42no—
237C2×C4×C31—42no—
238C2×C4×C31—42no—
239C2×C4×C31—42no—
240C2×C4×C31—82no—
241C2×C4×C31—82no—
242C2×C4×C31—81no—
243C2×C4×C31—81no—
244C2×C4×C31C2×C4211yescharacteristic
245C2×C4×C31C2×C22⋊C411yescharacteristic
246C2×C4×C31C2×C22⋊C411yescharacteristic
247C2×C4×C31#11711yescharacteristic
248C2×C4×C31#11711yescharacteristic
249C2×C4×C31#11711yescharacteristic
250C2×C4×C31C4×D411yescharacteristic
251C2×C4×C31C4×D411yescharacteristic
252C2×C4×C31C4×D411yescharacteristic
253C2×C4×C31C4×D411yescharacteristic
254C2×C4×C31C4×D411yescharacteristic
255C2×C4×C31C4×D411yescharacteristic
256C2×C4×C31C4×D411yescharacteristic
257C2×C4×C31C4×D411yescharacteristic
258C2×C4×C31C4×D411yescharacteristic
259C2×C4×C31C4×D411yescharacteristic
260C2×C4×C31#12111yescharacteristic
261C2×C4×C31#12111yescharacteristic
262C2×C4×C31#12111yescharacteristic
263C2×C4×C31#12111yescharacteristic
264C2×C4×C31#12311yescharacteristic
265C2×C4×C31#12311yescharacteristic
266C2×C4×C31#12411yescharacteristic
267C2×C4×C31#12411yescharacteristic
268C2×C4×C31#12411yescharacteristic
269C2×C4×C31#12611yescharacteristic
270C2×C4×C31#12611yescharacteristic
271C2×C4×C31#12711yescharacteristic
272C8×C31—42no—
273C8×C31—42no—
274C2×#421C2×C3111yescharacteristic
275C2×#417C2×C3111yescharacteristic
276C2×#410C2×C3111yescharacteristic
277#2274C2×C3111yescharacteristic, complemented
278C2×#344C2×C3111yescharacteristic, complemented
279#1736C2×C3111yescharacteristic, complemented
280C2×H(3,ℤ/4)C2×C3111yescharacteristic, complemented
281C22×C31—21no—
282C22×C31—21no—
283C22×C31—21no—
284C22×C31—21no—
285C22×C31—21no—
286C22×C31—21no—
287C22×C31—41no—
288C22×C31—41no—
289C22×C31—41no—
290C22×C31—41no—
291C22×C31—42no—
292C22×C31—82no—
293C22×C31—82no—
294C22×C31—81no—
295C22×C31—81no—
296C22×C31C4×C22⋊C411yescharacteristic
297C22×C31#38611yescharacteristic
298C22×C31#38611yescharacteristic
299C22×C31#38611yescharacteristic
300C22×C31#38811yescharacteristic
301C22×C31#38811yescharacteristic
302C22×C31#39011yescharacteristic
303C4×C31—44no—
304C4×C31—44no—
305C4×C31—44no—
306C4×C31—42no—
307C4×C31—82no—
308C4×C31C4×C22⋊C411yescharacteristic
309C4×C31C4×C22⋊C411yescharacteristic
310C4×C31#38811yescharacteristic
311C4×C31#38811yescharacteristic
312C4×C31#38811yescharacteristic
313C4×C31#38811yescharacteristic
314C4×C31#39011yescharacteristic
315C4×C31#39011yescharacteristic
316C22×#41C22×C3111yescharacteristic
317C22×M16C22×C3111yescharacteristic
318C22×M16C22×C3111yescharacteristic
319C2×#117C22×C3111yescharacteristic
320C2×#117C22×C3111yescharacteristic
321C2×#117C22×C3111yescharacteristic
322C22×C4⋊C4C22×C3111yescharacteristic
323#421—22no—
324#421C4×C3114yescomplemented
325C2×#104—22no—
326#417—22no—
327#417—22no—
328C2×#103C4×C3112yescomplemented
329C2×#102C4×C3112yescomplemented
330#410—22no—
331#410C4×C3114yescomplemented
332C2×#99—22no—
333C4×C4⋊C4—22no—
334C4×C4⋊C4—22no—
335C4×C22⋊C4—22no—
336#344—22no—
337#340—22no—
338H(3,ℤ/4)—22no—
339C2×C31—21no—
340C2×C31—21no—
341C2×C31—21no—
342C2×C31—21no—
343C2×C31—21no—
344C2×C31—21no—
345C2×C31—82no—
346C2×C31#143711yescharacteristic
347C2×C31#190411yescharacteristic
348C2×C31#190511yescharacteristic
