Finite Group Databasebeta

← C31×#13867 #21470626

Subgroups — C31×#13867 #21470626

Order 7936.

7150 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C24×D4×C31C211yesmaximal, characteristic
2C23×C22⋊C4×C31C211yesmaximal, characteristic, complemented
3C2×C31×#386C214yesmaximal, complemented
4C2×C31×#379C211yesmaximal, characteristic, complemented
5C2×C31×#379C212yesmaximal, complemented
6C31×#1438C214yesmaximal, complemented
7C31×C22⋊#95C212yesmaximal, complemented
8C26×C31C2211yescharacteristic
9C26×C31C2211yescharacteristic
10C23×D4×C31—24no—
11C23×D4×C31—24no—
12C23×D4×C31—216no—
13C23×D4×C31C418yescomplemented
14C23×D4×C31C2212yes—
15C23×D4×C31C2212yes—
16C24×C4×C31C2211yescharacteristic
17C24×C4×C31C2212yescomplemented
18C22×C22⋊C4×C31—24no—
19C22×C22⋊C4×C31—28no—
20C22×C22⋊C4×C31C2212yes—
21C22×C22⋊C4×C31C2212yes—
22C22×C22⋊C4×C31C2212yescomplemented
23C22×C22⋊C4×C31C2212yescomplemented
24C22×C22⋊C4×C31C2218yescomplemented
25C22×C22⋊C4×C31C2212yescomplemented
26C31×#386—216no—
27C31×#379—24no—
28C31×#379—28no—
29C2×C31×#95C2218yescomplemented
30C25×C31—24no—
31C25×C31—28no—
32C25×C31—28no—
33C25×C31—22no—
34C25×C31—22no—
35C25×C31—24no—
36C25×C31—24no—
37C25×C31—24no—
38C25×C31—44no—
39C25×C31—44no—
40C25×C31C2×C414yes—
41C25×C31C2×C414yescomplemented
42C25×C31D414yes—
43C25×C31D414yes—
44C25×C31C2311yescharacteristic
45C25×C31C2312yes—
46C25×C31C2311yescharacteristic
47C25×C31C2311yescharacteristic
48C22×D4×C31—264no—
49C22×D4×C31—264no—
50C22×D4×C31—232no—
51C22×D4×C31—24no—
52C22×D4×C31—24no—
53C22×D4×C31—28no—
54C22×D4×C31—24no—
55C22×D4×C31—28no—
56C22×D4×C31—24no—
57C22×D4×C31—264no—
58C22×D4×C31—28no—
59C22×D4×C31C2×C418yes—
60C22×D4×C31C2×C418yescomplemented
61C22×D4×C31D4116yes—
62C23×C4×C31—22no—
63C23×C4×C31—22no—
64C23×C4×C31—28no—
65C23×C4×C31—24no—
66C23×C4×C31—28no—
67C23×C4×C31—28no—
68C23×C4×C31—44no—
69C23×C4×C31C2×C414yescomplemented
70C23×C4×C31C2314yes—
71C23×C4×C31C2314yes—
72C23×C4×C31C2312yes—
73C2×C22⋊C4×C31—232no—
74C2×C22⋊C4×C31—24no—
75C2×C22⋊C4×C31—24no—
76C2×C22⋊C4×C31—24no—
77C2×C22⋊C4×C31—28no—
78C2×C22⋊C4×C31—24no—
79C2×C22⋊C4×C31—24no—
80C2×C22⋊C4×C31—24no—
81C2×C22⋊C4×C31—24no—
82C2×C22⋊C4×C31—216no—
83C2×C22⋊C4×C31—28no—
84C2×C22⋊C4×C31—24no—
85C2×C22⋊C4×C31—216no—
86C2×C22⋊C4×C31—48no—
87C2×C22⋊C4×C31—44no—
88C2×C22⋊C4×C31D418yescomplemented
89C31×#95—216no—
90C24×C31—22no—
91C24×C31—24no—
92C24×C31—22no—
93C24×C31—24no—
94C24×C31—24no—
95C24×C31—216no—
96C24×C31—216no—
97C24×C31—216no—
98C24×C31—28no—
99C24×C31—216no—
100C24×C31—216no—
101C24×C31—28no—
102C24×C31—216no—
103C24×C31—216no—
104C24×C31—216no—
