Finite Group Databasebeta

← C31×#18957 #21475716

Subgroups — C31×#18957 #21475716

Order 7936.

874 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
1C31×#3547C211yesmaximal, characteristic
2C31×#3238C211yesmaximal, characteristic, complemented
3C31×#3227C211yesmaximal, characteristic, complemented
4C31×#3209C211yesmaximal, characteristic, complemented
5C31×#1637C212yesmaximal, complemented
6C31×#1635C212yesmaximal, complemented
7C31×#1634C212yesmaximal, complemented
8C31×#1632C212yesmaximal, complemented
9C31×#1554C211yesmaximal, characteristic, complemented
10C31×#1496C211yesmaximal, characteristic, complemented
11C31×#1482C211yesmaximal, characteristic, complemented
12C31×#562—22no—
13C31×#555—21no—
14C31×#555C2211yescharacteristic
15C31×#554C2211yescharacteristic
16C31×#553—21no—
17C31×#552—21no—
18D4×Q8×C31C2211yescharacteristic
19C31×#548—21no—
20C31×#547C2211yescharacteristic
21C31×#544—21no—
22C31×#542—22no—
23C31×#540—21no—
24C31×#537—21no—
25C31×#536—21no—
26C31×#533C2211yescharacteristic
27C31×#532C2211yescharacteristic
28C2×C31×C4⋊Q8C2211yescharacteristic
29C31×#529C2211yescharacteristic
30C31×#501—21no—
31C31×#498—21no—
32C31×#494C2212yes—
33C31×#489—21no—
34C31×#487C2212yes—
35C31×#486—21no—
36C31×#483—21no—
37C31×#481—21no—
38C31×#481—21no—
39C31×#480—21no—
40C31×#477C2212yes—
41C31×#476C2212yes—
42C31×#476C2212yes—
43C31×#476C2212yes—
44C31×#462—22no—
45C31×#451—22no—
46C31×#432C2211yescharacteristic
47C31×#431C2211yescharacteristic
48C31×#431C2211yescharacteristic
49C31×#333C2212yes—
50C31×#332C2212yes—
51C31×Q8⋊C8C2212yes—
52C31×D4⋊C8C2212yes—
53C31×D4⋊C8C2212yes—
54C31×D4⋊C8C2212yes—
55C2×#41×C31—21no—
56C2×#41×C31—21no—
57C22×Q8×C31—21no—
58C22×Q8×C31—21no—
59C2×Q16×C31—24no—
60C31×C4⋊Q8—22no—
61C31×C4⋊Q8—21no—
62C31×C4⋊Q8—21no—
63C31×C4⋊Q8—21no—
64C31×C4⋊Q8—21no—
65C31×C4⋊Q8D411yescharacteristic
66C31×C4⋊Q8C2312yes—
67C31×C4⋊Q8C2311yescharacteristic
68C31×C4⋊Q8C2311yescharacteristic
69C31×#127D411yescharacteristic
70C31×#126—22no—
71C31×#126—21no—
72C31×#125—22no—
73C31×#125—21no—
74C31×#125—21no—
75C31×#125D411yescharacteristic
76C31×#124—22no—
77C31×#124—22no—
78C31×#124—22no—
79C31×#124—21no—
80C31×#124D411yescharacteristic
81C31×#123—22no—
82C31×#123—22no—
83C31×#123—22no—
84C31×#123—21no—
85C31×#123—22no—
86C31×C22⋊Q8—22no—
87C31×C22⋊Q8—22no—
88C31×C22⋊Q8—21no—
89C31×C22⋊Q8—22no—
90C31×C22⋊Q8—21no—
91C31×C22⋊Q8—21no—
92C31×C22⋊Q8—21no—
93C31×C22⋊Q8—22no—
94C31×C22⋊Q8D412yes—
95C31×C22⋊Q8D411yescharacteristic
96C31×#121—22no—
97C31×#121—22no—
98C31×#121—22no—
99C31×#121—22no—
100C31×#121—21no—
101C31×#121—21no—
102C31×#121—21no—
103C31×#121—21no—
104C31×#121D411yescharacteristic