349C2×#41—21no—
350C2×#41—21no—
351C2×#41—21no—
352C2×#41—21no—
353C2×#41—21no—
354C2×#41—21no—
355C2×#41—21no—
356C2×#41—21no—
357C2×#41—21no—
358C2×#41—21no—
359C2×#41C2×C4×C3112yes—
360C2×#41C2×C4×C3112yes—
361C22×Q8C2×C4×C3111yescharacteristic
362C22×D4—21no—
363C22×D4C2×C4×C3111yescharacteristic
364C23×C4—21no—
365C23×C4—21no—
366C23×C4C23×C3111yescharacteristic
367C2×M16—21no—
368C2×M16—21no—
369C2×M16—21no—
370C2×M16—21no—
371C2×M16—21no—
372C2×M16—21no—
373C2×M16—21no—
374C2×M16—21no—
375C2×M16—21no—
376C2×M16—21no—
377C2×M16—21no—
378C2×M16C2×C4×C3112yes—
379C22×C8—21no—
380C22×C8C2×C4×C3112yes—
381#117—21no—
382#117—21no—
383#117—21no—
384#117—21no—
385#117—21no—
386#117—21no—
387#117—21no—
388#117C2×C4×C3112yes—
389#117C2×C4×C3112yes—
390#117C2×C4×C3112yes—
391#117D4×C3112yes—
392#117D4×C3112yes—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4C2×C4×C3112yes—
400C2×C4⋊C4D4×C3112yes—
401C2×C4⋊C4D4×C3112yes—
402C2×C22⋊C4—21no—
403C2×C22⋊C4—21no—
404C2×C22⋊C4D4×C3112yes—
405C2×C42—21no—
406C2×C42—21no—
407C2×C42—24no—
408C2×C42D4×C3112yes—
409#104—28no—
410#103—24no—
411#102—24no—
412#99—28no—
413#95—24no—
414C31#1095411yescharacteristic, direct product factor, Sylow
415C24—21no—
416C24C2×D4×C3111yescharacteristic
417#41—22no—
418#41—22no—
419#41—22no—
420#41—22no—
421#41—22no—
422#41—22no—
423#41—22no—
424#41—22no—
425#41—41no—
426#41—41no—
427#41—41no—
428#41—41no—
429#41—41no—
430#41—41no—
431#41—41no—
432#41—41no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8C42×C3112yescomplemented
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—41no—
447C2×D4—41no—
448C2×D4—41no—
449C2×D4—41no—
450C2×D4C42×C3112yescomplemented
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—41no—
460C22×C4—41no—
461C22×C4—41no—
462C22×C4—41no—
463C22×C4—41no—
464C22×C4—82no—
465C22×C4C22×C4×C3111yescharacteristic
466C22×C4C22×C4×C3111yescharacteristic
467C22×C4C22×C4×C3111yescharacteristic
468C22×C4C2×D4×C3111yescharacteristic
469C22×C4C2×D4×C3111yescharacteristic
470C22×C4C2×D4×C3111yescharacteristic
471C22×C4C2×D4×C3111yescharacteristic
472C22×C4C2×D4×C3111yescharacteristic
473C22×C4#41×C3111yescharacteristic
474C22×C4#41×C3111yescharacteristic
475C22×C4#41×C3111yescharacteristic
476C22×C4#41×C3111yescharacteristic
477C22×C4#41×C3111yescharacteristic
478C22×C4#41×C3111yescharacteristic
479M16—22no—
480M16—22no—
481M16—41no—
482M16—41no—
483M16—41no—
484M16—41no—
485M16—41no—
486M16—41no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—41no—
492C2×C8—41no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—41no—
502C4⋊C4C22⋊C4×C3114yes—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—22no—
507C22⋊C4—41no—
508C22⋊C4C22⋊C4×C3114yes—
509C42—22no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—41no—
514C42—84no—
515C42C22⋊C4×C3114yes—
516C23—22no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—81no—
522C23—81no—
523C23C2×C22⋊C4×C3111yescharacteristic
524C23C4×D4×C3111yescharacteristic
525C23C4×D4×C3111yescharacteristic
526C23C31×#12011yescharacteristic
527C23C31×#12111yescharacteristic
528C23C31×#12111yescharacteristic
529C23C31×#12311yescharacteristic
530Q8—22no—
531Q8—22no—
