Finite Group Databasebeta

← C31×#18567 #21475326

Subgroups — C31×#18567 #21475326

Order 7936.

980 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3503C211yesmaximal, characteristic, complemented
2C31×#3254C211yesmaximal, characteristic, complemented
3C31×#3152C211yesmaximal, characteristic, complemented
4C31×#3106C211yesmaximal, characteristic
5C31×#1633C212yesmaximal, complemented
6C31×#1632C212yesmaximal, complemented
7C31×#1623C212yesmaximal, complemented
8C31×#1622C212yesmaximal, complemented
9C31×#1548C211yesmaximal, characteristic, complemented
10C31×#1489C211yesmaximal, characteristic, complemented
11C31×#1485C211yesmaximal, characteristic, complemented
12C2×C31×#137—22no—
13Q82×C31C2212yes—
14C31×#557—21no—
15C31×#554—22no—
16C31×#554C2212yes—
17D4×Q8×C31—22no—
18D4×Q8×C31—21no—
19C31×#544—22no—
20C31×#544—21no—
21C31×#533—21no—
22C31×#533C2211yescharacteristic
23C31×#532C2211yescharacteristic
24C2×C31×C4⋊Q8—21no—
25C2×C31×C4⋊Q8C2211yescharacteristic
26C31×#520—21no—
27C4×#41×C31C2211yescharacteristic
28C31×#497—22no—
29C31×#494C2212yes—
30C31×#492C2212yes—
31C31×#482—21no—
32C31×#477C2211yescharacteristic
33C31×#477C2212yes—
34C31×#476C2211yescharacteristic
35C31×#474—21no—
36C31×#468—22no—
37C31×#461—21no—
38C31×#460C2211yescharacteristic
39C31×#460C2212yes—
40C31×#459C2211yescharacteristic
41C31×#449—21no—
42C4×SD16×C31—21no—
43C4×D8×C31—21no—
44C31×#431C2211yescharacteristic
45C31×#423C2211yescharacteristic
46C4×M16×C31C2211yescharacteristic
47C31×#333C2212yes—
48C31×#333C2212yes—
49C31×Q8⋊C8C2212yes—
50C31×Q8⋊C8C2212yes—
51C31×Q8⋊C8C2212yes—
52C31×D4⋊C8C2212yes—
53C2×#41×C31—21no—
54C2×#41×C31D411yescharacteristic
55C22×Q8×C31—22no—
56C22×Q8×C31D412yes—
57C31×#137—48no—
58C2×Q16×C31—24no—
59C2×SD16×C31—21no—
60C2×SD16×C31—24no—
61C2×SD16×C31—22no—
62C2×D8×C31—21no—
63C2×M16×C31—21no—
64C2×M16×C31—21no—
65C31×C4⋊Q8—22no—
66C31×C4⋊Q8—22no—
67C31×C4⋊Q8—24no—
68C31×C4⋊Q8—22no—
69C31×C4⋊Q8—22no—
70C31×C4⋊Q8—21no—
71C31×C4⋊Q8—21no—
72C31×C4⋊Q8D411yescharacteristic
73C31×C4⋊Q8D412yes—
74C31×C4⋊Q8D411yescharacteristic
75C31×C4⋊Q8C2311yescharacteristic
76C31×C4⋊Q8C2312yes—
77C31×#127C2311yescharacteristic
78C31×#125—22no—
79C31×#125—24no—
80C31×#125—22no—
81C31×#125—21no—
82C31×#124—21no—
83C31×C22⋊Q8—22no—
84C31×C22⋊Q8—24no—
85C31×C22⋊Q8—24no—
86C31×C22⋊Q8—22no—
87C31×C22⋊Q8—24no—
88C31×C22⋊Q8—21no—
89C31×C22⋊Q8D412yes—
90C31×#121—21no—
91C31×#121—21no—
92C4×Q8×C31—22no—
93C4×Q8×C31—22no—
94C4×Q8×C31—22no—
95C4×Q8×C31—21no—
96C4×Q8×C31D411yescharacteristic
97C4×Q8×C31C2311yescharacteristic
98C4×Q8×C31C2312yes—
99C4×D4×C31—24no—
100C4×D4×C31—22no—
101C4×D4×C31—21no—
102C4×D4×C31—21no—
103C4×D4×C31—21no—
