Finite Group Databasebeta

← C31×#18217 #21474976

Subgroups — C31×#18217 #21474976

Order 7936.

932 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3493C211yesmaximal, characteristic
2C31×#3108C212yesmaximal, complemented
3C31×#2599C211yesmaximal, characteristic, complemented
4C31×#1612C212yesmaximal, complemented
5C31×#1611C211yesmaximal, characteristic, complemented
6C31×#1610C211yesmaximal, characteristic, complemented
7C31×#1606C212yesmaximal, complemented
8C31×#1605C211yesmaximal, characteristic, complemented
9C31×#1604C211yesmaximal, characteristic, complemented
10C31×#1521C212yesmaximal, complemented
11C31×#1458C211yesmaximal, characteristic, complemented
12C31×#556—22no—
13C31×#555—22no—
14C31×#553—22no—
15C31×#552—22no—
16C31×#544C2211yescharacteristic
17C31×#538—21no—
18C31×#538C2212yes—
19C31×#537C2211yescharacteristic
20C31×#529—21no—
21C31×#529C2212yes—
22C31×#525—21no—
23C31×#520—21no—
24C2×C4×Q8×C31C2211yescharacteristic
25C2×C31×#117C2211yescharacteristic
26C31×#483C2212yes—
27C31×#480C2212yes—
28C31×#479—22no—
29C31×#464—22no—
30C31×#451C2212yes—
31C31×#448C2212yes—
32C4×Q16×C31—22no—
33C4×SD16×C31—22no—
34C31×#431C2212yes—
35C31×#398C2211yescharacteristic
36C31×#397C2211yescharacteristic
37C31×#395C2211yescharacteristic
38C31×#393C2211yescharacteristic
39C31×#388—21no—
40C31×#387—21no—
41C31×#387—21no—
42C31×#385—21no—
43C31×#383C2211yescharacteristic
44C31×#328C2212yes—
45C31×#328C2212yes—
46C31×#327C2212yes—
47C31×#327C2212yes—
48C31×#324C2211yescharacteristic
49C31×#324C2211yescharacteristic
50C31×#323C2212yes—
51C2×#41×C31—21no—
52C22×Q8×C31C2311yescharacteristic
53C23×C4×C31D411yescharacteristic
54C2×Q16×C31—22no—
55C2×SD16×C31—22no—
56C31×C4⋊Q8—22no—
57C31×#126—22no—
58C31×#126—22no—
59C31×#126—22no—
60C31×#126—22no—
61C31×#125—21no—
62C31×#125—22no—
63C31×#125D412yes—
64C31×#124—21no—
65C31×#124—22no—
66C31×#124—22no—
67C31×#124D412yes—
68C31×#123—22no—
69C31×#123—22no—
70C31×#123—22no—
71C31×C22⋊Q8—21no—
72C31×C22⋊Q8—21no—
73C31×C22⋊Q8—21no—
74C31×C22⋊Q8—22no—
75C31×C22⋊Q8C2312yes—
76C31×#121—21no—
77C31×#121—21no—
78C31×#121—22no—
79C31×#121—22no—
80C31×#121—22no—
81C31×#121—21no—
82C31×#121C2312yes—
83C31×#120—22no—
84C4×Q8×C31—22no—
85C4×Q8×C31—21no—
86C4×Q8×C31—22no—
87C4×Q8×C31—22no—
88C4×Q8×C31D412yes—
89C4×D4×C31—22no—
90C4×D4×C31—22no—
91C4×D4×C31—22no—
92C4×D4×C31—21no—
93C4×D4×C31—22no—
94C4×D4×C31—22no—
95C4×D4×C31D412yes—
96C31×#117—22no—
97C31×#117—22no—
98C31×#117—21no—
99C31×#117—22no—
100C31×#117—22no—
101C31×#117—22no—
102C2×C4⋊C4×C31—22no—
103C2×C4⋊C4×C31—21no—
104C2×C4⋊C4×C31—21no—
105C2×C4⋊C4×C31—21no—
106C2×C4⋊C4×C31D411yescharacteristic
107C2×C4⋊C4×C31C2311yescharacteristic
108C2×C4⋊C4×C31C2311yescharacteristic
109C2×C22⋊C4×C31—21no—
110C2×C22⋊C4×C31—21no—
111C2×C22⋊C4×C31D411yescharacteristic
112C2×C22⋊C4×C31C2311yescharacteristic
113C2×C42×C31—21no—
114C2×C42×C31D411yescharacteristic
115C2×C42×C31C2311yescharacteristic
116C31×#107—22no—
117C31×#106—22no—
118C31×C4⋊C8—22no—
119C31×#103—22no—
120C31×#103—22no—
121C31×#103—22no—
122C31×#102—22no—
123C31×C22⋊C8C2312yes—
124C31×C22⋊C8C2312yes—
125C4×C8×C31—22no—
