Finite Group Databasebeta

← C31×#18984 #21475743

Subgroups — C31×#18984 #21475743

Order 7936.

990 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3529C211yesmaximal, characteristic, complemented
2C31×#3171C212yesmaximal, complemented
3C31×#3150C211yesmaximal, characteristic, complemented
4C31×#1661C212yesmaximal, complemented
5C31×#1660C211yesmaximal, characteristic, complemented
6C31×#1659C211yesmaximal, characteristic, complemented
7C31×#1657C211yesmaximal, characteristic, complemented
8C31×#1656C212yesmaximal, complemented
9C31×#1655C211yesmaximal, characteristic, complemented
10C31×#1525C211yesmaximal, characteristic, complemented
11C31×#1524C212yesmaximal, complemented
12C2×C31×#135—21no—
13C2×C31×#135—21no—
14C31×#563—21no—
15C31×#563C2211yescharacteristic
16C31×#562—22no—
17C31×#562C2212yes—
18C31×H(3,𝔽4)C2211yescharacteristic
19C31×#560—21no—
20C31×#560—22no—
21C31×#532—21no—
22C31×#532C2211yescharacteristic
23C31×#532C2211yescharacteristic, complemented
24C31×#529—22no—
25C31×#529C2212yes—
26C31×#525—21no—
27C31×#525—22no—
28C31×#501C2212yescomplemented
29C31×#497—21no—
30C31×#497C2211yescharacteristic
31C31×#497C2211yescharacteristic, complemented
32C31×#496—21no—
33C31×#496C2211yescharacteristic
34C31×#496C2212yes—
35C31×#496C2211yescharacteristic, complemented
36C31×#490C2212yescomplemented
37C31×#489C2212yes—
38C31×#469—22no—
39C31×#468—22no—
40C4×SD16×C31—22no—
41C4×D8×C31—22no—
42C4×M16×C31C2212yescomplemented
43C2×C31×#97C2211yescharacteristic, complemented
44C31×#330C2212yescomplemented
45C31×#330C2212yescomplemented
46C31×#329C2211yescharacteristic
47C31×#329C2212yes—
48C31×#329C2212yes—
49C31×#329C2211yescharacteristic
50C31×#329C2212yescomplemented
51C2×#41×C31—21no—
52C2×#41×C31—21no—
53C2×#41×C31D411yescharacteristic
54C2×#41×C31D411yescharacteristic
55C23×C4×C31—21no—
56C31×#135—44no—
57C31×#135—44no—
58C2×Q16×C31—21no—
59C2×Q16×C31—21no—
60C2×SD16×C31—22no—
61C2×SD16×C31—21no—
62C2×SD16×C31—21no—
63C2×SD16×C31—21no—
64C2×SD16×C31—21no—
65C2×D8×C31—22no—
66C2×D8×C31—21no—
67C2×D8×C31—21no—
68C2×M16×C31—22no—
69C22×C8×C31—21no—
70C31×C4⋊Q8—21no—
71C31×C4⋊Q8D411yescharacteristic
72C31×C4⋊Q8C2311yescharacteristic
73C31×#127—21no—
74C31×#127D411yescharacteristic
75C31×#127C2311yescharacteristic
76C31×#126—22no—
77C31×#126—22no—
78C31×#126—22no—
79C31×#125—22no—
80C31×#125C2312yes—
81C31×#124—21no—
82C31×#124—22no—
83C31×#124—21no—
84C31×#124—21no—
85C31×#124C2312yes—
86C31×#124C2311yescharacteristic
87C31×#124C2311yescharacteristic
88C31×#123—22no—
89C31×#123—21no—
90C31×#123—22no—
91C31×#123—21no—
92C31×#123—22no—
93C31×#123—22no—
94C31×#123—22no—
95C31×C22⋊Q8—22no—
96C31×C22⋊Q8—22no—
97C31×C22⋊Q8—22no—
