Finite Group Databasebeta

← C31×#18270 #21475029

Subgroups — C31×#18270 #21475029

Order 7936.

1000 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
1C31×#3536C211yesmaximal, characteristic
2C31×#3092C211yesmaximal, characteristic, complemented
3C2×C31×#480C211yesmaximal, characteristic, complemented
4C31×#2502C211yesmaximal, characteristic, complemented
5C31×#1610C214yesmaximal, complemented
6C31×#1609C214yesmaximal, complemented
7C31×#1507C211yesmaximal, characteristic, complemented
8C31×#1501C211yesmaximal, characteristic, complemented
9C31×#1472C211yesmaximal, characteristic, complemented
10C31×#562—22no—
11C31×#554—21no—
12C31×#548—21no—
13C31×#547—21no—
14C31×#546—21no—
15C31×#546—22no—
16C31×#543—21no—
17C31×#542—21no—
18C31×#539C2212yes—
19C31×#538—21no—
20C31×#537—21no—
21C31×#537C2212yes—
22C31×#525C2211yescharacteristic
23C2×C31×C22⋊Q8C2211yescharacteristic
24C2×C31×#121C2211yescharacteristic
25C22×C4⋊C4×C31C2211yescharacteristic
26C31×#484—21no—
27C31×#483C2212yescomplemented
28C31×#481—21no—
29C31×#480—22no—
30C31×#480C2214yes—
31C31×#480C2212yescomplemented
32C31×#428—21no—
33C2×C31×#107—21no—
34C31×#417—21no—
35C2×C31×#102—21no—
36C31×#407C2211yescharacteristic
37C2×C31×C22⋊C8C2211yescharacteristic
38C31×#398C2214yescomplemented
39C31×#391—21no—
40C31×#385—22no—
41C31×#384—21no—
42C4×C22⋊C4×C31C2211yescharacteristic
43C31×#328C2212yescomplemented
44C31×#327C2214yes—
45C31×#327C2212yescomplemented
46C31×#324C2214yescomplemented
47C2×#41×C31—21no—
48C2×#41×C31—22no—
49C2×#41×C31D411yescharacteristic
50C22×Q8×C31D411yescharacteristic
51C22×D4×C31—21no—
52C22×D4×C31D411yescharacteristic
53C23×C4×C31—21no—
54C23×C4×C31C2311yescharacteristic
55C2×M16×C31—21no—
56C22×C8×C31—21no—
57C31×C4⋊Q8—21no—
58C31×#126—22no—
59C31×#125—22no—
60C31×#124—22no—
61C31×#124—21no—
62C31×#124—22no—
63C31×#123—22no—
64C31×#123—22no—
65C31×#123—22no—
66C31×#123—21no—
67C31×#123—22no—
68C31×C22⋊Q8—21no—
69C31×C22⋊Q8—22no—
70C31×C22⋊Q8—22no—
71C31×C22⋊Q8—22no—
72C31×C22⋊Q8—22no—
73C31×C22⋊Q8—21no—
74C31×C22⋊Q8—21no—
75C31×C22⋊Q8—21no—
76C31×C22⋊Q8—21no—
77C31×C22⋊Q8—22no—
78C31×C22⋊Q8—22no—
79C31×C22⋊Q8D411yescharacteristic
80C31×C22⋊Q8C2311yescharacteristic
81C31×#121—21no—
82C31×#121—22no—
83C31×#121—22no—
84C31×#121—22no—
85C31×#121—22no—
86C31×#121—21no—
87C31×#121—21no—
88C31×#121—22no—
89C31×#121D412yes—
90C31×#121D411yescharacteristic
91C31×#121C2312yes—
92C31×#121C2311yescharacteristic
93C31×#120—22no—
94C31×#120—21no—
95C4×Q8×C31—21no—
96C4×D4×C31—22no—
97C4×D4×C31—21no—
98C4×D4×C31—21no—
99C4×D4×C31—21no—
100C4×D4×C31—22no—
101C31×#117D411yescharacteristic
102C2×C4⋊C4×C31—21no—
103C2×C4⋊C4×C31—21no—
104C2×C4⋊C4×C31—21no—