126C31×#95—21no—
127C31×#95—21no—
128C31×#95—21no—
129C31×#95—21no—
130C31×#95—21no—
131C31×#95—21no—
132C31×#95C2311yescharacteristic
133C31×#95C2311yescharacteristic
134C24×C31C2×D411yescharacteristic
135#41×C31—42no—
136#41×C31—42no—
137C2×Q8×C31—21no—
138C2×Q8×C31—21no—
139C2×Q8×C31—22no—
140C2×Q8×C31—42no—
141C2×Q8×C31C2×D412yes—
142C2×D4×C31—21no—
143C2×D4×C31—22no—
144C2×D4×C31—22no—
145C2×D4×C31—22no—
146C2×D4×C31—21no—
147C2×D4×C31C2×D412yes—
148C22×C4×C31—22no—
149C22×C4×C31—21no—
150C22×C4×C31—22no—
151C22×C4×C31—21no—
152C22×C4×C31—21no—
153C22×C4×C31—22no—
154C22×C4×C31—22no—
155C22×C4×C31—21no—
156C22×C4×C31—21no—
157C22×C4×C31—21no—
158C22×C4×C31—41no—
159C22×C4×C31C2×D411yescharacteristic
160C22×C4×C31C2×D411yescharacteristic
161C22×C4×C31C2×D411yescharacteristic
162C22×C4×C31C2×D411yescharacteristic
163C22×C4×C31C2×D411yescharacteristic
164C22×C4×C31C2411yescharacteristic
165Q16×C31—44no—
166SD16×C31—44no—
167C2×C8×C31—22no—
168C2×C8×C31—22no—
169C4⋊C4×C31—21no—
170C4⋊C4×C31—22no—
171C4⋊C4×C31—21no—
172C4⋊C4×C31—22no—
173C4⋊C4×C31—21no—
174C4⋊C4×C31—21no—
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—42no—
183C4⋊C4×C31—41no—
184C4⋊C4×C31—41no—
185C4⋊C4×C31—42no—
186C4⋊C4×C31C2×D412yes—
187C4⋊C4×C31C2×D412yes—
188C22⋊C4×C31—22no—
189C22⋊C4×C31—22no—
190C22⋊C4×C31—22no—
191C22⋊C4×C31—22no—
192C22⋊C4×C31—22no—
193C22⋊C4×C31—22no—
194C22⋊C4×C31—22no—
195C22⋊C4×C31—22no—
196C22⋊C4×C31—22no—
197C22⋊C4×C31—22no—
198C22⋊C4×C31—22no—
199C22⋊C4×C31—42no—
200C22⋊C4×C31—42no—
201C42×C31—22no—
202C42×C31—22no—
203C42×C31—21no—
204C42×C31—21no—
205C42×C31—22no—
206C42×C31—42no—
207C42×C31C2×D412yes—
208C42×C31C2×D412yes—
209#18217C3111yesmaximal, characteristic, direct product factor, Sylow
210C23×C31—21no—
211C23×C31—22no—
212C23×C31—21no—
213C23×C31—21no—
214C23×C31—41no—
215C23×C31—41no—
216C23×C31C22×D411yescharacteristic
217Q8×C31—24no—
218Q8×C31—42no—
219Q8×C31—42no—
220Q8×C31—42no—
221Q8×C31—81no—
222D4×C31—42no—
223D4×C31—42no—
224D4×C31—42no—
225D4×C31—42no—
226D4×C31—82no—
227D4×C31—81no—
228D4×C31—82no—
229C2×C4×C31—21no—
230C2×C4×C31—21no—
231C2×C4×C31—21no—
232C2×C4×C31—21no—
233C2×C4×C31—22no—
234C2×C4×C31—21no—
235C2×C4×C31—22no—
236C2×C4×C31—21no—
237C2×C4×C31—22no—
238C2×C4×C31—22no—
239C2×C4×C31—41no—
240C2×C4×C31—41no—
241C2×C4×C31—42no—
242C2×C4×C31—42no—
243C2×C4×C31—44no—
244C2×C4×C31—41no—
245C2×C4×C31—41no—
246C2×C4×C31—41no—
247C2×C4×C31—41no—
248C2×C4×C31—41no—
249C2×C4×C31—41no—
250C2×C4×C31—41no—
251C2×C4×C31—41no—
252C2×C4×C31—41no—
253C2×C4×C31—41no—
254C2×C4×C31—41no—
255C2×C4×C31—41no—
256C2×C4×C31—41no—
257C2×C4×C31—82no—
258C2×C4×C31—82no—
259C2×C4×C31#12012yes—
260C2×C4×C31#12011yescharacteristic
261C2×C4×C31#12011yescharacteristic
262C2×C4×C31#13512yes—
263C2×C4×C31C22×D412yes—
264C8×C31—42no—
265C8×C31—42no—
266#3493C2×C3111yescharacteristic
267#3108C2×C3112yescomplemented
268#2599C2×C3111yescharacteristic, complemented
269#1612C2×C3112yescomplemented
270#1611C2×C3111yescharacteristic, complemented
271#1610C2×C3111yescharacteristic, complemented
272#1606C2×C3112yescomplemented
273#1605C2×C3111yescharacteristic, complemented