105C24×C31—28no—
106C24×C31—22no—
107C24×C31—24no—
108C24×C31—22no—
109C24×C31—22no—
110C24×C31—24no—
111C24×C31—24no—
112C24×C31—22no—
113C24×C31—24no—
114C24×C31—24no—
115C24×C31—28no—
116C24×C31—216no—
117C24×C31—48no—
118C24×C31—48no—
119C24×C31—48no—
120C24×C31—48no—
121C24×C31—432no—
122C24×C31—432no—
123C24×C31—432no—
124C24×C31—432no—
125C24×C31—48no—
126C24×C31—48no—
127C24×C31—44no—
128C24×C31—48no—
129C24×C31—44no—
130C24×C31C22⋊C4116yescomplemented
131C24×C31C22×C412yes—
132C24×C31C22×C414yes—
133C24×C31C22×C412yes—
134C24×C31C22×C412yes—
135C24×C31C2×D412yes—
136C24×C31C2×D412yes—
137C24×C31C2×D418yes—
138C24×C31C2×D412yes—
139C24×C31C2×D418yes—
140C24×C31C2×D412yes—
141C24×C31C2×D412yes—
142C24×C31C2×D412yes—
143C24×C31C2×D414yes—
144C24×C31#4114yes—
145C24×C31C2411yescharacteristic
146C2×D4×C31—2256no—
147C2×D4×C31—232no—
148C2×D4×C31—232no—
149C2×D4×C31—216no—
150C2×D4×C31—216no—
151C2×D4×C31—216no—
152C2×D4×C31—216no—
153C2×D4×C31—232no—
154C2×D4×C31—232no—
155C2×D4×C31—216no—
156C2×D4×C31C22⋊C4132yescomplemented
157C22×C4×C31—216no—
158C22×C4×C31—216no—
159C22×C4×C31—28no—
160C22×C4×C31—22no—
161C22×C4×C31—22no—
162C22×C4×C31—24no—
163C22×C4×C31—216no—
164C22×C4×C31—216no—
165C22×C4×C31—28no—
166C22×C4×C31—24no—
167C22×C4×C31—24no—
168C22×C4×C31—28no—
169C22×C4×C31—28no—
170C22×C4×C31—24no—
171C22×C4×C31—24no—
172C22×C4×C31—24no—
173C22×C4×C31—28no—
174C22×C4×C31—432no—
175C22×C4×C31—48no—
176C22×C4×C31—48no—
177C22×C4×C31C22×C414yes—
178C22×C4×C31C2×D418yes—
179C22×C4×C31C2×D418yescomplemented
180C22⋊C4×C31—216no—
181C22⋊C4×C31—216no—
182C22⋊C4×C31—432no—
183C22⋊C4×C31—48no—
184C22⋊C4×C31—48no—
185C22⋊C4×C31—48no—
186#13867C3111yesmaximal, characteristic, direct product factor, Sylow
187C23×C31—24no—
188C23×C31—24no—
189C23×C31—24no—
190C23×C31—24no—
191C23×C31—22no—
192C23×C31—24no—
193C23×C31—24no—
194C23×C31—24no—
195C23×C31—22no—
196C23×C31—24no—
197C23×C31—24no—
198C23×C31—24no—
199C23×C31—24no—
200C23×C31—216no—
201C23×C31—216no—
202C23×C31—216no—
203C23×C31—216no—
204C23×C31—232no—
205C23×C31—24no—
206C23×C31—24no—
207C23×C31—24no—
208C23×C31—28no—
209C23×C31—24no—
210C23×C31—24no—
211C23×C31—28no—
212C23×C31—22no—
213C23×C31—24no—
214C23×C31—24no—
215C23×C31—28no—
216C23×C31—28no—
217C23×C31—24no—
218C23×C31—22no—
219C23×C31—24no—
220C23×C31—24no—
221C23×C31—216no—
222C23×C31—216no—
223C23×C31—232no—
224C23×C31—232no—
225C23×C31—416no—
226C23×C31—464no—
227C23×C31—464no—
228C23×C31—432no—
229C23×C31—432no—
230C23×C31—464no—
231C23×C31—416no—
232C23×C31—432no—
233C23×C31—432no—
234C23×C31—464no—
235C23×C31—42no—
236C23×C31—44no—