104C4×D4×C31—21no—
105C4×D4×C31D411yescharacteristic
106C4×D4×C31C2311yescharacteristic
107C31×#117—22no—
108C31×#117—21no—
109C31×#117—21no—
110C31×#117D411yescharacteristic
111C2×C4⋊C4×C31—22no—
112C2×C4⋊C4×C31—21no—
113C2×C4⋊C4×C31—21no—
114C2×C4⋊C4×C31—22no—
115C2×C42×C31—21no—
116C2×C42×C31C2311yescharacteristic
117C31×#107—21no—
118C31×#107—22no—
119C31×#106—21no—
120C31×C4⋊C8—21no—
121C31×C4⋊C8C2312yes—
122C31×C4⋊C8C2311yescharacteristic
123C31×C4⋊C8C2311yescharacteristic
124C31×#103—21no—
125C31×#103—22no—
126C31×#102—21no—
127C31×#102—21no—
128C31×#102—21no—
129C31×#102—22no—
130C31×C22⋊C8—21no—
131C31×#97—21no—
132C4×C8×C31—21no—
133C4×C8×C31C2312yes—
134#41×C31—22no—
135#41×C31—22no—
136#41×C31—44no—
137C2×Q8×C31—24no—
138C2×Q8×C31—22no—
139C2×Q8×C31—21no—
140C2×Q8×C31—21no—
141C2×Q8×C31—44no—
142C2×Q8×C31—84no—
143C2×Q8×C31C2×D412yes—
144C2×Q8×C31C2×D411yescharacteristic
145C2×Q8×C31C2×D414yes—
146C2×D4×C31—21no—
147C2×D4×C31—21no—
148C2×D4×C31—21no—
149C2×D4×C31—21no—
150C2×D4×C31—41no—
151C2×D4×C31C2×D411yescharacteristic
152C22×C4×C31—21no—
153C22×C4×C31—24no—
154C22×C4×C31—22no—
155C22×C4×C31—21no—
156C22×C4×C31—21no—
157C22×C4×C31—41no—
158C22×C4×C31C2×D411yescharacteristic
159C22×C4×C31C2×D411yescharacteristic
160C22×C4×C31C2×D411yescharacteristic
161Q16×C31—48no—
162SD16×C31—48no—
163SD16×C31—42no—
164SD16×C31—82no—
165D8×C31—42no—
166M16×C31—24no—
167M16×C31—81no—
168C2×C8×C31—22no—
169C2×C8×C31—21no—
170C2×C8×C31—21no—
171C2×C8×C31—22no—
172C4⋊C4×C31—24no—
173C4⋊C4×C31—24no—
174C4⋊C4×C31—24no—
175C4⋊C4×C31—24no—
176C4⋊C4×C31—22no—
177C4⋊C4×C31—21no—
178C4⋊C4×C31—22no—
179C4⋊C4×C31—21no—
180C4⋊C4×C31—21no—
181C4⋊C4×C31—21no—
182C4⋊C4×C31—22no—
183C4⋊C4×C31—22no—
184C4⋊C4×C31—22no—
185C4⋊C4×C31—21no—
186C4⋊C4×C31—22no—
187C4⋊C4×C31—21no—
188C4⋊C4×C31—21no—
189C4⋊C4×C31C2×D411yescharacteristic
190C4⋊C4×C31C2×D412yes—
191C4⋊C4×C31C2×D411yescharacteristic
192C22⋊C4×C31—24no—
193C22⋊C4×C31—22no—
194C22⋊C4×C31—21no—
195C22⋊C4×C31—21no—
196C22⋊C4×C31—21no—
197C22⋊C4×C31—41no—
198C22⋊C4×C31—41no—
199C42×C31—24no—
200C42×C31—21no—
201C42×C31—21no—
202C42×C31—21no—
203C42×C31—22no—
204C42×C31C2×D411yescharacteristic
205C42×C31C2×D411yescharacteristic
206C42×C31C2×D411yescharacteristic
207C42×C31C2411yescharacteristic
208#18567C3111yesmaximal, characteristic, direct product factor, Sylow
209C23×C31—21no—
210C23×C31—41no—
211C23×C31#12011yescharacteristic
212Q8×C31—21no—
213Q8×C31—22no—
214Q8×C31—22no—
215Q8×C31—44no—
216Q8×C31—42no—
217Q8×C31—44no—
218Q8×C31—42no—
219Q8×C31—84no—
220Q8×C31#13712yescomplemented
221D4×C31—24no—
222D4×C31—21no—
223D4×C31—42no—
224D4×C31—41no—
225D4×C31—41no—
