Finite Group Databasebeta

← C31×#18662 #21475421

Subgroups — C31×#18662 #21475421

Order 7936.

1260 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3496C211yesmaximal, characteristic, complemented
2C31×#3245C211yesmaximal, characteristic, complemented
3C31×#3169C211yesmaximal, characteristic, complemented
4C31×#3036C211yesmaximal, characteristic, complemented
5C31×#1647C212yesmaximal, complemented
6C31×#1639C212yesmaximal, complemented
7C31×#1625C212yesmaximal, complemented
8C31×#1619C212yesmaximal, complemented
9C31×#1547C211yesmaximal, characteristic, complemented
10C31×#1492C211yesmaximal, characteristic, complemented
11C2×C31×D4⋊C8C211yesmaximal, characteristic, complemented
12C2×C31×H(5,𝔽2)—21no—
13C2×C31×AGL(1,ℤ/8)—21no—
14C31×#547—21no—
15C31×#547C2212yes—
16C31×#546—22no—
17C31×#546—22no—
18D42×C31—21no—
19D42×C31C2212yes—
20C31×#537—21no—
21C31×#534—22no—
22C31×#532C2211yescharacteristic
23C31×#532C2211yescharacteristic
24C31×#532C2211yescharacteristic, complemented
25C2×C31×#121—22no—
26C2×C4×D4×C31C2211yescharacteristic
27C31×#500C2211yescharacteristic, complemented
28C31×#493C2211yescharacteristic, complemented
29C31×#486C2211yescharacteristic, complemented
30C31×#486C2211yescharacteristic, complemented
31C31×#474—21no—
32C31×#474C2212yescomplemented
33C31×#466—22no—
34C31×#466—22no—
35C31×#463—22no—
36C31×#461C2212yescomplemented
37C31×#459—21no—
38C31×#459C2212yescomplemented
39C31×#459C2212yescomplemented
40C4×D8×C31—22no—
41C31×#423C2211yescharacteristic, complemented
42C2×C31×C4⋊C8C2211yescharacteristic, complemented
43C31×#417—21no—
44C2×C4×C8×C31C2211yescharacteristic, complemented
45C31×#332C2212yescomplemented
46C31×#331C2212yescomplemented
47C31×D4⋊C8C2212yescomplemented
48C31×D4⋊C8C2212yescomplemented
49C31×D4⋊C8C2212yescomplemented
50C31×D4⋊C8C2212yescomplemented
51C31×H(5,𝔽2)—44no—
52C31×H(5,𝔽2)—44no—
53C2×#41×C31—22no—
54C2×#41×C31—22no—
55C2×#41×C31—21no—
56C2×#41×C31—21no—
57C2×#41×C31—21no—
58C22×D4×C31—24no—
59C22×D4×C31—22no—
60C22×D4×C31—22no—
61C22×D4×C31—22no—
62C22×D4×C31D411yescharacteristic
63C23×C4×C31—21no—
64C31×AGL(1,ℤ/8)—44no—
65C2×SD16×C31—22no—
66C2×D8×C31—22no—
67C2×D8×C31—22no—
68C2×D8×C31—22no—
69C2×M16×C31—21no—
70C22×C8×C31—21no—
71C22×C8×C31—21no—
72C31×C4⋊Q8D411yescharacteristic
73C31×C4⋊Q8C2311yescharacteristic
74C31×C4⋊Q8C2311yescharacteristic
75C31×#127D411yescharacteristic
76C31×#127C2311yescharacteristic
77C31×#127C2311yescharacteristic
78C31×#124—21no—
79C31×#124—21no—
80C31×#124—21no—
81C31×#123—24no—
82C31×#123—24no—
83C31×C22⋊Q8—22no—
84C31×C22⋊Q8—22no—