98C31×C22⋊Q8—22no—
99C31×C22⋊Q8D412yes—
100C31×#121—22no—
101C31×#121—22no—
102C31×#121—22no—
103C31×#121—22no—
104C31×#121—22no—
105C31×#121—22no—
106C31×#121—22no—
107C31×#121—22no—
108C31×#121D412yes—
109C31×#120—21no—
110C31×#120—22no—
111C31×#120—21no—
112C31×#120—22no—
113C4×Q8×C31D412yes—
114C4×D4×C31—22no—
115C4×D4×C31—22no—
116C4×D4×C31—22no—
117C4×D4×C31—22no—
118C4×D4×C31—22no—
119C4×D4×C31D412yes—
120C31×#117—21no—
121C31×#117—22no—
122C2×C42×C31—21no—
123C2×C42×C31C2311yescharacteristic
124C31×#107—22no—
125C31×#106—22no—
126C31×#103—22no—
127C31×#102—22no—
128C31×#102—22no—
129C31×#102—22no—
130C31×#97—21no—
131C31×#97C2311yescharacteristic
132C31×#97C2311yescharacteristic
133C31×#97C2312yes—
134C31×#97C2312yes—
135C4×C8×C31—22no—
136C24×C31—21no—
137#41×C31—42no—
138#41×C31—42no—
139#41×C31—42no—
140#41×C31—42no—
141#41×C31—82no—
142C2×Q8×C31—21no—
143C2×Q8×C31—21no—
144C2×Q8×C31C2×D412yes—
145C2×Q8×C31C2×D411yescharacteristic
146C2×Q8×C31C2×D411yescharacteristic
147C2×D4×C31—22no—
148C2×D4×C31—22no—
149C2×D4×C31—21no—
150C2×D4×C31—22no—
151C2×D4×C31—22no—
152C2×D4×C31—21no—
153C2×D4×C31—21no—
154C2×D4×C31—22no—
155C2×D4×C31—22no—
156C2×D4×C31—21no—
157C2×D4×C31—41no—
158C2×D4×C31C2×D412yes—
159C2×D4×C31C2×D411yescharacteristic
160C2×D4×C31C2×D411yescharacteristic
161C22×C4×C31—22no—
162C22×C4×C31—22no—
163C22×C4×C31—21no—
164C22×C4×C31—21no—
165C22×C4×C31—21no—
166C22×C4×C31—22no—
167C22×C4×C31—41no—
168C22×C4×C31—82no—
169C22×C4×C31C2×D411yescharacteristic
170C22×C4×C31C2×D412yes—
171Q16×C31—42no—
172Q16×C31—42no—
173SD16×C31—44no—
174SD16×C31—42no—
175SD16×C31—42no—
176SD16×C31—42no—
177SD16×C31—42no—
178D8×C31—44no—
179D8×C31—42no—
180D8×C31—42no—
181M16×C31—44no—
182C2×C8×C31—24no—
183C2×C8×C31—22no—
184C2×C8×C31—21no—
185C2×C8×C31—21no—
186C2×C8×C31—22no—
187C4⋊C4×C31—22no—
188C4⋊C4×C31—22no—
189C4⋊C4×C31—22no—
190C4⋊C4×C31—22no—
191C4⋊C4×C31—22no—
192C4⋊C4×C31—22no—
193C4⋊C4×C31—22no—
194C4⋊C4×C31—22no—
195C4⋊C4×C31—22no—
196C4⋊C4×C31C2×D412yes—
197C4⋊C4×C31C2×D412yes—
198C22⋊C4×C31—21no—
199C22⋊C4×C31—21no—
200C22⋊C4×C31—22no—
201C22⋊C4×C31—22no—
202C22⋊C4×C31—22no—
203C22⋊C4×C31—22no—
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—22no—
211C22⋊C4×C31—42no—
212C42×C31—21no—
213C42×C31—21no—
214C42×C31—22no—
215C42×C31C2×D411yescharacteristic
216C42×C31C2×D412yes—
217C42×C31C2411yescharacteristic
218#18984C3111yesmaximal, characteristic, direct product factor, Sylow
219C23×C31—21no—
220C23×C31—22no—
221C23×C31—21no—
222C23×C31—41no—
223C23×C31—81no—
224C23×C31—82no—
225C23×C31#12011yescharacteristic
226Q8×C31—41no—
227Q8×C31—41no—
228Q8×C31—42no—
229Q8×C31—41no—
230Q8×C31—41no—
231Q8×C31—81no—
232D4×C31—41no—
