Finite Group Databasebeta

← C31×#32916 #21489675

Subgroups — C31×#32916 #21489675

Order 7936.

1406 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#520C211yesmaximal, characteristic
2C2×C31×#462C211yesmaximal, characteristic, complemented
3C2×C31×#461C211yesmaximal, characteristic, complemented
4C2×C31×#452C212yesmaximal
5C31×#2621C211yesmaximal, characteristic, complemented
6C31×#1878C212yesmaximal, complemented
7C31×#1877C212yesmaximal, complemented
8C31×#1872C212yesmaximal, complemented
9C31×#1805C211yesmaximal, characteristic, complemented
10C31×#1804C211yesmaximal, characteristic, complemented
11C31×#1798C211yesmaximal, characteristic, complemented
12C22×#41×C31C2211yescharacteristic
13C22×Q16×C31C2212yes—
14C22×SD16×C31C2212yes—
15C2×C31×C4⋊Q8C2211yescharacteristic
16C2×C31×C22⋊Q8C2212yes—
17C31×#520—24no—
18C31×#520—24no—
19C2×C4×Q8×C31C2211yescharacteristic
20C2×C4×Q8×C31C2211yescharacteristic
21C2×C4×D4×C31—21no—
22C2×C4×D4×C31—21no—
23C2×C4×D4×C31C2211yescharacteristic
24C2×C4×D4×C31C2211yescharacteristic
25C2×C31×#117—21no—
26C2×C31×#117C2211yescharacteristic
27C22×C4⋊C4×C31—21no—
28C22×C4⋊C4×C31C2211yescharacteristic
29C31×#462—24no—
30C31×#461—24no—
31C31×#452—28no—
32C2×C31×C4⋊C8C2211yescharacteristic
33C2×C31×C4⋊C8C2211yescharacteristic
34C2×C31×#103C2212yes—
35C2×C31×#103C2212yes—
36C2×C31×#103C2212yes—
37C2×C31×#102C2212yes—
38C2×C31×C22⋊C8C2212yes—
39C31×#398—21no—
40C31×#398—21no—
41C31×#397—21no—
42C31×#397—21no—
43C31×#391C2212yes—
44C31×#382C2211yescharacteristic
45C31×#340C2212yes—
46C31×#340C2212yes—
47C31×#336C2212yes—
48C2×#41×C31—22no—
49C2×#41×C31—22no—
50C2×#41×C31—22no—
51C2×#41×C31—22no—
52C2×#41×C31—22no—
53C2×#41×C31D414yes—
54C22×Q8×C31C2312yes—
55C22×Q8×C31C2311yescharacteristic
56C22×D4×C31—21no—
57C22×D4×C31C2311yescharacteristic
58C23×C4×C31—21no—
59C23×C4×C31—21no—
60C23×C4×C31—21no—
61C23×C4×C31—21no—
62C23×C4×C31C2311yescharacteristic
63C2×Q16×C31—24no—
64C2×Q16×C31—24no—
65C2×Q16×C31—24no—
66C2×SD16×C31—24no—
67C2×SD16×C31—24no—
68C2×SD16×C31—24no—
69C22×C8×C31C2312yes—
70C22×C8×C31C2312yes—
71C31×C4⋊Q8—22no—
72C31×C4⋊Q8—22no—
73C31×C22⋊Q8—24no—
74C31×C22⋊Q8—24no—
75C4×Q8×C31—22no—
76C4×Q8×C31—22no—
77C4×Q8×C31—22no—
78C4×Q8×C31D414yes—
79C4×D4×C31—28no—
80C4×D4×C31—28no—
81C4×D4×C31—22no—
82C4×D4×C31—22no—
83C4×D4×C31—22no—
84C4×D4×C31D414yes—
85C31×#117—28no—
86C31×#117—22no—
87C31×#117—22no—
88C2×C4⋊C4×C31—28no—
89C2×C4⋊C4×C31—21no—
90C2×C4⋊C4×C31—21no—
91C2×C4⋊C4×C31—21no—
92C2×C4⋊C4×C31—22no—
93C2×C4⋊C4×C31—22no—
94C2×C4⋊C4×C31—21no—
95C2×C4⋊C4×C31—21no—
96C2×C4⋊C4×C31—22no—
97C2×C4⋊C4×C31D411yescharacteristic
98C2×C4⋊C4×C31D411yescharacteristic
99C2×C4⋊C4×C31C2312yes—
100C2×C4⋊C4×C31C2311yescharacteristic
101C2×C4⋊C4×C31C2311yescharacteristic