226D4×C31—41no—
227D4×C31—41no—
228D4×C31—82no—
229D4×C31#13712yescomplemented
230C2×C4×C31—24no—
231C2×C4×C31—21no—
232C2×C4×C31—21no—
233C2×C4×C31—21no—
234C2×C4×C31—21no—
235C2×C4×C31—21no—
236C2×C4×C31—22no—
237C2×C4×C31—22no—
238C2×C4×C31—21no—
239C2×C4×C31—21no—
240C2×C4×C31—24no—
241C2×C4×C31—22no—
242C2×C4×C31—41no—
243C2×C4×C31—41no—
244C2×C4×C31—42no—
245C2×C4×C31—41no—
246C2×C4×C31—82no—
247C2×C4×C31—84no—
248C2×C4×C31—82no—
249C2×C4×C31#12011yescharacteristic
250C2×C4×C31#12011yescharacteristic
251C2×C4×C31#12011yescharacteristic
252C2×C4×C31C22×D411yescharacteristic
253C2×C4×C31C22×D411yescharacteristic
254C2×C4×C31C22×D411yescharacteristic
255C8×C31—24no—
256C8×C31—41no—
257C8×C31—41no—
258C8×C31—42no—
259#3503C2×C3111yescharacteristic, complemented
260#3254C2×C3111yescharacteristic, complemented
261#3152C2×C3111yescharacteristic, complemented
262#3106C2×C3111yescharacteristic
263#1633C2×C3112yescomplemented
264#1632C2×C3112yescomplemented
265#1623C2×C3112yescomplemented
266#1622C2×C3112yescomplemented
267#1548C2×C3111yescharacteristic, complemented
268#1489C2×C3111yescharacteristic, complemented
269#1485C2×C3111yescharacteristic, complemented
270C22×C31—21no—
271C22×C31—22no—
272C22×C31—21no—
273C22×C31—21no—
274C22×C31—41no—
275C22×C31—41no—
276C22×C31—81no—
277C22×C31—81no—
278C22×C31—81no—
279C22×C31C2×#12011yescharacteristic
280C4×C31—21no—
281C4×C31—22no—
282C4×C31—21no—
283C4×C31—21no—
284C4×C31—41no—
285C4×C31—42no—
286C4×C31—41no—
287C4×C31—44no—
288C4×C31—42no—
289C4×C31—81no—
290C4×C31#45312yes—
291C4×C31C2×#13711yescharacteristic
292C4×C31C2×#13711yescharacteristic
293C4×C31#57711yescharacteristic
294C4×C31#57811yescharacteristic
295C2×#137—22no—
296Q82C22×C3112yes—
297#557—21no—
298#554—22no—
299#554C22×C3112yes—
300D4×Q8—22no—
301D4×Q8—21no—
302#544—22no—
303#544—21no—
304#533—21no—
305#533C22×C3111yescharacteristic
306#532C22×C3111yescharacteristic
307C2×C4⋊Q8—21no—
308C2×C4⋊Q8C22×C3111yescharacteristic
309#520—21no—
310C4×#41C22×C3111yescharacteristic
311#497—22no—
312#494C22×C3112yes—
313#492C22×C3112yes—
314#482—21no—
315#477C22×C3111yescharacteristic
316#477C22×C3112yes—
317#476C22×C3111yescharacteristic
318#474—21no—
319#468—22no—
320#461—21no—
321#460C22×C3111yescharacteristic
322#460C22×C3112yes—
323#459C22×C3111yescharacteristic
324#449—21no—
325C4×SD16—21no—
326C4×D8—21no—
327#431C22×C3111yescharacteristic
328#423C22×C3111yescharacteristic
329C4×M16C22×C3111yescharacteristic
330#333C22×C3112yes—
331#333C22×C3112yes—
332Q8⋊C8C22×C3112yes—
333Q8⋊C8C22×C3112yes—
334Q8⋊C8C22×C3112yes—
335D4⋊C8C22×C3112yes—
336C2×C31—41no—
337C2×C31—42no—
338C2×C31—161no—
339C2×C31#300411yescharacteristic