233D4×C31—41no—
234D4×C31—42no—
235D4×C31—41no—
236D4×C31—41no—
237D4×C31—42no—
238D4×C31—42no—
239D4×C31—82no—
240D4×C31—82no—
241D4×C31—81no—
242D4×C31—81no—
243D4×C31—82no—
244D4×C31—82no—
245D4×C31—82no—
246D4×C31—81no—
247D4×C31—82no—
248C2×C4×C31—22no—
249C2×C4×C31—22no—
250C2×C4×C31—22no—
251C2×C4×C31—21no—
252C2×C4×C31—21no—
253C2×C4×C31—22no—
254C2×C4×C31—42no—
255C2×C4×C31—42no—
256C2×C4×C31—44no—
257C2×C4×C31—42no—
258C2×C4×C31—41no—
259C2×C4×C31—82no—
260C2×C4×C31—82no—
261C2×C4×C31—81no—
262C2×C4×C31—82no—
263C2×C4×C31#12012yes—
264C2×C4×C31#12011yescharacteristic
265C2×C4×C31C22×D412yes—
266C2×C4×C31C22×D411yescharacteristic
267C8×C31—22no—
268C8×C31—22no—
269C8×C31—24no—
270C8×C31—42no—
271#3529C2×C3111yescharacteristic, complemented
272#3171C2×C3112yescomplemented
273#3150C2×C3111yescharacteristic, complemented
274#1661C2×C3112yescomplemented
275#1660C2×C3111yescharacteristic, complemented
276#1659C2×C3111yescharacteristic, complemented
277#1657C2×C3111yescharacteristic, complemented
278#1656C2×C3112yescomplemented
279#1655C2×C3111yescharacteristic, complemented
280#1525C2×C3111yescharacteristic, complemented
281#1524C2×C3112yescomplemented
282C22×C31—21no—
283C22×C31—22no—
284C22×C31—41no—
285C22×C31—42no—
286C22×C31—41no—
287C22×C31—42no—
288C22×C31—42no—
289C22×C31—42no—
290C22×C31—42no—
291C22×C31—84no—
292C22×C31—81no—
293C22×C31—82no—
294C22×C31C2×#12011yescharacteristic
295C4×C31—21no—
296C4×C31—22no—
297C4×C31—22no—
298C4×C31—41no—
299C4×C31—42no—
300C4×C31—41no—
301C4×C31—42no—
302C4×C31—42no—
303C4×C31—81no—
304C4×C31#45312yes—
305C2×#135—21no—
306C2×#135—21no—
307#563—21no—
308#563C22×C3111yescharacteristic
309#562—22no—
310#562C22×C3112yes—
311H(3,𝔽4)C22×C3111yescharacteristic
312#560—21no—
313#560—22no—
314#532—21no—
315#532C22×C3111yescharacteristic
316#532C22×C3111yescharacteristic, complemented
317#529—22no—
318#529C22×C3112yes—
319#525—21no—
320#525—22no—
321#501C22×C3112yescomplemented
322#497—21no—
323#497C22×C3111yescharacteristic
324#497C22×C3111yescharacteristic, complemented
325#496—21no—
326#496C22×C3111yescharacteristic
327#496C22×C3112yes—
328#496C22×C3111yescharacteristic, complemented
329#490C22×C3112yescomplemented
330#489C22×C3112yes—
331#469—22no—
332#468—22no—
333C4×SD16—22no—
334C4×D8—22no—
335C4×M16C22×C3112yescomplemented
336C2×#97C22×C3111yescharacteristic, complemented
337#330C22×C3112yescomplemented
338#330C22×C3112yescomplemented
339#329C22×C3111yescharacteristic
340#329C22×C3112yes—
341#329C22×C3112yes—
342#329C22×C3111yescharacteristic
343#329C22×C3112yescomplemented
344C2×C31—41no—
345C2×C31—81no—
346C2×C31—81no—
347C2×C31—82no—
348C2×C31—161no—
349C2×C31C2×#45311yescharacteristic
