Finite Group Databasebeta

← C2×C31×#1606 #21474872

Subgroups — C2×C31×#1606 #21474872

Order 7936.

1328 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#538C211yesmaximal, characteristic
2C2×C31×#451C211yesmaximal, characteristic, complemented
3C2×C31×#448C211yesmaximal, characteristic, complemented
4C2×C31×#393C211yesmaximal, characteristic, complemented
5C31×#1606C218yesmaximal, direct product factor
6C2×C31×#328C211yesmaximal, characteristic, complemented
7C2×C31×#327C211yesmaximal, characteristic, complemented
8C2×C31×#323C211yesmaximal, characteristic, complemented
9C23×Q8×C31C2211yescharacteristic
10C22×Q16×C31—21no—
11C22×SD16×C31—21no—
12C31×#538—22no—
13C31×#538—22no—
14C31×#538—22no—
15C31×#538C2214yes—
16C2×C31×#126—21no—
17C2×C31×#124—21no—
18C2×C31×#123—21no—
19C2×C31×C22⋊Q8C2211yescharacteristic
20C2×C31×#121—21no—
21C2×C31×#121C2211yescharacteristic
22C2×C31×#120—21no—
23C2×C4×D4×C31—21no—
24C22×C22⋊C4×C31—21no—
25C22×C22⋊C4×C31C2211yescharacteristic
26C31×#451—22no—
27C31×#451C2214yescomplemented
28C31×#448—22no—
29C31×#448C2214yescomplemented
30C2×C31×#103—21no—
31C2×C31×#103—21no—
32C2×C31×C22⋊C8C2211yescharacteristic
33C2×C31×C22⋊C8C2211yescharacteristic
34C31×#393—22no—
35C31×#393C2214yescomplemented
36C2×C31×#95—21no—
37C2×C31×#95C2211yescharacteristic
38C31×#328C2214yescomplemented
39C31×#327C2214yescomplemented
40C31×#323C2214yescomplemented
41C25×C31D411yescharacteristic
42C22×Q8×C31—21no—
43C22×Q8×C31—21no—
44C22×Q8×C31—21no—
45C22×Q8×C31—21no—
46C22×Q8×C31—21no—
47C22×Q8×C31—44no—
48C22×Q8×C31D411yescharacteristic
49C22×Q8×C31C2312yes—
50C22×D4×C31—21no—
51C22×D4×C31—21no—
52C22×D4×C31—21no—
53C22×D4×C31D411yescharacteristic
54C23×C4×C31—21no—
55C23×C4×C31—21no—
56C23×C4×C31—21no—
57C23×C4×C31D411yescharacteristic
58C23×C4×C31C2311yescharacteristic
59C2×Q16×C31—24no—
60C2×Q16×C31—44no—
61C2×SD16×C31—24no—
62C2×SD16×C31—44no—
63C22×C8×C31—21no—
64C22×C8×C31—21no—
65C31×#126—24no—
66C31×#126—24no—
67C31×#124—24no—
68C31×#124—24no—
69C31×#123—24no—
70C31×#123—24no—
71C31×C22⋊Q8—21no—
72C31×C22⋊Q8—21no—
73C31×C22⋊Q8D412yes—
74C31×C22⋊Q8C2312yes—
75C31×#121—24no—
76C31×#121—24no—
77C31×#121—21no—
78C31×#121—21no—
79C31×#121D412yes—
80C31×#121C2312yes—
81C31×#120—24no—
82C31×#120—24no—
83C4×D4×C31—24no—
84C4×D4×C31—24no—
85C2×C4⋊C4×C31—21no—
86C2×C4⋊C4×C31—21no—
87C2×C4⋊C4×C31D411yescharacteristic
88C2×C4⋊C4×C31D411yescharacteristic
89C2×C22⋊C4×C31—21no—
90C2×C22⋊C4×C31—21no—
91C2×C22⋊C4×C31—21no—
92C2×C22⋊C4×C31—21no—
93C2×C22⋊C4×C31—21no—
94C2×C22⋊C4×C31—21no—
95C2×C22⋊C4×C31—21no—
96C2×C22⋊C4×C31—21no—
97C2×C22⋊C4×C31—21no—
98C2×C22⋊C4×C31—24no—
99C2×C22⋊C4×C31—44no—
100C2×C22⋊C4×C31D412yes—
101C2×C22⋊C4×C31C2312yes—