105C2×C4⋊C4×C31—21no—
106C2×C4⋊C4×C31—21no—
107C2×C4⋊C4×C31—21no—
108C2×C4⋊C4×C31—24no—
109C2×C4⋊C4×C31D411yescharacteristic
110C2×C4⋊C4×C31D411yescharacteristic
111C2×C4⋊C4×C31C2312yes—
112C2×C4⋊C4×C31C2312yes—
113C2×C22⋊C4×C31—22no—
114C2×C22⋊C4×C31—21no—
115C2×C22⋊C4×C31—21no—
116C2×C22⋊C4×C31D412yes—
117C2×C42×C31—21no—
118C31×#107—24no—
119C31×#107—22no—
120C31×#106—22no—
121C31×#103—22no—
122C31×#102—24no—
123C31×#102—22no—
124C31×C22⋊C8—21no—
125C31×C22⋊C8—21no—
126C31×C22⋊C8C2312yes—
127C31×C22⋊C8C2312yes—
128C31×#95—24no—
129C31×#95C2312yes—
130C24×C31—21no—
131C24×C31C2×D411yescharacteristic
132#41×C31—42no—
133#41×C31—84no—
134#41×C31—82no—
135C2×Q8×C31—22no—
136C2×Q8×C31—21no—
137C2×Q8×C31—21no—
138C2×Q8×C31—42no—
139C2×Q8×C31C2×D411yescharacteristic
140C2×D4×C31—21no—
141C2×D4×C31—21no—
142C2×D4×C31—22no—
143C2×D4×C31—22no—
144C2×D4×C31—22no—
145C2×D4×C31—21no—
146C2×D4×C31—22no—
147C2×D4×C31—21no—
148C2×D4×C31—42no—
149C2×D4×C31—82no—
150C2×D4×C31C2×D411yescharacteristic
151C2×D4×C31C2×D412yes—
152C22×C4×C31—22no—
153C22×C4×C31—22no—
154C22×C4×C31—22no—
155C22×C4×C31—22no—
156C22×C4×C31—21no—
157C22×C4×C31—21no—
158C22×C4×C31—22no—
159C22×C4×C31—42no—
160C22×C4×C31—41no—
161C22×C4×C31—41no—
162C22×C4×C31—82no—
163C22×C4×C31C2×D412yes—
164C22×C4×C31C2×D411yescharacteristic
165C22×C4×C31C2×D411yescharacteristic
166C22×C4×C31C2×D411yescharacteristic
167C22×C4×C31C2×D412yes—
168C22×C4×C31C2×D411yescharacteristic
169C22×C4×C31C2×D411yescharacteristic
170C22×C4×C31C2411yescharacteristic
171M16×C31—42no—
172C2×C8×C31—22no—
173C2×C8×C31—22no—
174C2×C8×C31—42no—
175C4⋊C4×C31—22no—
176C4⋊C4×C31—22no—
177C4⋊C4×C31—22no—
178C4⋊C4×C31—22no—
179C4⋊C4×C31—21no—
180C4⋊C4×C31—21no—
181C4⋊C4×C31—21no—
182C4⋊C4×C31—21no—
183C4⋊C4×C31—22no—
184C4⋊C4×C31—22no—
185C4⋊C4×C31—21no—
186C4⋊C4×C31—22no—
187C4⋊C4×C31—22no—
188C4⋊C4×C31—41no—
189C4⋊C4×C31—41no—
190C4⋊C4×C31—84no—
191C4⋊C4×C31C2×D411yescharacteristic
192C4⋊C4×C31C2×D412yes—
193C4⋊C4×C31C2×D411yescharacteristic
194C22⋊C4×C31—24no—
195C22⋊C4×C31—22no—
196C22⋊C4×C31—22no—
197C22⋊C4×C31—22no—
198C22⋊C4×C31—22no—
199C22⋊C4×C31—22no—
200C22⋊C4×C31—21no—
201C22⋊C4×C31—22no—
202C22⋊C4×C31—22no—
203C22⋊C4×C31—21no—
204C22⋊C4×C31—21no—
205C22⋊C4×C31—22no—
206C22⋊C4×C31—21no—
207C22⋊C4×C31—22no—
208C22⋊C4×C31—21no—
209C42×C31—22no—
210C42×C31—21no—
211C42×C31—42no—
212#18270C3111yesmaximal, characteristic, direct product factor, Sylow
213C23×C31—22no—
214C23×C31—22no—
215C23×C31—22no—
216C23×C31—21no—
217C23×C31—41no—
218C23×C31—42no—
219C23×C31—81no—
220C23×C31—81no—
221C23×C31#12011yescharacteristic
222C23×C31#12011yescharacteristic
223C23×C31C22×D411yescharacteristic