85C31×#121—24no—
86C31×#121—24no—
87C31×#121—22no—
88C31×#121—22no—
89C31×#121—21no—
90C31×#121—21no—
91C31×#121—22no—
92C31×#121—22no—
93C31×#121—21no—
94C31×#121—21no—
95C31×#121D412yes—
96C31×#121D412yes—
97C31×#120—24no—
98C31×#120—24no—
99C4×D4×C31—21no—
100C4×D4×C31—21no—
101C4×D4×C31—21no—
102C4×D4×C31—21no—
103C4×D4×C31—21no—
104C4×D4×C31—21no—
105C4×D4×C31D412yes—
106C4×D4×C31D412yes—
107C4×D4×C31C2312yes—
108C4×D4×C31C2312yes—
109C2×C4⋊C4×C31—22no—
110C2×C4⋊C4×C31D411yescharacteristic
111C2×C22⋊C4×C31—24no—
112C2×C22⋊C4×C31—21no—
113C2×C42×C31C2311yescharacteristic
114C31×#107—22no—
115C31×#107—22no—
116C31×C4⋊C8—21no—
117C31×C4⋊C8—21no—
118C31×C4⋊C8C2312yes—
119C31×C4⋊C8C2312yes—
120C31×#103—22no—
121C31×#102—22no—
122C31×#102—22no—
123C31×#102—22no—
124C31×#102—22no—
125C31×#102—22no—
126C4×C8×C31—21no—
127C4×C8×C31C2311yescharacteristic
128C4×C8×C31C2311yescharacteristic
129C24×C31—21no—
130C24×C31—24no—
131#41×C31—44no—
132#41×C31—48no—
133#41×C31—44no—
134#41×C31—42no—
135#41×C31—42no—
136#41×C31—42no—
137#41×C31—42no—
138#41×C31—42no—
139#41×C31—42no—
140C2×Q8×C31—21no—
141C2×Q8×C31—21no—
142C2×Q8×C31—21no—
143C2×D4×C31—24no—
144C2×D4×C31—24no—
145C2×D4×C31—22no—
146C2×D4×C31—22no—
147C2×D4×C31—24no—
148C2×D4×C31—21no—
149C2×D4×C31—21no—
150C2×D4×C31—21no—
151C2×D4×C31—22no—
152C2×D4×C31—21no—
153C2×D4×C31—21no—
154C2×D4×C31—21no—
155C2×D4×C31—21no—
156C2×D4×C31—21no—
157C2×D4×C31—44no—
158C2×D4×C31—48no—
159C2×D4×C31—416no—
160C2×D4×C31—44no—
161C2×D4×C31—41no—
162C2×D4×C31—82no—
163C2×D4×C31—82no—
164C2×D4×C31D814yescomplemented
165C2×D4×C31C2×D412yes—
166C2×D4×C31C2×D412yes—
167C2×D4×C31C2×D412yes—
168C22×C4×C31—24no—
169C22×C4×C31—21no—
170C22×C4×C31—22no—
171C22×C4×C31—22no—
172C22×C4×C31—22no—
173C22×C4×C31—22no—
174C22×C4×C31—41no—
175C22×C4×C31—81no—
176C22×C4×C31—81no—
177C22×C4×C31C2×D411yescharacteristic
178C22×C4×C31C2×D411yescharacteristic
179C22×C4×C31C2×D411yescharacteristic
180SD16×C31—44no—
181D8×C31—44no—
182D8×C31—44no—
183D8×C31—82no—
184M16×C31—42no—
185C2×C8×C31—21no—
186C2×C8×C31—21no—
187C2×C8×C31—22no—
188C2×C8×C31—22no—
189C2×C8×C31—42no—
190C2×C8×C31—81no—
191C4⋊C4×C31—24no—
192C4⋊C4×C31—21no—
193C4⋊C4×C31—21no—
194C4⋊C4×C31—21no—
195C4⋊C4×C31—21no—
196C4⋊C4×C31—21no—
197C4⋊C4×C31C2×D412yes—
198C4⋊C4×C31C2×D412yes—
199C4⋊C4×C31C2×D412yes—
200C22⋊C4×C31—24no—
201C22⋊C4×C31—24no—
202C22⋊C4×C31—22no—
203C22⋊C4×C31—22no—
204C22⋊C4×C31—22no—
205C22⋊C4×C31—22no—
206C22⋊C4×C31—22no—
207C22⋊C4×C31—22no—
208C22⋊C4×C31—41no—
209C22⋊C4×C31—41no—
210C42×C31C2×D411yescharacteristic