102C2×C22⋊C4×C31—21no—
103C2×C22⋊C4×C31—21no—
104C2×C22⋊C4×C31—22no—
105C2×C22⋊C4×C31D411yescharacteristic
106C2×C22⋊C4×C31D411yescharacteristic
107C2×C42×C31—21no—
108C2×C42×C31—21no—
109C2×C42×C31C2311yescharacteristic
110C2×C42×C31C2311yescharacteristic
111C31×C4⋊C8—22no—
112C31×C4⋊C8—22no—
113C31×#103—24no—
114C31×#103—24no—
115C31×#103—24no—
116C31×#102—24no—
117C31×C22⋊C8—24no—
118C31×#95—21no—
119C31×#95—21no—
120C31×#95—22no—
121C31×#95—22no—
122C24×C31—21no—
123C24×C31C2×D411yescharacteristic
124#41×C31—28no—
125#41×C31—28no—
126#41×C31—28no—
127#41×C31—28no—
128C2×Q8×C31—21no—
129C2×Q8×C31—21no—
130C2×Q8×C31—21no—
131C2×Q8×C31—22no—
132C2×Q8×C31—22no—
133C2×Q8×C31—22no—
134C2×Q8×C31—22no—
135C2×Q8×C31—22no—
136C2×Q8×C31—22no—
137C2×Q8×C31C2×D412yes—
138C2×Q8×C31C2×D412yes—
139C2×Q8×C31#4112yes—
140C2×D4×C31—24no—
141C2×D4×C31—24no—
142C2×D4×C31—24no—
143C2×D4×C31—21no—
144C2×D4×C31—21no—
145C2×D4×C31—21no—
146C2×D4×C31C2×D412yes—
147C2×D4×C31C2×D412yes—
148C2×D4×C31#4112yes—
149C22×C4×C31—24no—
150C22×C4×C31—24no—
151C22×C4×C31—21no—
152C22×C4×C31—24no—
153C22×C4×C31—21no—
154C22×C4×C31—21no—
155C22×C4×C31—21no—
156C22×C4×C31—21no—
157C22×C4×C31—21no—
158C22×C4×C31—21no—
159C22×C4×C31—24no—
160C22×C4×C31—24no—
161C22×C4×C31—24no—
162C22×C4×C31—22no—
163C22×C4×C31—22no—
164C22×C4×C31—82no—
165C22×C4×C31—82no—
166C22×C4×C31—82no—
167C22×C4×C31C2×D411yescharacteristic
168C22×C4×C31C2×D411yescharacteristic
169C22×C4×C31C2×D411yescharacteristic
170C22×C4×C31C2×D411yescharacteristic
171C22×C4×C31C2×D411yescharacteristic
172C22×C4×C31C2×D412yes—
173C22×C4×C31#4112yes—
174C22×C4×C31C2411yescharacteristic
175Q16×C31—48no—
176SD16×C31—48no—
177C2×C8×C31—22no—
178C2×C8×C31—22no—
179C2×C8×C31—22no—
180C2×C8×C31—22no—
181C2×C8×C31—22no—
182C2×C8×C31—22no—
183C4⋊C4×C31—24no—
184C4⋊C4×C31—24no—
185C4⋊C4×C31—22no—
186C4⋊C4×C31—24no—
187C4⋊C4×C31—22no—
188C4⋊C4×C31—21no—
189C4⋊C4×C31—21no—
190C4⋊C4×C31—22no—
191C4⋊C4×C31—22no—
192C4⋊C4×C31—42no—
193C4⋊C4×C31—42no—
194C4⋊C4×C31—44no—
195C4⋊C4×C31C2×D412yes—
196C4⋊C4×C31C2×D412yes—
197C22⋊C4×C31—24no—
198C22⋊C4×C31—24no—
199C22⋊C4×C31—22no—
200C22⋊C4×C31—22no—
201C22⋊C4×C31—44no—
202C42×C31—24no—
203C42×C31—24no—
204C42×C31—21no—
205C42×C31—21no—
206C42×C31C2×D412yes—
207C42×C31C2×D412yes—
208#32916C3111yesmaximal, characteristic, direct product factor, Sylow
209C23×C31—21no—
210C23×C31—22no—
211C23×C31—22no—
212C23×C31—21no—
213C23×C31—22no—
214C23×C31—22no—
215C23×C31—22no—
216C23×C31—42no—
217C23×C31—42no—
218C23×C31#12112yes—
219C23×C31C22×D411yescharacteristic
220Q8×C31—22no—
221Q8×C31—22no—
222Q8×C31—22no—
223Q8×C31—42no—
224Q8×C31—42no—
225Q8×C31—42no—
226Q8×C31—42no—
227Q8×C31#12114yes—
228D4×C31—216no—
229D4×C31—22no—
230D4×C31—22no—
