Finite Group Databasebeta

← C31×#20925 #21477684

Subgroups — C31×#20925 #21477684

Order 7936.

2234 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C31×AGL(1,ℤ/8)C211yesmaximal, characteristic
2C22×C4×D4×C31C211yesmaximal, characteristic, complemented
3C2×C31×#417C211yesmaximal, characteristic, complemented
4C2×C31×#418C211yesmaximal, characteristic, complemented
5C31×#2297C211yesmaximal, characteristic, complemented
6C31×#1879C214yesmaximal, complemented
7C31×#1877C214yesmaximal, complemented
8C31×#1736C211yesmaximal, characteristic, complemented
9C31×#1728C211yesmaximal, characteristic, complemented
10C22×#41×C31C2211yescharacteristic
11C23×D4×C31C2211yescharacteristic
12C24×C4×C31—21no—
13C2×C31×AGL(1,ℤ/8)—24no—
14C2×C31×AGL(1,ℤ/8)—24no—
15C2×C31×AGL(1,ℤ/8)C418yescomplemented
16C22×SD16×C31C2212yes—
17C22×D8×C31C2212yes—
18C22×M16×C31C2211yescharacteristic
19C22×M16×C31C2211yescharacteristic
20C2×C4×D4×C31—24no—
21C2×C4×D4×C31—24no—
22C2×C4×D4×C31—22no—
23C2×C4×D4×C31C2214yes—
24C2×C31×#117C2211yescharacteristic
25C22×C4⋊C4×C31C2211yescharacteristic
26C22×C22⋊C4×C31—21no—
27C22×C42×C31C2211yescharacteristic
28C31×#418—24no—
29C31×#417—24no—
30C2×C31×#103C2212yescomplemented
31C2×C31×#102C2214yes—
32C2×C31×#102C2212yescomplemented
33C31×#391C2212yescomplemented
34C31×#390C2212yescomplemented
35C31×#386—21no—
36C31×#386—21no—
37C4×C22⋊C4×C31—22no—
38C31×#340C2214yescomplemented
39C31×#336C2214yescomplemented
40C25×C31—21no—
41C2×#41×C31—22no—
42C2×#41×C31—22no—
43C2×#41×C31—22no—
44C2×#41×C31—22no—
45C2×#41×C31—22no—
46C2×#41×C31C2×C414yescomplemented
47C22×Q8×C31C2311yescharacteristic
48C22×D4×C31—21no—
49C22×D4×C31—21no—
50C22×D4×C31—22no—
51C22×D4×C31—22no—
52C22×D4×C31—22no—
53C22×D4×C31C2×C414yes—
54C22×D4×C31D414yes—
55C22×D4×C31D414yes—
56C22×D4×C31C2311yescharacteristic
57C22×D4×C31C2312yes—
58C23×C4×C31—28no—
59C23×C4×C31—21no—
60C23×C4×C31—24no—
61C23×C4×C31—21no—
62C23×C4×C31—48no—
63C23×C4×C31D411yescharacteristic
64C23×C4×C31D411yescharacteristic
65C23×C4×C31C2311yescharacteristic
66C31×AGL(1,ℤ/8)—232no—
67C2×SD16×C31—24no—
68C2×SD16×C31—24no—
69C2×SD16×C31C2×C418yescomplemented
70C2×D8×C31—24no—
71C2×D8×C31—24no—
72C2×D8×C31C2×C418yescomplemented
73C2×M16×C31—22no—
74C2×M16×C31—22no—
75C2×M16×C31—22no—
76C2×M16×C31—22no—
77C2×M16×C31—22no—
78C2×M16×C31C2×C414yescomplemented
79C22×C8×C31C2312yes—
80C22×C8×C31C2312yes—
81C4×D4×C31—216no—
82C4×D4×C31—28no—
83C4×D4×C31—28no—
84C31×#117—22no—
85C31×#117—22no—
86C2×C4⋊C4×C31—21no—
87C2×C4⋊C4×C31—22no—
88C2×C4⋊C4×C31D414yes—
89C2×C4⋊C4×C31C2312yes—
90C2×C4⋊C4×C31C2312yes—
91C2×C22⋊C4×C31—28no—
92C2×C22⋊C4×C31—24no—
