Finite Group Databasebeta

← C31×#18407 #21475166

Subgroups — C31×#18407 #21475166

Order 7936.

904 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3493C211yesmaximal, characteristic
2C31×#3232C212yesmaximal, complemented
3C31×#2611C211yesmaximal, characteristic, complemented
4C31×#1618C212yesmaximal, complemented
5C31×#1617C211yesmaximal, characteristic, complemented
6C31×#1616C211yesmaximal, characteristic, complemented
7C31×#1612C212yesmaximal, complemented
8C31×#1611C211yesmaximal, characteristic, complemented
9C31×#1610C211yesmaximal, characteristic, complemented
10C31×#1500C212yesmaximal, 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—
22C2×C31×#125C2211yescharacteristic
23C31×#525—21no—
24C31×#520—21no—
25C2×C31×#117C2211yescharacteristic
26C31×#499—22no—
27C31×#488—22no—
28C31×#484C2212yes—
29C31×#483C2212yes—
30C31×#481C2212yes—
31C31×#480C2212yes—
32C31×#479—22no—
33C31×#478—22no—
34C31×#431C2212yes—
35C31×#400C2211yescharacteristic
36C31×#399C2211yescharacteristic
37C31×#398C2211yescharacteristic
38C31×#397C2211yescharacteristic
39C31×#389—21no—
40C31×#388—21no—
41C31×#387—21no—
42C31×#385—21no—
43C31×#376C2211yescharacteristic
44C31×#328C2212yes—
45C31×#328C2212yes—
46C31×#327C2212yes—
47C31×#327C2212yes—
48C31×#324C2211yescharacteristic
49C31×#324C2211yescharacteristic
50C31×#323C2212yes—
51C2×#41×C31—21no—
52C23×C4×C31D411yescharacteristic
53C31×C4⋊Q8—22no—
54C31×#126—22no—
55C31×#126—22no—
56C31×#126—22no—
57C31×#126—22no—
58C31×#125—21no—
59C31×#125—22no—
60C31×#125—22no—
61C31×#125—22no—
62C31×#125D412yes—
63C31×#124—21no—
64C31×#124—22no—
65C31×#124—22no—
66C31×#124D412yes—
67C31×#123—22no—
68C31×#123—22no—
69C31×#123—22no—
70C31×C22⋊Q8—21no—
71C31×C22⋊Q8—21no—
72C31×C22⋊Q8—21no—
73C31×C22⋊Q8—22no—
74C31×C22⋊Q8C2312yes—
75C31×#121—21no—
76C31×#121—21no—
77C31×#121—22no—
78C31×#121—22no—
79C31×#121—22no—
80C31×#121—21no—
81C31×#121C2312yes—
82C31×#120—22no—
83C4×Q8×C31—22no—
84C4×Q8×C31—21no—
85C4×Q8×C31D412yes—
86C4×D4×C31—22no—
87C4×D4×C31—22no—
88C4×D4×C31—22no—
89C4×D4×C31—21no—
90C4×D4×C31—22no—
91C4×D4×C31—22no—
92C4×D4×C31D412yes—
93C31×#117—22no—
94C31×#117—22no—
95C31×#117—21no—
96C31×#117—22no—
97C31×#117—22no—
98C31×#117—22no—
99C2×C4⋊C4×C31—22no—
100C2×C4⋊C4×C31—21no—
101C2×C4⋊C4×C31—21no—
102C2×C4⋊C4×C31—21no—
103C2×C4⋊C4×C31—21no—
104C2×C4⋊C4×C31D411yescharacteristic
105C2×C4⋊C4×C31C2311yescharacteristic
106C2×C4⋊C4×C31C2311yescharacteristic
107C2×C4⋊C4×C31C2311yescharacteristic
108C2×C22⋊C4×C31—21no—
109C2×C22⋊C4×C31—21no—
110C2×C22⋊C4×C31D411yescharacteristic
111C2×C22⋊C4×C31C2311yescharacteristic
112C2×C42×C31—21no—
113C2×C42×C31D411yescharacteristic
114C2×C42×C31C2311yescharacteristic
115C31×#107—22no—
116C31×#107—22no—
117C31×#106—22no—
118C31×#106—22no—
119C31×C4⋊C8—22no—
120C31×#103—22no—
121C31×#103—22no—
122C31×#102—22no—
123C31×#102—22no—
124C31×C22⋊C8C2312yes—
125C31×C22⋊C8C2312yes—
126C4×C8×C31—22no—