102C2×C42×C31—21no—
103C31×#103—24no—
104C31×#103—24no—
105C31×C22⋊C8—21no—
106C31×C22⋊C8—21no—
107C31×C22⋊C8C2312yes—
108C31×C22⋊C8C2312yes—
109C31×#95—24no—
110C31×#95—21no—
111C31×#95C2312yes—
112C24×C31—21no—
113C24×C31—21no—
114C24×C31—22no—
115C24×C31—21no—
116C24×C31—22no—
117C24×C31—42no—
118C24×C31—42no—
119C24×C31C2×D411yescharacteristic
120C24×C31C2×D412yes—
121C2×Q8×C31—21no—
122C2×Q8×C31—24no—
123C2×Q8×C31—24no—
124C2×Q8×C31—22no—
125C2×Q8×C31—22no—
126C2×Q8×C31—44no—
127C2×Q8×C31—42no—
128C2×Q8×C31—42no—
129C2×Q8×C31—44no—
130C2×Q8×C31—42no—
131C2×Q8×C31—44no—
132C2×Q8×C31—44no—
133C2×Q8×C31—41no—
134C2×Q8×C31—41no—
135C2×Q8×C31C2×D412yes—
136C2×D4×C31—22no—
137C2×D4×C31—22no—
138C2×D4×C31—24no—
139C2×D4×C31—24no—
140C2×D4×C31—21no—
141C2×D4×C31—42no—
142C2×D4×C31—42no—
143C2×D4×C31—42no—
144C2×D4×C31—82no—
145C2×D4×C31—82no—
146C2×D4×C31C2×D412yes—
147C22×C4×C31—24no—
148C22×C4×C31—21no—
149C22×C4×C31—21no—
150C22×C4×C31—21no—
151C22×C4×C31—21no—
152C22×C4×C31—22no—
153C22×C4×C31—24no—
154C22×C4×C31—21no—
155C22×C4×C31—22no—
156C22×C4×C31—42no—
157C22×C4×C31—42no—
158C22×C4×C31—41no—
159C22×C4×C31—41no—
160C22×C4×C31—42no—
161C22×C4×C31—42no—
162C22×C4×C31—42no—
163C22×C4×C31—42no—
164C22×C4×C31—82no—
165C22×C4×C31C2×D411yescharacteristic
166C22×C4×C31C2×D411yescharacteristic
167C22×C4×C31C2×D412yes—
168C22×C4×C31C2×D411yescharacteristic
169C22×C4×C31C2×D411yescharacteristic
170C22×C4×C31C2×D411yescharacteristic
171C22×C4×C31C2411yescharacteristic
172Q16×C31—48no—
173SD16×C31—48no—
174C2×C8×C31—22no—
175C2×C8×C31—22no—
176C2×C8×C31—42no—
177C2×C8×C31—42no—
178C4⋊C4×C31—24no—
179C4⋊C4×C31—22no—
180C4⋊C4×C31—21no—
181C4⋊C4×C31—21no—
182C4⋊C4×C31—22no—
183C4⋊C4×C31C2×D412yes—
184C4⋊C4×C31C2×D412yes—
185C22⋊C4×C31—24no—
186C22⋊C4×C31—24no—
187C22⋊C4×C31—24no—
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—48no—
197C42×C31—24no—
198C2×#1606C3111yesmaximal, characteristic, direct product factor, Sylow
199C23×C31—22no—
200C23×C31—22no—
201C23×C31—22no—
202C23×C31—22no—
203C23×C31—22no—
204C23×C31—22no—
205C23×C31—42no—
206C23×C31—42no—
207C23×C31—48no—
208C23×C31—42no—
209C23×C31—42no—
210C23×C31—42no—
211C23×C31—42no—
212C23×C31—42no—
213C23×C31—42no—
214C23×C31—42no—
215C23×C31—81no—
216C23×C31—81no—
217C23×C31—84no—
218C23×C31#12011yescharacteristic
219C23×C31#12011yescharacteristic
220C23×C31#13512yes—
221C23×C31#13712yes—
222C23×C31C22×D411yescharacteristic
223Q8×C31—24no—
224Q8×C31—24no—
225Q8×C31—42no—
226Q8×C31—42no—
227Q8×C31—42no—
228Q8×C31—44no—
229Q8×C31—44no—
230Q8×C31—81no—
231D4×C31—42no—
232D4×C31—42no—
233D4×C31—42no—
234D4×C31—44no—
235D4×C31—81no—
236D4×C31—84no—
237D4×C31—84no—
238C2×C4×C31—22no—
239C2×C4×C31—22no—