93C2×C22⋊C4×C31—21no—
94C2×C22⋊C4×C31—22no—
95C2×C22⋊C4×C31—22no—
96C2×C42×C31—22no—
97C2×C42×C31—22no—
98C2×C42×C31—21no—
99C2×C42×C31D412yes—
100C2×C42×C31C2312yes—
101C31×#103—24no—
102C31×#102—28no—
103C31×#102—24no—
104C31×#95—22no—
105C31×#95—22no—
106C24×C31—24no—
107C24×C31—24no—
108C24×C31—24no—
109C24×C31—22no—
110C24×C31—21no—
111C24×C31—44no—
112C24×C31—44no—
113C24×C31C2×D411yescharacteristic
114#41×C31—28no—
115#41×C31—28no—
116#41×C31—44no—
117#41×C31—44no—
118C2×Q8×C31—21no—
119C2×Q8×C31—21no—
120C2×Q8×C31—21no—
121C2×Q8×C31—21no—
122C2×Q8×C31—21no—
123C2×Q8×C31C22×C412yescomplemented
124C2×D4×C31—24no—
125C2×D4×C31—28no—
126C2×D4×C31—24no—
127C2×D4×C31—24no—
128C2×D4×C31—28no—
129C2×D4×C31—24no—
130C2×D4×C31—28no—
131C2×D4×C31—22no—
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—42no—
141C2×D4×C31C22⋊C4116yescomplemented
142C2×D4×C31C22×C414yes—
143C2×D4×C31C22×C412yescomplemented
144C2×D4×C31C2×D414yes—
145C2×D4×C31C2×D414yes—
146C22×C4×C31—22no—
147C22×C4×C31—24no—
148C22×C4×C31—216no—
149C22×C4×C31—22no—
150C22×C4×C31—24no—
151C22×C4×C31—21no—
152C22×C4×C31—22no—
153C22×C4×C31—24no—
154C22×C4×C31—22no—
155C22×C4×C31—42no—
156C22×C4×C31—432no—
157C22×C4×C31—48no—
158C22×C4×C31—44no—
159C22×C4×C31—42no—
160C22×C4×C31—42no—
161C22×C4×C31—48no—
162C22×C4×C31C22×C412yes—
163C22×C4×C31C2×D412yes—
164C22×C4×C31C2×D412yes—
165C22×C4×C31C2×D412yes—
166C22×C4×C31C2×D412yes—
167C22×C4×C31C2×D411yescharacteristic
168C22×C4×C31C2×D412yes—
169C22×C4×C31#4112yes—
170C22×C4×C31#4112yes—
171C22×C4×C31C2411yescharacteristic
172SD16×C31—216no—
173D8×C31—216no—
174M16×C31—28no—
175M16×C31—44no—
176C2×C8×C31—22no—
177C2×C8×C31—22no—
178C2×C8×C31—22no—
179C2×C8×C31—22no—
180C2×C8×C31—22no—
181C2×C8×C31C22×C414yescomplemented
182C4⋊C4×C31—24no—
183C4⋊C4×C31—22no—
184C4⋊C4×C31—22no—
185C4⋊C4×C31—22no—
186C4⋊C4×C31C2×D414yes—
187C22⋊C4×C31—216no—
188C22⋊C4×C31—44no—
189C22⋊C4×C31—42no—
190C22⋊C4×C31—44no—
191C42×C31—24no—
192C42×C31—22no—
193C42×C31—22no—
194C42×C31—42no—
195#20925C3111yesmaximal, characteristic, direct product factor, Sylow
196C23×C31—22no—
197C23×C31—22no—
198C23×C31—24no—
199C23×C31—24no—
200C23×C31—24no—
201C23×C31—216no—
202C23×C31—22no—
203C23×C31—48no—
204C23×C31—416no—
205C23×C31—48no—
206C23×C31—416no—
207C23×C31—44no—
208C23×C31—44no—
209C23×C31—41no—
210C23×C31—41no—
211C23×C31—42no—
212C23×C31—42no—
213C23×C31C4×D412yes—
214C23×C31#12012yes—
215C23×C31#12112yes—
216C23×C31C22×D411yescharacteristic
217Q8×C31—22no—
218Q8×C31—22no—
219Q8×C31—41no—
220Q8×C31—41no—
221D4×C31—22no—
222D4×C31—24no—
223D4×C31—22no—
224D4×C31—24no—
225D4×C31—24no—
226D4×C31—216no—