105C31×#121D411yescharacteristic
106C31×#121D411yescharacteristic
107C4×Q8×C31—21no—
108C4×Q8×C31D412yes—
109C4×Q8×C31C2311yescharacteristic
110C4×D4×C31—22no—
111C4×D4×C31—21no—
112C4×D4×C31—21no—
113C4×D4×C31—21no—
114C4×D4×C31—21no—
115C4×D4×C31C2311yescharacteristic
116C4×D4×C31C2311yescharacteristic
117C4×D4×C31C2311yescharacteristic
118C31×#117—22no—
119C31×#117—21no—
120C31×#117—21no—
121C31×#117—21no—
122C31×#117—21no—
123C2×C4⋊C4×C31—21no—
124C2×C4⋊C4×C31—21no—
125C2×C4⋊C4×C31—21no—
126C2×C4⋊C4×C31—21no—
127C2×C42×C31C2311yescharacteristic
128C31×#107—22no—
129C31×#106—22no—
130C31×#106—22no—
131C31×#106—22no—
132C31×C4⋊C8—21no—
133C31×C4⋊C8C2312yes—
134C31×C4⋊C8C2312yes—
135C31×#103—22no—
136C31×#103—24no—
137C31×#102—22no—
138C31×#102—22no—
139C31×#102—22no—
140C31×C22⋊C8—21no—
141C31×C22⋊C8—21no—
142C31×C22⋊C8—21no—
143C31×#97—21no—
144C4×C8×C31—21no—
145C4×C8×C31C2312yes—
146#41×C31—82no—
147#41×C31—82no—
148C2×Q8×C31—22no—
149C2×Q8×C31—21no—
150C2×Q8×C31—21no—
151C2×Q8×C31—22no—
152C2×Q8×C31—41no—
153C2×Q8×C31—82no—
154C2×Q8×C31—82no—
155C2×Q8×C31C2×D411yescharacteristic
156C2×Q8×C31C2×D412yes—
157C2×D4×C31—22no—
158C2×D4×C31—22no—
159C2×D4×C31—21no—
160C2×D4×C31—21no—
161C2×D4×C31—21no—
162C2×D4×C31—21no—
163C2×D4×C31—21no—
164C2×D4×C31C2×D411yescharacteristic
165C2×D4×C31C2×D411yescharacteristic
166C2×D4×C31C2×D411yescharacteristic
167C22×C4×C31—22no—
168C22×C4×C31—22no—
169C22×C4×C31—22no—
170C22×C4×C31—21no—
171C22×C4×C31—21no—
172C22×C4×C31—21no—
173C22×C4×C31—41no—
174C22×C4×C31C2×D411yescharacteristic
175C22×C4×C31C2×D411yescharacteristic
176C22×C4×C31C2×D411yescharacteristic
177Q16×C31—84no—
178C2×C8×C31—22no—
179C2×C8×C31—22no—
180C2×C8×C31—22no—
181C4⋊C4×C31—22no—
182C4⋊C4×C31—22no—
183C4⋊C4×C31—22no—
184C4⋊C4×C31—22no—
185C4⋊C4×C31—21no—
186C4⋊C4×C31—21no—
187C4⋊C4×C31—21no—
188C4⋊C4×C31—21no—
189C4⋊C4×C31—22no—
190C4⋊C4×C31—22no—
191C4⋊C4×C31—21no—
192C4⋊C4×C31—21no—
193C4⋊C4×C31—21no—
194C4⋊C4×C31—21no—
195C4⋊C4×C31—22no—
196C4⋊C4×C31—22no—
197C4⋊C4×C31—41no—
198C4⋊C4×C31—41no—
199C4⋊C4×C31C2×D412yes—
200C4⋊C4×C31C2×D411yescharacteristic
201C4⋊C4×C31C2×D411yescharacteristic
202C4⋊C4×C31C2×D411yescharacteristic
203C4⋊C4×C31C2×D411yescharacteristic
204C22⋊C4×C31—22no—
205C22⋊C4×C31—22no—
206C22⋊C4×C31—22no—
207C22⋊C4×C31—22no—
208C22⋊C4×C31—22no—
209C22⋊C4×C31—22no—
210C22⋊C4×C31—21no—
211C22⋊C4×C31—21no—
212C22⋊C4×C31—21no—
213C22⋊C4×C31—21no—
214C22⋊C4×C31—21no—
215C22⋊C4×C31—21no—
216C22⋊C4×C31—21no—
217C22⋊C4×C31—21no—
218C22⋊C4×C31—21no—
219C42×C31—22no—
220C42×C31—22no—
221C42×C31—21no—
222C42×C31C2×D411yescharacteristic
223C42×C31C2×D411yescharacteristic
224C42×C31C2×D411yescharacteristic