211C42×C31C2×D411yescharacteristic
212C42×C31C2×D411yescharacteristic
213C42×C31C2411yescharacteristic
214#18662C3111yesmaximal, characteristic, direct product factor, Sylow
215C23×C31—22no—
216C23×C31—24no—
217C23×C31—22no—
218C23×C31—22no—
219C23×C31—48no—
220C23×C31—41no—
221C23×C31—81no—
222C23×C31—81no—
223C23×C31—84no—
224C23×C31—84no—
225C23×C31#12011yescharacteristic
226Q8×C31—41no—
227Q8×C31—41no—
228Q8×C31—42no—
229Q8×C31—41no—
230Q8×C31—41no—
231D4×C31—22no—
232D4×C31—22no—
233D4×C31—22no—
234D4×C31—24no—
235D4×C31—42no—
236D4×C31—42no—
237D4×C31—48no—
238D4×C31—48no—
239D4×C31—44no—
240D4×C31—44no—
241D4×C31—41no—
242D4×C31—41no—
243D4×C31—41no—
244D4×C31—41no—
245D4×C31—41no—
246D4×C31—41no—
247D4×C31—41no—
248D4×C31—41no—
249D4×C31—42no—
250D4×C31—81no—
251D4×C31—84no—
252D4×C31—82no—
253D4×C31C2×D814yes—
254C2×C4×C31—21no—
255C2×C4×C31—21no—
256C2×C4×C31—22no—
257C2×C4×C31—22no—
258C2×C4×C31—22no—
259C2×C4×C31—41no—
260C2×C4×C31—48no—
261C2×C4×C31—42no—
262C2×C4×C31—42no—
263C2×C4×C31—42no—
264C2×C4×C31—42no—
265C2×C4×C31—44no—
266C2×C4×C31—41no—
267C2×C4×C31—82no—
268C2×C4×C31—81no—
269C2×C4×C31—82no—
270C2×C4×C31#12011yescharacteristic
271C2×C4×C31#12011yescharacteristic
272C2×C4×C31#12011yescharacteristic
273C2×C4×C31C2×D812yes—
274C2×C4×C31AGL(1,ℤ/8)12yes—
275C2×C4×C31C22×D411yescharacteristic
276C2×C4×C31C22×D411yescharacteristic
277C2×C4×C31C22×D411yescharacteristic
278C8×C31—22no—
279C8×C31—41no—
280C8×C31—42no—
281C8×C31—42no—
282#3496C2×C3111yescharacteristic, complemented
283#3245C2×C3111yescharacteristic, complemented
284#3169C2×C3111yescharacteristic, complemented
285#3036C2×C3111yescharacteristic, complemented
286#1647C2×C3112yescomplemented
287#1639C2×C3112yescomplemented
288#1625C2×C3112yescomplemented
289#1619C2×C3112yescomplemented
290#1547C2×C3111yescharacteristic, complemented
291#1492C2×C3111yescharacteristic, complemented
292C2×D4⋊C8C2×C3111yescharacteristic, complemented
293C22×C31—21no—
294C22×C31—21no—
295C22×C31—24no—
296C22×C31—42no—
297C22×C31—42no—
298C22×C31—42no—
299C22×C31—42no—
300C22×C31—42no—
301C22×C31—42no—
302C22×C31—42no—
303C22×C31—42no—
304C22×C31—84no—
305C22×C31—81no—
306C22×C31—81no—
307C22×C31—81no—
308C22×C31—168no—
309C22×C31#44712yes—
310C22×C31C2×#12011yescharacteristic
311C4×C31—21no—
312C4×C31—21no—
313C4×C31—21no—
314C4×C31—42no—
315C4×C31—42no—
316C4×C31—42no—
317C4×C31—81no—
318C4×C31#44712yes—
319C4×C31C22×D811yescharacteristic
320C4×C31C2×AGL(1,ℤ/8)11yescharacteristic
321C4×C31#57611yescharacteristic
322C4×C31#57711yescharacteristic
323C2×H(5,𝔽2)—21no—