240C2×C4×C31—24no—
241C2×C4×C31—22no—
242C2×C4×C31—22no—
243C2×C4×C31—22no—
244C2×C4×C31—22no—
245C2×C4×C31—41no—
246C2×C4×C31—41no—
247C2×C4×C31—41no—
248C2×C4×C31—41no—
249C2×C4×C31—42no—
250C2×C4×C31—44no—
251C2×C4×C31—42no—
252C2×C4×C31—42no—
253C2×C4×C31—42no—
254C2×C4×C31—42no—
255C2×C4×C31—44no—
256C2×C4×C31—42no—
257C2×C4×C31—42no—
258C2×C4×C31—42no—
259C2×C4×C31—42no—
260C2×C4×C31—42no—
261C2×C4×C31—81no—
262C2×C4×C31—84no—
263C2×C4×C31—84no—
264C2×C4×C31#12011yescharacteristic
265C2×C4×C31#12011yescharacteristic
266C2×C4×C31C22×D411yescharacteristic
267C2×C4×C31C22×D411yescharacteristic
268C8×C31—42no—
269C8×C31—42no—
270C2×#538C2×C3111yescharacteristic
271C2×#451C2×C3111yescharacteristic, complemented
272C2×#448C2×C3111yescharacteristic, complemented
273C2×#393C2×C3111yescharacteristic, complemented
274#1606C2×C3118yesdirect product factor
275C2×#328C2×C3111yescharacteristic, complemented
276C2×#327C2×C3111yescharacteristic, complemented
277C2×#323C2×C3111yescharacteristic, complemented
278C22×C31—24no—
279C22×C31—24no—
280C22×C31—21no—
281C22×C31—21no—
282C22×C31—21no—
283C22×C31—21no—
284C22×C31—44no—
285C22×C31—42no—
286C22×C31—42no—
287C22×C31—42no—
288C22×C31—42no—
289C22×C31—42no—
290C22×C31—42no—
291C22×C31—82no—
292C22×C31—84no—
293C22×C31—84no—
294C22×C31—84no—
295C22×C31#45212yes—
296C22×C31#45212yes—
297C22×C31#45512yes—
298C22×C31B(4,𝔽2)12yes—
299C22×C31C2×#12011yescharacteristic
300C22×C31C2×#13511yescharacteristic
301C22×C31C2×#13711yescharacteristic
302C4×C31—21no—
303C4×C31—21no—
304C4×C31—21no—
305C4×C31—21no—
306C4×C31—42no—
307C4×C31—42no—
308C4×C31—42no—
309C4×C31—44no—
310C4×C31—42no—
311C4×C31—82no—
312C23×Q8C22×C3111yescharacteristic
313C22×Q16—21no—
314C22×SD16—21no—
315#538—22no—
316#538—22no—
317#538—22no—
318#538C22×C3114yes—
319C2×#126—21no—
320C2×#124—21no—
321C2×#123—21no—
322C2×C22⋊Q8C22×C3111yescharacteristic
323C2×#121—21no—
324C2×#121C22×C3111yescharacteristic
325C2×#120—21no—
326C2×C4×D4—21no—
327C22×C22⋊C4—21no—
328C22×C22⋊C4C22×C3111yescharacteristic
329#451—22no—
330#451C22×C3114yescomplemented
331#448—22no—
332#448C22×C3114yescomplemented
333C2×#103—21no—
334C2×#103—21no—
335C2×C22⋊C8C22×C3111yescharacteristic
336C2×C22⋊C8C22×C3111yescharacteristic
337#393—22no—
338#393C22×C3114yescomplemented
339C2×#95—21no—
340C2×#95C22×C3111yescharacteristic
341#328C22×C3114yescomplemented
342#327C22×C3114yescomplemented
343#323C22×C3114yescomplemented
344C2×C31—21no—
345C2×C31—21no—
346C2×C31—22no—
347C2×C31—82no—
348C2×C31—82no—
349C2×C31#160614yesdirect product factor
350C2×C31C2×#45211yescharacteristic
351C2×C31C2×#45511yescharacteristic
352C2×C31C2×B(4,𝔽2)11yescharacteristic
353C25D4×C3111yescharacteristic
354C22×Q8—21no—