225C42×C31C2411yescharacteristic
226#18957C3111yesmaximal, characteristic, direct product factor, Sylow
227C23×C31—21no—
228C23×C31—21no—
229C23×C31—21no—
230C23×C31#12011yescharacteristic
231Q8×C31—22no—
232Q8×C31—42no—
233Q8×C31—44no—
234Q8×C31—42no—
235Q8×C31—81no—
236Q8×C31—82no—
237Q8×C31—82no—
238D4×C31—22no—
239D4×C31—22no—
240D4×C31—22no—
241D4×C31—81no—
242D4×C31—81no—
243D4×C31—81no—
244D4×C31—81no—
245D4×C31—81no—
246D4×C31—82no—
247D4×C31—82no—
248C2×C4×C31—22no—
249C2×C4×C31—22no—
250C2×C4×C31—21no—
251C2×C4×C31—22no—
252C2×C4×C31—21no—
253C2×C4×C31—21no—
254C2×C4×C31—21no—
255C2×C4×C31—22no—
256C2×C4×C31—21no—
257C2×C4×C31—41no—
258C2×C4×C31—42no—
259C2×C4×C31—42no—
260C2×C4×C31—42no—
261C2×C4×C31—42no—
262C2×C4×C31—82no—
263C2×C4×C31—82no—
264C2×C4×C31—82no—
265C2×C4×C31—82no—
266C2×C4×C31#12011yescharacteristic
267C2×C4×C31#12011yescharacteristic
268C2×C4×C31#12011yescharacteristic
269C2×C4×C31C22×D411yescharacteristic
270C2×C4×C31C22×D411yescharacteristic
271C2×C4×C31C22×D411yescharacteristic
272C8×C31—42no—
273C8×C31—42no—
274C8×C31—42no—
275#3547C2×C3111yescharacteristic
276#3238C2×C3111yescharacteristic, complemented
277#3227C2×C3111yescharacteristic, complemented
278#3209C2×C3111yescharacteristic, complemented
279#1637C2×C3112yescomplemented
280#1635C2×C3112yescomplemented
281#1634C2×C3112yescomplemented
282#1632C2×C3112yescomplemented
283#1554C2×C3111yescharacteristic, complemented
284#1496C2×C3111yescharacteristic, complemented
285#1482C2×C3111yescharacteristic, complemented
286C22×C31—21no—
287C22×C31—21no—
288C22×C31—21no—
289C22×C31—41no—
290C22×C31—41no—
291C22×C31—41no—
292C22×C31—41no—
293C22×C31—41no—
294C22×C31—41no—
295C22×C31—41no—
296C22×C31—41no—
297C22×C31—41no—
298C22×C31C2×#12011yescharacteristic
299C4×C31—21no—
300C4×C31—21no—
301C4×C31—21no—
302C4×C31—21no—
303C4×C31—41no—
304C4×C31—41no—
305C4×C31—42no—
306C4×C31—41no—
307C4×C31—41no—
308C4×C31—41no—
309C4×C31—42no—
310C4×C31—82no—
311C4×C31#57712yes—
312C4×C31#57812yes—
313#562—22no—
314#555—21no—
315#555C22×C3111yescharacteristic
316#554C22×C3111yescharacteristic
317#553—21no—
318#552—21no—
319D4×Q8C22×C3111yescharacteristic
320#548—21no—
321#547C22×C3111yescharacteristic
322#544—21no—
323#542—22no—
324#540—21no—
325#537—21no—
326#536—21no—
327#533C22×C3111yescharacteristic
328#532C22×C3111yescharacteristic
329C2×C4⋊Q8C22×C3111yescharacteristic
330#529C22×C3111yescharacteristic
331#501—21no—
332#498—21no—
333#494C22×C3112yes—
334#489—21no—
335#487C22×C3112yes—
336#486—21no—
337#483—21no—
338#481—21no—
339#481—21no—
340#480—21no—
341#477C22×C3112yes—
342#476C22×C3112yes—
343#476C22×C3112yes—
344#476C22×C3112yes—