340C2×C31#301311yescharacteristic
341C2×C31C2×#45311yescharacteristic
342C2×#41—21no—
343C2×#41D4×C3111yescharacteristic
344C22×Q8—22no—
345C22×Q8D4×C3112yes—
346#137—48no—
347C2×Q16—24no—
348C2×SD16—24no—
349C2×SD16—21no—
350C2×SD16—22no—
351C2×D8—21no—
352C2×M16—21no—
353C2×M16—21no—
354C4⋊Q8—22no—
355C4⋊Q8—22no—
356C4⋊Q8—24no—
357C4⋊Q8—22no—
358C4⋊Q8—22no—
359C4⋊Q8—21no—
360C4⋊Q8—21no—
361C4⋊Q8D4×C3112yes—
362C4⋊Q8D4×C3111yescharacteristic
363C4⋊Q8D4×C3111yescharacteristic
364C4⋊Q8C23×C3111yescharacteristic
365C4⋊Q8C23×C3112yes—
366#127C23×C3111yescharacteristic
367#125—22no—
368#125—24no—
369#125—22no—
370#125—21no—
371#124—21no—
372C22⋊Q8—22no—
373C22⋊Q8—24no—
374C22⋊Q8—24no—
375C22⋊Q8—22no—
376C22⋊Q8—24no—
377C22⋊Q8—21no—
378C22⋊Q8D4×C3112yes—
379#121—21no—
380#121—21no—
381C4×Q8—22no—
382C4×Q8—22no—
383C4×Q8—22no—
384C4×Q8—21no—
385C4×Q8D4×C3111yescharacteristic
386C4×Q8C23×C3111yescharacteristic
387C4×Q8C23×C3112yes—
388C4×D4—24no—
389C4×D4—22no—
390C4×D4—21no—
391C4×D4—21no—
392C4×D4—21no—
393C4×D4—21no—
394C4×D4D4×C3111yescharacteristic
395C4×D4C23×C3111yescharacteristic
396#117—22no—
397#117—21no—
398#117—21no—
399#117D4×C3111yescharacteristic
400C2×C4⋊C4—22no—
401C2×C4⋊C4—21no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4—22no—
404C2×C42—21no—
405C2×C42C23×C3111yescharacteristic
406#107—21no—
407#107—22no—
408#106—21no—
409C4⋊C8—21no—
410C4⋊C8C23×C3112yes—
411C4⋊C8C23×C3111yescharacteristic
412C4⋊C8C23×C3111yescharacteristic
413#103—21no—
414#103—22no—
415#102—21no—
416#102—21no—
417#102—21no—
418#102—22no—
419C22⋊C8—21no—
420#97—21no—
421C4×C8—21no—
422C4×C8C23×C3112yes—
423C31#1856711yescharacteristic, direct product factor, Sylow
424#41—22no—
425#41—22no—
426#41—44no—
427C2×Q8—24no—
428C2×Q8—22no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—44no—
432C2×Q8—84no—
433C2×Q8C2×D4×C3112yes—
434C2×Q8C2×D4×C3111yescharacteristic
435C2×Q8C2×D4×C3114yes—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—41no—
441C2×D4C2×D4×C3111yescharacteristic
442C22×C4—21no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—41no—
448C22×C4C2×D4×C3111yescharacteristic
449C22×C4C2×D4×C3111yescharacteristic
450C22×C4C2×D4×C3111yescharacteristic
451Q16—48no—
452SD16—48no—
453SD16—42no—
454SD16—82no—
455D8—42no—
456M16—24no—
457M16—81no—
458C2×C8—22no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—22no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—21no—
468C4⋊C4—22no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—21no—
476C4⋊C4—22no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4C2×D4×C3111yescharacteristic
480C4⋊C4C2×D4×C3112yes—
481C4⋊C4C2×D4×C3111yescharacteristic