227D4×C31—48no—
228D4×C31—41no—
229D4×C31—41no—
230D4×C31C2×C22⋊C418yes—
231C2×C4×C31—28no—
232C2×C4×C31—22no—
233C2×C4×C31—22no—
234C2×C4×C31—22no—
235C2×C4×C31—22no—
236C2×C4×C31—24no—
237C2×C4×C31—22no—
238C2×C4×C31—22no—
239C2×C4×C31—432no—
240C2×C4×C31—44no—
241C2×C4×C31—48no—
242C2×C4×C31—41no—
243C2×C4×C31—41no—
244C2×C4×C31—44no—
245C2×C4×C31—48no—
246C2×C4×C31—42no—
247C2×C4×C31—42no—
248C2×C4×C31—42no—
249C2×C4×C31C2×C22⋊C414yes—
250C2×C4×C31C4×D414yes—
251C2×C4×C31C4×D412yes—
252C2×C4×C31#12012yes—
253C2×C4×C31#12114yes—
254C2×C4×C31#12112yes—
255C2×C4×C31#12314yes—
256C2×C4×C31AGL(1,ℤ/8)14yescomplemented
257C2×C4×C31C23×C411yescharacteristic
258C2×C4×C31C22×D411yescharacteristic
259C2×C4×C31C22×D411yescharacteristic
260C2×C4×C31C22×D411yescharacteristic
261C2×C4×C31C2×#4111yescharacteristic
262C2×C4×C31C2×#4111yescharacteristic
263C8×C31—24no—
264C8×C31—42no—
265C22×AGL(1,ℤ/8)C2×C3111yescharacteristic
266C22×C4×D4C2×C3111yescharacteristic, complemented
267C2×#417C2×C3111yescharacteristic, complemented
268C2×#418C2×C3111yescharacteristic, complemented
269#2297C2×C3111yescharacteristic, complemented
270#1879C2×C3114yescomplemented
271#1877C2×C3114yescomplemented
272#1736C2×C3111yescharacteristic, complemented
273#1728C2×C3111yescharacteristic, complemented
274C22×C31—24no—
275C22×C31—22no—
276C22×C31—22no—
277C22×C31—24no—
278C22×C31—28no—
279C22×C31—432no—
280C22×C31—48no—
281C22×C31—44no—
282C22×C31—44no—
283C22×C31—48no—
284C22×C31—44no—
285C22×C31—42no—
286C22×C31—81no—
287C22×C31—81no—
288C22×C31—82no—
289C22×C31#38614yes—
290C22×C31C2×C4×D411yescharacteristic
291C22×C31C2×#12011yescharacteristic
292C22×C31C2×#12111yescharacteristic
293C22×C31C2×AGL(1,ℤ/8)12yes—
294C22×C31#57512yes—
295C4×C31—24no—
296C4×C31—44no—
297C4×C31—42no—
298C4×C31—42no—
299C4×C31—82no—
300C4×C31#38614yes—
301C4×C31C22×C22⋊C411yescharacteristic
302C4×C31C2×C4×D411yescharacteristic
303C4×C31C2×#12111yescharacteristic
304C4×C31C2×#12311yescharacteristic
305C22×#41C22×C3111yescharacteristic
306C23×D4C22×C3111yescharacteristic
307C24×C4—21no—
308C2×AGL(1,ℤ/8)—24no—
309C2×AGL(1,ℤ/8)—24no—
310C2×AGL(1,ℤ/8)C4×C3118yescomplemented
311C22×SD16C22×C3112yes—
312C22×D8C22×C3112yes—
313C22×M16C22×C3111yescharacteristic
314C22×M16C22×C3111yescharacteristic
315C2×C4×D4—24no—
316C2×C4×D4—24no—
317C2×C4×D4—22no—
318C2×C4×D4C22×C3114yes—
319C2×#117C22×C3111yescharacteristic
320C22×C4⋊C4C22×C3111yescharacteristic
321C22×C22⋊C4—21no—
322C22×C42C22×C3111yescharacteristic
323#418—24no—
324#417—24no—
325C2×#103C22×C3112yescomplemented
326C2×#102C22×C3114yes—
327C2×#102C22×C3112yescomplemented
328#391C22×C3112yescomplemented