237C23×C31—42no—
238C23×C31—44no—
239C23×C31—44no—
240C23×C31—48no—
241C23×C31—48no—
242C23×C31—48no—
243C23×C31—48no—
244C23×C31—416no—
245C23×C31C2×C22⋊C414yes—
246C23×C31C2×C22⋊C414yes—
247C23×C31C2×C22⋊C418yes—
248C23×C31C2×C22⋊C414yes—
249C23×C31C2×C22⋊C418yescomplemented
250C23×C31C4×D418yes—
251C23×C31#12018yes—
252C23×C31#12018yes—
253C23×C31#12014yes—
254C23×C31#12118yes—
255C23×C31#12314yes—
256C23×C31C23×C411yescharacteristic
257C23×C31C22×D411yescharacteristic
258C23×C31C22×D411yescharacteristic
259C23×C31C22×D414yes—
260C23×C31C22×D412yes—
261C23×C31C2×#4112yes—
262C23×C31H(5,𝔽2)14yes—
263D4×C31—264no—
264D4×C31—264no—
265C2×C4×C31—232no—
266C2×C4×C31—24no—
267C2×C4×C31—24no—
268C2×C4×C31—28no—
269C2×C4×C31—28no—
270C2×C4×C31—28no—
271C2×C4×C31—216no—
272C2×C4×C31—432no—
273C2×C4×C31—44no—
274C2×C4×C31—44no—
275C2×C4×C31—48no—
276C2×C4×C31—44no—
277C2×C4×C31—44no—
278C2×C4×C31—44no—
279C2×C4×C31—48no—
280C2×C4×C31—48no—
281C2×C4×C31C2×C22⋊C418yescomplemented
282C24×D4C2×C3111yescharacteristic
283C23×C22⋊C4C2×C3111yescharacteristic, complemented
284C2×#386C2×C3114yescomplemented
285C2×#379C2×C3111yescharacteristic, complemented
286C2×#379C2×C3112yescomplemented
287#1438C2×C3114yescomplemented
288C22⋊#95C2×C3112yescomplemented
289C22×C31—24no—
290C22×C31—24no—
291C22×C31—216no—
292C22×C31—216no—
293C22×C31—28no—
294C22×C31—28no—
295C22×C31—28no—
296C22×C31—28no—
297C22×C31—216no—
298C22×C31—216no—
299C22×C31—24no—
300C22×C31—24no—
301C22×C31—24no—
302C22×C31—216no—
303C22×C31—24no—
304C22×C31—28no—
305C22×C31—24no—
306C22×C31—464no—
307C22×C31—464no—
308C22×C31—44no—
309C22×C31—48no—
310C22×C31—48no—
311C22×C31—44no—
312C22×C31—48no—
313C22×C31—44no—
314C22×C31—48no—
315C22×C31—48no—
316C22×C31—416no—
317C22×C31—44no—
318C22×C31#37918yescomplemented
319C22×C31#38618yescomplemented
320C22×C31C22×C22⋊C411yescharacteristic
321C22×C31C22×C22⋊C412yes—
322C22×C31C2×C4×D412yes—
323C22×C31#51812yes—
324C22×C31C2×#12011yescharacteristic
325C22×C31C2×#12012yes—
326C22×C31C2×#12012yes—
327C22×C31C2×#12112yes—
328C22×C31C2×#12311yescharacteristic
329C22×C31#53412yes—
330C22×C31#53412yes—
331C22×C31D4218yes—
332C22×C31#54618yes—
333C4×C31—28no—
334C4×C31—48no—
335C4×C31—84no—
336C26C22×C3111yescharacteristic
337C26C22×C3111yescharacteristic
338C23×D4—24no—
339C23×D4—24no—
340C23×D4—216no—
341C23×D4C4×C3118yescomplemented
342C23×D4C22×C3112yes—
343C23×D4C22×C3112yes—
344C24×C4C22×C3111yescharacteristic
345C24×C4C22×C3112yescomplemented
346C22×C22⋊C4—24no—
347C22×C22⋊C4—28no—
348C22×C22⋊C4C22×C3112yes—
349C22×C22⋊C4C22×C3112yes—
