Finite Group Databasebeta

← C31×#18247 #21475006

Subgroups — C31×#18247 #21475006

Order 7936.

852 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3563C211yesmaximal, characteristic
2C31×#3117C211yesmaximal, characteristic
3C31×#3104C211yesmaximal, characteristic, complemented
4C31×#2634C211yesmaximal, characteristic, complemented
5C31×#1617C212yesmaximal, complemented
6C31×#1615C212yesmaximal
7C31×#1605C212yesmaximal, complemented
8C31×#1603C212yesmaximal
9C31×#1521C211yesmaximal, characteristic
10C31×#1515C211yesmaximal, characteristic, complemented
11C31×#1470C211yesmaximal, characteristic, complemented
12C31×#564—21no—
13C31×#563—21no—
14C31×#557—21no—
15C31×#554—21no—
16C31×#552—21no—
17C31×#552—21no—
18C31×#552—22no—
19C31×#544—21no—
20C31×#544C2212yes—
21C31×#541—21no—
22C31×#541—22no—
23C31×#539C2212yes—
24C31×#533C2211yescharacteristic
25C31×#529C2211yescharacteristic
26C2×C31×#125C2211yescharacteristic
27C2×C4×Q8×C31C2211yescharacteristic
28C31×#484C2212yes—
29C31×#484C2211yescharacteristic
30C31×#481C2211yescharacteristic
31C31×#479—21no—
32C31×#477—21no—
33C31×#464—21no—
34C31×#462—21no—
35C31×#451C2212yes—
36C31×#451C2211yescharacteristic
37C31×#448C2211yescharacteristic
38C31×#441—21no—
39C31×#440—21no—
40C4×Q16×C31—22no—
41C31×#432C2211yescharacteristic
42C31×#431C2211yescharacteristic
43C31×#400—21no—
44C31×#400C2212yes—
45C31×#398—21no—
46C31×#395C2212yes—
47C31×#387—22no—
48C31×#383C2211yescharacteristic
49C31×#328C2212yes—
50C31×#328C2212yes—
51C31×#328C2212yes—
52C31×#327C2212yes—
53C31×#324C2212yes—
54C31×#324C2212yes—
55C22×Q8×C31C2311yescharacteristic
56C2×Q16×C31—22no—
57C2×Q16×C31—21no—
58C2×SD16×C31—21no—
59C31×C4⋊Q8—21no—
60C31×C4⋊Q8—22no—
61C31×C4⋊Q8—22no—
62C31×C4⋊Q8D411yescharacteristic
63C31×C4⋊Q8D411yescharacteristic
64C31×#126—22no—
65C31×#126—22no—
66C31×#126—22no—
67C31×#126—22no—
68C31×#126—21no—
69C31×#125—22no—
70C31×#125—22no—
71C31×#125—22no—
72C31×#125—22no—
73C31×#125—21no—
74C31×#125—22no—
75C31×#125—22no—
76C31×#125—22no—
77C31×#125—22no—
78C31×#125—21no—
79C31×#125—21no—
80C31×#125—21no—
81C31×#125—21no—
82C31×#125—21no—
83C31×#125D411yescharacteristic
84C31×#124D411yescharacteristic
85C31×#123—22no—
86C31×#123—21no—
87C31×#123—22no—
88C31×C22⋊Q8—21no—
89C31×C22⋊Q8—21no—
90C31×C22⋊Q8—22no—
91C31×C22⋊Q8—22no—
92C31×C22⋊Q8—22no—
93C31×C22⋊Q8—21no—
94C31×C22⋊Q8—22no—
95C31×C22⋊Q8C2311yescharacteristic
96C31×C22⋊Q8C2312yes—
97C31×#121C2311yescharacteristic
98C4×Q8×C31—21no—
99C4×Q8×C31—22no—
100C4×Q8×C31—21no—
101C4×Q8×C31—21no—
102C4×Q8×C31—21no—
103C4×Q8×C31—21no—
104C4×Q8×C31—21no—
105C4×Q8×C31—21no—
106C4×Q8×C31D412yes—
107C4×Q8×C31D411yescharacteristic
108C4×D4×C31—21no—
109C4×D4×C31—22no—
110C4×D4×C31D411yescharacteristic
111C31×#117—22no—
112C31×#117—21no—
113C31×#117—21no—
114C31×#117—22no—
115C31×#117—21no—
116C2×C4⋊C4×C31—21no—
117C2×C4⋊C4×C31—22no—
118C2×C4⋊C4×C31—21no—
119C2×C4⋊C4×C31D411yescharacteristic
120C2×C4⋊C4×C31D411yescharacteristic
121C2×C4⋊C4×C31D412yes—
122C2×C4⋊C4×C31C2311yescharacteristic
123C2×C4⋊C4×C31C2312yes—
124C2×C42×C31—21no—
125C2×C42×C31C2311yescharacteristic
