Finite Group Databasebeta

← C31×#25616 #21482375

Subgroups — C31×#25616 #21482375

Order 7936.

1240 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3511C211yesmaximal, characteristic, complemented
2C31×#3151C211yesmaximal, characteristic, complemented
3C31×#3038C211yesmaximal, characteristic, complemented
4C31×#2858C211yesmaximal, characteristic, complemented
5C31×#2006C214yesmaximal, complemented
6C31×#1997C211yesmaximal, characteristic, complemented
7C31×#1962C211yesmaximal, characteristic, complemented
8C31×#1883C214yesmaximal, complemented
9C4×C31×#104C211yesmaximal, characteristic, complemented
10C2×C31×AGL(1,ℤ/8)C2214yes—
11C2×C31×AGL(1,ℤ/8)C2212yes—
12C31×#556—21no—
13C31×#555—21no—
14C31×#554—21no—
15C31×#553—22no—
16C31×#548—22no—
17C31×#547—21no—
18C31×#542—22no—
19C31×#537C2212yes—
20C31×#533—21no—
21C31×#532—21no—
22C31×#532C2211yescharacteristic
23C2×C31×C4⋊Q8C2211yescharacteristic
24C2×C31×#127C2211yescharacteristic
25C2×C31×#124—22no—
26C31×#520C2212yes—
27C4×#41×C31C2211yescharacteristic
28C31×#496—22no—
29C31×#493—21no—
30C31×#492—21no—
31C31×#460—21no—
32C31×#460—21no—
33C31×#459—21no—
34C31×#459—21no—
35C31×#423C2211yescharacteristic
36C2×C31×#104C2212yes—
37C2×C31×#104C2214yes—
38C31×#418C2212yes—
39C4×M16×C31C2211yescharacteristic
40C31×#390—22no—
41C31×#390C2214yes—
42C43×C31C2211yescharacteristic
43C31×#343C2214yes—
44C31×#339C2212yes—
45C2×#41×C31—22no—
46C2×#41×C31C2311yescharacteristic
47C2×#41×C31C2312yes—
48C22×Q8×C31D412yes—
49C22×D4×C31—21no—
50C22×D4×C31—21no—
51C22×D4×C31C2312yes—
52C31×AGL(1,ℤ/8)—28no—
53C31×AGL(1,ℤ/8)—44no—
54C31×AGL(1,ℤ/8)—44no—
55C2×SD16×C31—22no—
56C2×SD16×C31—24no—
57C2×D8×C31—22no—
58C2×D8×C31—24no—
59C2×M16×C31C2312yes—
60C2×M16×C31C2312yes—
61C31×C4⋊Q8—21no—
62C31×C4⋊Q8—21no—
63C31×C4⋊Q8—21no—
64C31×C4⋊Q8—21no—
65C31×C4⋊Q8D411yescharacteristic
66C31×#127—21no—
67C31×#127—21no—
68C31×#127—21no—
69C31×#127—21no—
70C31×#127D411yescharacteristic
71C31×#126—24no—
72C31×#125—22no—
73C31×#124—22no—
74C31×#124—24no—
75C31×#124—48no—
76C31×#124D412yes—
77C31×#123—24no—
78C31×C22⋊Q8—22no—
79C31×C22⋊Q8—24no—
80C31×C22⋊Q8—22no—
81C31×C22⋊Q8—22no—
82C31×#121—22no—
83C31×#121—22no—
84C31×#121—22no—
85C31×#121—21no—
86C31×#121—21no—
87C31×#121—24no—
88C4×Q8×C31—22no—
89C4×Q8×C31—22no—
90C4×Q8×C31D411yescharacteristic
91C4×Q8×C31D411yescharacteristic
92C4×D4×C31—22no—
93C4×D4×C31—22no—
94C4×D4×C31—21no—
95C4×D4×C31—21no—
96C4×D4×C31—24no—
97C4×D4×C31—21no—
98C4×D4×C31—21no—
99C4×D4×C31D411yescharacteristic
