Finite Group Databasebeta

← C31×#18149 #21474908

Subgroups — C31×#18149 #21474908

Order 7936.

1074 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
1C2×C31×#538C211yesmaximal, characteristic
2C31×#3095C211yesmaximal, characteristic, complemented
3C31×#3094C211yesmaximal, characteristic, complemented
4C31×#2482C211yesmaximal, characteristic, complemented
5C31×#1618C214yesmaximal, complemented
6C31×#1612C214yesmaximal, complemented
7C31×#1511C211yesmaximal, characteristic, complemented
8C31×#1510C211yesmaximal, characteristic, complemented
9C31×#1462C211yesmaximal, characteristic, complemented
10C31×#538—22no—
11C31×#538—22no—
12C31×#538—22no—
13C31×#538C2214yes—
14C2×C31×#126—21no—
15C2×C31×#124—21no—
16C2×C31×#123—21no—
17C2×C31×C22⋊Q8C2211yescharacteristic
18C2×C31×#121—21no—
19C2×C31×#121C2211yescharacteristic
20C2×C31×#120—21no—
21C2×C4×D4×C31—21no—
22C2×C31×#117C2211yescharacteristic
23C22×C22⋊C4×C31C2211yescharacteristic
24C31×#485—22no—
25C31×#484C2212yescomplemented
26C31×#483C2212yescomplemented
27C31×#482—22no—
28C31×#481C2212yescomplemented
29C31×#480C2212yescomplemented
30C31×#428—21no—
31C31×#428—21no—
32C31×#419—21no—
33C31×#418—21no—
34C31×#407C2211yescharacteristic
35C31×#407C2211yescharacteristic
36C31×#399C2212yescomplemented
37C31×#397C2212yescomplemented
38C31×#388—22no—
39C31×#385—21no—
40C31×#381C2211yescharacteristic
41C31×#379—21no—
42C31×#328C2214yescomplemented
43C31×#327C2214yescomplemented
44C31×#323C2214yescomplemented
45C25×C31D411yescharacteristic
46C22×Q8×C31D411yescharacteristic
47C22×D4×C31—21no—
48C22×D4×C31—21no—
49C22×D4×C31—21no—
50C22×D4×C31D411yescharacteristic
51C23×C4×C31—21no—
52C23×C4×C31D411yescharacteristic
53C23×C4×C31C2311yescharacteristic
54C2×M16×C31—21no—
55C2×M16×C31—21no—
56C31×#126—24no—
57C31×#126—24no—
58C31×#124—24no—
59C31×#124—24no—
60C31×#123—24no—
61C31×#123—24no—
62C31×C22⋊Q8—21no—
63C31×C22⋊Q8—21no—
64C31×C22⋊Q8D412yes—
65C31×C22⋊Q8C2312yes—
66C31×#121—24no—
67C31×#121—24no—
68C31×#121—21no—
69C31×#121—21no—
70C31×#121D412yes—
71C31×#121C2312yes—
72C31×#120—24no—
73C31×#120—24no—
74C4×D4×C31—24no—
75C4×D4×C31—24no—
76C31×#117—21no—
77C31×#117—21no—
78C31×#117—21no—
79C31×#117—21no—
80C2×C4⋊C4×C31—21no—
81C2×C4⋊C4×C31—21no—
82C2×C4⋊C4×C31—22no—
83C2×C4⋊C4×C31D411yescharacteristic
84C2×C4⋊C4×C31D411yescharacteristic
85C2×C4⋊C4×C31C2311yescharacteristic
86C2×C4⋊C4×C31C2311yescharacteristic
87C2×C22⋊C4×C31—21no—
88C2×C22⋊C4×C31—21no—
89C2×C22⋊C4×C31—21no—
90C2×C22⋊C4×C31—21no—
91C2×C22⋊C4×C31—21no—
92C2×C22⋊C4×C31—21no—
93C2×C22⋊C4×C31—21no—
94C2×C22⋊C4×C31—21no—
95C2×C22⋊C4×C31—21no—