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×C31C2×D412yes—
140C2×D4×C31—21no—
141C2×D4×C31—22no—
142C2×D4×C31—22no—
143C2×D4×C31—22no—
144C2×D4×C31—21no—
145C2×D4×C31C2×D412yes—
146C22×C4×C31—22no—
147C22×C4×C31—21no—
148C22×C4×C31—22no—
149C22×C4×C31—21no—
150C22×C4×C31—21no—
151C22×C4×C31—22no—
152C22×C4×C31—22no—
153C22×C4×C31—21no—
154C22×C4×C31—21no—
155C22×C4×C31—21no—
156C22×C4×C31—41no—
157C22×C4×C31C2×D411yescharacteristic
158C22×C4×C31C2×D411yescharacteristic
159C22×C4×C31C2×D411yescharacteristic
160C22×C4×C31C2×D411yescharacteristic
161C22×C4×C31C2×D411yescharacteristic
162C22×C4×C31C2411yescharacteristic
163C2×C8×C31—22no—
164C2×C8×C31—22no—
165C4⋊C4×C31—21no—
166C4⋊C4×C31—22no—
167C4⋊C4×C31—21no—
168C4⋊C4×C31—22no—
169C4⋊C4×C31—21no—
170C4⋊C4×C31—21no—
171C4⋊C4×C31—22no—
172C4⋊C4×C31—22no—
173C4⋊C4×C31—22no—
174C4⋊C4×C31—22no—
175C4⋊C4×C31—22no—
176C4⋊C4×C31—22no—
177C4⋊C4×C31—22no—
178C4⋊C4×C31—22no—
179C4⋊C4×C31—42no—
180C4⋊C4×C31—41no—
181C4⋊C4×C31—41no—
182C4⋊C4×C31—42no—
183C4⋊C4×C31—42no—
184C4⋊C4×C31C2×D412yes—
185C4⋊C4×C31C2×D412yes—
186C22⋊C4×C31—22no—
187C22⋊C4×C31—22no—
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—42no—
198C22⋊C4×C31—42no—
199C42×C31—22no—
200C42×C31—22no—
201C42×C31—21no—
202C42×C31—21no—
203C42×C31—22no—
204C42×C31—42no—
205C42×C31C2×D412yes—
206C42×C31C2×D412yes—
207#18407C3111yesmaximal, characteristic, direct product factor, Sylow
208C23×C31—21no—
209C23×C31—22no—
210C23×C31—21no—
211C23×C31—21no—
212C23×C31—41no—
213C23×C31—41no—
214C23×C31C22×D411yescharacteristic
215Q8×C31—42no—
216Q8×C31—42no—
217Q8×C31—81no—
218D4×C31—42no—
219D4×C31—42no—
220D4×C31—42no—
221D4×C31—42no—
222D4×C31—82no—
223D4×C31—81no—
224D4×C31—82no—
225C2×C4×C31—21no—
226C2×C4×C31—21no—
227C2×C4×C31—21no—
228C2×C4×C31—21no—
229C2×C4×C31—22no—
230C2×C4×C31—21no—
231C2×C4×C31—22no—
232C2×C4×C31—21no—
233C2×C4×C31—22no—
234C2×C4×C31—22no—
235C2×C4×C31—41no—
236C2×C4×C31—41no—
237C2×C4×C31—42no—
238C2×C4×C31—42no—
239C2×C4×C31—44no—
240C2×C4×C31—41no—
241C2×C4×C31—41no—
242C2×C4×C31—41no—
243C2×C4×C31—41no—
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—82no—
254C2×C4×C31—82no—
255C2×C4×C31#12011yescharacteristic
256C2×C4×C31#12011yescharacteristic
257C2×C4×C31#12012yes—
258C2×C4×C31#13512yes—
259C2×C4×C31C22×D412yes—
260C8×C31—42no—
261C8×C31—42no—
262#3493C2×C3111yescharacteristic
263#3232C2×C3112yescomplemented
264#2611C2×C3111yescharacteristic, complemented
265#1618C2×C3112yescomplemented
266#1617C2×C3111yescharacteristic, complemented
267#1616C2×C3111yescharacteristic, complemented
268#1612C2×C3112yescomplemented
269#1611C2×C3111yescharacteristic, complemented
270#1610C2×C3111yescharacteristic, complemented
271#1500C2×C3112yescomplemented
272#1458C2×C3111yescharacteristic, complemented
273C22×C31—21no—
274C22×C31—21no—
275C22×C31—42no—
276C22×C31—41no—
277C22×C31—42no—
278C22×C31—42no—
279C22×C31—41no—
280C22×C31—42no—
