Finite Group Databasebeta

← C31×#18163 #21474922

Subgroups — C31×#18163 #21474922

Order 7936.

1316 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3490C211yesmaximal, characteristic
2C31×#3004C212yesmaximal, complemented
3C2×C31×#393C211yesmaximal, characteristic, complemented
4C31×#1606C214yesmaximal, complemented
5C31×#1602C212yesmaximal, complemented
6C31×#1601C212yesmaximal, complemented
7C31×#1504C212yesmaximal, complemented
8C31×#1462C211yesmaximal, characteristic, complemented
9C23×Q8×C31C2211yescharacteristic, complemented
10C2×C31×#137—22no—
11C31×#551—22no—
12C31×#551—22no—
13C31×#546—22no—
14C31×#546—22no—
15C31×#543C2211yescharacteristic
16C31×#538—21no—
17C31×#538C2212yes—
18C31×#534C2211yescharacteristic
19C31×#525—21no—
20C31×#525C2212yes—
21C2×C31×#120—21no—
22C31×#518—21no—
23C22×C22⋊C4×C31—21no—
24C22×C22⋊C4×C31C2211yescharacteristic
25C31×#451—22no—
26C31×#451C2214yescomplemented
27C31×#448—22no—
28C31×#448C2214yescomplemented
29C31×#419—22no—
30C31×#407C2212yescomplemented
31C31×#393—21no—
32C31×#393—21no—
33C31×#393C2212yes—
34C31×#393C2212yescomplemented
35C2×C31×#95—21no—
36C2×C31×#95C2211yescharacteristic, complemented
37C31×#328C2214yescomplemented
38C31×#327C2214yescomplemented
39C31×#323C2212yes—
40C31×#323C2212yescomplemented
41C25×C31D411yescharacteristic
42C2×#41×C31—21no—
43C2×#41×C31D412yes—
44C22×Q8×C31—21no—
45C22×Q8×C31—22no—
46C22×Q8×C31—22no—
47C22×Q8×C31—44no—
48C22×Q8×C31C2312yes—
49C22×D4×C31—21no—
50C22×D4×C31—21no—
51C22×D4×C31—21no—
52C23×C4×C31—21no—
53C23×C4×C31—21no—
54C23×C4×C31D411yescharacteristic
55C23×C4×C31C2311yescharacteristic
56C31×#137—48no—
57C2×Q16×C31—24no—
58C2×SD16×C31—24no—
59C2×M16×C31—22no—
60C31×#126—22no—
61C31×#126—22no—
62C31×#124—22no—
63C31×#124—22no—
64C31×#123—22no—
65C31×#123—22no—
66C31×#123—22no—
67C31×#123—22no—
68C31×#123—22no—
69C31×C22⋊Q8—21no—
70C31×C22⋊Q8—22no—
71C31×C22⋊Q8—21no—
72C31×C22⋊Q8—22no—
73C31×C22⋊Q8D412yes—
74C31×C22⋊Q8C2312yes—
75C31×#121—21no—
76C31×#121—22no—
77C31×#121—22no—
78C31×#121—21no—
79C31×#121D412yes—
80C31×#121C2312yes—
81C31×#120—22no—
82C31×#120—22no—
83C31×#120—22no—
84C31×#120—22no—
85C31×#120—22no—
86C4×D4×C31—22no—
87C4×D4×C31—22no—
88C4×D4×C31—22no—
89C4×D4×C31—22no—
90C4×D4×C31—22no—
91C4×D4×C31—22no—
92C31×#117—22no—
93C31×#117—21no—
94C31×#117D412yes—
95C2×C22⋊C4×C31—22no—
96C2×C22⋊C4×C31—21no—
97C2×C22⋊C4×C31—21no—
98C2×C22⋊C4×C31—21no—
99C2×C22⋊C4×C31—22no—
100C2×C22⋊C4×C31—22no—
101C2×C22⋊C4×C31—22no—
102C2×C22⋊C4×C31—22no—
103C2×C22⋊C4×C31—44no—
104C2×C22⋊C4×C31D411yescharacteristic
105C2×C22⋊C4×C31D411yescharacteristic
106C2×C22⋊C4×C31C2311yescharacteristic
107C2×C22⋊C4×C31C2311yescharacteristic
108C31×#103—24no—
109C31×#103—24no—
110C31×C22⋊C8—22no—
111C31×C22⋊C8C2314yes—
