Finite Group Databasebeta

← C31×#18222 #21474981

Subgroups — C31×#18222 #21474981

Order 7936.

1078 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
1C31×#3506C211yesmaximal, characteristic, complemented
2C31×#3101C211yesmaximal, characteristic, complemented
3C31×#3100C211yesmaximal, characteristic, complemented
4C31×#2535C211yesmaximal, characteristic, complemented
5C31×#1612C214yesmaximal, complemented
6C31×#1600C214yesmaximal, complemented
7C31×#1518C211yesmaximal, characteristic, complemented
8C31×#1517C211yesmaximal, characteristic, complemented
9C31×#1460C211yesmaximal, characteristic, complemented
10C31×#559—22no—
11C31×#556—21no—
12C31×#555—21no—
13C31×#553—22no—
14C31×#550—21no—
15C31×#547—21no—
16C31×#538C2214yes—
17C31×#535—21no—
18C31×#533C2211yescharacteristic
19C31×#532C2211yescharacteristic
20C31×#525—22no—
21C2×C31×#121—21no—
22C31×#520—21no—
23C2×C4×D4×C31C2211yescharacteristic
24C2×C4×D4×C31C2211yescharacteristic
25C31×#483C2212yes—
26C31×#480C2212yes—
27C31×#476—21no—
28C31×#474—21no—
29C31×#461—21no—
30C31×#459—21no—
31C31×#450C2212yes—
32C31×#447C2212yes—
33C4×SD16×C31—22no—
34C4×D8×C31—22no—
35C31×#431C2211yescharacteristic
36C31×#431C2211yescharacteristic
37C31×#397C2212yes—
38C31×#392C2212yes—
39C31×#386—22no—
40C31×#382C2211yescharacteristic
41C31×#380—21no—
42C31×#328C2214yes—
43C31×#327C2214yes—
44C31×#323C2214yes—
45C2×#41×C31—22no—
46C2×#41×C31—21no—
47C22×D4×C31—22no—
48C22×D4×C31D411yescharacteristic
49C22×D4×C31C2311yescharacteristic
50C23×C4×C31—21no—
51C23×C4×C31D412yes—
52C2×SD16×C31—22no—
53C2×D8×C31—22no—
54C31×C4⋊Q8D411yescharacteristic, complemented
55C31×C4⋊Q8D411yescharacteristic, complemented
56C31×C4⋊Q8D411yescharacteristic, complemented
57C31×#127—22no—
58C31×#127D411yescharacteristic, complemented
59C31×#126—24no—
60C31×#125—21no—
61C31×#124—22no—
62C31×#124—24no—
63C31×#124—21no—
64C31×#123—24no—
65C31×#123—24no—
66C31×C22⋊Q8—22no—
67C31×C22⋊Q8—21no—
68C31×C22⋊Q8C2312yes—
69C31×#121—22no—
70C31×#121—22no—
71C31×#121—22no—
72C31×#121—24no—
73C31×#121—21no—
74C31×#121—44no—
75C31×#121C2312yes—
76C31×#120—24no—
77C4×Q8×C31—22no—
78C4×Q8×C31—21no—
79C4×Q8×C31D412yescomplemented
80C4×D4×C31—22no—
81C4×D4×C31—24no—
82C4×D4×C31—24no—
83C4×D4×C31—21no—
84C4×D4×C31—21no—
85C4×D4×C31—21no—
86C4×D4×C31—21no—
87C4×D4×C31—21no—
88C4×D4×C31—21no—
89C4×D4×C31—21no—
90C4×D4×C31—21no—
91C4×D4×C31—21no—
92C4×D4×C31D412yescomplemented
93C31×#117—22no—
94C31×#117—22no—
95C31×#117—21no—
96C2×C4⋊C4×C31—22no—
97C2×C4⋊C4×C31—21no—
98C2×C4⋊C4×C31D411yescharacteristic
99C2×C4⋊C4×C31C2311yescharacteristic
100C2×C22⋊C4×C31—22no—
101C2×C22⋊C4×C31—21no—
102C2×C22⋊C4×C31—22no—
103C2×C22⋊C4×C31C2312yes—
104C2×C42×C31C2311yescharacteristic
105C31×#107—22no—
106C31×#106—22no—
107C31×C4⋊C8—21no—
108C31×C4⋊C8—21no—
109C31×#103—22no—
110C31×#102—22no—
111C31×#102—22no—
112C31×#102—22no—
113C31×C22⋊C8C2312yes—
114C31×C22⋊C8C2312yes—
115C4×C8×C31—21no—
116C4×C8×C31—21no—
117C31×#95—22no—
118C31×#95—21no—
119C31×#95C2312yes—
120C24×C31—21no—
121C24×C31C2×D412yes—
122#41×C31—42no—
123#41×C31—42no—
124#41×C31—84no—
125C2×Q8×C31—21no—
126C2×Q8×C31—21no—
127C2×Q8×C31C2×D412yes—
128C2×D4×C31—22no—
129C2×D4×C31—24no—
130C2×D4×C31—24no—