126C31×#107—22no—
127C31×#107—21no—
128C31×#106—21no—
129C31×C4⋊C8—21no—
130C31×C4⋊C8—21no—
131C31×#103—22no—
132C31×#103—21no—
133C31×#103—22no—
134C31×#103—21no—
135C31×#103—21no—
136C31×#102—21no—
137C31×C22⋊C8C2312yes—
138C31×C22⋊C8C2311yescharacteristic
139C31×C22⋊C8C2311yescharacteristic
140C31×#97—21no—
141C4×C8×C31—21no—
142C31×#95—22no—
143C31×#95—21no—
144C31×#95—22no—
145C31×#95—22no—
146C31×#95—21no—
147C31×#95C2312yes—
148C2×Q8×C31—22no—
149C2×Q8×C31—21no—
150C2×Q8×C31—21no—
151C2×Q8×C31—21no—
152C2×Q8×C31—42no—
153C2×Q8×C31C2×D412yes—
154C2×Q8×C31C2×D411yescharacteristic
155C2×D4×C31—21no—
156C2×D4×C31C2×D411yescharacteristic
157C22×C4×C31—21no—
158C22×C4×C31—22no—
159C22×C4×C31—21no—
160C22×C4×C31—22no—
161C22×C4×C31—21no—
162C22×C4×C31C2×D412yes—
163C22×C4×C31C2×D411yescharacteristic
164C22×C4×C31C2×D412yes—
165C22×C4×C31C2×D411yescharacteristic
166C22×C4×C31C2411yescharacteristic
167Q16×C31—42no—
168Q16×C31—44no—
169SD16×C31—42no—
170C2×C8×C31—22no—
171C2×C8×C31—21no—
172C2×C8×C31—21no—
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—22no—
180C4⋊C4×C31—22no—
181C4⋊C4×C31—21no—
182C4⋊C4×C31—21no—
183C4⋊C4×C31—21no—
184C4⋊C4×C31—22no—
185C4⋊C4×C31—21no—
186C4⋊C4×C31—22no—
187C4⋊C4×C31—21no—
188C4⋊C4×C31—21no—
189C4⋊C4×C31—21no—
190C4⋊C4×C31—21no—
191C4⋊C4×C31—21no—
192C4⋊C4×C31—21no—
193C4⋊C4×C31—22no—
194C4⋊C4×C31—22no—
195C4⋊C4×C31—22no—
196C4⋊C4×C31—21no—
197C4⋊C4×C31—22no—
198C4⋊C4×C31—21no—
199C4⋊C4×C31—21no—
200C4⋊C4×C31—44no—
201C4⋊C4×C31—41no—
202C4⋊C4×C31—41no—
203C4⋊C4×C31C2×D411yescharacteristic
204C4⋊C4×C31C2×D411yescharacteristic
205C4⋊C4×C31C2×D412yes—
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—
215C42×C31—22no—
216C42×C31—22no—
217C42×C31—22no—
218C42×C31—22no—
219C42×C31—21no—
220C42×C31—41no—
221C42×C31—41no—
222C42×C31C2×D411yescharacteristic
223C42×C31C2×D411yescharacteristic
224C42×C31C2×D411yescharacteristic
225C42×C31C2×D411yescharacteristic
226#18247C3111yesmaximal, characteristic, direct product factor, Sylow
227C23×C31—21no—
228C23×C31C22×D411yescharacteristic
229Q8×C31—22no—
230Q8×C31—22no—
231Q8×C31—41no—
232Q8×C31—42no—
233Q8×C31—42no—
234Q8×C31—41no—
235Q8×C31—41no—
236Q8×C31—82no—
237D4×C31—41no—
238D4×C31—42no—
239C2×C4×C31—22no—
240C2×C4×C31—22no—
241C2×C4×C31—22no—
242C2×C4×C31—21no—
243C2×C4×C31—21no—
244C2×C4×C31—22no—
245C2×C4×C31—21no—
246C2×C4×C31—22no—
247C2×C4×C31—22no—
248C2×C4×C31—24no—
249C2×C4×C31—41no—
250C2×C4×C31—41no—
251C2×C4×C31—42no—
252C2×C4×C31—41no—
253C2×C4×C31—41no—
254C2×C4×C31—42no—
255C2×C4×C31—42no—
256C2×C4×C31—42no—
257C2×C4×C31—42no—
258C2×C4×C31—41no—
259C2×C4×C31—42no—
260C2×C4×C31—82no—
261C2×C4×C31#12011yescharacteristic
262C2×C4×C31#12011yescharacteristic
263C2×C4×C31#12011yescharacteristic
264C2×C4×C31#12011yescharacteristic
265C2×C4×C31C22×D411yescharacteristic
266C2×C4×C31C22×D411yescharacteristic
267C8×C31—41no—
268C8×C31—41no—
269C8×C31—42no—
270#3563C2×C3111yescharacteristic
271#3117C2×C3111yescharacteristic
272#3104C2×C3111yescharacteristic, complemented
273#2634C2×C3111yescharacteristic, complemented
274#1617C2×C3112yescomplemented