350C22×C22⋊C4C22×C3112yescomplemented
351C22×C22⋊C4C22×C3112yescomplemented
352C22×C22⋊C4C22×C3118yescomplemented
353C22×C22⋊C4C22×C3112yescomplemented
354#386—216no—
355#379—24no—
356#379—28no—
357C2×#95C22×C3118yescomplemented
358C2×C31—28no—
359C2×C31—28no—
360C2×C31—44no—
361C2×C31—48no—
362C2×C31C2×#37911yescharacteristic
363C2×C31C2×#38611yescharacteristic
364C2×C31#230811yescharacteristic
365C2×C31D4×C22⋊C414yes—
366C2×C31#240312yes—
367C2×C31#240512yes—
368C2×C31#241014yes—
369C25—24no—
370C25—28no—
371C25—28no—
372C25—22no—
373C25—22no—
374C25—24no—
375C25—24no—
376C25—24no—
377C25—44no—
378C25—44no—
379C25C2×C4×C3114yes—
380C25C2×C4×C3114yescomplemented
381C25D4×C3114yes—
382C25D4×C3114yes—
383C25C23×C3111yescharacteristic
384C25C23×C3112yes—
385C25C23×C3111yescharacteristic
386C25C23×C3111yescharacteristic
387C22×D4—264no—
388C22×D4—264no—
389C22×D4—232no—
390C22×D4—24no—
391C22×D4—24no—
392C22×D4—28no—
393C22×D4—24no—
394C22×D4—28no—
395C22×D4—24no—
396C22×D4—264no—
397C22×D4—28no—
398C22×D4C2×C4×C3118yes—
399C22×D4C2×C4×C3118yescomplemented
400C22×D4D4×C31116yes—
401C23×C4—22no—
402C23×C4—22no—
403C23×C4—28no—
404C23×C4—24no—
405C23×C4—28no—
406C23×C4—28no—
407C23×C4—44no—
408C23×C4C2×C4×C3114yescomplemented
409C23×C4C23×C3114yes—
410C23×C4C23×C3114yes—
411C23×C4C23×C3112yes—
412C2×C22⋊C4—232no—
413C2×C22⋊C4—24no—
414C2×C22⋊C4—24no—
415C2×C22⋊C4—24no—
416C2×C22⋊C4—28no—
417C2×C22⋊C4—24no—
418C2×C22⋊C4—24no—
419C2×C22⋊C4—24no—
420C2×C22⋊C4—24no—
421C2×C22⋊C4—216no—
422C2×C22⋊C4—28no—
423C2×C22⋊C4—24no—
424C2×C22⋊C4—216no—
425C2×C22⋊C4—48no—
426C2×C22⋊C4—44no—
427C2×C22⋊C4D4×C3118yescomplemented
428#95—216no—
429C31#1386711yescharacteristic, direct product factor, Sylow
430C24—22no—
431C24—24no—
432C24—22no—
433C24—24no—
434C24—24no—
435C24—216no—
436C24—216no—
437C24—216no—
438C24—28no—
439C24—216no—
440C24—216no—
441C24—28no—
442C24—216no—
443C24—216no—
444C24—216no—
445C24—28no—
446C24—22no—
447C24—24no—
448C24—22no—
449C24—22no—
450C24—24no—
451C24—24no—
452C24—22no—
453C24—24no—
454C24—24no—
455C24—28no—
456C24—216no—
457C24—48no—
458C24—48no—
459C24—48no—
460C24—48no—
461C24—432no—
462C24—432no—
463C24—432no—
464C24—432no—
465C24—48no—
466C24—48no—
467C24—44no—
468C24—48no—
469C24—44no—
470C24C22⋊C4×C31116yescomplemented
471C24C22×C4×C3112yes—
472C24C22×C4×C3114yes—
473C24C22×C4×C3112yes—
474C24C22×C4×C3112yes—
475C24C2×D4×C3112yes—
476C24C2×D4×C3112yes—
477C24C2×D4×C3118yes—
478C24C2×D4×C3112yes—
479C24C2×D4×C3118yes—
480C24C2×D4×C3112yes—
481C24C2×D4×C3112yes—
482C24C2×D4×C3112yes—
483C24C2×D4×C3114yes—
484C24#41×C3114yes—
485C24C24×C3111yescharacteristic