324C2×AGL(1,ℤ/8)—21no—
325#547—21no—
326#547C22×C3112yes—
327#546—22no—
328#546—22no—
329D42—21no—
330D42C22×C3112yes—
331#537—21no—
332#534—22no—
333#532C22×C3111yescharacteristic
334#532C22×C3111yescharacteristic
335#532C22×C3111yescharacteristic, complemented
336C2×#121—22no—
337C2×C4×D4C22×C3111yescharacteristic
338#500C22×C3111yescharacteristic, complemented
339#493C22×C3111yescharacteristic, complemented
340#486C22×C3111yescharacteristic, complemented
341#486C22×C3111yescharacteristic, complemented
342#474—21no—
343#474C22×C3112yescomplemented
344#466—22no—
345#466—22no—
346#463—22no—
347#461C22×C3112yescomplemented
348#459—21no—
349#459C22×C3112yescomplemented
350#459C22×C3112yescomplemented
351C4×D8—22no—
352#423C22×C3111yescharacteristic, complemented
353C2×C4⋊C8C22×C3111yescharacteristic, complemented
354#417—21no—
355C2×C4×C8C22×C3111yescharacteristic, complemented
356#332C22×C3112yescomplemented
357#331C22×C3112yescomplemented
358D4⋊C8C22×C3112yescomplemented
359D4⋊C8C22×C3112yescomplemented
360D4⋊C8C22×C3112yescomplemented
361D4⋊C8C22×C3112yescomplemented
362C2×C31—22no—
363C2×C31—44no—
364C2×C31—82no—
365C2×C31—82no—
366C2×C31—161no—
367C2×C31C2×#44711yescharacteristic
368C2×C31#301211yescharacteristic
369C2×C31#301811yescharacteristic
370H(5,𝔽2)—44no—
371H(5,𝔽2)—44no—
372C2×#41—22no—
373C2×#41—22no—
374C2×#41—21no—
375C2×#41—21no—
376C2×#41—21no—
377C22×D4—24no—
378C22×D4—22no—
379C22×D4—22no—
380C22×D4—22no—
381C22×D4D4×C3111yescharacteristic
382C23×C4—21no—
383AGL(1,ℤ/8)—44no—
384C2×SD16—22no—
385C2×D8—22no—
386C2×D8—22no—
387C2×D8—22no—
388C2×M16—21no—
389C22×C8—21no—
390C22×C8—21no—
391C4⋊Q8D4×C3111yescharacteristic
392C4⋊Q8C23×C3111yescharacteristic
393C4⋊Q8C23×C3111yescharacteristic
394#127D4×C3111yescharacteristic
395#127C23×C3111yescharacteristic
396#127C23×C3111yescharacteristic
397#124—21no—
398#124—21no—
399#124—21no—
400#123—24no—
401#123—24no—
402C22⋊Q8—22no—
403C22⋊Q8—22no—
404#121—24no—
405#121—24no—
406#121—22no—
407#121—22no—
408#121—21no—
409#121—21no—
410#121—22no—
411#121—22no—
412#121—21no—
413#121—21no—
414#121D4×C3112yes—
415#121D4×C3112yes—
416#120—24no—
417#120—24no—
418C4×D4—21no—
419C4×D4—21no—
420C4×D4—21no—
421C4×D4—21no—
422C4×D4—21no—
423C4×D4—21no—
424C4×D4D4×C3112yes—
425C4×D4D4×C3112yes—
426C4×D4C23×C3112yes—
427C4×D4C23×C3112yes—
428C2×C4⋊C4—22no—
429C2×C4⋊C4D4×C3111yescharacteristic
430C2×C22⋊C4—24no—
431C2×C22⋊C4—21no—
432C2×C42C23×C3111yescharacteristic
433#107—22no—
434#107—22no—
435C4⋊C8—21no—
436C4⋊C8—21no—
437C4⋊C8C23×C3112yes—
438C4⋊C8C23×C3112yes—
439#103—22no—