224Q8×C31—41no—
225Q8×C31—42no—
226Q8×C31—81no—
227Q8×C31—82no—
228D4×C31—42no—
229D4×C31—41no—
230D4×C31—44no—
231D4×C31—42no—
232D4×C31—42no—
233D4×C31—42no—
234D4×C31—82no—
235D4×C31—81no—
236D4×C31—82no—
237D4×C31—82no—
238C2×C4×C31—22no—
239C2×C4×C31—22no—
240C2×C4×C31—22no—
241C2×C4×C31—24no—
242C2×C4×C31—21no—
243C2×C4×C31—22no—
244C2×C4×C31—21no—
245C2×C4×C31—22no—
246C2×C4×C31—21no—
247C2×C4×C31—22no—
248C2×C4×C31—24no—
249C2×C4×C31—41no—
250C2×C4×C31—41no—
251C2×C4×C31—41no—
252C2×C4×C31—42no—
253C2×C4×C31—42no—
254C2×C4×C31—42no—
255C2×C4×C31—42no—
256C2×C4×C31—42no—
257C2×C4×C31—44no—
258C2×C4×C31—81no—
259C2×C4×C31—82no—
260C2×C4×C31—82no—
261C2×C4×C31—82no—
262C2×C4×C31—82no—
263C2×C4×C31—82no—
264C2×C4×C31#12011yescharacteristic
265C2×C4×C31#12011yescharacteristic
266C2×C4×C31C22×D411yescharacteristic
267C2×C4×C31C22×D411yescharacteristic
268C8×C31—42no—
269C8×C31—42no—
270#3536C2×C3111yescharacteristic
271#3092C2×C3111yescharacteristic, complemented
272C2×#480C2×C3111yescharacteristic, complemented
273#2502C2×C3111yescharacteristic, complemented
274#1610C2×C3114yescomplemented
275#1609C2×C3114yescomplemented
276#1507C2×C3111yescharacteristic, complemented
277#1501C2×C3111yescharacteristic, complemented
278#1472C2×C3111yescharacteristic, complemented
279C22×C31—21no—
280C22×C31—21no—
281C22×C31—21no—
282C22×C31—21no—
283C22×C31—21no—
284C22×C31—21no—
285C22×C31—21no—
286C22×C31—44no—
287C22×C31—41no—
288C22×C31—42no—
289C22×C31—41no—
290C22×C31—42no—
291C22×C31—41no—
292C22×C31—42no—
293C22×C31—162no—
294C22×C31#45612yes—
295C22×C31C2×#12011yescharacteristic
296C22×C31#57612yes—
297C4×C31—21no—
298C4×C31—21no—
299C4×C31—21no—
300C4×C31—44no—
301C4×C31—42no—
302C4×C31—42no—
303C4×C31—41no—
304C4×C31—41no—
305C4×C31—41no—
306C4×C31—42no—
307C4×C31—82no—
308C4×C31#45812yes—
309#562—22no—
310#554—21no—
311#548—21no—
312#547—21no—
313#546—21no—
314#546—22no—
315#543—21no—
316#542—21no—
317#539C22×C3112yes—
318#538—21no—
319#537—21no—
320#537C22×C3112yes—
321#525C22×C3111yescharacteristic
322C2×C22⋊Q8C22×C3111yescharacteristic
323C2×#121C22×C3111yescharacteristic
324C22×C4⋊C4C22×C3111yescharacteristic
325#484—21no—
326#483C22×C3112yescomplemented
327#481—21no—
328#480—22no—
329#480C22×C3114yes—
330#480C22×C3112yescomplemented
331#428—21no—
332C2×#107—21no—
333#417—21no—
334C2×#102—21no—
335#407C22×C3111yescharacteristic
336C2×C22⋊C8C22×C3111yescharacteristic
337#398C22×C3114yescomplemented
338#391—21no—
339#385—22no—
340#384—21no—
341C4×C22⋊C4C22×C3111yescharacteristic
342#328C22×C3112yescomplemented
343#327C22×C3114yes—
344#327C22×C3112yescomplemented
345#324C22×C3114yescomplemented
346C2×C31—22no—
347C2×C31—22no—