100C4×D4×C31D411yescharacteristic
101C31×#117—22no—
102C31×#117—22no—
103C31×#117—21no—
104C31×#117C2311yescharacteristic
105C2×C4⋊C4×C31—22no—
106C2×C4⋊C4×C31D412yes—
107C2×C4⋊C4×C31C2312yes—
108C2×C22⋊C4×C31—28no—
109C2×C42×C31—21no—
110C2×C42×C31—21no—
111C2×C42×C31—21no—
112C2×C42×C31C2312yes—
113C2×C42×C31C2311yescharacteristic
114C31×C4⋊C8—21no—
115C31×C4⋊C8—21no—
116C31×#104—24no—
117C31×#104—44no—
118C31×#102—22no—
119C31×#102—22no—
120C31×#97—21no—
121C4×C8×C31—21no—
122C24×C31—22no—
123#41×C31—22no—
124#41×C31—24no—
125#41×C31—24no—
126#41×C31—22no—
127#41×C31—84no—
128C2×Q8×C31—22no—
129C2×Q8×C31—22no—
130C2×Q8×C31—44no—
131C2×Q8×C31C2×D412yes—
132C2×Q8×C31C2×D411yescharacteristic
133C2×D4×C31—21no—
134C2×D4×C31—22no—
135C2×D4×C31—24no—
136C2×D4×C31—21no—
137C2×D4×C31—21no—
138C2×D4×C31—22no—
139C2×D4×C31—22no—
140C2×D4×C31—24no—
141C2×D4×C31—24no—
142C2×D4×C31—44no—
143C2×D4×C31—44no—
144C2×D4×C31—41no—
145C2×D4×C31—41no—
146C2×D4×C31—41no—
147C2×D4×C31—41no—
148C2×D4×C31C2×D412yes—
149C2×D4×C31C2×D411yescharacteristic
150C22×C4×C31—22no—
151C22×C4×C31—24no—
152C22×C4×C31—21no—
153C22×C4×C31—21no—
154C22×C4×C31—21no—
155C22×C4×C31—22no—
156C22×C4×C31C2×D411yescharacteristic
157C22×C4×C31C2×D411yescharacteristic
158C22×C4×C31C2×D414yes—
159C22×C4×C31C2411yescharacteristic
160SD16×C31—48no—
161SD16×C31—44no—
162D8×C31—48no—
163D8×C31—44no—
164M16×C31—22no—
165M16×C31—24no—
166M16×C31#4114yescomplemented
167C2×C8×C31—22no—
168C2×C8×C31—22no—
169C4⋊C4×C31—24no—
170C4⋊C4×C31—24no—
171C4⋊C4×C31—22no—
172C4⋊C4×C31—22no—
173C4⋊C4×C31—21no—
174C4⋊C4×C31—21no—
175C4⋊C4×C31—21no—
176C4⋊C4×C31—21no—
177C4⋊C4×C31—22no—
178C4⋊C4×C31—22no—
179C4⋊C4×C31C2×D411yescharacteristic
180C4⋊C4×C31C2×D411yescharacteristic
181C22⋊C4×C31—24no—
182C22⋊C4×C31—24no—
183C22⋊C4×C31—24no—
184C22⋊C4×C31—21no—
185C22⋊C4×C31—21no—
186C22⋊C4×C31—21no—
187C22⋊C4×C31—22no—
188C22⋊C4×C31—22no—
189C22⋊C4×C31—416no—
190C42×C31—24no—
191C42×C31—24no—
192C42×C31—21no—
193C42×C31—21no—
194C42×C31—21no—
195C42×C31—22no—
196C42×C31—22no—
197C42×C31—42no—
198C42×C31C2×D411yescharacteristic
199C42×C31C2×D411yescharacteristic
200C42×C31C2×D411yescharacteristic
201C42×C31C2×D411yescharacteristic
202#25616C3111yesmaximal, characteristic, direct product factor, Sylow
203C23×C31—21no—
204C23×C31—22no—
205C23×C31—44no—
206C23×C31—44no—
207C23×C31—42no—
208C23×C31—44no—
209C23×C31C22×D411yescharacteristic
210Q8×C31—21no—
211Q8×C31—21no—
212Q8×C31—24no—
213Q8×C31—44no—
214Q8×C31—82no—
215D4×C31—21no—
216D4×C31—21no—