96C2×C22⋊C4×C31—22no—
97C2×C22⋊C4×C31—21no—
98C2×C22⋊C4×C31—21no—
99C2×C22⋊C4×C31—22no—
100C2×C22⋊C4×C31D412yes—
101C2×C22⋊C4×C31C2312yes—
102C2×C42×C31—21no—
103C2×C42×C31—21no—
104C31×#107—22no—
105C31×#107—22no—
106C31×#106—22no—
107C31×#106—22no—
108C31×#103—22no—
109C31×#103—22no—
110C31×#102—22no—
111C31×#102—22no—
112C31×C22⋊C8—21no—
113C31×C22⋊C8—21no—
114C31×C22⋊C8C2312yes—
115C31×C22⋊C8C2312yes—
116C31×#95—22no—
117C31×#95—21no—
118C31×#95C2312yes—
119C24×C31—21no—
120C24×C31—21no—
121C24×C31—21no—
122C24×C31—42no—
123C24×C31—42no—
124C24×C31—81no—
125C24×C31C2×D411yescharacteristic
126C24×C31C2×D412yes—
127C2×Q8×C31—21no—
128C2×Q8×C31—42no—
129C2×Q8×C31C2×D412yes—
130C2×D4×C31—22no—
131C2×D4×C31—22no—
132C2×D4×C31—24no—
133C2×D4×C31—24no—
134C2×D4×C31—21no—
135C2×D4×C31—42no—
136C2×D4×C31—44no—
137C2×D4×C31—82no—
138C2×D4×C31—82no—
139C2×D4×C31C2×D412yes—
140C22×C4×C31—21no—
141C22×C4×C31—21no—
142C22×C4×C31—21no—
143C22×C4×C31—21no—
144C22×C4×C31—24no—
145C22×C4×C31—21no—
146C22×C4×C31—22no—
147C22×C4×C31—21no—
148C22×C4×C31—21no—
149C22×C4×C31—42no—
150C22×C4×C31—81no—
151C22×C4×C31—82no—
152C22×C4×C31C2×D411yescharacteristic
153C22×C4×C31C2×D411yescharacteristic
154C22×C4×C31C2×D412yes—
155C22×C4×C31C2×D411yescharacteristic
156C22×C4×C31C2×D411yescharacteristic
157C22×C4×C31C2×D411yescharacteristic
158C22×C4×C31C2411yescharacteristic
159M16×C31—42no—
160M16×C31—42no—
161C2×C8×C31—22no—
162C2×C8×C31—22no—
163C4⋊C4×C31—24no—
164C4⋊C4×C31—22no—
165C4⋊C4×C31—21no—
166C4⋊C4×C31—21no—
167C4⋊C4×C31—22no—
168C4⋊C4×C31—21no—
169C4⋊C4×C31—21no—
170C4⋊C4×C31—21no—
171C4⋊C4×C31—21no—
172C4⋊C4×C31—82no—
173C4⋊C4×C31C2×D412yes—
174C4⋊C4×C31C2×D412yes—
175C22⋊C4×C31—24no—
176C22⋊C4×C31—24no—
177C22⋊C4×C31—24no—
178C22⋊C4×C31—22no—
179C22⋊C4×C31—22no—
180C22⋊C4×C31—22no—
181C22⋊C4×C31—22no—
182C22⋊C4×C31—22no—
183C22⋊C4×C31—22no—
184C22⋊C4×C31—22no—
185C22⋊C4×C31—22no—
186C22⋊C4×C31—42no—
187C22⋊C4×C31—44no—
188C22⋊C4×C31—41no—
189C22⋊C4×C31—41no—
190C22⋊C4×C31—82no—
191C42×C31—24no—
192C42×C31—41no—
193C42×C31—41no—
194#18149C3111yesmaximal, characteristic, direct product factor, Sylow
195C23×C31—22no—
196C23×C31—22no—
197C23×C31—22no—
198C23×C31—22no—
199C23×C31—22no—
200C23×C31—22no—
201C23×C31—41no—
202C23×C31—42no—
203C23×C31—42no—
204C23×C31—42no—
205C23×C31—42no—
206C23×C31—42no—
207C23×C31—42no—
208C23×C31—42no—
209C23×C31—42no—
210C23×C31—42no—
211C23×C31—81no—
212C23×C31—81no—
213C23×C31—88no—
214C23×C31#12011yescharacteristic
215C23×C31#12011yescharacteristic
216C23×C31C22×D411yescharacteristic
217Q8×C31—42no—
218Q8×C31—81no—
219D4×C31—42no—