486C2×D4—2256no—
487C2×D4—232no—
488C2×D4—232no—
489C2×D4—216no—
490C2×D4—216no—
491C2×D4—216no—
492C2×D4—216no—
493C2×D4—232no—
494C2×D4—232no—
495C2×D4—216no—
496C2×D4C22⋊C4×C31132yescomplemented
497C22×C4—216no—
498C22×C4—216no—
499C22×C4—28no—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—24no—
503C22×C4—216no—
504C22×C4—216no—
505C22×C4—28no—
506C22×C4—24no—
507C22×C4—24no—
508C22×C4—28no—
509C22×C4—28no—
510C22×C4—24no—
511C22×C4—24no—
512C22×C4—24no—
513C22×C4—28no—
514C22×C4—432no—
515C22×C4—48no—
516C22×C4—48no—
517C22×C4C22×C4×C3114yes—
518C22×C4C2×D4×C3118yes—
519C22×C4C2×D4×C3118yescomplemented
520C22⋊C4—216no—
521C22⋊C4—216no—
522C22⋊C4—432no—
523C22⋊C4—48no—
524C22⋊C4—48no—
525C22⋊C4—48no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—22no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—22no—
535C23—24no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—216no—
540C23—216no—
541C23—216no—
542C23—216no—
543C23—232no—
544C23—24no—
545C23—24no—
546C23—24no—
547C23—28no—
548C23—24no—
549C23—24no—
550C23—28no—
551C23—22no—
552C23—24no—
553C23—24no—
554C23—28no—
555C23—28no—
556C23—24no—
557C23—22no—
558C23—24no—
559C23—24no—
560C23—216no—
561C23—216no—
562C23—232no—
563C23—232no—
564C23—416no—
565C23—464no—
566C23—464no—
567C23—432no—
568C23—432no—
569C23—464no—
570C23—416no—
571C23—432no—
572C23—432no—
573C23—464no—
574C23—42no—
575C23—44no—
576C23—42no—
577C23—44no—
578C23—44no—
579C23—48no—
580C23—48no—
581C23—48no—
582C23—48no—
583C23—416no—
584C23C2×C22⋊C4×C3114yes—
585C23C2×C22⋊C4×C3114yes—
586C23C2×C22⋊C4×C3118yes—
587C23C2×C22⋊C4×C3114yes—
588C23C2×C22⋊C4×C3118yescomplemented
589C23C4×D4×C3118yes—
590C23C31×#12018yes—
591C23C31×#12018yes—
592C23C31×#12014yes—
593C23C31×#12118yes—
594C23C31×#12314yes—
595C23C23×C4×C3111yescharacteristic
596C23C22×D4×C3111yescharacteristic
597C23C22×D4×C3111yescharacteristic
598C23C22×D4×C3114yes—
599C23C22×D4×C3112yes—
600C23C2×#41×C3112yes—
601C23C31×H(5,𝔽2)14yes—
602D4—264no—
603D4—264no—
604C2×C4—232no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—28no—
608C2×C4—28no—
609C2×C4—28no—
610C2×C4—216no—
611C2×C4—432no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—48no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—48no—
619C2×C4—48no—
620C2×C4C2×C22⋊C4×C3118yescomplemented
621C22—24no—
622C22—24no—
623C22—216no—
624C22—216no—
625C22—28no—
626C22—28no—
627C22—28no—
628C22—28no—
629C22—216no—
630C22—216no—
631C22—24no—
632C22—24no—
633C22—24no—
634C22—216no—
635C22—24no—
636C22—28no—
637C22—24no—
638C22—464no—
639C22—464no—
640C22—44no—
641C22—48no—
642C22—48no—