217D4×C31—24no—
218D4×C31—42no—
219D4×C31—44no—
220D4×C31—42no—
221D4×C31—42no—
222D4×C31—44no—
223D4×C31—42no—
224D4×C31—42no—
225D4×C31—82no—
226D4×C31—82no—
227D4×C31—84no—
228D4×C31—82no—
229D4×C31—82no—
230C2×C4×C31—22no—
231C2×C4×C31—22no—
232C2×C4×C31—24no—
233C2×C4×C31—21no—
234C2×C4×C31—22no—
235C2×C4×C31—21no—
236C2×C4×C31—21no—
237C2×C4×C31—24no—
238C2×C4×C31—42no—
239C2×C4×C31—42no—
240C2×C4×C31—42no—
241C2×C4×C31—42no—
242C2×C4×C31—42no—
243C2×C4×C31—42no—
244C2×C4×C31—42no—
245C2×C4×C31—44no—
246C2×C4×C31—84no—
247C2×C4×C31#12011yescharacteristic
248C2×C4×C31#12011yescharacteristic
249C2×C4×C31#12011yescharacteristic
250C2×C4×C31#12011yescharacteristic
251C2×C4×C31C22×D411yescharacteristic
252C2×C4×C31C22×D411yescharacteristic
253C2×C4×C31C2×#4112yes—
254C2×C4×C31H(5,𝔽2)12yes—
255C8×C31—24no—
256C8×C31—42no—
257#3511C2×C3111yescharacteristic, complemented
258#3151C2×C3111yescharacteristic, complemented
259#3038C2×C3111yescharacteristic, complemented
260#2858C2×C3111yescharacteristic, complemented
261#2006C2×C3114yescomplemented
262#1997C2×C3111yescharacteristic, complemented
263#1962C2×C3111yescharacteristic, complemented
264#1883C2×C3114yescomplemented
265C4×#104C2×C3111yescharacteristic, complemented
266C22×C31—21no—
267C22×C31—21no—
268C22×C31—42no—
269C22×C31—41no—
270C22×C31—41no—
271C22×C31—42no—
272C22×C31—41no—
273C22×C31—42no—
274C22×C31—88no—
275C22×C31—82no—
276C22×C31—82no—
277C22×C31—82no—
278C22×C31C2×#12011yescharacteristic
279C22×C31#55012yes—
280C4×C31—21no—
281C4×C31—21no—
282C4×C31—44no—
283C4×C31—42no—
284C4×C31—41no—
285C4×C31—41no—
286C4×C31—44no—
287C4×C31—41no—
288C4×C31#45312yes—
289C4×C31#45412yes—
290C4×C31#53211yescharacteristic
291C4×C31#53511yescharacteristic
292C4×C31#54612yes—
293C2×AGL(1,ℤ/8)C22×C3114yes—
294C2×AGL(1,ℤ/8)C22×C3112yes—
295#556—21no—
296#555—21no—
297#554—21no—
298#553—22no—
299#548—22no—
300#547—21no—
301#542—22no—
302#537C22×C3112yes—
303#533—21no—
304#532—21no—
305#532C22×C3111yescharacteristic
306C2×C4⋊Q8C22×C3111yescharacteristic
307C2×#127C22×C3111yescharacteristic
308C2×#124—22no—
309#520C22×C3112yes—
310C4×#41C22×C3111yescharacteristic
311#496—22no—
312#493—21no—
313#492—21no—
314#460—21no—
315#460—21no—
316#459—21no—
317#459—21no—
318#423C22×C3111yescharacteristic
319C2×#104C22×C3112yes—
320C2×#104C22×C3114yes—
321#418C22×C3112yes—
322C4×M16C22×C3111yescharacteristic
323#390—22no—
324#390C22×C3114yes—
325C43C22×C3111yescharacteristic
326#343C22×C3114yes—
327#339C22×C3112yes—
328C2×C31—22no—
329C2×C31—81no—