220D4×C31—44no—
221D4×C31—44no—
222D4×C31—81no—
223D4×C31—84no—
224D4×C31—84no—
225C2×C4×C31—22no—
226C2×C4×C31—22no—
227C2×C4×C31—22no—
228C2×C4×C31—22no—
229C2×C4×C31—22no—
230C2×C4×C31—22no—
231C2×C4×C31—41no—
232C2×C4×C31—41no—
233C2×C4×C31—41no—
234C2×C4×C31—41no—
235C2×C4×C31—42no—
236C2×C4×C31—42no—
237C2×C4×C31—42no—
238C2×C4×C31—42no—
239C2×C4×C31—42no—
240C2×C4×C31—42no—
241C2×C4×C31—42no—
242C2×C4×C31—42no—
243C2×C4×C31—41no—
244C2×C4×C31—41no—
245C2×C4×C31—42no—
246C2×C4×C31—42no—
247C2×C4×C31—81no—
248C2×C4×C31—84no—
249C2×C4×C31#12011yescharacteristic
250C2×C4×C31#12011yescharacteristic
251C2×C4×C31C22×D411yescharacteristic
252C2×C4×C31C22×D411yescharacteristic
253C8×C31—42no—
254C8×C31—42no—
255C2×#538C2×C3111yescharacteristic
256#3095C2×C3111yescharacteristic, complemented
257#3094C2×C3111yescharacteristic, complemented
258#2482C2×C3111yescharacteristic, complemented
259#1618C2×C3114yescomplemented
260#1612C2×C3114yescomplemented
261#1511C2×C3111yescharacteristic, complemented
262#1510C2×C3111yescharacteristic, complemented
263#1462C2×C3111yescharacteristic, complemented
264C22×C31—21no—
265C22×C31—21no—
266C22×C31—21no—
267C22×C31—21no—
268C22×C31—21no—
269C22×C31—21no—
270C22×C31—21no—
271C22×C31—44no—
272C22×C31—44no—
273C22×C31—42no—
274C22×C31—42no—
275C22×C31—42no—
276C22×C31—42no—
277C22×C31—42no—
278C22×C31—42no—
279C22×C31—82no—
280C22×C31—84no—
281C22×C31—84no—
282C22×C31—84no—
283C22×C31#45812yes—
284C22×C31C2×#12011yescharacteristic
285C22×C31#57512yes—
286C4×C31—21no—
287C4×C31—21no—
288C4×C31—21no—
289C4×C31—21no—
290C4×C31—42no—
291C4×C31—42no—
292C4×C31—42no—
293C4×C31—42no—
294C4×C31—82no—
295C4×C31—81no—
296C4×C31—81no—
297#538—22no—
298#538—22no—
299#538—22no—
300#538C22×C3114yes—
301C2×#126—21no—
302C2×#124—21no—
303C2×#123—21no—
304C2×C22⋊Q8C22×C3111yescharacteristic
305C2×#121—21no—
306C2×#121C22×C3111yescharacteristic
307C2×#120—21no—
308C2×C4×D4—21no—
309C2×#117C22×C3111yescharacteristic
310C22×C22⋊C4C22×C3111yescharacteristic
311#485—22no—
312#484C22×C3112yescomplemented
313#483C22×C3112yescomplemented
314#482—22no—
315#481C22×C3112yescomplemented
316#480C22×C3112yescomplemented
317#428—21no—
318#428—21no—
319#419—21no—
320#418—21no—
321#407C22×C3111yescharacteristic
322#407C22×C3111yescharacteristic
323#399C22×C3112yescomplemented
324#397C22×C3112yescomplemented
325#388—22no—
326#385—21no—
327#381C22×C3111yescharacteristic
328#379—21no—
329#328C22×C3114yescomplemented
330#327C22×C3114yescomplemented
331#323C22×C3114yescomplemented
332C2×C31—22no—
333C2×C31—22no—
334C2×C31—41no—
335C2×C31—82no—
336C2×C31—82no—