131C2×D4×C31—21no—
132C2×D4×C31—21no—
133C2×D4×C31—21no—
134C2×D4×C31—21no—
135C2×D4×C31—22no—
136C2×D4×C31—22no—
137C2×D4×C31—21no—
138C2×D4×C31—21no—
139C2×D4×C31—42no—
140C2×D4×C31—44no—
141C2×D4×C31—44no—
142C2×D4×C31—82no—
143C2×D4×C31C2×D412yes—
144C22×C4×C31—24no—
145C22×C4×C31—22no—
146C22×C4×C31—22no—
147C22×C4×C31—22no—
148C22×C4×C31—22no—
149C22×C4×C31—21no—
150C22×C4×C31—41no—
151C22×C4×C31—41no—
152C22×C4×C31—42no—
153C22×C4×C31—42no—
154C22×C4×C31—82no—
155C22×C4×C31C2×D412yes—
156C22×C4×C31C2×D411yescharacteristic
157C22×C4×C31C2×D411yescharacteristic
158C22×C4×C31C2411yescharacteristic
159SD16×C31—44no—
160D8×C31—44no—
161C2×C8×C31—22no—
162C2×C8×C31—22no—
163C4⋊C4×C31—24no—
164C4⋊C4×C31—21no—
165C4⋊C4×C31—21no—
166C4⋊C4×C31—21no—
167C4⋊C4×C31—21no—
168C4⋊C4×C31—21no—
169C4⋊C4×C31—21no—
170C4⋊C4×C31—22no—
171C4⋊C4×C31—22no—
172C4⋊C4×C31—21no—
173C4⋊C4×C31—21no—
174C4⋊C4×C31—42no—
175C4⋊C4×C31C2×D412yes—
176C4⋊C4×C31C2×D412yes—
177C22⋊C4×C31—24no—
178C22⋊C4×C31—24no—
179C22⋊C4×C31—24no—
180C22⋊C4×C31—22no—
181C22⋊C4×C31—22no—
182C22⋊C4×C31—22no—
183C22⋊C4×C31—22no—
184C22⋊C4×C31—22no—
185C22⋊C4×C31—44no—
186C22⋊C4×C31—41no—
187C22⋊C4×C31—41no—
188C22⋊C4×C31—82no—
189C42×C31—24no—
190C42×C31—22no—
191C42×C31C2×D411yescharacteristic
192C42×C31C2×D411yescharacteristic
193C42×C31C2×D411yescharacteristic
194C42×C31C2×D411yescharacteristic
195#18222C3111yesmaximal, characteristic, direct product factor, Sylow
196C23×C31—22no—
197C23×C31—22no—
198C23×C31—22no—
199C23×C31—22no—
200C23×C31—22no—
201C23×C31—42no—
202C23×C31—42no—
203C23×C31—41no—
204C23×C31—41no—
205C23×C31—41no—
206C23×C31—41no—
207C23×C31—41no—
208C23×C31—41no—
209C23×C31C22×D411yescharacteristic
210Q8×C31—42no—
211Q8×C31—42no—
212Q8×C31—42no—
213D4×C31—22no—
214D4×C31—22no—
215D4×C31—42no—
216D4×C31—41no—
217D4×C31—41no—
218D4×C31—42no—
219D4×C31—42no—
220D4×C31—42no—
221D4×C31—42no—
222D4×C31—42no—
223D4×C31—42no—
224D4×C31—84no—
225D4×C31—84no—
226D4×C31—88no—
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—44no—
234C2×C4×C31—44no—
235C2×C4×C31—42no—
236C2×C4×C31—42no—
237C2×C4×C31—42no—
238C2×C4×C31—41no—
239C2×C4×C31—41no—
240C2×C4×C31—41no—
241C2×C4×C31—42no—
242C2×C4×C31—82no—
243C2×C4×C31—84no—
244C2×C4×C31—82no—
245C2×C4×C31—82no—
246C2×C4×C31#12011yescharacteristic
247C2×C4×C31#12011yescharacteristic
248C2×C4×C31#12011yescharacteristic
249C2×C4×C31#12011yescharacteristic
250C2×C4×C31#13512yes—
251C2×C4×C31AGL(1,ℤ/8)12yes—
252C2×C4×C31C22×D411yescharacteristic
253C2×C4×C31C22×D411yescharacteristic
254C8×C31—42no—
255C8×C31—42no—
256#3506C2×C3111yescharacteristic, complemented
257#3101C2×C3111yescharacteristic, complemented
258#3100C2×C3111yescharacteristic, complemented
259#2535C2×C3111yescharacteristic, complemented
260#1612C2×C3114yescomplemented
261#1600C2×C3114yescomplemented
262#1518C2×C3111yescharacteristic, complemented
263#1517C2×C3111yescharacteristic, complemented
264#1460C2×C3111yescharacteristic, complemented
265C22×C31—21no—
266C22×C31—21no—
267C22×C31—42no—
268C22×C31—44no—
269C22×C31—42no—
270C22×C31—42no—
271C22×C31—42no—
272C22×C31—42no—
273C22×C31—42no—
274C22×C31—42no—
275C22×C31—42no—
276C22×C31—42no—
277C22×C31—82no—
278C22×C31—81no—