440#102—22no—
441#102—22no—
442#102—22no—
443#102—22no—
444#102—22no—
445C4×C8—21no—
446C4×C8C23×C3111yescharacteristic
447C4×C8C23×C3111yescharacteristic
448C31#1866211yescharacteristic, direct product factor, Sylow
449C24—21no—
450C24—24no—
451#41—48no—
452#41—44no—
453#41—44no—
454#41—42no—
455#41—42no—
456#41—42no—
457#41—42no—
458#41—42no—
459#41—42no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×D4—24no—
464C2×D4—24no—
465C2×D4—22no—
466C2×D4—22no—
467C2×D4—24no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—22no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—48no—
478C2×D4—44no—
479C2×D4—416no—
480C2×D4—44no—
481C2×D4—41no—
482C2×D4—82no—
483C2×D4—82no—
484C2×D4D8×C3114yescomplemented
485C2×D4C2×D4×C3112yes—
486C2×D4C2×D4×C3112yes—
487C2×D4C2×D4×C3112yes—
488C22×C4—24no—
489C22×C4—21no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—41no—
495C22×C4—81no—
496C22×C4—81no—
497C22×C4C2×D4×C3111yescharacteristic
498C22×C4C2×D4×C3111yescharacteristic
499C22×C4C2×D4×C3111yescharacteristic
500SD16—44no—
501D8—44no—
502D8—44no—
503D8—82no—
504M16—42no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—42no—
510C2×C8—81no—
511C4⋊C4—24no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4C2×D4×C3112yes—
518C4⋊C4C2×D4×C3112yes—
519C4⋊C4C2×D4×C3112yes—
520C22⋊C4—24no—
521C22⋊C4—24no—
522C22⋊C4—22no—
523C22⋊C4—22no—
524C22⋊C4—22no—
525C22⋊C4—22no—
526C22⋊C4—22no—
527C22⋊C4—22no—
528C22⋊C4—41no—
529C22⋊C4—41no—
530C42C2×D4×C3111yescharacteristic
531C42C2×D4×C3111yescharacteristic
532C42C2×D4×C3111yescharacteristic
533C42C24×C3111yescharacteristic
534C23—22no—
535C23—24no—
536C23—22no—
537C23—22no—
538C23—48no—
539C23—41no—
540C23—81no—
541C23—81no—
542C23—84no—
543C23—84no—
544C23C31×#12011yescharacteristic
545Q8—42no—
546Q8—41no—
547Q8—41no—
548Q8—41no—
549Q8—41no—
550D4—24no—
551D4—22no—
552D4—22no—
553D4—22no—
554D4—42no—
555D4—42no—
556D4—48no—
557D4—48no—
558D4—44no—
559D4—44no—
560D4—42no—
561D4—41no—
562D4—41no—
563D4—41no—
564D4—41no—
565D4—41no—
566D4—41no—
567D4—41no—
568D4—41no—
569D4—81no—
570D4—84no—
571D4—82no—
572D4C2×D8×C3114yes—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—41no—
579C2×C4—48no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—82no—
587C2×C4—81no—
588C2×C4—82no—
589C2×C4C31×#12011yescharacteristic
590C2×C4C31×#12011yescharacteristic
591C2×C4C31×#12011yescharacteristic
592C2×C4C2×D8×C3112yes—
593C2×C4C31×AGL(1,ℤ/8)12yes—
594C2×C4C22×D4×C3111yescharacteristic
595C2×C4C22×D4×C3111yescharacteristic
596C2×C4C22×D4×C3111yescharacteristic
597C8—22no—
598C8—41no—
599C8—42no—