348C2×C31—41no—
349C2×C31—81no—
350C2×C31—82no—
351C2×C31#300811yescharacteristic
352C2×C31C2×#45611yescharacteristic
353C2×C31C2×#45811yescharacteristic
354C2×#41—21no—
355C2×#41—22no—
356C2×#41D4×C3111yescharacteristic
357C22×Q8D4×C3111yescharacteristic
358C22×D4—21no—
359C22×D4D4×C3111yescharacteristic
360C23×C4—21no—
361C23×C4C23×C3111yescharacteristic
362C2×M16—21no—
363C22×C8—21no—
364C4⋊Q8—21no—
365#126—22no—
366#125—22no—
367#124—22no—
368#124—21no—
369#124—22no—
370#123—22no—
371#123—22no—
372#123—22no—
373#123—21no—
374#123—22no—
375C22⋊Q8—21no—
376C22⋊Q8—22no—
377C22⋊Q8—22no—
378C22⋊Q8—22no—
379C22⋊Q8—22no—
380C22⋊Q8—21no—
381C22⋊Q8—21no—
382C22⋊Q8—21no—
383C22⋊Q8—21no—
384C22⋊Q8—22no—
385C22⋊Q8—22no—
386C22⋊Q8D4×C3111yescharacteristic
387C22⋊Q8C23×C3111yescharacteristic
388#121—21no—
389#121—22no—
390#121—22no—
391#121—22no—
392#121—22no—
393#121—21no—
394#121—21no—
395#121—22no—
396#121D4×C3112yes—
397#121D4×C3111yescharacteristic
398#121C23×C3112yes—
399#121C23×C3111yescharacteristic
400#120—22no—
401#120—21no—
402C4×Q8—21no—
403C4×D4—22no—
404C4×D4—21no—
405C4×D4—21no—
406C4×D4—21no—
407C4×D4—22no—
408#117D4×C3111yescharacteristic
409C2×C4⋊C4—21no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—24no—
416C2×C4⋊C4D4×C3111yescharacteristic
417C2×C4⋊C4D4×C3111yescharacteristic
418C2×C4⋊C4C23×C3112yes—
419C2×C4⋊C4C23×C3112yes—
420C2×C22⋊C4—22no—
421C2×C22⋊C4—21no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4D4×C3112yes—
424C2×C42—21no—
425#107—24no—
426#107—22no—
427#106—22no—
428#103—22no—
429#102—24no—
430#102—22no—
431C22⋊C8—21no—
432C22⋊C8—21no—
433C22⋊C8C23×C3112yes—
434C22⋊C8C23×C3112yes—
435#95—24no—
436#95C23×C3112yes—
437C31#1827011yescharacteristic, direct product factor, Sylow
438C24—21no—
439C24C2×D4×C3111yescharacteristic
440#41—42no—
441#41—84no—
442#41—82no—
443C2×Q8—22no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—42no—
447C2×Q8C2×D4×C3111yescharacteristic
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—21no—
454C2×D4—22no—
455C2×D4—21no—
456C2×D4—42no—
457C2×D4—82no—
458C2×D4C2×D4×C3111yescharacteristic
459C2×D4C2×D4×C3112yes—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—42no—
468C22×C4—41no—
469C22×C4—41no—
470C22×C4—82no—
471C22×C4C2×D4×C3112yes—
472C22×C4C2×D4×C3111yescharacteristic
473C22×C4C2×D4×C3111yescharacteristic
474C22×C4C2×D4×C3111yescharacteristic
475C22×C4C2×D4×C3112yes—
476C22×C4C2×D4×C3111yescharacteristic
477C22×C4C2×D4×C3111yescharacteristic
478C22×C4C24×C3111yescharacteristic
479M16—42no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—42no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—21no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—41no—