350C2×C31#301812yes—
351C2×#41—21no—
352C2×#41—21no—
353C2×#41D4×C3111yescharacteristic
354C2×#41D4×C3111yescharacteristic
355C23×C4—21no—
356#135—44no—
357#135—44no—
358C2×Q16—21no—
359C2×Q16—21no—
360C2×SD16—21no—
361C2×SD16—21no—
362C2×SD16—22no—
363C2×SD16—21no—
364C2×SD16—21no—
365C2×D8—21no—
366C2×D8—21no—
367C2×D8—22no—
368C2×M16—22no—
369C22×C8—21no—
370C4⋊Q8—21no—
371C4⋊Q8D4×C3111yescharacteristic
372C4⋊Q8C23×C3111yescharacteristic
373#127—21no—
374#127D4×C3111yescharacteristic
375#127C23×C3111yescharacteristic
376#126—22no—
377#126—22no—
378#126—22no—
379#125—22no—
380#125C23×C3112yes—
381#124—21no—
382#124—22no—
383#124—21no—
384#124—21no—
385#124C23×C3112yes—
386#124C23×C3111yescharacteristic
387#124C23×C3111yescharacteristic
388#123—22no—
389#123—21no—
390#123—22no—
391#123—21no—
392#123—22no—
393#123—22no—
394#123—22no—
395C22⋊Q8—22no—
396C22⋊Q8—22no—
397C22⋊Q8—22no—
398C22⋊Q8—22no—
399C22⋊Q8D4×C3112yes—
400#121—22no—
401#121—22no—
402#121—22no—
403#121—22no—
404#121—22no—
405#121—22no—
406#121—22no—
407#121—22no—
408#121D4×C3112yes—
409#120—21no—
410#120—22no—
411#120—21no—
412#120—22no—
413C4×Q8D4×C3112yes—
414C4×D4—22no—
415C4×D4—22no—
416C4×D4—22no—
417C4×D4—22no—
418C4×D4—22no—
419C4×D4D4×C3112yes—
420#117—21no—
421#117—22no—
422C2×C42—21no—
423C2×C42C23×C3111yescharacteristic
424#107—22no—
425#106—22no—
426#103—22no—
427#102—22no—
428#102—22no—
429#102—22no—
430#97—21no—
431#97C23×C3112yes—
432#97C23×C3112yes—
433#97C23×C3111yescharacteristic
434#97C23×C3111yescharacteristic
435C4×C8—22no—
436C31#1898411yescharacteristic, direct product factor, Sylow
437C24—21no—
438#41—42no—
439#41—42no—
440#41—42no—
441#41—42no—
442#41—82no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8C2×D4×C3112yes—
446C2×Q8C2×D4×C3111yescharacteristic
447C2×Q8C2×D4×C3111yescharacteristic
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—21no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—22no—
456C2×D4—22no—
457C2×D4—21no—
458C2×D4—41no—
459C2×D4C2×D4×C3112yes—
460C2×D4C2×D4×C3111yescharacteristic
461C2×D4C2×D4×C3111yescharacteristic
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—22no—
468C22×C4—41no—
469C22×C4—82no—
470C22×C4C2×D4×C3111yescharacteristic
471C22×C4C2×D4×C3112yes—
472Q16—42no—
473Q16—42no—
474SD16—44no—
475SD16—42no—
476SD16—42no—
477SD16—42no—
478SD16—42no—
479D8—44no—
480D8—42no—
481D8—42no—
482M16—44no—
483C2×C8—24no—
484C2×C8—22no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4C2×D4×C3112yes—
498C4⋊C4C2×D4×C3112yes—
499C22⋊C4—21no—
500C22⋊C4—21no—
501C22⋊C4—22no—
502C22⋊C4—22no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—22no—
507C22⋊C4—22no—
508C22⋊C4—22no—
509C22⋊C4—22no—