112C31×#95—22no—
113C31×#95—22no—
114C31×#95—21no—
115C31×#95C2312yes—
116C24×C31—21no—
117C24×C31—22no—
118C24×C31—21no—
119C24×C31—21no—
120C24×C31—41no—
121C24×C31—42no—
122C24×C31—42no—
123C24×C31C2×D411yescharacteristic
124C24×C31C2×D411yescharacteristic
125C24×C31C2×D411yescharacteristic
126#41×C31—44no—
127#41×C31—82no—
128C2×Q8×C31—21no—
129C2×Q8×C31—28no—
130C2×Q8×C31—24no—
131C2×Q8×C31—44no—
132C2×Q8×C31—44no—
133C2×Q8×C31—44no—
134C2×Q8×C31—48no—
135C2×Q8×C31—42no—
136C2×Q8×C31C2×D412yes—
137C2×D4×C31—22no—
138C2×D4×C31—22no—
139C2×D4×C31—22no—
140C2×D4×C31—22no—
141C2×D4×C31—21no—
142C2×D4×C31—22no—
143C2×D4×C31—22no—
144C2×D4×C31—42no—
145C2×D4×C31—42no—
146C2×D4×C31—82no—
147C2×D4×C31—82no—
148C2×D4×C31C2×D412yes—
149C22×C4×C31—21no—
150C22×C4×C31—24no—
151C22×C4×C31—21no—
152C22×C4×C31—22no—
153C22×C4×C31—21no—
154C22×C4×C31—21no—
155C22×C4×C31—22no—
156C22×C4×C31—21no—
157C22×C4×C31—22no—
158C22×C4×C31—22no—
159C22×C4×C31—42no—
160C22×C4×C31—42no—
161C22×C4×C31—41no—
162C22×C4×C31—41no—
163C22×C4×C31—42no—
164C22×C4×C31—42no—
165C22×C4×C31—42no—
166C22×C4×C31—42no—
167C22×C4×C31C2×D412yes—
168C22×C4×C31C2×D411yescharacteristic
169C22×C4×C31C2×D411yescharacteristic
170C22×C4×C31C2×D411yescharacteristic
171C22×C4×C31C2×D412yes—
172C22×C4×C31C2411yescharacteristic
173Q16×C31—48no—
174SD16×C31—48no—
175M16×C31—44no—
176C2×C8×C31—24no—
177C4⋊C4×C31—22no—
178C4⋊C4×C31—21no—
179C4⋊C4×C31—21no—
180C4⋊C4×C31—22no—
181C4⋊C4×C31—22no—
182C4⋊C4×C31—22no—
183C4⋊C4×C31C2×D412yes—
184C4⋊C4×C31C2×D412yes—
185C22⋊C4×C31—22no—
186C22⋊C4×C31—22no—
187C22⋊C4×C31—22no—
188C22⋊C4×C31—22no—
189C22⋊C4×C31—22no—
190C22⋊C4×C31—22no—
191C22⋊C4×C31—22no—
192C22⋊C4×C31—22no—
193C22⋊C4×C31—22no—
194C22⋊C4×C31—22no—
195C22⋊C4×C31—22no—
196C22⋊C4×C31—22no—
197C22⋊C4×C31—22no—
198C22⋊C4×C31—22no—
199C22⋊C4×C31—44no—
200C22⋊C4×C31—44no—
201C42×C31—22no—
202C42×C31—22no—
203#18163C3111yesmaximal, characteristic, direct product factor, Sylow
204C23×C31—21no—
205C23×C31—21no—
206C23×C31—22no—
207C23×C31—22no—
208C23×C31—21no—
209C23×C31—21no—
210C23×C31—21no—
211C23×C31—22no—
212C23×C31—21no—
213C23×C31—21no—
214C23×C31—42no—
215C23×C31—42no—
216C23×C31—41no—
217C23×C31—41no—
218C23×C31—41no—
219C23×C31—41no—
220C23×C31—48no—
221C23×C31—42no—
222C23×C31—42no—
223C23×C31—42no—
224C23×C31—81no—
225C23×C31—81no—
226C23×C31—84no—
227C23×C31—81no—
228C23×C31—81no—
229C23×C31#12011yescharacteristic
230C23×C31#12011yescharacteristic
231C23×C31#13712yescomplemented
232C23×C31C22×D411yescharacteristic
233Q8×C31—28no—
234Q8×C31—42no—
235Q8×C31—44no—
236Q8×C31—48no—
237Q8×C31—81no—
238D4×C31—42no—
239D4×C31—44no—
240D4×C31—42no—
241D4×C31—42no—
242D4×C31—44no—
243D4×C31—44no—
244D4×C31—82no—
245D4×C31—82no—
246D4×C31—81no—
247C2×C4×C31—24no—