231D4×C31—22no—
232D4×C31#12114yes—
233C2×C4×C31—216no—
234C2×C4×C31—22no—
235C2×C4×C31—21no—
236C2×C4×C31—22no—
237C2×C4×C31—21no—
238C2×C4×C31—22no—
239C2×C4×C31—22no—
240C2×C4×C31—22no—
241C2×C4×C31—22no—
242C2×C4×C31—22no—
243C2×C4×C31—22no—
244C2×C4×C31—22no—
245C2×C4×C31—21no—
246C2×C4×C31—21no—
247C2×C4×C31—22no—
248C2×C4×C31—22no—
249C2×C4×C31—22no—
250C2×C4×C31—22no—
251C2×C4×C31—42no—
252C2×C4×C31—42no—
253C2×C4×C31—42no—
254C2×C4×C31—42no—
255C2×C4×C31—42no—
256C2×C4×C31—42no—
257C2×C4×C31—42no—
258C2×C4×C31—42no—
259C2×C4×C31—84no—
260C2×C4×C31—84no—
261C2×C4×C31—84no—
262C2×C4×C31#12012yes—
263C2×C4×C31#12012yes—
264C2×C4×C31#12112yes—
265C2×C4×C31C22×D411yescharacteristic
266C2×C4×C31C22×D411yescharacteristic
267C2×C4×C31C22×D411yescharacteristic
268C2×C4×C31C2×#4111yescharacteristic
269C2×C4×C31C2×#4111yescharacteristic
270C2×C4×C31C2×#4111yescharacteristic
271C8×C31—42no—
272C8×C31—42no—
273C2×#520C2×C3111yescharacteristic
274C2×#462C2×C3111yescharacteristic, complemented
275C2×#461C2×C3111yescharacteristic, complemented
276C2×#452C2×C3112yes—
277#2621C2×C3111yescharacteristic, complemented
278#1878C2×C3112yescomplemented
279#1877C2×C3112yescomplemented
280#1872C2×C3112yescomplemented
281#1805C2×C3111yescharacteristic, complemented
282#1804C2×C3111yescharacteristic, complemented
283#1798C2×C3111yescharacteristic, complemented
284C22×C31—22no—
285C22×C31—24no—
286C22×C31—24no—
287C22×C31—42no—
288C22×C31—44no—
289C22×C31—41no—
290C22×C31—42no—
291C22×C31—41no—
292C22×C31—42no—
293C22×C31—44no—
294C22×C31C2×#12011yescharacteristic
295C22×C31C2×#12111yescharacteristic
296C22×C31#52511yescharacteristic
297C22×C31#57712yes—
298C22×C31#57812yes—
299C4×C31—22no—
300C4×C31—24no—
301C4×C31—22no—
302C4×C31—22no—
303C4×C31—41no—
304C4×C31—42no—
305C4×C31—42no—
306C4×C31—41no—
307C4×C31—82no—
308C4×C31—82no—
309C4×C31C2×#12111yescharacteristic
310C4×C31C2×#12111yescharacteristic
311C4×C31#52511yescharacteristic
312C4×C31#52511yescharacteristic
313C22×#41C22×C3111yescharacteristic
314C22×Q16C22×C3112yes—
315C22×SD16C22×C3112yes—
316C2×C4⋊Q8C22×C3111yescharacteristic
317C2×C22⋊Q8C22×C3112yes—
318#520—24no—
319#520—24no—
320C2×C4×Q8C22×C3111yescharacteristic
321C2×C4×Q8C22×C3111yescharacteristic
322C2×C4×D4—21no—
323C2×C4×D4—21no—
324C2×C4×D4C22×C3111yescharacteristic
325C2×C4×D4C22×C3111yescharacteristic
326C2×#117—21no—
327C2×#117C22×C3111yescharacteristic
328C22×C4⋊C4—21no—
329C22×C4⋊C4C22×C3111yescharacteristic
330#462—24no—
331#461—24no—
332#452—28no—
333C2×C4⋊C8C22×C3111yescharacteristic
334C2×C4⋊C8C22×C3111yescharacteristic
335C2×#103C22×C3112yes—
336C2×#103C22×C3112yes—
337C2×#103C22×C3112yes—
338C2×#102C22×C3112yes—
339C2×C22⋊C8C22×C3112yes—
340#398—21no—
341#398—21no—
342#397—21no—