497C4⋊C4—41no—
498C4⋊C4—84no—
499C4⋊C4C2×D4×C3111yescharacteristic
500C4⋊C4C2×D4×C3112yes—
501C4⋊C4C2×D4×C3111yescharacteristic
502C22⋊C4—24no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—22no—
507C22⋊C4—22no—
508C22⋊C4—21no—
509C22⋊C4—22no—
510C22⋊C4—22no—
511C22⋊C4—21no—
512C22⋊C4—21no—
513C22⋊C4—22no—
514C22⋊C4—21no—
515C22⋊C4—22no—
516C22⋊C4—21no—
517C42—22no—
518C42—21no—
519C42—42no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—21no—
524C23—41no—
525C23—42no—
526C23—81no—
527C23—81no—
528C23C31×#12011yescharacteristic
529C23C31×#12011yescharacteristic
530C23C22×D4×C3111yescharacteristic
531Q8—41no—
532Q8—42no—
533Q8—81no—
534Q8—82no—
535D4—42no—
536D4—41no—
537D4—42no—
538D4—44no—
539D4—42no—
540D4—42no—
541D4—82no—
542D4—81no—
543D4—82no—
544D4—82no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—81no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4C31×#12011yescharacteristic
572C2×C4C31×#12011yescharacteristic
573C2×C4C22×D4×C3111yescharacteristic
574C2×C4C22×D4×C3111yescharacteristic
575C8—42no—
576C8—42no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—21no—
584C22—44no—
585C22—41no—
586C22—42no—
587C22—41no—
588C22—42no—
589C22—41no—
590C22—42no—
591C22—162no—
592C22C31×#45612yes—
593C22C2×C31×#12011yescharacteristic
594C22C31×#57612yes—
595C4—21no—
596C4—21no—
597C4—21no—
598C4—44no—
599C4—42no—
600C4—42no—
601C4—41no—
602C4—41no—
603C4—41no—
604C4—42no—
605C4—82no—
606C4C31×#45812yes—
607C2—22no—
608C2—22no—
609C2—41no—
610C2—82no—
611C2—81no—
612C2C31×#300811yescharacteristic
613C2C2×C31×#45611yescharacteristic
614C2C2×C31×#45811yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×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×#3092C2semidirect
3. C2×C31×#480C2semidirect
4. C31×#2502C2semidirect
5. C31×#1610C2semidirect
6. C31×#1609C2semidirect
7. C31×#1507C2semidirect
8. C31×#1501C2semidirect
9. C31×#1472C2semidirect
27. C31×#483C22semidirect
31. C31×#480C22semidirect
38. C31×#398C22semidirect
43. C31×#328C22semidirect
45. C31×#327C22semidirect
46. C31×#324C22semidirect
212. #18270C31direct (N × H)
271. #3092C2×C31semidirect
272. C2×#480C2×C31semidirect
273. #2502C2×C31semidirect
274. #1610C2×C31semidirect
275. #1609C2×C31semidirect
276. #1507C2×C31semidirect
277. #1501C2×C31semidirect
278. #1472C2×C31semidirect
326. #483C22×C31semidirect
330. #480C22×C31semidirect
337. #398C22×C31semidirect
342. #328C22×C31semidirect
344. #327C22×C31semidirect
345. #324C22×C31semidirect
437. C31#18270direct (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×#30922
C2×C31×#4802
C31×#25022
C31×#16102
C31×#16092
C31×#15072
C31×#15012
C31×#14722
#1827031