Finite Group Databasebeta

← C31×#8480 #21465239

Subgroups — C31×#8480 #21465239

Order 7936.

828 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3536C211yesmaximal, characteristic
2C31×#1996C211yesmaximal, characteristic, complemented
3C31×#1901C211yesmaximal, characteristic, complemented
4C31×#1895C211yesmaximal, characteristic, complemented
5C31×#1871C211yesmaximal, characteristic, complemented
6C31×#1513C211yesmaximal, characteristic, complemented
7C31×#1507C211yesmaximal, characteristic, complemented
8C22×M16×C31C2211yescharacteristic
9C31×#562—22no—
10C31×#554—21no—
11C31×#548—21no—
12C31×#547—21no—
13C31×#546—21no—
14C31×#546—22no—
15C31×#543—21no—
16C31×#542—21no—
17C31×#539C412yescomplemented
18C31×#538—21no—
19C31×#537—21no—
20C31×#537C412yescomplemented
21C31×#525C2211yescharacteristic
22C2×C31×C22⋊Q8C2211yescharacteristic
23C2×C31×#121C2211yescharacteristic
24C2×C31×#104—21no—
25C2×C31×#102—21no—
26C2×C31×#101—21no—
27C31×#407C2211yescharacteristic
28C31×#380C2211yescharacteristic
29C4×C22⋊C4×C31C2211yescharacteristic
30C31×#343—21no—
31C31×#342—22no—
32C31×#340—21no—
33C31×#339—21no—
34C31×#328—21no—
35C31×#327—22no—
36C31×#327—21no—
37C2×#41×C31—21no—
38C2×#41×C31—22no—
39C2×#41×C31D411yescharacteristic
40C22×Q8×C31D411yescharacteristic
41C22×D4×C31—21no—
42C22×D4×C31D411yescharacteristic
43C23×C4×C31C2311yescharacteristic
44C2×M16×C31—21no—
45C2×M16×C31—21no—
46C2×M16×C31—21no—
47C2×M16×C31—42no—
48C22×C8×C31—21no—
49C31×C4⋊Q8—21no—
50C31×#126—22no—
51C31×#125—22no—
52C31×#124—22no—
53C31×#124—21no—
54C31×#124—22no—
55C31×#123—22no—
56C31×#123—22no—
57C31×#123—22no—
58C31×#123—22no—
59C31×#123—21no—
60C31×C22⋊Q8—21no—
61C31×C22⋊Q8—22no—
62C31×C22⋊Q8—22no—
63C31×C22⋊Q8—22no—
64C31×C22⋊Q8—22no—
65C31×C22⋊Q8—21no—
66C31×C22⋊Q8—21no—
67C31×C22⋊Q8—21no—
68C31×C22⋊Q8—21no—
69C31×C22⋊Q8—22no—
70C31×C22⋊Q8—22no—
71C31×C22⋊Q8C2×C411yescharacteristic
72C31×C22⋊Q8D411yescharacteristic
73C31×#121—21no—
74C31×#121—22no—
75C31×#121—22no—
76C31×#121—22no—
77C31×#121—22no—
78C31×#121—22no—
79C31×#121—21no—
80C31×#121—21no—
81C31×#121C2×C412yes—
82C31×#121C2×C411yescharacteristic
83C31×#121D412yes—
84C31×#121D411yescharacteristic
85C31×#120—22no—
86C31×#120—21no—
87C4×Q8×C31—21no—
88C4×D4×C31—22no—
89C4×D4×C31—21no—
90C4×D4×C31—22no—
91C4×D4×C31—21no—
92C4×D4×C31—21no—
93C31×#117D411yescharacteristic
94C2×C4⋊C4×C31—21no—
95C2×C4⋊C4×C31—21no—
96C2×C4⋊C4×C31C2×C412yes—
97C2×C4⋊C4×C31D411yescharacteristic
98C2×C4⋊C4×C31D411yescharacteristic
99C2×C22⋊C4×C31—22no—
100C2×C22⋊C4×C31—21no—
101C2×C22⋊C4×C31—21no—
102C2×C22⋊C4×C31—21no—
103C2×C22⋊C4×C31D412yes—
104C2×C42×C31—21no—