279C22×C31—81no—
280C22×C31—81no—
281C22×C31—81no—
282C22×C31C2×#12011yescharacteristic
283C22×C31C2×#13511yescharacteristic
284C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
285C4×C31—21no—
286C4×C31—21no—
287C4×C31—21no—
288C4×C31—21no—
289C4×C31—42no—
290C4×C31—42no—
291C4×C31—42no—
292C4×C31—42no—
293C4×C31—82no—
294C4×C31—81no—
295C4×C31#44912yes—
296C4×C31#44912yes—
297#559—22no—
298#556—21no—
299#555—21no—
300#553—22no—
301#550—21no—
302#547—21no—
303#538C22×C3114yes—
304#535—21no—
305#533C22×C3111yescharacteristic
306#532C22×C3111yescharacteristic
307#525—22no—
308C2×#121—21no—
309#520—21no—
310C2×C4×D4C22×C3111yescharacteristic
311C2×C4×D4C22×C3111yescharacteristic
312#483C22×C3112yes—
313#480C22×C3112yes—
314#476—21no—
315#474—21no—
316#461—21no—
317#459—21no—
318#450C22×C3112yes—
319#447C22×C3112yes—
320C4×SD16—22no—
321C4×D8—22no—
322#431C22×C3111yescharacteristic
323#431C22×C3111yescharacteristic
324#397C22×C3112yes—
325#392C22×C3112yes—
326#386—22no—
327#382C22×C3111yescharacteristic
328#380—21no—
329#328C22×C3114yes—
330#327C22×C3114yes—
331#323C22×C3114yes—
332C2×C31—21no—
333C2×C31—21no—
334C2×C31—82no—
335C2×C31—82no—
336C2×C31—82no—
337C2×C31C2×#44911yescharacteristic
338C2×C31#301911yescharacteristic
339C2×C31#302511yescharacteristic
340C2×#41—22no—
341C2×#41—21no—
342C22×D4—22no—
343C22×D4D4×C3111yescharacteristic
344C22×D4C23×C3111yescharacteristic
345C23×C4—21no—
346C23×C4D4×C3112yes—
347C2×SD16—22no—
348C2×D8—22no—
349C4⋊Q8D4×C3111yescharacteristic, complemented
350C4⋊Q8D4×C3111yescharacteristic, complemented
351C4⋊Q8D4×C3111yescharacteristic, complemented
352#127—22no—
353#127D4×C3111yescharacteristic, complemented
354#126—24no—
355#125—21no—
356#124—22no—
357#124—24no—
358#124—21no—
359#123—24no—
360#123—24no—
361C22⋊Q8—22no—
362C22⋊Q8—21no—
363C22⋊Q8C23×C3112yes—
364#121—22no—
365#121—22no—
366#121—22no—
367#121—24no—
368#121—21no—
369#121—44no—
370#121C23×C3112yes—
371#120—24no—
372C4×Q8—22no—
373C4×Q8—21no—
374C4×Q8D4×C3112yescomplemented
375C4×D4—22no—
376C4×D4—24no—
377C4×D4—24no—
378C4×D4—21no—
379C4×D4—21no—
380C4×D4—21no—
381C4×D4—21no—
382C4×D4—21no—
383C4×D4—21no—
384C4×D4—21no—
385C4×D4—21no—
386C4×D4—21no—
387C4×D4D4×C3112yescomplemented
388#117—22no—
389#117—22no—
390#117—21no—
391C2×C4⋊C4—22no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4D4×C3111yescharacteristic
394C2×C4⋊C4C23×C3111yescharacteristic
395C2×C22⋊C4—22no—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—22no—
398C2×C22⋊C4C23×C3112yes—
399C2×C42C23×C3111yescharacteristic
400#107—22no—
401#106—22no—
402C4⋊C8—21no—
403C4⋊C8—21no—
404#103—22no—
405#102—22no—
406#102—22no—
407#102—22no—
408C22⋊C8C23×C3112yes—
409C22⋊C8C23×C3112yes—
410C4×C8—21no—
411C4×C8—21no—
412#95—22no—
413#95—21no—
414#95C23×C3112yes—
415C31#1822211yescharacteristic, direct product factor, Sylow
416C24—21no—
417C24C2×D4×C3112yes—
418#41—42no—
419#41—42no—
420#41—84no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8C2×D4×C3112yes—
424C2×D4—22no—
425C2×D4—24no—
426C2×D4—24no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—42no—
436C2×D4—44no—
437C2×D4—44no—
438C2×D4—82no—
439C2×D4C2×D4×C3112yes—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—82no—