643C22—44no—
644C22—48no—
645C22—44no—
646C22—48no—
647C22—48no—
648C22—416no—
649C22—44no—
650C22C31×#37918yescomplemented
651C22C31×#38618yescomplemented
652C22C22×C22⋊C4×C3111yescharacteristic
653C22C22×C22⋊C4×C3112yes—
654C22C2×C4×D4×C3112yes—
655C22C31×#51812yes—
656C22C2×C31×#12011yescharacteristic
657C22C2×C31×#12012yes—
658C22C2×C31×#12012yes—
659C22C2×C31×#12112yes—
660C22C2×C31×#12311yescharacteristic
661C22C31×#53412yes—
662C22C31×#53412yes—
663C22D42×C3118yes—
664C22C31×#54618yes—
665C4—28no—
666C4—48no—
667C4—84no—
668C2—28no—
669C2—28no—
670C2—44no—
671C2—48no—
672C2C2×C31×#37911yescharacteristic
673C2C2×C31×#38611yescharacteristic
674C2C31×#230811yescharacteristic
675C2D4×C22⋊C4×C3114yes—
676C2C31×#240312yes—
677C2C31×#240512yes—
678C2C31×#241014yes—
subgroupG/H
center, socleC24×C31C24
commutatorC23C23×C4×C31
Frattini, cosocleC24C24×C31
prime radicalC24×D4×C31C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×C22⋊C4×C31C2semidirect
3. C2×C31×#386C2semidirect
4. C2×C31×#379C2semidirect
5. C2×C31×#379C2semidirect
6. C31×#1438C2semidirect
7. C31×C22⋊#95C2semidirect
13. C23×D4×C31C4semidirect
17. C24×C4×C31C22semidirect
22. C22×C22⋊C4×C31C22semidirect
23. C22×C22⋊C4×C31C22semidirect
24. C22×C22⋊C4×C31C22semidirect
25. C22×C22⋊C4×C31C22semidirect
29. C2×C31×#95C22semidirect
41. C25×C31C2×C4semidirect
60. C22×D4×C31C2×C4semidirect
69. C23×C4×C31C2×C4semidirect
88. C2×C22⋊C4×C31D4semidirect
130. C24×C31C22⋊C4semidirect
156. C2×D4×C31C22⋊C4semidirect
179. C22×C4×C31C2×D4semidirect
186. #13867C31direct (N × H)
249. C23×C31C2×C22⋊C4semidirect
281. C2×C4×C31C2×C22⋊C4semidirect
283. C23×C22⋊C4C2×C31semidirect
284. C2×#386C2×C31semidirect
285. C2×#379C2×C31semidirect
286. C2×#379C2×C31semidirect
287. #1438C2×C31semidirect
288. C22⋊#95C2×C31semidirect
318. C22×C31#379semidirect
319. C22×C31#386semidirect
341. C23×D4C4×C31semidirect
345. C24×C4C22×C31semidirect
350. C22×C22⋊C4C22×C31semidirect
351. C22×C22⋊C4C22×C31semidirect
352. C22×C22⋊C4C22×C31semidirect
353. C22×C22⋊C4C22×C31semidirect
357. C2×#95C22×C31semidirect
380. C25C2×C4×C31semidirect
399. C22×D4C2×C4×C31semidirect
408. C23×C4C2×C4×C31semidirect
427. C2×C22⋊C4D4×C31semidirect
429. C31#13867direct (N × H)
470. C24C22⋊C4×C31semidirect
496. C2×D4C22⋊C4×C31semidirect
519. C22×C4C2×D4×C31semidirect
588. C23C2×C22⋊C4×C31semidirect
620. C2×C4C2×C22⋊C4×C31semidirect
650. C22C31×#379semidirect
651. C22C31×#386semidirect

Maximal subgroups

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

subgroupindex
C24×D4×C312
C23×C22⋊C4×C312
C2×C31×#3862
C2×C31×#3792
C31×#14382
C31×C22⋊#952
#1386731