337C2×C31#300511yescharacteristic
338C2×C31#301911yescharacteristic
339C2×C31C2×#45811yescharacteristic
340C25D4×C3111yescharacteristic
341C22×Q8D4×C3111yescharacteristic
342C22×D4—21no—
343C22×D4—21no—
344C22×D4—21no—
345C22×D4D4×C3111yescharacteristic
346C23×C4—21no—
347C23×C4D4×C3111yescharacteristic
348C23×C4C23×C3111yescharacteristic
349C2×M16—21no—
350C2×M16—21no—
351#126—24no—
352#126—24no—
353#124—24no—
354#124—24no—
355#123—24no—
356#123—24no—
357C22⋊Q8—21no—
358C22⋊Q8—21no—
359C22⋊Q8D4×C3112yes—
360C22⋊Q8C23×C3112yes—
361#121—24no—
362#121—24no—
363#121—21no—
364#121—21no—
365#121D4×C3112yes—
366#121C23×C3112yes—
367#120—24no—
368#120—24no—
369C4×D4—24no—
370C4×D4—24no—
371#117—21no—
372#117—21no—
373#117—21no—
374#117—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—22no—
378C2×C4⋊C4D4×C3111yescharacteristic
379C2×C4⋊C4D4×C3111yescharacteristic
380C2×C4⋊C4C23×C3111yescharacteristic
381C2×C4⋊C4C23×C3111yescharacteristic
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C22⋊C4—21no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—21no—
391C2×C22⋊C4—21no—
392C2×C22⋊C4—21no—
393C2×C22⋊C4—22no—
394C2×C22⋊C4—22no—
395C2×C22⋊C4D4×C3112yes—
396C2×C22⋊C4C23×C3112yes—
397C2×C42—21no—
398C2×C42—21no—
399#107—22no—
400#107—22no—
401#106—22no—
402#106—22no—
403#103—22no—
404#103—22no—
405#102—22no—
406#102—22no—
407C22⋊C8—21no—
408C22⋊C8—21no—
409C22⋊C8C23×C3112yes—
410C22⋊C8C23×C3112yes—
411#95—21no—
412#95—22no—
413#95C23×C3112yes—
414C31#1814911yescharacteristic, direct product factor, Sylow
415C24—21no—
416C24—21no—
417C24—21no—
418C24—42no—
419C24—42no—
420C24—81no—
421C24C2×D4×C3111yescharacteristic
422C24C2×D4×C3112yes—
423C2×Q8—21no—
424C2×Q8—42no—
425C2×Q8C2×D4×C3112yes—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—24no—
429C2×D4—24no—
430C2×D4—21no—
431C2×D4—42no—
432C2×D4—44no—
433C2×D4—82no—
434C2×D4—82no—
435C2×D4C2×D4×C3112yes—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—24no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—42no—
446C22×C4—81no—
447C22×C4—82no—
448C22×C4C2×D4×C3111yescharacteristic
449C22×C4C2×D4×C3111yescharacteristic
450C22×C4C2×D4×C3112yes—
451C22×C4C2×D4×C3111yescharacteristic
452C22×C4C2×D4×C3111yescharacteristic
453C22×C4C2×D4×C3111yescharacteristic
454C22×C4C24×C3111yescharacteristic
455M16—42no—
456M16—42no—
457C2×C8—22no—
458C2×C8—22no—
459C4⋊C4—24no—
460C4⋊C4—22no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—22no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—82no—
469C4⋊C4C2×D4×C3112yes—
470C4⋊C4C2×D4×C3112yes—
471C22⋊C4—24no—
472C22⋊C4—24no—