275#1615C2×C3112yes—
276#1605C2×C3112yescomplemented
277#1603C2×C3112yes—
278#1521C2×C3111yescharacteristic
279#1515C2×C3111yescharacteristic, complemented
280#1470C2×C3111yescharacteristic, complemented
281C22×C31—21no—
282C22×C31—21no—
283C22×C31—41no—
284C22×C31—41no—
285C22×C31—41no—
286C22×C31C2×#12011yescharacteristic
287C22×C31#57812yes—
288C4×C31—21no—
289C4×C31—21no—
290C4×C31—21no—
291C4×C31—21no—
292C4×C31—21no—
293C4×C31—21no—
294C4×C31—42no—
295C4×C31—41no—
296C4×C31—41no—
297C4×C31—42no—
298C4×C31—42no—
299C4×C31—42no—
300C4×C31—42no—
301C4×C31—41no—
302C4×C31—82no—
303C4×C31—81no—
304#564—21no—
305#563—21no—
306#557—21no—
307#554—21no—
308#552—21no—
309#552—22no—
310#552—21no—
311#544—21no—
312#544C22×C3112yes—
313#541—21no—
314#541—22no—
315#539C22×C3112yes—
316#533C22×C3111yescharacteristic
317#529C22×C3111yescharacteristic
318C2×#125C22×C3111yescharacteristic
319C2×C4×Q8C22×C3111yescharacteristic
320#484C22×C3112yes—
321#484C22×C3111yescharacteristic
322#481C22×C3111yescharacteristic
323#479—21no—
324#477—21no—
325#464—21no—
326#462—21no—
327#451C22×C3112yes—
328#451C22×C3111yescharacteristic
329#448C22×C3111yescharacteristic
330#441—21no—
331#440—21no—
332C4×Q16—22no—
333#432C22×C3111yescharacteristic
334#431C22×C3111yescharacteristic
335#400—21no—
336#400C22×C3112yes—
337#398—21no—
338#395C22×C3112yes—
339#387—22no—
340#383C22×C3111yescharacteristic
341#328C22×C3112yes—
342#328C22×C3112yes—
343#328C22×C3112yes—
344#327C22×C3112yes—
345#324C22×C3112yes—
346#324C22×C3112yes—
347C2×C31—22no—
348C2×C31—81no—
349C2×C31#301111yescharacteristic
350C2×C31#302011yescharacteristic
351C2×C31#302511yescharacteristic
352C22×Q8C23×C3111yescharacteristic
353C2×Q16—22no—
354C2×Q16—21no—
355C2×SD16—21no—
356C4⋊Q8—21no—
357C4⋊Q8—22no—
358C4⋊Q8—22no—
359C4⋊Q8D4×C3111yescharacteristic
360C4⋊Q8D4×C3111yescharacteristic
361#126—22no—
362#126—22no—
363#126—22no—
364#126—22no—
365#126—21no—
366#125—22no—
367#125—22no—
368#125—22no—
369#125—22no—
370#125—21no—
371#125—22no—
372#125—22no—
373#125—22no—
374#125—22no—
375#125—21no—
376#125—21no—
377#125—21no—
378#125—21no—
379#125—21no—
380#125D4×C3111yescharacteristic
381#124D4×C3111yescharacteristic
382#123—22no—
383#123—21no—
384#123—22no—
385C22⋊Q8—21no—
386C22⋊Q8—21no—
387C22⋊Q8—22no—
388C22⋊Q8—22no—
389C22⋊Q8—22no—
390C22⋊Q8—21no—
391C22⋊Q8—22no—
392C22⋊Q8C23×C3111yescharacteristic
393C22⋊Q8C23×C3112yes—
394#121C23×C3111yescharacteristic
395C4×Q8—21no—
396C4×Q8—22no—
397C4×Q8—21no—
398C4×Q8—21no—
399C4×Q8—21no—
400C4×Q8—21no—
401C4×Q8—21no—
402C4×Q8—21no—
403C4×Q8D4×C3111yescharacteristic
404C4×Q8D4×C3112yes—
405C4×D4—21no—
406C4×D4—22no—
407C4×D4D4×C3111yescharacteristic
408#117—22no—
409#117—21no—
410#117—21no—
411#117—22no—
412#117—21no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—22no—
416C2×C4⋊C4D4×C3112yes—
417C2×C4⋊C4D4×C3111yescharacteristic
418C2×C4⋊C4D4×C3111yescharacteristic
419C2×C4⋊C4C23×C3111yescharacteristic
420C2×C4⋊C4C23×C3112yes—
421C2×C42—21no—
422C2×C42C23×C3111yescharacteristic
423#107—22no—
424#107—21no—
425#106—21no—
426C4⋊C8—21no—
427C4⋊C8—21no—
428#103—22no—
429#103—21no—