105C31×#104—42no—
106C31×#102—42no—
107C31×#101—42no—
108C31×C22⋊C8—21no—
109C31×C22⋊C8—21no—
110C31×#95—21no—
111C31×#95—21no—
112C24×C31—21no—
113C24×C31C2×D411yescharacteristic
114#41×C31—41no—
115#41×C31—41no—
116#41×C31—84no—
117#41×C31—81no—
118#41×C31—81no—
119C2×Q8×C31—22no—
120C2×Q8×C31—21no—
121C2×Q8×C31—21no—
122C2×Q8×C31—41no—
123C2×Q8×C31—41no—
124C2×Q8×C31C22⋊C411yescharacteristic
125C2×D4×C31—21no—
126C2×D4×C31—22no—
127C2×D4×C31—21no—
128C2×D4×C31—22no—
129C2×D4×C31—22no—
130C2×D4×C31—21no—
131C2×D4×C31—22no—
132C2×D4×C31—21no—
133C2×D4×C31—41no—
134C2×D4×C31—41no—
135C2×D4×C31—82no—
136C2×D4×C31C22⋊C411yescharacteristic
137C2×D4×C31C22⋊C412yes—
138C22×C4×C31—22no—
139C22×C4×C31—22no—
140C22×C4×C31—22no—
141C22×C4×C31—21no—
142C22×C4×C31—21no—
143C22×C4×C31—41no—
144C22×C4×C31—41no—
145C22×C4×C31—41no—
146C22×C4×C31—41no—
147C22×C4×C31C22⋊C412yes—
148C22×C4×C31C22⋊C412yes—
149C22×C4×C31C22×C411yescharacteristic
150C22×C4×C31C2×D411yescharacteristic
151C22×C4×C31C2×D411yescharacteristic
152C22×C4×C31C2×D411yescharacteristic
153C22×C4×C31C2×D411yescharacteristic
154C22×C4×C31C2×D411yescharacteristic
155M16×C31—22no—
156M16×C31—22no—
157M16×C31—41no—
158M16×C31—41no—
159M16×C31—41no—
160M16×C31—41no—
161C2×C8×C31—22no—
162C2×C8×C31—41no—
163C2×C8×C31—41no—
164C2×C8×C31—41no—
165C4⋊C4×C31—22no—
166C4⋊C4×C31—22no—
167C4⋊C4×C31—22no—
168C4⋊C4×C31—22no—
169C4⋊C4×C31—22no—
170C4⋊C4×C31—22no—
171C4⋊C4×C31—21no—
172C4⋊C4×C31—21no—
173C4⋊C4×C31—21no—
174C4⋊C4×C31—21no—
175C4⋊C4×C31—21no—
176C4⋊C4×C31—41no—
177C4⋊C4×C31—41no—
178C4⋊C4×C31C22⋊C412yes—
179C4⋊C4×C31C22⋊C411yescharacteristic
180C4⋊C4×C31C22⋊C411yescharacteristic
181C22⋊C4×C31—22no—
182C22⋊C4×C31—22no—
183C22⋊C4×C31—22no—
184C22⋊C4×C31—22no—
185C22⋊C4×C31—22no—
186C22⋊C4×C31—21no—
187C22⋊C4×C31—21no—
188C22⋊C4×C31—22no—
189C22⋊C4×C31—22no—
190C22⋊C4×C31—21no—
191C22⋊C4×C31—22no—
192C22⋊C4×C31—21no—
193C22⋊C4×C31—22no—
194C22⋊C4×C31—21no—
195C22⋊C4×C31—22no—
196C22⋊C4×C31—41no—
197C22⋊C4×C31—42no—
198C42×C31—22no—
199C42×C31—21no—
200C42×C31—41no—
201C42×C31—41no—
202#8480C3111yesmaximal, characteristic, direct product factor, Sylow
203C23×C31—21no—
204C23×C31—21no—
205C23×C31—22no—
206C23×C31—21no—
207C23×C31—41no—
208C23×C31—41no—
209C23×C31—41no—
210C23×C31—41no—
211C23×C31—81no—
212C23×C31—81no—
213C23×C31C2×C22⋊C411yescharacteristic
214C23×C31#12011yescharacteristic
215C23×C31#12011yescharacteristic
216Q8×C31—41no—
217Q8×C31—41no—
218Q8×C31—41no—
219Q8×C31—81no—
220Q8×C31—82no—
221D4×C31—42no—
222D4×C31—41no—
223D4×C31—44no—
224D4×C31—42no—
225D4×C31—41no—
226D4×C31—41no—
227D4×C31—41no—