330C2×C31—82no—
331C2×C31—162no—
332C2×C31#243111yescharacteristic
333C2×C31C2×#45311yescharacteristic
334C2×C31C2×#45411yescharacteristic
335C2×#41—22no—
336C2×#41C23×C3111yescharacteristic
337C2×#41C23×C3112yes—
338C22×Q8D4×C3112yes—
339C22×D4—21no—
340C22×D4—21no—
341C22×D4C23×C3112yes—
342AGL(1,ℤ/8)—28no—
343AGL(1,ℤ/8)—44no—
344AGL(1,ℤ/8)—44no—
345C2×SD16—22no—
346C2×SD16—24no—
347C2×D8—22no—
348C2×D8—24no—
349C2×M16C23×C3112yes—
350C2×M16C23×C3112yes—
351C4⋊Q8—21no—
352C4⋊Q8—21no—
353C4⋊Q8—21no—
354C4⋊Q8—21no—
355C4⋊Q8D4×C3111yescharacteristic
356#127—21no—
357#127—21no—
358#127—21no—
359#127—21no—
360#127D4×C3111yescharacteristic
361#126—24no—
362#125—22no—
363#124—22no—
364#124—24no—
365#124—48no—
366#124D4×C3112yes—
367#123—24no—
368C22⋊Q8—22no—
369C22⋊Q8—24no—
370C22⋊Q8—22no—
371C22⋊Q8—22no—
372#121—22no—
373#121—22no—
374#121—22no—
375#121—21no—
376#121—21no—
377#121—24no—
378C4×Q8—22no—
379C4×Q8—22no—
380C4×Q8D4×C3111yescharacteristic
381C4×Q8D4×C3111yescharacteristic
382C4×D4—22no—
383C4×D4—22no—
384C4×D4—21no—
385C4×D4—21no—
386C4×D4—24no—
387C4×D4—21no—
388C4×D4—21no—
389C4×D4D4×C3111yescharacteristic
390C4×D4D4×C3111yescharacteristic
391#117—22no—
392#117—22no—
393#117—21no—
394#117C23×C3111yescharacteristic
395C2×C4⋊C4—22no—
396C2×C4⋊C4D4×C3112yes—
397C2×C4⋊C4C23×C3112yes—
398C2×C22⋊C4—28no—
399C2×C42—21no—
400C2×C42—21no—
401C2×C42—21no—
402C2×C42C23×C3112yes—
403C2×C42C23×C3111yescharacteristic
404C4⋊C8—21no—
405C4⋊C8—21no—
406#104—24no—
407#104—44no—
408#102—22no—
409#102—22no—
410#97—21no—
411C4×C8—21no—
412C31#2561611yescharacteristic, direct product factor, Sylow
413C24—22no—
414#41—22no—
415#41—24no—
416#41—24no—
417#41—22no—
418#41—84no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—44no—
422C2×Q8C2×D4×C3112yes—
423C2×Q8C2×D4×C3111yescharacteristic
424C2×D4—21no—
425C2×D4—22no—
426C2×D4—24no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—24no—
432C2×D4—24no—
433C2×D4—44no—
434C2×D4—44no—
435C2×D4—41no—
436C2×D4—41no—
437C2×D4—41no—
438C2×D4—41no—
439C2×D4C2×D4×C3112yes—
440C2×D4C2×D4×C3111yescharacteristic
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4C2×D4×C3111yescharacteristic
448C22×C4C2×D4×C3111yescharacteristic
449C22×C4C2×D4×C3114yes—
450C22×C4C24×C3111yescharacteristic
451SD16—48no—
452SD16—44no—
453D8—48no—
454D8—44no—
455M16—22no—
456M16—24no—
457M16#41×C3114yescomplemented
458C2×C8—22no—
459C2×C8—22no—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4C2×D4×C3111yescharacteristic
471C4⋊C4C2×D4×C3111yescharacteristic