343#397—21no—
344#391C22×C3112yes—
345#382C22×C3111yescharacteristic
346#340C22×C3112yes—
347#340C22×C3112yes—
348#336C22×C3112yes—
349C2×C31—42no—
350C2×C31—44no—
351C2×C31#284711yescharacteristic
352C2×C31#300911yescharacteristic
353C2×C31#301111yescharacteristic
354C2×C31#304612yes—
355C2×C31#312012yes—
356C2×#41—22no—
357C2×#41—22no—
358C2×#41—22no—
359C2×#41—22no—
360C2×#41—22no—
361C2×#41D4×C3114yes—
362C22×Q8C23×C3112yes—
363C22×Q8C23×C3111yescharacteristic
364C22×D4—21no—
365C22×D4C23×C3111yescharacteristic
366C23×C4—21no—
367C23×C4—21no—
368C23×C4—21no—
369C23×C4—21no—
370C23×C4C23×C3111yescharacteristic
371C2×Q16—24no—
372C2×Q16—24no—
373C2×Q16—24no—
374C2×SD16—24no—
375C2×SD16—24no—
376C2×SD16—24no—
377C22×C8C23×C3112yes—
378C22×C8C23×C3112yes—
379C4⋊Q8—22no—
380C4⋊Q8—22no—
381C22⋊Q8—24no—
382C22⋊Q8—24no—
383C4×Q8—22no—
384C4×Q8—22no—
385C4×Q8—22no—
386C4×Q8D4×C3114yes—
387C4×D4—28no—
388C4×D4—28no—
389C4×D4—22no—
390C4×D4—22no—
391C4×D4—22no—
392C4×D4D4×C3114yes—
393#117—28no—
394#117—22no—
395#117—22no—
396C2×C4⋊C4—28no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—22no—
401C2×C4⋊C4—22no—
402C2×C4⋊C4—22no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4D4×C3111yescharacteristic
406C2×C4⋊C4D4×C3111yescharacteristic
407C2×C4⋊C4C23×C3112yes—
408C2×C4⋊C4C23×C3111yescharacteristic
409C2×C4⋊C4C23×C3111yescharacteristic
410C2×C22⋊C4—21no—
411C2×C22⋊C4—21no—
412C2×C22⋊C4—22no—
413C2×C22⋊C4D4×C3111yescharacteristic
414C2×C22⋊C4D4×C3111yescharacteristic
415C2×C42—21no—
416C2×C42—21no—
417C2×C42C23×C3111yescharacteristic
418C2×C42C23×C3111yescharacteristic
419C4⋊C8—22no—
420C4⋊C8—22no—
421#103—24no—
422#103—24no—
423#103—24no—
424#102—24no—
425C22⋊C8—24no—
426#95—21no—
427#95—21no—
428#95—22no—
429#95—22no—
430C31#3291611yescharacteristic, direct product factor, Sylow
431C24—21no—
432C24C2×D4×C3111yescharacteristic
433#41—28no—
434#41—28no—
435#41—28no—
436#41—28no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—22no—
441C2×Q8—22no—
442C2×Q8—22no—
443C2×Q8—22no—
444C2×Q8—22no—
445C2×Q8—22no—
446C2×Q8C2×D4×C3112yes—
447C2×Q8C2×D4×C3112yes—
448C2×Q8#41×C3112yes—
449C2×D4—24no—
450C2×D4—24no—
451C2×D4—24no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4C2×D4×C3112yes—
456C2×D4C2×D4×C3112yes—
457C2×D4#41×C3112yes—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—21no—
461C22×C4—24no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—82no—
474C22×C4—82no—
475C22×C4—82no—
476C22×C4C2×D4×C3111yescharacteristic
477C22×C4C2×D4×C3111yescharacteristic
478C22×C4C2×D4×C3111yescharacteristic
479C22×C4C2×D4×C3111yescharacteristic
480C22×C4C2×D4×C3111yescharacteristic
481C22×C4C2×D4×C3112yes—
482C22×C4#41×C3112yes—
483C22×C4C24×C3111yescharacteristic
484Q16—48no—