532Q8—41no—
533Q8—41no—
534D4—22no—
535D4—22no—
536D4—42no—
537D4—42no—
538D4—42no—
539D4—42no—
540D4—41no—
541D4—41no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—81no—
561C2×C4—81no—
562C2×C4C2×C42×C3111yescharacteristic
563C2×C4C2×C22⋊C4×C3111yescharacteristic
564C2×C4C2×C22⋊C4×C3111yescharacteristic
565C2×C4C31×#11711yescharacteristic
566C2×C4C31×#11711yescharacteristic
567C2×C4C31×#11711yescharacteristic
568C2×C4C4×D4×C3111yescharacteristic
569C2×C4C4×D4×C3111yescharacteristic
570C2×C4C4×D4×C3111yescharacteristic
571C2×C4C4×D4×C3111yescharacteristic
572C2×C4C4×D4×C3111yescharacteristic
573C2×C4C4×D4×C3111yescharacteristic
574C2×C4C4×D4×C3111yescharacteristic
575C2×C4C4×D4×C3111yescharacteristic
576C2×C4C4×D4×C3111yescharacteristic
577C2×C4C4×D4×C3111yescharacteristic
578C2×C4C31×#12111yescharacteristic
579C2×C4C31×#12111yescharacteristic
580C2×C4C31×#12111yescharacteristic
581C2×C4C31×#12111yescharacteristic
582C2×C4C31×#12311yescharacteristic
583C2×C4C31×#12311yescharacteristic
584C2×C4C31×#12411yescharacteristic
585C2×C4C31×#12411yescharacteristic
586C2×C4C31×#12411yescharacteristic
587C2×C4C31×#12611yescharacteristic
588C2×C4C31×#12611yescharacteristic
589C2×C4C31×#12711yescharacteristic
590C8—42no—
591C8—42no—
592C22—21no—
593C22—21no—
594C22—21no—
595C22—21no—
596C22—21no—
597C22—21no—
598C22—41no—
599C22—41no—
600C22—41no—
601C22—41no—
602C22—42no—
603C22—82no—
604C22—82no—
605C22—81no—
606C22—81no—
607C22C4×C22⋊C4×C3111yescharacteristic
608C22C31×#38611yescharacteristic
609C22C31×#38611yescharacteristic
610C22C31×#38611yescharacteristic
611C22C31×#38811yescharacteristic
612C22C31×#38811yescharacteristic
613C22C31×#39011yescharacteristic
614C4—44no—
615C4—44no—
616C4—44no—
617C4—42no—
618C4—82no—
619C4C4×C22⋊C4×C3111yescharacteristic
620C4C4×C22⋊C4×C3111yescharacteristic
621C4C31×#38811yescharacteristic
622C4C31×#38811yescharacteristic
623C4C31×#38811yescharacteristic
624C4C31×#38811yescharacteristic
625C4C31×#39011yescharacteristic
626C4C31×#39011yescharacteristic
627C2—21no—
628C2—21no—
629C2—21no—
630C2—21no—
631C2—21no—
632C2—21no—
633C2—82no—
634C2C31×#143711yescharacteristic
635C2C31×#190411yescharacteristic
636C2C31×#190511yescharacteristic
subgroupG/H
center, socleC22×C31#386
commutatorC2×C4C2×C42×C31
Frattini, cosocleC23×C4C23×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
4. C31×#2274C2semidirect
5. C2×C31×#344C2semidirect
6. C31×#1736C2semidirect
7. C2×C31×H(3,ℤ/4)C2semidirect
16. C31×#421C4semidirect
20. C2×C31×#103C4semidirect
21. C2×C31×#102C4semidirect
23. C31×#410C4semidirect
119. C2×Q8×C31C42semidirect
131. C2×D4×C31C42semidirect
197. #10954C31direct (N × H)
277. #2274C2×C31semidirect
278. C2×#344C2×C31semidirect
279. #1736C2×C31semidirect
280. C2×H(3,ℤ/4)C2×C31semidirect
324. #421C4×C31semidirect
328. C2×#103C4×C31semidirect
329. C2×#102C4×C31semidirect
331. #410C4×C31semidirect
414. C31#10954direct (N × H)
438. C2×Q8C42×C31semidirect
450. C2×D4C42×C31semidirect

Maximal subgroups

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

subgroupindex
C2×C31×#4212
C2×C31×#4172
C2×C31×#4102
C31×#22742
C2×C31×#3442
C31×#17362
C2×C31×H(3,ℤ/4)2
#1095431