Finite Group Databasebeta

← C31×#18185 #21474944

Subgroups — C31×#18185 #21474944

Order 7936.

944 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3493C211yesmaximal, characteristic
2C31×#3117C212yesmaximal, complemented
3C31×#2396C211yesmaximal, characteristic, complemented
4C31×#1618C212yesmaximal, complemented
5C31×#1617C211yesmaximal, characteristic, complemented
6C31×#1616C211yesmaximal, characteristic, complemented
7C31×#1606C212yesmaximal, complemented
8C31×#1605C211yesmaximal, characteristic, complemented
9C31×#1604C211yesmaximal, characteristic, complemented
10C31×#1509C212yesmaximal, complemented
11C31×#1464C211yesmaximal, 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×#484C2212yes—
27C31×#481C2212yes—
28C31×#479—22no—
29C31×#464—22no—
30C31×#451C2212yes—
31C31×#448C2212yes—
32C31×#441—22no—
33C31×#440—22no—
34C31×#432C2212yes—
35C31×#400C2211yescharacteristic
36C31×#399C2211yescharacteristic
37C31×#398—21no—
38C31×#396—21no—
39C31×#395C2211yescharacteristic
40C31×#393C2211yescharacteristic
41C4×C4⋊C4×C31—21no—
42C4×C4⋊C4×C31C2211yescharacteristic
43C4×C22⋊C4×C31—21no—
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×C31—21no—
107C2×C4⋊C4×C31D411yescharacteristic
108C2×C4⋊C4×C31C2311yescharacteristic
109C2×C4⋊C4×C31C2311yescharacteristic
110C2×C22⋊C4×C31—21no—
111C2×C22⋊C4×C31—21no—
112C2×C22⋊C4×C31D411yescharacteristic
113C2×C22⋊C4×C31C2311yescharacteristic
114C2×C42×C31—21no—
115C2×C42×C31—21no—
116C2×C42×C31D411yescharacteristic
117C2×C42×C31C2311yescharacteristic
118C31×#107—22no—
119C31×#106—22no—
120C31×C4⋊C8—22no—
121C31×#103—22no—
122C31×#103—22no—
123C31×#103—22no—
124C31×#102—22no—
125C31×C22⋊C8C2312yes—
126C31×C22⋊C8C2312yes—
127C31×#97—22no—
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—21no—
151C22×C4×C31—22no—
152C22×C4×C31—21no—
153C22×C4×C31—21no—
154C22×C4×C31—22no—
155C22×C4×C31—22no—
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—42no—
184C4⋊C4×C31—42no—
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×C31—41no—
208C42×C31—41no—
209C42×C31C2×D412yes—
210C42×C31C2×D412yes—
211#18185C3111yesmaximal, characteristic, direct product factor, Sylow
212C23×C31—21no—
213C23×C31—22no—
214C23×C31—21no—
215C23×C31—21no—
216C23×C31—41no—
217C23×C31—41no—
218C23×C31C22×D411yescharacteristic
219Q8×C31—24no—
220Q8×C31—42no—
221Q8×C31—42no—
222Q8×C31—42no—
223Q8×C31—81no—
224D4×C31—42no—
225D4×C31—42no—
226D4×C31—42no—
227D4×C31—42no—
228D4×C31—82no—
229D4×C31—81no—
230D4×C31—82no—
231C2×C4×C31—21no—
232C2×C4×C31—21no—
233C2×C4×C31—21no—
234C2×C4×C31—21no—
235C2×C4×C31—22no—
236C2×C4×C31—22no—
237C2×C4×C31—21no—
238C2×C4×C31—22no—
239C2×C4×C31—21no—
240C2×C4×C31—22no—
241C2×C4×C31—22no—
242C2×C4×C31—41no—
243C2×C4×C31—41no—
244C2×C4×C31—42no—
245C2×C4×C31—42no—
246C2×C4×C31—44no—
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—41no—
258C2×C4×C31—41no—
259C2×C4×C31—82no—
260C2×C4×C31—82no—
261C2×C4×C31#12011yescharacteristic
262C2×C4×C31#12011yescharacteristic
263C2×C4×C31#12012yes—
264C2×C4×C31#13712yes—
265C2×C4×C31C22×D412yes—
266C8×C31—42no—
267C8×C31—42no—
268#3493C2×C3111yescharacteristic
269#3117C2×C3112yescomplemented
270#2396C2×C3111yescharacteristic, complemented