281C22×C31—41no—
282C22×C31—81no—
283C22×C31C2×#12011yescharacteristic
284C22×C31C2×#13511yescharacteristic
285C22×C31#57511yescharacteristic
286C4×C31—22no—
287C4×C31—21no—
288C4×C31—21no—
289C4×C31—22no—
290C4×C31—22no—
291C4×C31—42no—
292C4×C31—42no—
293C4×C31—41no—
294C4×C31—42no—
295C4×C31—41no—
296C4×C31—42no—
297C4×C31—81no—
298C4×C31—81no—
299C4×C31—81no—
300#556—22no—
301#555—22no—
302#553—22no—
303#552—22no—
304#544C22×C3111yescharacteristic
305#538—21no—
306#538C22×C3112yes—
307#537C22×C3111yescharacteristic
308#529—21no—
309#529C22×C3112yes—
310C2×#125C22×C3111yescharacteristic
311#525—21no—
312#520—21no—
313C2×#117C22×C3111yescharacteristic
314#499—22no—
315#488—22no—
316#484C22×C3112yes—
317#483C22×C3112yes—
318#481C22×C3112yes—
319#480C22×C3112yes—
320#479—22no—
321#478—22no—
322#431C22×C3112yes—
323#400C22×C3111yescharacteristic
324#399C22×C3111yescharacteristic
325#398C22×C3111yescharacteristic
326#397C22×C3111yescharacteristic
327#389—21no—
328#388—21no—
329#387—21no—
330#385—21no—
331#376C22×C3111yescharacteristic
332#328C22×C3112yes—
333#328C22×C3112yes—
334#327C22×C3112yes—
335#327C22×C3112yes—
336#324C22×C3111yescharacteristic
337#324C22×C3111yescharacteristic
338#323C22×C3112yes—
339C2×C31—21no—
340C2×C31—21no—
341C2×C31—82no—
342C2×C31—81no—
343C2×C31#300211yescharacteristic
344C2×C31#301811yescharacteristic
345C2×C31#302811yescharacteristic
346C2×#41—21no—
347C23×C4D4×C3111yescharacteristic
348C4⋊Q8—22no—
349#126—22no—
350#126—22no—
351#126—22no—
352#126—22no—
353#125—21no—
354#125—22no—
355#125—22no—
356#125—22no—
357#125D4×C3112yes—
358#124—21no—
359#124—22no—
360#124—22no—
361#124D4×C3112yes—
362#123—22no—
363#123—22no—
364#123—22no—
365C22⋊Q8—21no—
366C22⋊Q8—21no—
367C22⋊Q8—21no—
368C22⋊Q8—22no—
369C22⋊Q8C23×C3112yes—
370#121—21no—
371#121—21no—
372#121—22no—
373#121—22no—
374#121—22no—
375#121—21no—
376#121C23×C3112yes—
377#120—22no—
378C4×Q8—22no—
379C4×Q8—21no—
380C4×Q8D4×C3112yes—
381C4×D4—22no—
382C4×D4—22no—
383C4×D4—22no—
384C4×D4—21no—
385C4×D4—22no—
386C4×D4—22no—
387C4×D4D4×C3112yes—
388#117—22no—
389#117—22no—
390#117—21no—
391#117—22no—
392#117—22no—
393#117—22no—
394C2×C4⋊C4—22no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4D4×C3111yescharacteristic
400C2×C4⋊C4C23×C3111yescharacteristic
401C2×C4⋊C4C23×C3111yescharacteristic
402C2×C4⋊C4C23×C3111yescharacteristic
403C2×C22⋊C4—21no—
404C2×C22⋊C4—21no—
405C2×C22⋊C4D4×C3111yescharacteristic
406C2×C22⋊C4C23×C3111yescharacteristic
407C2×C42—21no—
408C2×C42D4×C3111yescharacteristic
409C2×C42C23×C3111yescharacteristic
410#107—22no—
411#107—22no—
412#106—22no—
413#106—22no—
414C4⋊C8—22no—
415#103—22no—
416#103—22no—
417#102—22no—
418#102—22no—
419C22⋊C8C23×C3112yes—
420C22⋊C8C23×C3112yes—
421C4×C8—22no—
422#95—21no—
423#95—21no—
424#95—21no—
425#95—21no—
426#95—21no—
427#95C23×C3111yescharacteristic
428#95C23×C3111yescharacteristic
429C31#1840711yescharacteristic, direct product factor, Sylow