600C8—42no—
601C22—21no—
602C22—21no—
603C22—24no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—84no—
613C22—81no—
614C22—81no—
615C22—81no—
616C22—168no—
617C22C31×#44712yes—
618C22C2×C31×#12011yescharacteristic
619C4—21no—
620C4—21no—
621C4—21no—
622C4—42no—
623C4—42no—
624C4—42no—
625C4—81no—
626C4C31×#44712yes—
627C4C22×D8×C3111yescharacteristic
628C4C2×C31×AGL(1,ℤ/8)11yescharacteristic
629C4C31×#57611yescharacteristic
630C4C31×#57711yescharacteristic
631C2—22no—
632C2—44no—
633C2—82no—
634C2—82no—
635C2—161no—
636C2C2×C31×#44711yescharacteristic
637C2C31×#301211yescharacteristic
638C2C31×#301811yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC42C24×C31
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×#3496C2semidirect
2. C31×#3245C2semidirect
3. C31×#3169C2semidirect
4. C31×#3036C2semidirect
5. C31×#1647C2semidirect
6. C31×#1639C2semidirect
7. C31×#1625C2semidirect
8. C31×#1619C2semidirect
9. C31×#1547C2semidirect
10. C31×#1492C2semidirect
11. C2×C31×D4⋊C8C2semidirect
24. C31×#532C22semidirect
27. C31×#500C22semidirect
28. C31×#493C22semidirect
29. C31×#486C22semidirect
30. C31×#486C22semidirect
32. C31×#474C22semidirect
36. C31×#461C22semidirect
38. C31×#459C22semidirect
39. C31×#459C22semidirect
41. C31×#423C22semidirect
42. C2×C31×C4⋊C8C22semidirect
44. C2×C4×C8×C31C22semidirect
45. C31×#332C22semidirect
46. C31×#331C22semidirect
47. C31×D4⋊C8C22semidirect
48. C31×D4⋊C8C22semidirect
49. C31×D4⋊C8C22semidirect
50. C31×D4⋊C8C22semidirect
164. C2×D4×C31D8semidirect
214. #18662C31direct (N × H)
282. #3496C2×C31semidirect
283. #3245C2×C31semidirect
284. #3169C2×C31semidirect
285. #3036C2×C31semidirect
286. #1647C2×C31semidirect
287. #1639C2×C31semidirect
288. #1625C2×C31semidirect
289. #1619C2×C31semidirect
290. #1547C2×C31semidirect
291. #1492C2×C31semidirect
292. C2×D4⋊C8C2×C31semidirect
335. #532C22×C31semidirect
338. #500C22×C31semidirect
339. #493C22×C31semidirect
340. #486C22×C31semidirect
341. #486C22×C31semidirect
343. #474C22×C31semidirect
347. #461C22×C31semidirect
349. #459C22×C31semidirect
350. #459C22×C31semidirect
352. #423C22×C31semidirect
353. C2×C4⋊C8C22×C31semidirect
355. C2×C4×C8C22×C31semidirect
356. #332C22×C31semidirect
357. #331C22×C31semidirect
358. D4⋊C8C22×C31semidirect
359. D4⋊C8C22×C31semidirect
360. D4⋊C8C22×C31semidirect
361. D4⋊C8C22×C31semidirect
448. C31#18662direct (N × H)
484. C2×D4D8×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#34962
C31×#32452
C31×#31692
C31×#30362
C31×#16472
C31×#16392
C31×#16252
C31×#16192
C31×#15472
C31×#14922
C2×C31×D4⋊C82
#1866231