482C22⋊C4—24no—
483C22⋊C4—22no—
484C22⋊C4—21no—
485C22⋊C4—21no—
486C22⋊C4—21no—
487C22⋊C4—41no—
488C22⋊C4—41no—
489C42—24no—
490C42—21no—
491C42—21no—
492C42—21no—
493C42—22no—
494C42C2×D4×C3111yescharacteristic
495C42C2×D4×C3111yescharacteristic
496C42C2×D4×C3111yescharacteristic
497C42C24×C3111yescharacteristic
498C23—21no—
499C23—41no—
500C23C31×#12011yescharacteristic
501Q8—21no—
502Q8—22no—
503Q8—22no—
504Q8—44no—
505Q8—42no—
506Q8—44no—
507Q8—42no—
508Q8—84no—
509Q8C31×#13712yescomplemented
510D4—24no—
511D4—21no—
512D4—42no—
513D4—41no—
514D4—41no—
515D4—41no—
516D4—41no—
517D4—82no—
518D4C31×#13712yescomplemented
519C2×C4—24no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—42no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—82no—
536C2×C4—84no—
537C2×C4—82no—
538C2×C4C31×#12011yescharacteristic
539C2×C4C31×#12011yescharacteristic
540C2×C4C31×#12011yescharacteristic
541C2×C4C22×D4×C3111yescharacteristic
542C2×C4C22×D4×C3111yescharacteristic
543C2×C4C22×D4×C3111yescharacteristic
544C8—24no—
545C8—41no—
546C8—41no—
547C8—42no—
548C22—21no—
549C22—22no—
550C22—21no—
551C22—21no—
552C22—41no—
553C22—41no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22C2×C31×#12011yescharacteristic
558C4—21no—
559C4—22no—
560C4—21no—
561C4—21no—
562C4—41no—
563C4—42no—
564C4—41no—
565C4—44no—
566C4—42no—
567C4—81no—
568C4C31×#45312yes—
569C4C2×C31×#13711yescharacteristic
570C4C2×C31×#13711yescharacteristic
571C4C31×#57711yescharacteristic
572C4C31×#57811yescharacteristic
573C2—41no—
574C2—42no—
575C2—161no—
576C2C31×#300411yescharacteristic
577C2C31×#301311yescharacteristic
578C2C2×C31×#45311yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC42C24×C31
prime radicalC31×#3106C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#3503C2semidirect
2. C31×#3254C2semidirect
3. C31×#3152C2semidirect
5. C31×#1633C2semidirect
6. C31×#1632C2semidirect
7. C31×#1623C2semidirect
8. C31×#1622C2semidirect
9. C31×#1548C2semidirect
10. C31×#1489C2semidirect
11. C31×#1485C2semidirect
208. #18567C31direct (N × H)
220. Q8×C31#137semidirect
229. D4×C31#137semidirect
259. #3503C2×C31semidirect
260. #3254C2×C31semidirect
261. #3152C2×C31semidirect
263. #1633C2×C31semidirect
264. #1632C2×C31semidirect
265. #1623C2×C31semidirect
266. #1622C2×C31semidirect
267. #1548C2×C31semidirect
268. #1489C2×C31semidirect
269. #1485C2×C31semidirect
423. C31#18567direct (N × H)
509. Q8C31×#137semidirect
518. D4C31×#137semidirect

Maximal subgroups

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

subgroupindex
C31×#35032
C31×#32542
C31×#31522
C31×#31062
C31×#16332
C31×#16322
C31×#16232
C31×#16222
C31×#15482
C31×#14892
C31×#14852
#1856731