355C22×Q8—21no—
356C22×Q8—21no—
357C22×Q8—21no—
358C22×Q8—21no—
359C22×Q8—44no—
360C22×Q8D4×C3111yescharacteristic
361C22×Q8C23×C3112yes—
362C22×D4—21no—
363C22×D4—21no—
364C22×D4—21no—
365C22×D4D4×C3111yescharacteristic
366C23×C4—21no—
367C23×C4—21no—
368C23×C4—21no—
369C23×C4D4×C3111yescharacteristic
370C23×C4C23×C3111yescharacteristic
371C2×Q16—24no—
372C2×Q16—44no—
373C2×SD16—24no—
374C2×SD16—44no—
375C22×C8—21no—
376C22×C8—21no—
377#126—24no—
378#126—24no—
379#124—24no—
380#124—24no—
381#123—24no—
382#123—24no—
383C22⋊Q8—21no—
384C22⋊Q8—21no—
385C22⋊Q8D4×C3112yes—
386C22⋊Q8C23×C3112yes—
387#121—24no—
388#121—24no—
389#121—21no—
390#121—21no—
391#121D4×C3112yes—
392#121C23×C3112yes—
393#120—24no—
394#120—24no—
395C4×D4—24no—
396C4×D4—24no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4D4×C3111yescharacteristic
400C2×C4⋊C4D4×C3111yescharacteristic
401C2×C22⋊C4—21no—
402C2×C22⋊C4—21no—
403C2×C22⋊C4—21no—
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C22⋊C4—21no—
409C2×C22⋊C4—21no—
410C2×C22⋊C4—24no—
411C2×C22⋊C4—44no—
412C2×C22⋊C4D4×C3112yes—
413C2×C22⋊C4C23×C3112yes—
414C2×C42—21no—
415#103—24no—
416#103—24no—
417C22⋊C8—21no—
418C22⋊C8—21no—
419C22⋊C8C23×C3112yes—
420C22⋊C8C23×C3112yes—
421#95—24no—
422#95—21no—
423#95C23×C3112yes—
424C31C2×#160611yescharacteristic, direct product factor, Sylow
425C24—21no—
426C24—21no—
427C24—22no—
428C24—21no—
429C24—22no—
430C24—42no—
431C24—42no—
432C24C2×D4×C3111yescharacteristic
433C24C2×D4×C3112yes—
434C2×Q8—21no—
435C2×Q8—24no—
436C2×Q8—24no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—44no—
440C2×Q8—42no—
441C2×Q8—42no—
442C2×Q8—44no—
443C2×Q8—42no—
444C2×Q8—44no—
445C2×Q8—44no—
446C2×Q8—41no—
447C2×Q8—41no—
448C2×Q8C2×D4×C3112yes—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—24no—
452C2×D4—24no—
453C2×D4—21no—
454C2×D4—42no—
455C2×D4—42no—
456C2×D4—42no—
457C2×D4—82no—
458C2×D4—82no—
459C2×D4C2×D4×C3112yes—
460C22×C4—24no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—22no—
466C22×C4—24no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—41no—
472C22×C4—41no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—82no—
478C22×C4C2×D4×C3111yescharacteristic
479C22×C4C2×D4×C3111yescharacteristic
480C22×C4C2×D4×C3112yes—
481C22×C4C2×D4×C3111yescharacteristic
482C22×C4C2×D4×C3111yescharacteristic
483C22×C4C2×D4×C3111yescharacteristic
484C22×C4C24×C3111yescharacteristic
485Q16—48no—
486SD16—48no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—42no—
490C2×C8—42no—
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—22no—
496C4⋊C4C2×D4×C3112yes—
497C4⋊C4C2×D4×C3112yes—
498C22⋊C4—24no—
499C22⋊C4—24no—
500C22⋊C4—24no—
501C22⋊C4—22no—