248C2×C4×C31—24no—
249C2×C4×C31—22no—
250C2×C4×C31—21no—
251C2×C4×C31—22no—
252C2×C4×C31—22no—
253C2×C4×C31—21no—
254C2×C4×C31—42no—
255C2×C4×C31—41no—
256C2×C4×C31—41no—
257C2×C4×C31—42no—
258C2×C4×C31—44no—
259C2×C4×C31—42no—
260C2×C4×C31—42no—
261C2×C4×C31—42no—
262C2×C4×C31—44no—
263C2×C4×C31—41no—
264C2×C4×C31—41no—
265C2×C4×C31—42no—
266C2×C4×C31—42no—
267C2×C4×C31—42no—
268C2×C4×C31—82no—
269C2×C4×C31—81no—
270C2×C4×C31—81no—
271C2×C4×C31—81no—
272C2×C4×C31—82no—
273C2×C4×C31—84no—
274C2×C4×C31#12012yes—
275C2×C4×C31C22×D412yes—
276C8×C31—44no—
277#3490C2×C3111yescharacteristic
278#3004C2×C3112yescomplemented
279C2×#393C2×C3111yescharacteristic, complemented
280#1606C2×C3114yescomplemented
281#1602C2×C3112yescomplemented
282#1601C2×C3112yescomplemented
283#1504C2×C3112yescomplemented
284#1462C2×C3111yescharacteristic, complemented
285C22×C31—24no—
286C22×C31—24no—
287C22×C31—21no—
288C22×C31—21no—
289C22×C31—21no—
290C22×C31—21no—
291C22×C31—21no—
292C22×C31—21no—
293C22×C31—22no—
294C22×C31—42no—
295C22×C31—42no—
296C22×C31—41no—
297C22×C31—41no—
298C22×C31—42no—
299C22×C31—41no—
300C22×C31—42no—
301C22×C31—41no—
302C22×C31—42no—
303C22×C31—41no—
304C22×C31—81no—
305C22×C31—84no—
306C22×C31—84no—
307C22×C31—81no—
308C22×C31—162no—
309C22×C31#45512yes—
310C22×C31B(4,𝔽2)12yes—
311C22×C31C2×#12011yescharacteristic
312C22×C31C2×#13711yescharacteristic
313C22×C31#57511yescharacteristic
314C4×C31—22no—
315C4×C31—22no—
316C4×C31—42no—
317C4×C31—41no—
318C4×C31—41no—
319C4×C31—42no—
320C4×C31—44no—
321C4×C31—42no—
322C4×C31—82no—
323C23×Q8C22×C3111yescharacteristic, complemented
324C2×#137—22no—
325#551—22no—
326#551—22no—
327#546—22no—
328#546—22no—
329#543C22×C3111yescharacteristic
330#538—21no—
331#538C22×C3112yes—
332#534C22×C3111yescharacteristic
333#525—21no—
334#525C22×C3112yes—
335C2×#120—21no—
336#518—21no—
337C22×C22⋊C4—21no—
338C22×C22⋊C4C22×C3111yescharacteristic
339#451—22no—
340#451C22×C3114yescomplemented
341#448—22no—
342#448C22×C3114yescomplemented
343#419—22no—
344#407C22×C3112yescomplemented
345#393—21no—
346#393—21no—
347#393C22×C3112yes—
348#393C22×C3112yescomplemented
349C2×#95—21no—
350C2×#95C22×C3111yescharacteristic, complemented
351#328C22×C3114yescomplemented
352#327C22×C3114yescomplemented
353#323C22×C3112yes—
354#323C22×C3112yescomplemented
355C2×C31—21no—
356C2×C31—21no—
357C2×C31—22no—
358C2×C31—22no—
359C2×C31—42no—
360C2×C31—81no—
361C2×C31—82no—
362C2×C31#300711yescharacteristic
363C2×C31C2×#45511yescharacteristic
364C2×C31C2×B(4,𝔽2)11yescharacteristic
365C25D4×C3111yescharacteristic
366C2×#41—21no—
367C2×#41D4×C3112yes—
368C22×Q8—21no—
369C22×Q8—22no—
370C22×Q8—22no—
371C22×Q8—44no—
372C22×Q8C23×C3112yes—
373C22×D4—21no—
374C22×D4—21no—
375C22×D4—21no—
376C23×C4—21no—