451C22×C4C2×D4×C3112yes—
452C22×C4C2×D4×C3111yescharacteristic
453C22×C4C2×D4×C3111yescharacteristic
454C22×C4C24×C3111yescharacteristic
455SD16—44no—
456D8—44no—
457C2×C8—22no—
458C2×C8—22no—
459C4⋊C4—24no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—42no—
471C4⋊C4C2×D4×C3112yes—
472C4⋊C4C2×D4×C3112yes—
473C22⋊C4—24no—
474C22⋊C4—24no—
475C22⋊C4—24no—
476C22⋊C4—22no—
477C22⋊C4—22no—
478C22⋊C4—22no—
479C22⋊C4—22no—
480C22⋊C4—22no—
481C22⋊C4—44no—
482C22⋊C4—41no—
483C22⋊C4—41no—
484C22⋊C4—82no—
485C42—24no—
486C42—22no—
487C42C2×D4×C3111yescharacteristic
488C42C2×D4×C3111yescharacteristic
489C42C2×D4×C3111yescharacteristic
490C42C2×D4×C3111yescharacteristic
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—42no—
497C23—42no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23C22×D4×C3111yescharacteristic
505Q8—42no—
506Q8—42no—
507Q8—42no—
508D4—22no—
509D4—22no—
510D4—42no—
511D4—41no—
512D4—41no—
513D4—42no—
514D4—42no—
515D4—42no—
516D4—42no—
517D4—42no—
518D4—42no—
519D4—84no—
520D4—84no—
521D4—88no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—82no—
538C2×C4—84no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4C31×#12011yescharacteristic
542C2×C4C31×#12011yescharacteristic
543C2×C4C31×#12011yescharacteristic
544C2×C4C31×#12011yescharacteristic
545C2×C4C31×#13512yes—
546C2×C4C31×AGL(1,ℤ/8)12yes—
547C2×C4C22×D4×C3111yescharacteristic
548C2×C4C22×D4×C3111yescharacteristic
549C8—42no—
550C8—42no—
551C22—21no—
552C22—21no—
553C22—42no—
554C22—44no—
555C22—42no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22—42no—
562C22—42no—
563C22—82no—
564C22—81no—
565C22—81no—
566C22—81no—
567C22—81no—
568C22C2×C31×#12011yescharacteristic
569C22C2×C31×#13511yescharacteristic
570C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
571C4—21no—
572C4—21no—
573C4—21no—
574C4—21no—
575C4—42no—
576C4—42no—
577C4—42no—
578C4—42no—
579C4—82no—
580C4—81no—
581C4C31×#44912yes—
582C4C31×#44912yes—
583C2—21no—
584C2—21no—
585C2—82no—
586C2—82no—
587C2—82no—
588C2C2×C31×#44911yescharacteristic
589C2C31×#301911yescharacteristic
590C2C31×#302511yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
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
1. C31×#3506C2semidirect
2. C31×#3101C2semidirect
3. C31×#3100C2semidirect
4. C31×#2535C2semidirect
5. C31×#1612C2semidirect
6. C31×#1600C2semidirect
7. C31×#1518C2semidirect
8. C31×#1517C2semidirect
9. C31×#1460C2semidirect
54. C31×C4⋊Q8D4semidirect
55. C31×C4⋊Q8D4semidirect
56. C31×C4⋊Q8D4semidirect
58. C31×#127D4semidirect
79. C4×Q8×C31D4semidirect
92. C4×D4×C31D4semidirect
195. #18222C31direct (N × H)
256. #3506C2×C31semidirect
257. #3101C2×C31semidirect
258. #3100C2×C31semidirect
259. #2535C2×C31semidirect
260. #1612C2×C31semidirect
261. #1600C2×C31semidirect
262. #1518C2×C31semidirect
263. #1517C2×C31semidirect
264. #1460C2×C31semidirect
349. C4⋊Q8D4×C31semidirect
350. C4⋊Q8D4×C31semidirect
351. C4⋊Q8D4×C31semidirect
353. #127D4×C31semidirect
374. C4×Q8D4×C31semidirect
387. C4×D4D4×C31semidirect
415. C31#18222direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35062
C31×#31012
C31×#31002
C31×#25352
C31×#16122
C31×#16002
C31×#15182
C31×#15172
C31×#14602
#1822231