485SD16—48no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—22no—
495C4⋊C4—24no—
496C4⋊C4—22no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—44no—
504C4⋊C4C2×D4×C3112yes—
505C4⋊C4C2×D4×C3112yes—
506C22⋊C4—24no—
507C22⋊C4—24no—
508C22⋊C4—22no—
509C22⋊C4—22no—
510C22⋊C4—44no—
511C42—24no—
512C42—24no—
513C42—21no—
514C42—21no—
515C42C2×D4×C3112yes—
516C42C2×D4×C3112yes—
517C23—21no—
518C23—22no—
519C23—22no—
520C23—21no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—42no—
525C23—42no—
526C23C31×#12112yes—
527C23C22×D4×C3111yescharacteristic
528Q8—22no—
529Q8—22no—
530Q8—22no—
531Q8—42no—
532Q8—42no—
533Q8—42no—
534Q8—42no—
535Q8C31×#12114yes—
536D4—216no—
537D4—22no—
538D4—22no—
539D4—22no—
540D4C31×#12114yes—
541C2×C4—216no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—84no—
568C2×C4—84no—
569C2×C4—84no—
570C2×C4C31×#12012yes—
571C2×C4C31×#12012yes—
572C2×C4C31×#12112yes—
573C2×C4C22×D4×C3111yescharacteristic
574C2×C4C22×D4×C3111yescharacteristic
575C2×C4C22×D4×C3111yescharacteristic
576C2×C4C2×#41×C3111yescharacteristic
577C2×C4C2×#41×C3111yescharacteristic
578C2×C4C2×#41×C3111yescharacteristic
579C8—42no—
580C8—42no—
581C22—22no—
582C22—24no—
583C22—24no—
584C22—42no—
585C22—44no—
586C22—41no—
587C22—42no—
588C22—41no—
589C22—42no—
590C22—44no—
591C22C2×C31×#12011yescharacteristic
592C22C2×C31×#12111yescharacteristic
593C22C31×#52511yescharacteristic
594C22C31×#57712yes—
595C22C31×#57812yes—
596C4—22no—
597C4—24no—
598C4—22no—
599C4—22no—
600C4—41no—
601C4—42no—
602C4—42no—
603C4—41no—
604C4—82no—
605C4—82no—
606C4C2×C31×#12111yescharacteristic
607C4C2×C31×#12111yescharacteristic
608C4C31×#52511yescharacteristic
609C4C31×#52511yescharacteristic
610C2—42no—
611C2—44no—
612C2C31×#284711yescharacteristic
613C2C31×#300911yescharacteristic
614C2C31×#301111yescharacteristic
615C2C31×#304612yes—
616C2C31×#312012yes—
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC22×#41×C31C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C2×C31×#117 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×C31×#462C2semidirect
3. C2×C31×#461C2semidirect
5. C31×#2621C2semidirect
6. C31×#1878C2semidirect
7. C31×#1877C2semidirect
8. C31×#1872C2semidirect
9. C31×#1805C2semidirect
10. C31×#1804C2semidirect
11. C31×#1798C2semidirect
208. #32916C31direct (N × H)
274. C2×#462C2×C31semidirect
275. C2×#461C2×C31semidirect
277. #2621C2×C31semidirect
278. #1878C2×C31semidirect
279. #1877C2×C31semidirect
280. #1872C2×C31semidirect
281. #1805C2×C31semidirect
282. #1804C2×C31semidirect
283. #1798C2×C31semidirect
430. C31#32916direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5202
C2×C31×#4622
C2×C31×#4612
C2×C31×#4522
C31×#26212
C31×#18782
C31×#18772
C31×#18722
C31×#18052
C31×#18042
C31×#17982
#3291631