377C23×C4—21no—
378C23×C4D4×C3111yescharacteristic
379C23×C4C23×C3111yescharacteristic
380#137—48no—
381C2×Q16—24no—
382C2×SD16—24no—
383C2×M16—22no—
384#126—22no—
385#126—22no—
386#124—22no—
387#124—22no—
388#123—22no—
389#123—22no—
390#123—22no—
391#123—22no—
392#123—22no—
393C22⋊Q8—21no—
394C22⋊Q8—22no—
395C22⋊Q8—21no—
396C22⋊Q8—22no—
397C22⋊Q8D4×C3112yes—
398C22⋊Q8C23×C3112yes—
399#121—21no—
400#121—22no—
401#121—22no—
402#121—21no—
403#121D4×C3112yes—
404#121C23×C3112yes—
405#120—22no—
406#120—22no—
407#120—22no—
408#120—22no—
409#120—22no—
410C4×D4—22no—
411C4×D4—22no—
412C4×D4—22no—
413C4×D4—22no—
414C4×D4—22no—
415C4×D4—22no—
416#117—22no—
417#117—21no—
418#117D4×C3112yes—
419C2×C22⋊C4—22no—
420C2×C22⋊C4—21no—
421C2×C22⋊C4—21no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—22no—
424C2×C22⋊C4—22no—
425C2×C22⋊C4—22no—
426C2×C22⋊C4—22no—
427C2×C22⋊C4—44no—
428C2×C22⋊C4D4×C3111yescharacteristic
429C2×C22⋊C4D4×C3111yescharacteristic
430C2×C22⋊C4C23×C3111yescharacteristic
431C2×C22⋊C4C23×C3111yescharacteristic
432#103—24no—
433#103—24no—
434C22⋊C8—22no—
435C22⋊C8C23×C3114yes—
436#95—22no—
437#95—22no—
438#95—21no—
439#95C23×C3112yes—
440C31#1816311yescharacteristic, direct product factor, Sylow
441C24—21no—
442C24—22no—
443C24—21no—
444C24—21no—
445C24—41no—
446C24—42no—
447C24—42no—
448C24C2×D4×C3111yescharacteristic
449C24C2×D4×C3111yescharacteristic
450C24C2×D4×C3111yescharacteristic
451#41—44no—
452#41—82no—
453C2×Q8—21no—
454C2×Q8—28no—
455C2×Q8—24no—
456C2×Q8—44no—
457C2×Q8—44no—
458C2×Q8—44no—
459C2×Q8—48no—
460C2×Q8—42no—
461C2×Q8C2×D4×C3112yes—
462C2×D4—22no—
463C2×D4—22no—
464C2×D4—22no—
465C2×D4—22no—
466C2×D4—21no—
467C2×D4—22no—
468C2×D4—22no—
469C2×D4—42no—
470C2×D4—42no—
471C2×D4—82no—
472C2×D4—82no—
473C2×D4C2×D4×C3112yes—
474C22×C4—21no—
475C22×C4—24no—
476C22×C4—21no—
477C22×C4—22no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—22no—
481C22×C4—21no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—41no—
487C22×C4—41no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—42no—
491C22×C4—42no—
492C22×C4C2×D4×C3112yes—
493C22×C4C2×D4×C3111yescharacteristic
494C22×C4C2×D4×C3111yescharacteristic
495C22×C4C2×D4×C3111yescharacteristic
496C22×C4C2×D4×C3112yes—
497C22×C4C24×C3111yescharacteristic
498Q16—48no—
499SD16—48no—
500M16—44no—
501C2×C8—24no—
502C4⋊C4—22no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4C2×D4×C3112yes—
509C4⋊C4C2×D4×C3112yes—
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—22no—
513C22⋊C4—22no—
514C22⋊C4—22no—
515C22⋊C4—22no—
516C22⋊C4—22no—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—22no—
522C22⋊C4—22no—
523C22⋊C4—22no—
524C22⋊C4—44no—
525C22⋊C4—44no—
526C42—22no—
527C42—22no—
528C23—22no—