502C22⋊C4—22no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—22no—
507C22⋊C4—22no—
508C22⋊C4—22no—
509C22⋊C4—48no—
510C42—24no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—42no—
518C23—42no—
519C23—48no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—81no—
528C23—81no—
529C23—84no—
530C23C31×#12011yescharacteristic
531C23C31×#12011yescharacteristic
532C23C31×#13512yes—
533C23C31×#13712yes—
534C23C22×D4×C3111yescharacteristic
535Q8—24no—
536Q8—24no—
537Q8—42no—
538Q8—42no—
539Q8—42no—
540Q8—44no—
541Q8—44no—
542Q8—81no—
543D4—42no—
544D4—44no—
545D4—42no—
546D4—42no—
547D4—81no—
548D4—84no—
549D4—84no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—81no—
574C2×C4—84no—
575C2×C4—84no—
576C2×C4C31×#12011yescharacteristic
577C2×C4C31×#12011yescharacteristic
578C2×C4C22×D4×C3111yescharacteristic
579C2×C4C22×D4×C3111yescharacteristic
580C8—42no—
581C8—42no—
582C22—24no—
583C22—24no—
584C22—21no—
585C22—21no—
586C22—21no—
587C22—21no—
588C22—44no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22—82no—
596C22—84no—
597C22—84no—
598C22—84no—
599C22C31×#45212yes—
600C22C31×#45212yes—
601C22C31×#45512yes—
602C22C31×B(4,𝔽2)12yes—
603C22C2×C31×#12011yescharacteristic
604C22C2×C31×#13511yescharacteristic
605C22C2×C31×#13711yescharacteristic
606C4—21no—
607C4—21no—
608C4—21no—
609C4—21no—
610C4—42no—
611C4—42no—
612C4—42no—
613C4—44no—
614C4—42no—
615C4—82no—
616C2—21no—
617C2—21no—
618C2—22no—
619C2—82no—
620C2—82no—
621C2C31×#160614yesdirect product factor
622C2C2×C31×#45211yescharacteristic
623C2C2×C31×#45511yescharacteristic
624C2C2×C31×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC23×C31#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC2×C31×#538C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×#451C2semidirect
3. C2×C31×#448C2semidirect
4. C2×C31×#393C2semidirect
5. C31×#1606C2direct (N × H)
6. C2×C31×#328C2semidirect
7. C2×C31×#327C2semidirect
8. C2×C31×#323C2semidirect
27. C31×#451C22semidirect
29. C31×#448C22semidirect
35. C31×#393C22semidirect
38. C31×#328C22semidirect
39. C31×#327C22semidirect
40. C31×#323C22semidirect
198. C2×#1606C31direct (N × H)
271. C2×#451C2×C31semidirect
272. C2×#448C2×C31semidirect
273. C2×#393C2×C31semidirect
274. #1606C2×C31direct (N × H)
275. C2×#328C2×C31semidirect
276. C2×#327C2×C31semidirect
277. C2×#323C2×C31semidirect
330. #451C22×C31semidirect
332. #448C22×C31semidirect
338. #393C22×C31semidirect
341. #328C22×C31semidirect
342. #327C22×C31semidirect
343. #323C22×C31semidirect
349. C2×C31#1606direct (N × H)
424. C31C2×#1606direct (N × H)
621. C2C31×#1606direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5382
C2×C31×#4512
C2×C31×#4482
C2×C31×#3932
C31×#16062
C2×C31×#3282
C2×C31×#3272
C2×C31×#3232
C2×#160631