228D4×C31—41no—
229D4×C31—82no—
230D4×C31—81no—
231D4×C31—82no—
232D4×C31—82no—
233C2×C4×C31—21no—
234C2×C4×C31—21no—
235C2×C4×C31—21no—
236C2×C4×C31—21no—
237C2×C4×C31—21no—
238C2×C4×C31—21no—
239C2×C4×C31—21no—
240C2×C4×C31—22no—
241C2×C4×C31—21no—
242C2×C4×C31—22no—
243C2×C4×C31—21no—
244C2×C4×C31—22no—
245C2×C4×C31—41no—
246C2×C4×C31—41no—
247C2×C4×C31—41no—
248C2×C4×C31—41no—
249C2×C4×C31—41no—
250C2×C4×C31—42no—
251C2×C4×C31—42no—
252C2×C4×C31—42no—
253C2×C4×C31—81no—
254C2×C4×C31—82no—
255C2×C4×C31—82no—
256C2×C4×C31—82no—
257C2×C4×C31—81no—
258C2×C4×C31—81no—
259C2×C4×C31—81no—
260C2×C4×C31—81no—
261C2×C4×C31—81no—
262C2×C4×C31C2×C22⋊C411yescharacteristic
263C2×C4×C31C2×C22⋊C411yescharacteristic
264C2×C4×C31#12011yescharacteristic
265C2×C4×C31#12011yescharacteristic
266C8×C31—42no—
267C8×C31—81no—
268#3536C2×C3111yescharacteristic
269#1996C2×C3111yescharacteristic, complemented
270#1901C2×C3111yescharacteristic, complemented
271#1895C2×C3111yescharacteristic, complemented
272#1871C2×C3111yescharacteristic, complemented
273#1513C2×C3111yescharacteristic, complemented
274#1507C2×C3111yescharacteristic, complemented
275C22×C31—21no—
276C22×C31—21no—
277C22×C31—21no—
278C22×C31—21no—
279C22×C31—21no—
280C22×C31—21no—
281C22×C31—21no—
282C22×C31—21no—
283C22×C31—41no—
284C22×C31—42no—
285C22×C31—41no—
286C22×C31—42no—
287C22×C31—41no—
288C22×C31—42no—
289C22×C31—81no—
290C22×C31—81no—
291C22×C31—161no—
292C22×C31—161no—
293C22×C31#37911yescharacteristic
294C22×C31#45411yescharacteristic
295C22×C31#45611yescharacteristic
296C4×C31—21no—
297C4×C31—21no—
298C4×C31—21no—
299C4×C31—41no—
300C4×C31—42no—
301C4×C31—42no—
302C4×C31—41no—
303C4×C31—41no—
304C4×C31—42no—
305C4×C31—82no—
306C4×C31—161no—
307C4×C31B(4,𝔽2)11yescharacteristic
308C4×C31#45811yescharacteristic
309C22×M16C22×C3111yescharacteristic
310#562—22no—
311#554—21no—
312#548—21no—
313#547—21no—
314#546—21no—
315#546—22no—
316#543—21no—
317#542—21no—
318#539C4×C3112yescomplemented
319#538—21no—
320#537—21no—
321#537C4×C3112yescomplemented
322#525C22×C3111yescharacteristic
323C2×C22⋊Q8C22×C3111yescharacteristic
324C2×#121C22×C3111yescharacteristic
325C2×#104—21no—
326C2×#102—21no—
327C2×#101—21no—
328#407C22×C3111yescharacteristic
329#380C22×C3111yescharacteristic
330C4×C22⋊C4C22×C3111yescharacteristic
331#343—21no—
332#342—22no—
333#340—21no—
334#339—21no—
335#328—21no—
336#327—22no—
337#327—21no—
338C2×C31—21no—
339C2×C31—21no—
340C2×C31—41no—
341C2×C31—41no—
342C2×C31—81no—
343C2×C31—82no—
344C2×C31#179011yescharacteristic
345C2×C31#179311yescharacteristic
346C2×C31#179511yescharacteristic
347C2×#41—21no—