271#1618C2×C3112yescomplemented
272#1617C2×C3111yescharacteristic, complemented
273#1616C2×C3111yescharacteristic, complemented
274#1606C2×C3112yescomplemented
275#1605C2×C3111yescharacteristic, complemented
276#1604C2×C3111yescharacteristic, complemented
277#1509C2×C3112yescomplemented
278#1464C2×C3111yescharacteristic, complemented
279C22×C31—21no—
280C22×C31—21no—
281C22×C31—42no—
282C22×C31—41no—
283C22×C31—42no—
284C22×C31—42no—
285C22×C31—41no—
286C22×C31—42no—
287C22×C31—41no—
288C22×C31—81no—
289C22×C31C2×#12011yescharacteristic
290C22×C31C2×#13711yescharacteristic
291C22×C31#57511yescharacteristic
292C4×C31—22no—
293C4×C31—21no—
294C4×C31—21no—
295C4×C31—22no—
296C4×C31—22no—
297C4×C31—42no—
298C4×C31—42no—
299C4×C31—41no—
300C4×C31—42no—
301C4×C31—42no—
302C4×C31—41no—
303C4×C31—42no—
304C4×C31—81no—
305C4×C31—81no—
306#556—22no—
307#555—22no—
308#553—22no—
309#552—22no—
310#544C22×C3111yescharacteristic
311#538—21no—
312#538C22×C3112yes—
313#537C22×C3111yescharacteristic
314#529—21no—
315#529C22×C3112yes—
316#525—21no—
317#520—21no—
318C2×C4×Q8C22×C3111yescharacteristic
319C2×#117C22×C3111yescharacteristic
320#484C22×C3112yes—
321#481C22×C3112yes—
322#479—22no—
323#464—22no—
324#451C22×C3112yes—
325#448C22×C3112yes—
326#441—22no—
327#440—22no—
328#432C22×C3112yes—
329#400C22×C3111yescharacteristic
330#399C22×C3111yescharacteristic
331#398—21no—
332#396—21no—
333#395C22×C3111yescharacteristic
334#393C22×C3111yescharacteristic
335C4×C4⋊C4—21no—
336C4×C4⋊C4C22×C3111yescharacteristic
337C4×C22⋊C4—21no—
338#328C22×C3112yes—
339#328C22×C3112yes—
340#327C22×C3112yes—
341#327C22×C3112yes—
342#324C22×C3111yescharacteristic
343#324C22×C3111yescharacteristic
344#323C22×C3112yes—
345C2×C31—21no—
346C2×C31—21no—
347C2×C31—82no—
348C2×C31—81no—
349C2×C31#300711yescharacteristic
350C2×C31#301811yescharacteristic
351C2×C31#302511yescharacteristic
352C2×#41—21no—
353C22×Q8C23×C3111yescharacteristic
354C23×C4D4×C3111yescharacteristic
355C2×Q16—22no—
356C2×SD16—22no—
357C4⋊Q8—22no—
358#126—22no—
359#126—22no—
360#126—22no—
361#126—22no—
362#125—21no—
363#125—22no—
364#125D4×C3112yes—
365#124—21no—
366#124—22no—
367#124—22no—
368#124D4×C3112yes—
369#123—22no—
370#123—22no—
371#123—22no—
372C22⋊Q8—21no—
373C22⋊Q8—21no—
374C22⋊Q8—21no—
375C22⋊Q8—22no—
376C22⋊Q8C23×C3112yes—
377#121—21no—
378#121—21no—
379#121—22no—
380#121—22no—
381#121—22no—
382#121—21no—
383#121C23×C3112yes—
384#120—22no—
385C4×Q8—22no—
386C4×Q8—21no—
387C4×Q8—22no—
388C4×Q8—22no—
389C4×Q8D4×C3112yes—
390C4×D4—22no—
391C4×D4—22no—
392C4×D4—22no—
393C4×D4—21no—
394C4×D4—22no—
395C4×D4—22no—
396C4×D4D4×C3112yes—
397#117—22no—
398#117—22no—
399#117—21no—
400#117—22no—
401#117—22no—
402#117—22no—
403C2×C4⋊C4—22no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4D4×C3111yescharacteristic
409C2×C4⋊C4C23×C3111yescharacteristic
410C2×C4⋊C4C23×C3111yescharacteristic
411C2×C22⋊C4—21no—
412C2×C22⋊C4—21no—
413C2×C22⋊C4D4×C3111yescharacteristic
414C2×C22⋊C4C23×C3111yescharacteristic
415C2×C42—21no—
416C2×C42—21no—
417C2×C42D4×C3111yescharacteristic
418C2×C42C23×C3111yescharacteristic
419#107—22no—
420#106—22no—
421C4⋊C8—22no—
422#103—22no—
423#103—22no—
424#103—22no—