430C24C2×D4×C3111yescharacteristic
431#41—42no—
432#41—42no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8C2×D4×C3112yes—
436C2×D4—21no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4—21no—
441C2×D4C2×D4×C3112yes—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—41no—
453C22×C4C2×D4×C3111yescharacteristic
454C22×C4C2×D4×C3111yescharacteristic
455C22×C4C2×D4×C3111yescharacteristic
456C22×C4C2×D4×C3111yescharacteristic
457C22×C4C2×D4×C3111yescharacteristic
458C22×C4C24×C3111yescharacteristic
459C2×C8—22no—
460C2×C8—22no—
461C4⋊C4—21no—
462C4⋊C4—22no—
463C4⋊C4—21no—
464C4⋊C4—22no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—42no—
476C4⋊C4—41no—
477C4⋊C4—41no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4C2×D4×C3112yes—
481C4⋊C4C2×D4×C3112yes—
482C22⋊C4—22no—
483C22⋊C4—22no—
484C22⋊C4—22no—
485C22⋊C4—22no—
486C22⋊C4—22no—
487C22⋊C4—22no—
488C22⋊C4—22no—
489C22⋊C4—22no—
490C22⋊C4—22no—
491C22⋊C4—22no—
492C22⋊C4—22no—
493C22⋊C4—42no—
494C22⋊C4—42no—
495C42—22no—
496C42—22no—
497C42—21no—
498C42—21no—
499C42—22no—
500C42—42no—
501C42C2×D4×C3112yes—
502C42C2×D4×C3112yes—
503C23—21no—
504C23—22no—
505C23—21no—
506C23—21no—
507C23—41no—
508C23—41no—
509C23C22×D4×C3111yescharacteristic
510Q8—42no—
511Q8—42no—
512Q8—81no—
513D4—42no—
514D4—42no—
515D4—42no—
516D4—42no—
517D4—82no—
518D4—81no—
519D4—82no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4C31×#12011yescharacteristic
551C2×C4C31×#12011yescharacteristic
552C2×C4C31×#12012yes—
553C2×C4C31×#13512yes—
554C2×C4C22×D4×C3112yes—
555C8—42no—
556C8—42no—
557C22—21no—
558C22—21no—
559C22—42no—
560C22—41no—
561C22—42no—
562C22—42no—
563C22—41no—
564C22—42no—
565C22—41no—
566C22—81no—
567C22C2×C31×#12011yescharacteristic
568C22C2×C31×#13511yescharacteristic
569C22C31×#57511yescharacteristic
570C4—22no—
571C4—21no—
572C4—21no—
573C4—22no—
574C4—22no—
575C4—42no—
576C4—42no—
577C4—41no—
578C4—42no—
579C4—41no—
580C4—42no—
581C4—81no—
582C4—81no—
583C4—81no—
584C2—21no—
585C2—21no—
586C2—81no—
587C2—82no—
588C2C31×#300211yescharacteristic
589C2C31×#301811yescharacteristic
590C2C31×#302811yescharacteristic
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×#3232C2semidirect
3. C31×#2611C2semidirect
4. C31×#1618C2semidirect
5. C31×#1617C2semidirect
6. C31×#1616C2semidirect
7. C31×#1612C2semidirect
8. C31×#1611C2semidirect
9. C31×#1610C2semidirect
10. C31×#1500C2semidirect
11. C31×#1458C2semidirect
207. #18407C31direct (N × H)
263. #3232C2×C31semidirect
264. #2611C2×C31semidirect
265. #1618C2×C31semidirect
266. #1617C2×C31semidirect
267. #1616C2×C31semidirect
268. #1612C2×C31semidirect
269. #1611C2×C31semidirect
270. #1610C2×C31semidirect
271. #1500C2×C31semidirect
272. #1458C2×C31semidirect
429. C31#18407direct (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×#32322
C31×#26112
C31×#16182
C31×#16172
C31×#16162
C31×#16122
C31×#16112
C31×#16102
C31×#15002
C31×#14582
#1840731