473C22⋊C4—24no—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—22no—
477C22⋊C4—22no—
478C22⋊C4—22no—
479C22⋊C4—22no—
480C22⋊C4—22no—
481C22⋊C4—22no—
482C22⋊C4—42no—
483C22⋊C4—44no—
484C22⋊C4—41no—
485C22⋊C4—41no—
486C22⋊C4—82no—
487C42—24no—
488C42—41no—
489C42—41no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—41no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—81no—
507C23—81no—
508C23—88no—
509C23C31×#12011yescharacteristic
510C23C31×#12011yescharacteristic
511C23C22×D4×C3111yescharacteristic
512Q8—42no—
513Q8—81no—
514D4—42no—
515D4—44no—
516D4—44no—
517D4—81no—
518D4—84no—
519D4—84no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—81no—
543C2×C4—84no—
544C2×C4C31×#12011yescharacteristic
545C2×C4C31×#12011yescharacteristic
546C2×C4C22×D4×C3111yescharacteristic
547C2×C4C22×D4×C3111yescharacteristic
548C8—42no—
549C8—42no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—44no—
558C22—44no—
559C22—42no—
560C22—42no—
561C22—42no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—82no—
566C22—84no—
567C22—84no—
568C22—84no—
569C22C31×#45812yes—
570C22C2×C31×#12011yescharacteristic
571C22C31×#57512yes—
572C4—21no—
573C4—21no—
574C4—21no—
575C4—21no—
576C4—42no—
577C4—42no—
578C4—42no—
579C4—42no—
580C4—82no—
581C4—81no—
582C4—81no—
583C2—22no—
584C2—22no—
585C2—41no—
586C2—82no—
587C2—82no—
588C2C31×#300511yescharacteristic
589C2C31×#301911yescharacteristic
590C2C2×C31×#45811yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC2×C31×#538C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#3095C2semidirect
3. C31×#3094C2semidirect
4. C31×#2482C2semidirect
5. C31×#1618C2semidirect
6. C31×#1612C2semidirect
7. C31×#1511C2semidirect
8. C31×#1510C2semidirect
9. C31×#1462C2semidirect
25. C31×#484C22semidirect
26. C31×#483C22semidirect
28. C31×#481C22semidirect
29. C31×#480C22semidirect
36. C31×#399C22semidirect
37. C31×#397C22semidirect
42. C31×#328C22semidirect
43. C31×#327C22semidirect
44. C31×#323C22semidirect
194. #18149C31direct (N × H)
256. #3095C2×C31semidirect
257. #3094C2×C31semidirect
258. #2482C2×C31semidirect
259. #1618C2×C31semidirect
260. #1612C2×C31semidirect
261. #1511C2×C31semidirect
262. #1510C2×C31semidirect
263. #1462C2×C31semidirect
312. #484C22×C31semidirect
313. #483C22×C31semidirect
315. #481C22×C31semidirect
316. #480C22×C31semidirect
323. #399C22×C31semidirect
324. #397C22×C31semidirect
329. #328C22×C31semidirect
330. #327C22×C31semidirect
331. #323C22×C31semidirect
414. C31#18149direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5382
C31×#30952
C31×#30942
C31×#24822
C31×#16182
C31×#16122
C31×#15112
C31×#15102
C31×#14622
#1814931