348C2×#41—22no—
349C2×#41D4×C3111yescharacteristic
350C22×Q8D4×C3111yescharacteristic
351C22×D4—21no—
352C22×D4D4×C3111yescharacteristic
353C23×C4C23×C3111yescharacteristic
354C2×M16—21no—
355C2×M16—21no—
356C2×M16—21no—
357C2×M16—42no—
358C22×C8—21no—
359C4⋊Q8—21no—
360#126—22no—
361#125—22no—
362#124—22no—
363#124—21no—
364#124—22no—
365#123—22no—
366#123—22no—
367#123—22no—
368#123—22no—
369#123—21no—
370C22⋊Q8—21no—
371C22⋊Q8—22no—
372C22⋊Q8—22no—
373C22⋊Q8—22no—
374C22⋊Q8—22no—
375C22⋊Q8—21no—
376C22⋊Q8—21no—
377C22⋊Q8—21no—
378C22⋊Q8—21no—
379C22⋊Q8—22no—
380C22⋊Q8—22no—
381C22⋊Q8C2×C4×C3111yescharacteristic
382C22⋊Q8D4×C3111yescharacteristic
383#121—21no—
384#121—22no—
385#121—22no—
386#121—22no—
387#121—22no—
388#121—22no—
389#121—21no—
390#121—21no—
391#121C2×C4×C3112yes—
392#121C2×C4×C3111yescharacteristic
393#121D4×C3112yes—
394#121D4×C3111yescharacteristic
395#120—22no—
396#120—21no—
397C4×Q8—21no—
398C4×D4—22no—
399C4×D4—21no—
400C4×D4—22no—
401C4×D4—21no—
402C4×D4—21no—
403#117D4×C3111yescharacteristic
404C2×C4⋊C4—21no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4C2×C4×C3112yes—
407C2×C4⋊C4D4×C3111yescharacteristic
408C2×C4⋊C4D4×C3111yescharacteristic
409C2×C22⋊C4—22no—
410C2×C22⋊C4—21no—
411C2×C22⋊C4—21no—
412C2×C22⋊C4—21no—
413C2×C22⋊C4D4×C3112yes—
414C2×C42—21no—
415#104—42no—
416#102—42no—
417#101—42no—
418C22⋊C8—21no—
419C22⋊C8—21no—
420#95—21no—
421#95—21no—
422C31#848011yescharacteristic, direct product factor, Sylow
423C24—21no—
424C24C2×D4×C3111yescharacteristic
425#41—41no—
426#41—41no—
427#41—84no—
428#41—81no—
429#41—81no—
430C2×Q8—22no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—41no—
434C2×Q8—41no—
435C2×Q8C22⋊C4×C3111yescharacteristic
436C2×D4—21no—
437C2×D4—22no—
438C2×D4—21no—
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—21no—
442C2×D4—22no—
443C2×D4—21no—
444C2×D4—41no—
445C2×D4—41no—
446C2×D4—82no—
447C2×D4C22⋊C4×C3111yescharacteristic
448C2×D4C22⋊C4×C3112yes—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—41no—
455C22×C4—41no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4C22⋊C4×C3112yes—
459C22×C4C22⋊C4×C3112yes—
460C22×C4C22×C4×C3111yescharacteristic
461C22×C4C2×D4×C3111yescharacteristic
462C22×C4C2×D4×C3111yescharacteristic
463C22×C4C2×D4×C3111yescharacteristic
464C22×C4C2×D4×C3111yescharacteristic
465C22×C4C2×D4×C3111yescharacteristic
466M16—22no—
467M16—22no—
468M16—41no—
469M16—41no—
470M16—41no—
471M16—41no—
472C2×C8—22no—
473C2×C8—41no—
474C2×C8—41no—
475C2×C8—41no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—41no—
488C4⋊C4—41no—
489C4⋊C4C22⋊C4×C3112yes—
490C4⋊C4C22⋊C4×C3111yescharacteristic