472C22⋊C4—24no—
473C22⋊C4—24no—
474C22⋊C4—24no—
475C22⋊C4—21no—
476C22⋊C4—21no—
477C22⋊C4—21no—
478C22⋊C4—22no—
479C22⋊C4—22no—
480C22⋊C4—416no—
481C42—24no—
482C42—24no—
483C42—21no—
484C42—21no—
485C42—21no—
486C42—22no—
487C42—22no—
488C42—42no—
489C42C2×D4×C3111yescharacteristic
490C42C2×D4×C3111yescharacteristic
491C42C2×D4×C3111yescharacteristic
492C42C2×D4×C3111yescharacteristic
493C23—21no—
494C23—22no—
495C23—44no—
496C23—44no—
497C23—42no—
498C23—44no—
499C23C22×D4×C3111yescharacteristic
500Q8—21no—
501Q8—21no—
502Q8—24no—
503Q8—44no—
504Q8—82no—
505D4—21no—
506D4—21no—
507D4—24no—
508D4—42no—
509D4—44no—
510D4—42no—
511D4—42no—
512D4—42no—
513D4—42no—
514D4—44no—
515D4—82no—
516D4—82no—
517D4—84no—
518D4—82no—
519D4—82no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—24no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—84no—
537C2×C4C31×#12011yescharacteristic
538C2×C4C31×#12011yescharacteristic
539C2×C4C31×#12011yescharacteristic
540C2×C4C31×#12011yescharacteristic
541C2×C4C22×D4×C3111yescharacteristic
542C2×C4C22×D4×C3111yescharacteristic
543C2×C4C2×#41×C3112yes—
544C2×C4C31×H(5,𝔽2)12yes—
545C8—24no—
546C8—42no—
547C22—21no—
548C22—21no—
549C22—42no—
550C22—41no—
551C22—41no—
552C22—42no—
553C22—41no—
554C22—42no—
555C22—88no—
556C22—82no—
557C22—82no—
558C22—82no—
559C22C2×C31×#12011yescharacteristic
560C22C31×#55012yes—
561C4—21no—
562C4—21no—
563C4—44no—
564C4—42no—
565C4—41no—
566C4—41no—
567C4—44no—
568C4—41no—
569C4C31×#45312yes—
570C4C31×#45412yes—
571C4C31×#53211yescharacteristic
572C4C31×#53511yescharacteristic
573C4C31×#54612yes—
574C2—22no—
575C2—81no—
576C2—82no—
577C2—162no—
578C2C31×#243111yescharacteristic
579C2C2×C31×#45311yescharacteristic
580C2C2×C31×#45411yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#3511C2semidirect
2. C31×#3151C2semidirect
3. C31×#3038C2semidirect
4. C31×#2858C2semidirect
5. C31×#2006C2semidirect
6. C31×#1997C2semidirect
7. C31×#1962C2semidirect
8. C31×#1883C2semidirect
9. C4×C31×#104C2semidirect
166. M16×C31#41semidirect
202. #25616C31direct (N × H)
257. #3511C2×C31semidirect
258. #3151C2×C31semidirect
259. #3038C2×C31semidirect
260. #2858C2×C31semidirect
261. #2006C2×C31semidirect
262. #1997C2×C31semidirect
263. #1962C2×C31semidirect
264. #1883C2×C31semidirect
265. C4×#104C2×C31semidirect
412. C31#25616direct (N × H)
457. M16#41×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#35112
C31×#31512
C31×#30382
C31×#28582
C31×#20062
C31×#19972
C31×#19622
C31×#18832
C4×C31×#1042
#2561631