345#462—22no—
346#451—22no—
347#432C22×C3111yescharacteristic
348#431C22×C3111yescharacteristic
349#431C22×C3111yescharacteristic
350#333C22×C3112yes—
351#332C22×C3112yes—
352Q8⋊C8C22×C3112yes—
353D4⋊C8C22×C3112yes—
354D4⋊C8C22×C3112yes—
355D4⋊C8C22×C3112yes—
356C2×C31—41no—
357C2×C31—81no—
358C2×C31—81no—
359C2×C31—81no—
360C2×C31#300911yescharacteristic
361C2×C31#301111yescharacteristic
362C2×C31#302111yescharacteristic
363C2×#41—21no—
364C2×#41—21no—
365C22×Q8—21no—
366C22×Q8—21no—
367C2×Q16—24no—
368C4⋊Q8—22no—
369C4⋊Q8—21no—
370C4⋊Q8—21no—
371C4⋊Q8—21no—
372C4⋊Q8—21no—
373C4⋊Q8D4×C3111yescharacteristic
374C4⋊Q8C23×C3112yes—
375C4⋊Q8C23×C3111yescharacteristic
376C4⋊Q8C23×C3111yescharacteristic
377#127D4×C3111yescharacteristic
378#126—22no—
379#126—21no—
380#125—22no—
381#125—21no—
382#125—21no—
383#125D4×C3111yescharacteristic
384#124—22no—
385#124—22no—
386#124—22no—
387#124—21no—
388#124D4×C3111yescharacteristic
389#123—22no—
390#123—22no—
391#123—22no—
392#123—21no—
393#123—22no—
394C22⋊Q8—22no—
395C22⋊Q8—22no—
396C22⋊Q8—21no—
397C22⋊Q8—22no—
398C22⋊Q8—21no—
399C22⋊Q8—21no—
400C22⋊Q8—21no—
401C22⋊Q8—22no—
402C22⋊Q8D4×C3111yescharacteristic
403C22⋊Q8D4×C3112yes—
404#121—22no—
405#121—22no—
406#121—22no—
407#121—22no—
408#121—21no—
409#121—21no—
410#121—21no—
411#121—21no—
412#121D4×C3111yescharacteristic
413#121D4×C3111yescharacteristic
414#121D4×C3111yescharacteristic
415C4×Q8—21no—
416C4×Q8D4×C3112yes—
417C4×Q8C23×C3111yescharacteristic
418C4×D4—22no—
419C4×D4—21no—
420C4×D4—21no—
421C4×D4—21no—
422C4×D4—21no—
423C4×D4C23×C3111yescharacteristic
424C4×D4C23×C3111yescharacteristic
425C4×D4C23×C3111yescharacteristic
426#117—22no—
427#117—21no—
428#117—21no—
429#117—21no—
430#117—21no—
431C2×C4⋊C4—21no—
432C2×C4⋊C4—21no—
433C2×C4⋊C4—21no—
434C2×C4⋊C4—21no—
435C2×C42C23×C3111yescharacteristic
436#107—22no—
437#106—22no—
438#106—22no—
439#106—22no—
440C4⋊C8—21no—
441C4⋊C8C23×C3112yes—
442C4⋊C8C23×C3112yes—
443#103—22no—
444#103—24no—
445#102—22no—
446#102—22no—
447#102—22no—
448C22⋊C8—21no—
449C22⋊C8—21no—
450C22⋊C8—21no—
451#97—21no—
452C4×C8—21no—
453C4×C8C23×C3112yes—
454C31#1895711yescharacteristic, direct product factor, Sylow
455#41—82no—
456#41—82no—
457C2×Q8—22no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—22no—
461C2×Q8—41no—
462C2×Q8—82no—
463C2×Q8—82no—
464C2×Q8C2×D4×C3111yescharacteristic
465C2×Q8C2×D4×C3112yes—
466C2×D4—22no—
467C2×D4—22no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4C2×D4×C3111yescharacteristic
474C2×D4C2×D4×C3111yescharacteristic
475C2×D4C2×D4×C3111yescharacteristic
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—41no—
483C22×C4C2×D4×C3111yescharacteristic