529C23—22no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—22no—
536C23—21no—
537C23—21no—
538C23—42no—
539C23—42no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—48no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—81no—
549C23—81no—
550C23—84no—
551C23—81no—
552C23—81no—
553C23C31×#12011yescharacteristic
554C23C31×#12011yescharacteristic
555C23C31×#13712yescomplemented
556C23C22×D4×C3111yescharacteristic
557Q8—28no—
558Q8—42no—
559Q8—44no—
560Q8—48no—
561Q8—81no—
562D4—42no—
563D4—44no—
564D4—44no—
565D4—44no—
566D4—42no—
567D4—42no—
568D4—82no—
569D4—82no—
570D4—81no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—82no—
593C2×C4—81no—
594C2×C4—81no—
595C2×C4—81no—
596C2×C4—82no—
597C2×C4—84no—
598C2×C4C31×#12012yes—
599C2×C4C22×D4×C3112yes—
600C8—44no—
601C22—24no—
602C22—24no—
603C22—21no—
604C22—21no—
605C22—21no—
606C22—21no—
607C22—21no—
608C22—21no—
609C22—22no—
610C22—42no—
611C22—42no—
612C22—41no—
613C22—41no—
614C22—42no—
615C22—41no—
616C22—42no—
617C22—41no—
618C22—42no—
619C22—41no—
620C22—81no—
621C22—84no—
622C22—84no—
623C22—81no—
624C22—162no—
625C22C31×#45512yes—
626C22C31×B(4,𝔽2)12yes—
627C22C2×C31×#12011yescharacteristic
628C22C2×C31×#13711yescharacteristic
629C22C31×#57511yescharacteristic
630C4—22no—
631C4—22no—
632C4—42no—
633C4—41no—
634C4—41no—
635C4—42no—
636C4—44no—
637C4—42no—
638C4—82no—
639C2—21no—
640C2—21no—
641C2—22no—
642C2—22no—
643C2—42no—
644C2—81no—
645C2—82no—
646C2C31×#300711yescharacteristic
647C2C2×C31×#45511yescharacteristic
648C2C2×C31×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#3490C2
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×#3004C2semidirect
3. C2×C31×#393C2semidirect
4. C31×#1606C2semidirect
5. C31×#1602C2semidirect
6. C31×#1601C2semidirect
7. C31×#1504C2semidirect
8. C31×#1462C2semidirect
9. C23×Q8×C31C22semidirect
26. C31×#451C22semidirect
28. C31×#448C22semidirect
30. C31×#407C22semidirect
34. C31×#393C22semidirect
36. C2×C31×#95C22semidirect
37. C31×#328C22semidirect
38. C31×#327C22semidirect
40. C31×#323C22semidirect
203. #18163C31direct (N × H)
231. C23×C31#137semidirect
278. #3004C2×C31semidirect
279. C2×#393C2×C31semidirect
280. #1606C2×C31semidirect
281. #1602C2×C31semidirect
282. #1601C2×C31semidirect
283. #1504C2×C31semidirect
284. #1462C2×C31semidirect
323. C23×Q8C22×C31semidirect
340. #451C22×C31semidirect
342. #448C22×C31semidirect
344. #407C22×C31semidirect
348. #393C22×C31semidirect
350. C2×#95C22×C31semidirect
351. #328C22×C31semidirect
352. #327C22×C31semidirect
354. #323C22×C31semidirect
440. C31#18163direct (N × H)
555. C23C31×#137semidirect

Maximal subgroups

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

subgroupindex
C31×#34902
C31×#30042
C2×C31×#3932
C31×#16062
C31×#16022
C31×#16012
C31×#15042
C31×#14622
#1816331