491C4⋊C4C22⋊C4×C3111yescharacteristic
492C22⋊C4—22no—
493C22⋊C4—22no—
494C22⋊C4—22no—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—21no—
498C22⋊C4—21no—
499C22⋊C4—22no—
500C22⋊C4—22no—
501C22⋊C4—21no—
502C22⋊C4—22no—
503C22⋊C4—21no—
504C22⋊C4—22no—
505C22⋊C4—21no—
506C22⋊C4—22no—
507C22⋊C4—41no—
508C22⋊C4—42no—
509C42—22no—
510C42—21no—
511C42—41no—
512C42—41no—
513C23—21no—
514C23—21no—
515C23—22no—
516C23—21no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—81no—
522C23—81no—
523C23C2×C22⋊C4×C3111yescharacteristic
524C23C31×#12011yescharacteristic
525C23C31×#12011yescharacteristic
526Q8—41no—
527Q8—41no—
528Q8—41no—
529Q8—81no—
530Q8—82no—
531D4—42no—
532D4—41no—
533D4—44no—
534D4—42no—
535D4—41no—
536D4—41no—
537D4—41no—
538D4—41no—
539D4—82no—
540D4—81no—
541D4—82no—
542D4—82no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—81no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—81no—
568C2×C4—81no—
569C2×C4—81no—
570C2×C4—81no—
571C2×C4—81no—
572C2×C4C2×C22⋊C4×C3111yescharacteristic
573C2×C4C2×C22⋊C4×C3111yescharacteristic
574C2×C4C31×#12011yescharacteristic
575C2×C4C31×#12011yescharacteristic
576C8—42no—
577C8—81no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—21no—
584C22—21no—
585C22—21no—
586C22—41no—
587C22—42no—
588C22—41no—
589C22—42no—
590C22—41no—
591C22—42no—
592C22—81no—
593C22—81no—
594C22—161no—
595C22—161no—
596C22C31×#37911yescharacteristic
597C22C31×#45411yescharacteristic
598C22C31×#45611yescharacteristic
599C4—21no—
600C4—21no—
601C4—21no—
602C4—42no—
603C4—42no—
604C4—41no—
605C4—41no—
606C4—42no—
607C4—41no—
608C4—82no—
609C4—161no—
610C4C31×B(4,𝔽2)11yescharacteristic
611C4C31×#45811yescharacteristic
612C2—21no—
613C2—21no—
614C2—41no—
615C2—41no—
616C2—82no—
617C2—81no—
618C2C31×#179011yescharacteristic
619C2C31×#179311yescharacteristic
620C2C31×#179511yescharacteristic
subgroupG/H
center, socleC22×C31#379
commutatorC22×C4C22×C4×C31
Frattini, cosocleC23×C4C23×C31
prime radicalC31×#3536C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#1996C2semidirect
3. C31×#1901C2semidirect
4. C31×#1895C2semidirect
5. C31×#1871C2semidirect
6. C31×#1513C2semidirect
7. C31×#1507C2semidirect
17. C31×#539C4semidirect
20. C31×#537C4semidirect
202. #8480C31direct (N × H)
269. #1996C2×C31semidirect
270. #1901C2×C31semidirect
271. #1895C2×C31semidirect
272. #1871C2×C31semidirect
273. #1513C2×C31semidirect
274. #1507C2×C31semidirect
318. #539C4×C31semidirect
321. #537C4×C31semidirect
422. C31#8480direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35362
C31×#19962
C31×#19012
C31×#18952
C31×#18712
C31×#15132
C31×#15072
#848031