484C22×C4C2×D4×C3111yescharacteristic
485C22×C4C2×D4×C3111yescharacteristic
486Q16—84no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—41no—
507C4⋊C4—41no—
508C4⋊C4C2×D4×C3112yes—
509C4⋊C4C2×D4×C3111yescharacteristic
510C4⋊C4C2×D4×C3111yescharacteristic
511C4⋊C4C2×D4×C3111yescharacteristic
512C4⋊C4C2×D4×C3111yescharacteristic
513C22⋊C4—22no—
514C22⋊C4—22no—
515C22⋊C4—22no—
516C22⋊C4—22no—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—21no—
520C22⋊C4—21no—
521C22⋊C4—21no—
522C22⋊C4—21no—
523C22⋊C4—21no—
524C22⋊C4—21no—
525C22⋊C4—21no—
526C22⋊C4—21no—
527C22⋊C4—21no—
528C42—22no—
529C42—22no—
530C42—21no—
531C42C2×D4×C3111yescharacteristic
532C42C2×D4×C3111yescharacteristic
533C42C2×D4×C3111yescharacteristic
534C42C24×C3111yescharacteristic
535C23—21no—
536C23—21no—
537C23—21no—
538C23C31×#12011yescharacteristic
539Q8—22no—
540Q8—42no—
541Q8—44no—
542Q8—42no—
543Q8—81no—
544Q8—82no—
545Q8—82no—
546D4—22no—
547D4—22no—
548D4—22no—
549D4—81no—
550D4—81no—
551D4—81no—
552D4—81no—
553D4—81no—
554D4—82no—
555D4—82no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4C31×#12011yescharacteristic
575C2×C4C31×#12011yescharacteristic
576C2×C4C31×#12011yescharacteristic
577C2×C4C22×D4×C3111yescharacteristic
578C2×C4C22×D4×C3111yescharacteristic
579C2×C4C22×D4×C3111yescharacteristic
580C8—42no—
581C8—42no—
582C8—42no—
583C22—21no—
584C22—21no—
585C22—21no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22C2×C31×#12011yescharacteristic
596C4—21no—
597C4—21no—
598C4—21no—
599C4—21no—
600C4—41no—
601C4—41no—
602C4—42no—
603C4—41no—
604C4—41no—
605C4—41no—
606C4—42no—
607C4—82no—
608C4C31×#57712yes—
609C4C31×#57812yes—
610C2—41no—
611C2—81no—
612C2—81no—
613C2—81no—
614C2C31×#300911yescharacteristic
615C2C31×#301111yescharacteristic
616C2C31×#302111yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC42C24×C31
prime radicalC31×#3547C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#3238C2semidirect
3. C31×#3227C2semidirect
4. C31×#3209C2semidirect
5. C31×#1637C2semidirect
6. C31×#1635C2semidirect
7. C31×#1634C2semidirect
8. C31×#1632C2semidirect
9. C31×#1554C2semidirect
10. C31×#1496C2semidirect
11. C31×#1482C2semidirect
226. #18957C31direct (N × H)
276. #3238C2×C31semidirect
277. #3227C2×C31semidirect
278. #3209C2×C31semidirect
279. #1637C2×C31semidirect
280. #1635C2×C31semidirect
281. #1634C2×C31semidirect
282. #1632C2×C31semidirect
283. #1554C2×C31semidirect
284. #1496C2×C31semidirect
285. #1482C2×C31semidirect
454. C31#18957direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35472
C31×#32382
C31×#32272
C31×#32092
C31×#16372
C31×#16352
C31×#16342
C31×#16322
C31×#15542
C31×#14962
C31×#14822
#1895731