430#103—22no—
431#103—21no—
432#103—21no—
433#102—21no—
434C22⋊C8C23×C3112yes—
435C22⋊C8C23×C3111yescharacteristic
436C22⋊C8C23×C3111yescharacteristic
437#97—21no—
438C4×C8—21no—
439#95—21no—
440#95—21no—
441#95—22no—
442#95—22no—
443#95—22no—
444#95C23×C3112yes—
445C31#1824711yescharacteristic, direct product factor, Sylow
446C2×Q8—22no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—42no—
451C2×Q8C2×D4×C3112yes—
452C2×Q8C2×D4×C3111yescharacteristic
453C2×D4—21no—
454C2×D4C2×D4×C3111yescharacteristic
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4C2×D4×C3112yes—
461C22×C4C2×D4×C3111yescharacteristic
462C22×C4C2×D4×C3112yes—
463C22×C4C2×D4×C3111yescharacteristic
464C22×C4C24×C3111yescharacteristic
465Q16—42no—
466Q16—44no—
467SD16—42no—
468C2×C8—22no—
469C2×C8—21no—
470C2×C8—21no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—22no—
483C4⋊C4—21no—
484C4⋊C4—22no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—21no—
495C4⋊C4—22no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—44no—
499C4⋊C4—41no—
500C4⋊C4—41no—
501C4⋊C4C2×D4×C3111yescharacteristic
502C4⋊C4C2×D4×C3111yescharacteristic
503C4⋊C4C2×D4×C3112yes—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—22no—
507C22⋊C4—22no—
508C22⋊C4—21no—
509C22⋊C4—21no—
510C22⋊C4—21no—
511C22⋊C4—21no—
512C22⋊C4—21no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—21no—
518C42—41no—
519C42—41no—
520C42C2×D4×C3111yescharacteristic
521C42C2×D4×C3111yescharacteristic
522C42C2×D4×C3111yescharacteristic
523C42C2×D4×C3111yescharacteristic
524C23—21no—
525C23C22×D4×C3111yescharacteristic
526Q8—22no—
527Q8—22no—
528Q8—41no—
529Q8—42no—
530Q8—42no—
531Q8—41no—
532Q8—41no—
533Q8—82no—
534D4—41no—
535D4—42no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—82no—
558C2×C4C31×#12011yescharacteristic
559C2×C4C31×#12011yescharacteristic
560C2×C4C31×#12011yescharacteristic
561C2×C4C31×#12011yescharacteristic
562C2×C4C22×D4×C3111yescharacteristic
563C2×C4C22×D4×C3111yescharacteristic
564C8—41no—
565C8—41no—
566C8—42no—
567C22—21no—
568C22—21no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22C2×C31×#12011yescharacteristic
573C22C31×#57812yes—
574C4—21no—
575C4—21no—
576C4—21no—
577C4—21no—
578C4—21no—
579C4—21no—
580C4—42no—
581C4—41no—
582C4—41no—
583C4—42no—
584C4—42no—
585C4—42no—
586C4—42no—
587C4—41no—
588C4—82no—
589C4—81no—
590C2—22no—
591C2—81no—
592C2C31×#301111yescharacteristic
593C2C31×#302011yescharacteristic
594C2C31×#302511yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#121C23
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
3. C31×#3104C2semidirect
4. C31×#2634C2semidirect
5. C31×#1617C2semidirect
7. C31×#1605C2semidirect
10. C31×#1515C2semidirect
11. C31×#1470C2semidirect
226. #18247C31direct (N × H)
272. #3104C2×C31semidirect
273. #2634C2×C31semidirect
274. #1617C2×C31semidirect
276. #1605C2×C31semidirect
279. #1515C2×C31semidirect
280. #1470C2×C31semidirect
445. C31#18247direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35632
C31×#31172
C31×#31042
C31×#26342
C31×#16172
C31×#16152
C31×#16052
C31×#16032
C31×#15212
C31×#15152
C31×#14702
#1824731