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—42no—
513C42—21no—
514C42—21no—
515C42—22no—
516C42C2×D4×C3112yes—
517C42C2×D4×C3111yescharacteristic
518C42C24×C3111yescharacteristic
519C23—21no—
520C23—22no—
521C23—21no—
522C23—41no—
523C23—81no—
524C23—82no—
525C23C31×#12011yescharacteristic
526Q8—41no—
527Q8—41no—
528Q8—42no—
529Q8—41no—
530Q8—41no—
531Q8—81no—
532D4—41no—
533D4—41no—
534D4—42no—
535D4—41no—
536D4—41no—
537D4—42no—
538D4—42no—
539D4—82no—
540D4—82no—
541D4—81no—
542D4—81no—
543D4—82no—
544D4—82no—
545D4—82no—
546D4—81no—
547D4—82no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—81no—
562C2×C4—82no—
563C2×C4C31×#12012yes—
564C2×C4C31×#12011yescharacteristic
565C2×C4C22×D4×C3112yes—
566C2×C4C22×D4×C3111yescharacteristic
567C8—22no—
568C8—22no—
569C8—24no—
570C8—42no—
571C22—21no—
572C22—22no—
573C22—41no—
574C22—42no—
575C22—41no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—84no—
581C22—81no—
582C22—82no—
583C22C2×C31×#12011yescharacteristic
584C4—21no—
585C4—22no—
586C4—22no—
587C4—41no—
588C4—42no—
589C4—41no—
590C4—42no—
591C4—42no—
592C4—81no—
593C4C31×#45312yes—
594C2—41no—
595C2—81no—
596C2—82no—
597C2—81no—
598C2—161no—
599C2C2×C31×#45311yescharacteristic
600C2C31×#301812yes—
subgroupG/H
centerC2×C4×C31#120
commutator, Frattini, cosocleC42C24×C31
socleC22×C31C2×#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#3529C2semidirect
2. C31×#3171C2semidirect
3. C31×#3150C2semidirect
4. C31×#1661C2semidirect
5. C31×#1660C2semidirect
6. C31×#1659C2semidirect
7. C31×#1657C2semidirect
8. C31×#1656C2semidirect
9. C31×#1655C2semidirect
10. C31×#1525C2semidirect
11. C31×#1524C2semidirect
23. C31×#532C22semidirect
28. C31×#501C22semidirect
31. C31×#497C22semidirect
35. C31×#496C22semidirect
36. C31×#490C22semidirect
42. C4×M16×C31C22semidirect
43. C2×C31×#97C22semidirect
44. C31×#330C22semidirect
45. C31×#330C22semidirect
50. C31×#329C22semidirect
218. #18984C31direct (N × H)
271. #3529C2×C31semidirect
272. #3171C2×C31semidirect
273. #3150C2×C31semidirect
274. #1661C2×C31semidirect
275. #1660C2×C31semidirect
276. #1659C2×C31semidirect
277. #1657C2×C31semidirect
278. #1656C2×C31semidirect
279. #1655C2×C31semidirect
280. #1525C2×C31semidirect
281. #1524C2×C31semidirect
316. #532C22×C31semidirect
321. #501C22×C31semidirect
324. #497C22×C31semidirect
328. #496C22×C31semidirect
329. #490C22×C31semidirect
335. C4×M16C22×C31semidirect
336. C2×#97C22×C31semidirect
337. #330C22×C31semidirect
338. #330C22×C31semidirect
343. #329C22×C31semidirect
436. C31#18984direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35292
C31×#31712
C31×#31502
C31×#16612
C31×#16602
C31×#16592
C31×#16572
C31×#16562
C31×#16552
C31×#15252
C31×#15242
#1898431