274#1604C2×C3111yescharacteristic, complemented
275#1521C2×C3112yescomplemented
276#1458C2×C3111yescharacteristic, complemented
277C22×C31—21no—
278C22×C31—21no—
279C22×C31—42no—
280C22×C31—41no—
281C22×C31—42no—
282C22×C31—42no—
283C22×C31—41no—
284C22×C31—42no—
285C22×C31—41no—
286C22×C31—81no—
287C22×C31C2×#12011yescharacteristic
288C22×C31C2×#13511yescharacteristic
289C22×C31#57511yescharacteristic
290C4×C31—22no—
291C4×C31—21no—
292C4×C31—21no—
293C4×C31—22no—
294C4×C31—22no—
295C4×C31—42no—
296C4×C31—41no—
297C4×C31—42no—
298C4×C31—42no—
299C4×C31—41no—
300C4×C31—42no—
301C4×C31—81no—
302C4×C31—81no—
303C4×C31—81no—
304#556—22no—
305#555—22no—
306#553—22no—
307#552—22no—
308#544C22×C3111yescharacteristic
309#538—21no—
310#538C22×C3112yes—
311#537C22×C3111yescharacteristic
312#529—21no—
313#529C22×C3112yes—
314#525—21no—
315#520—21no—
316C2×C4×Q8C22×C3111yescharacteristic
317C2×#117C22×C3111yescharacteristic
318#483C22×C3112yes—
319#480C22×C3112yes—
320#479—22no—
321#464—22no—
322#451C22×C3112yes—
323#448C22×C3112yes—
324C4×Q16—22no—
325C4×SD16—22no—
326#431C22×C3112yes—
327#398C22×C3111yescharacteristic
328#397C22×C3111yescharacteristic
329#395C22×C3111yescharacteristic
330#393C22×C3111yescharacteristic
331#388—21no—
332#387—21no—
333#387—21no—
334#385—21no—
335#383C22×C3111yescharacteristic
336#328C22×C3112yes—
337#328C22×C3112yes—
338#327C22×C3112yes—
339#327C22×C3112yes—
340#324C22×C3111yescharacteristic
341#324C22×C3111yescharacteristic
342#323C22×C3112yes—
343C2×C31—21no—
344C2×C31—21no—
345C2×C31—82no—
346C2×C31—81no—
347C2×C31#300211yescharacteristic
348C2×C31#302211yescharacteristic
349C2×C31#302511yescharacteristic
350C2×#41—21no—
351C22×Q8C23×C3111yescharacteristic
352C23×C4D4×C3111yescharacteristic
353C2×Q16—22no—
354C2×SD16—22no—
355C4⋊Q8—22no—
356#126—22no—
357#126—22no—
358#126—22no—
359#126—22no—
360#125—21no—
361#125—22no—
362#125D4×C3112yes—
363#124—21no—
364#124—22no—
365#124—22no—
366#124D4×C3112yes—
367#123—22no—
368#123—22no—
369#123—22no—
370C22⋊Q8—21no—
371C22⋊Q8—21no—
372C22⋊Q8—21no—
373C22⋊Q8—22no—
374C22⋊Q8C23×C3112yes—
375#121—21no—
376#121—21no—
377#121—22no—
378#121—22no—
379#121—22no—
380#121—21no—
381#121C23×C3112yes—
382#120—22no—
383C4×Q8—22no—
384C4×Q8—21no—
385C4×Q8—22no—
386C4×Q8—22no—
387C4×Q8D4×C3112yes—
388C4×D4—22no—
389C4×D4—22no—
390C4×D4—22no—
391C4×D4—21no—
392C4×D4—22no—
393C4×D4—22no—
394C4×D4D4×C3112yes—
395#117—22no—
396#117—22no—
397#117—21no—
398#117—22no—
399#117—22no—
400#117—22no—
401C2×C4⋊C4—22no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4D4×C3111yescharacteristic
406C2×C4⋊C4C23×C3111yescharacteristic
407C2×C4⋊C4C23×C3111yescharacteristic
408C2×C22⋊C4—21no—
409C2×C22⋊C4—21no—
410C2×C22⋊C4D4×C3111yescharacteristic
411C2×C22⋊C4C23×C3111yescharacteristic
412C2×C42—21no—
413C2×C42D4×C3111yescharacteristic
414C2×C42C23×C3111yescharacteristic
415#107—22no—
416#106—22no—
417C4⋊C8—22no—
418#103—22no—
419#103—22no—
420#103—22no—
421#102—22no—
422C22⋊C8C23×C3112yes—
423C22⋊C8C23×C3112yes—
424C4×C8—22no—
425#95—21no—
426#95—21no—
427#95—21no—
428#95—21no—
429#95—21no—
430#95—21no—
431#95C23×C3111yescharacteristic