425#102—22no—
426C22⋊C8C23×C3112yes—
427C22⋊C8C23×C3112yes—
428#97—22no—
429#95—21no—
430#95—21no—
431#95—21no—
432#95—21no—
433#95C23×C3111yescharacteristic
434#95C23×C3111yescharacteristic
435C31#1818511yescharacteristic, direct product factor, Sylow
436C24C2×D4×C3111yescharacteristic
437#41—42no—
438#41—42no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—22no—
442C2×Q8—42no—
443C2×Q8C2×D4×C3112yes—
444C2×D4—21no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—21no—
449C2×D4C2×D4×C3112yes—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—41no—
461C22×C4C2×D4×C3111yescharacteristic
462C22×C4C2×D4×C3111yescharacteristic
463C22×C4C2×D4×C3111yescharacteristic
464C22×C4C2×D4×C3111yescharacteristic
465C22×C4C2×D4×C3111yescharacteristic
466C22×C4C24×C3111yescharacteristic
467Q16—44no—
468SD16—44no—
469C2×C8—22no—
470C2×C8—22no—
471C4⋊C4—21no—
472C4⋊C4—22no—
473C4⋊C4—21no—
474C4⋊C4—22no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4C2×D4×C3112yes—
489C4⋊C4C2×D4×C3112yes—
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—22no—
500C22⋊C4—22no—
501C22⋊C4—42no—
502C22⋊C4—42no—
503C42—22no—
504C42—22no—
505C42—21no—
506C42—21no—
507C42—22no—
508C42—42no—
509C42—41no—
510C42—41no—
511C42C2×D4×C3112yes—
512C42C2×D4×C3112yes—
513C23—21no—
514C23—22no—
515C23—21no—
516C23—21no—
517C23—41no—
518C23—41no—
519C23C22×D4×C3111yescharacteristic
520Q8—24no—
521Q8—42no—
522Q8—42no—
523Q8—42no—
524Q8—81no—
525D4—42no—
526D4—42no—
527D4—42no—
528D4—42no—
529D4—82no—
530D4—81no—
531D4—82no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—44no—
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—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—82no—
561C2×C4—82no—
562C2×C4C31×#12011yescharacteristic
563C2×C4C31×#12011yescharacteristic
564C2×C4C31×#12012yes—
565C2×C4C31×#13712yes—
566C2×C4C22×D4×C3112yes—
567C8—42no—
568C8—42no—
569C22—21no—
570C22—21no—
571C22—42no—
572C22—41no—
573C22—42no—
574C22—42no—
575C22—41no—
576C22—42no—
577C22—41no—
578C22—81no—
579C22C2×C31×#12011yescharacteristic
580C22C2×C31×#13711yescharacteristic
581C22C31×#57511yescharacteristic
582C4—22no—
583C4—21no—
584C4—21no—
585C4—22no—
586C4—22no—
587C4—42no—
588C4—42no—
589C4—41no—
590C4—42no—
591C4—42no—
592C4—41no—
593C4—42no—
594C4—81no—
595C4—81no—
596C2—21no—
597C2—21no—
598C2—81no—
599C2—82no—
600C2C31×#300711yescharacteristic
601C2C31×#301811yescharacteristic
602C2C31×#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×#3117C2semidirect
3. C31×#2396C2semidirect
4. C31×#1618C2semidirect
5. C31×#1617C2semidirect
6. C31×#1616C2semidirect
7. C31×#1606C2semidirect
8. C31×#1605C2semidirect
9. C31×#1604C2semidirect
10. C31×#1509C2semidirect
11. C31×#1464C2semidirect
211. #18185C31direct (N × H)
269. #3117C2×C31semidirect
270. #2396C2×C31semidirect
271. #1618C2×C31semidirect
272. #1617C2×C31semidirect
273. #1616C2×C31semidirect
274. #1606C2×C31semidirect
275. #1605C2×C31semidirect
276. #1604C2×C31semidirect
277. #1509C2×C31semidirect
278. #1464C2×C31semidirect
435. C31#18185direct (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×#31172
C31×#23962
C31×#16182
C31×#16172
C31×#16162
C31×#16062
C31×#16052
C31×#16042
C31×#15092
C31×#14642
#1818531