432#95C23×C3111yescharacteristic
433C31#1821711yescharacteristic, direct product factor, Sylow
434C24C2×D4×C3111yescharacteristic
435#41—42no—
436#41—42no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—22no—
440C2×Q8—42no—
441C2×Q8C2×D4×C3112yes—
442C2×D4—21no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—21no—
447C2×D4C2×D4×C3112yes—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—41no—
459C22×C4C2×D4×C3111yescharacteristic
460C22×C4C2×D4×C3111yescharacteristic
461C22×C4C2×D4×C3111yescharacteristic
462C22×C4C2×D4×C3111yescharacteristic
463C22×C4C2×D4×C3111yescharacteristic
464C22×C4C24×C3111yescharacteristic
465Q16—44no—
466SD16—44no—
467C2×C8—22no—
468C2×C8—22no—
469C4⋊C4—21no—
470C4⋊C4—22no—
471C4⋊C4—21no—
472C4⋊C4—22no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—42no—
483C4⋊C4—41no—
484C4⋊C4—41no—
485C4⋊C4—42no—
486C4⋊C4C2×D4×C3112yes—
487C4⋊C4C2×D4×C3112yes—
488C22⋊C4—22no—
489C22⋊C4—22no—
490C22⋊C4—22no—
491C22⋊C4—22no—
492C22⋊C4—22no—
493C22⋊C4—22no—
494C22⋊C4—22no—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—22no—
498C22⋊C4—22no—
499C22⋊C4—42no—
500C22⋊C4—42no—
501C42—22no—
502C42—22no—
503C42—21no—
504C42—21no—
505C42—22no—
506C42—42no—
507C42C2×D4×C3112yes—
508C42C2×D4×C3112yes—
509C23—21no—
510C23—22no—
511C23—21no—
512C23—21no—
513C23—41no—
514C23—41no—
515C23C22×D4×C3111yescharacteristic
516Q8—24no—
517Q8—42no—
518Q8—42no—
519Q8—42no—
520Q8—81no—
521D4—42no—
522D4—42no—
523D4—42no—
524D4—42no—
525D4—82no—
526D4—81no—
527D4—82no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4C31×#12012yes—
559C2×C4C31×#12011yescharacteristic
560C2×C4C31×#12011yescharacteristic
561C2×C4C31×#13512yes—
562C2×C4C22×D4×C3112yes—
563C8—42no—
564C8—42no—
565C22—21no—
566C22—21no—
567C22—42no—
568C22—41no—
569C22—42no—
570C22—42no—
571C22—41no—
572C22—42no—
573C22—41no—
574C22—81no—
575C22C2×C31×#12011yescharacteristic
576C22C2×C31×#13511yescharacteristic
577C22C31×#57511yescharacteristic
578C4—22no—
579C4—21no—
580C4—21no—
581C4—22no—
582C4—22no—
583C4—42no—
584C4—41no—
585C4—42no—
586C4—42no—
587C4—41no—
588C4—42no—
589C4—81no—
590C4—81no—
591C4—81no—
592C2—21no—
593C2—21no—
594C2—81no—
595C2—82no—
596C2C31×#300211yescharacteristic
597C2C31×#302211yescharacteristic
598C2C31×#302511yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#3493C2
LCSG ▷ C22×C4 ▷ 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×#3108C2semidirect
3. C31×#2599C2semidirect
4. C31×#1612C2semidirect
5. C31×#1611C2semidirect
6. C31×#1610C2semidirect
7. C31×#1606C2semidirect
8. C31×#1605C2semidirect
9. C31×#1604C2semidirect
10. C31×#1521C2semidirect
11. C31×#1458C2semidirect
209. #18217C31direct (N × H)
267. #3108C2×C31semidirect
268. #2599C2×C31semidirect
269. #1612C2×C31semidirect
270. #1611C2×C31semidirect
271. #1610C2×C31semidirect
272. #1606C2×C31semidirect
273. #1605C2×C31semidirect
274. #1604C2×C31semidirect
275. #1521C2×C31semidirect
276. #1458C2×C31semidirect
433. C31#18217direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#34932
C31×#31082
C31×#25992
C31×#16122
C31×#16112
C31×#16102
C31×#16062
C31×#16052
C31×#16042
C31×#15212
C31×#14582
#1821731