329#390C22×C3112yescomplemented
330#386—21no—
331#386—21no—
332C4×C22⋊C4—22no—
333#340C22×C3114yescomplemented
334#336C22×C3114yescomplemented
335C2×C31—24no—
336C2×C31—48no—
337C2×C31—82no—
338C2×C31C2×#38611yescharacteristic
339C2×C31C4×AGL(1,ℤ/8)12yes—
340C2×C31#300311yescharacteristic
341C2×C31#300511yescharacteristic
342C2×C31#303612yes—
343C25—21no—
344C2×#41—22no—
345C2×#41—22no—
346C2×#41—22no—
347C2×#41—22no—
348C2×#41—22no—
349C2×#41C2×C4×C3114yescomplemented
350C22×Q8C23×C3111yescharacteristic
351C22×D4—21no—
352C22×D4—21no—
353C22×D4—22no—
354C22×D4—22no—
355C22×D4—22no—
356C22×D4C2×C4×C3114yes—
357C22×D4D4×C3114yes—
358C22×D4D4×C3114yes—
359C22×D4C23×C3111yescharacteristic
360C22×D4C23×C3112yes—
361C23×C4—28no—
362C23×C4—21no—
363C23×C4—24no—
364C23×C4—21no—
365C23×C4—48no—
366C23×C4D4×C3111yescharacteristic
367C23×C4D4×C3111yescharacteristic
368C23×C4C23×C3111yescharacteristic
369AGL(1,ℤ/8)—232no—
370C2×SD16—24no—
371C2×SD16—24no—
372C2×SD16C2×C4×C3118yescomplemented
373C2×D8—24no—
374C2×D8—24no—
375C2×D8C2×C4×C3118yescomplemented
376C2×M16—22no—
377C2×M16—22no—
378C2×M16—22no—
379C2×M16—22no—
380C2×M16—22no—
381C2×M16C2×C4×C3114yescomplemented
382C22×C8C23×C3112yes—
383C22×C8C23×C3112yes—
384C4×D4—216no—
385C4×D4—28no—
386C4×D4—28no—
387#117—22no—
388#117—22no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—22no—
391C2×C4⋊C4D4×C3114yes—
392C2×C4⋊C4C23×C3112yes—
393C2×C4⋊C4C23×C3112yes—
394C2×C22⋊C4—28no—
395C2×C22⋊C4—24no—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—22no—
398C2×C22⋊C4—22no—
399C2×C42—22no—
400C2×C42—22no—
401C2×C42—21no—
402C2×C42D4×C3112yes—
403C2×C42C23×C3112yes—
404#103—24no—
405#102—28no—
406#102—24no—
407#95—22no—
408#95—22no—
409C31#2092511yescharacteristic, direct product factor, Sylow
410C24—24no—
411C24—24no—
412C24—24no—
413C24—22no—
414C24—21no—
415C24—44no—
416C24—44no—
417C24C2×D4×C3111yescharacteristic
418#41—28no—
419#41—28no—
420#41—44no—
421#41—44no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8C22×C4×C3112yescomplemented
428C2×D4—24no—
429C2×D4—28no—
430C2×D4—24no—
431C2×D4—24no—
432C2×D4—28no—
433C2×D4—24no—
434C2×D4—28no—
435C2×D4—22no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—42no—
444C2×D4—42no—
445C2×D4C22⋊C4×C31116yescomplemented
446C2×D4C22×C4×C3114yes—
447C2×D4C22×C4×C3112yescomplemented
448C2×D4C2×D4×C3114yes—
449C2×D4C2×D4×C3114yes—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—216no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—42no—
460C22×C4—432no—
461C22×C4—48no—
462C22×C4—44no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—48no—
466C22×C4C22×C4×C3112yes—
467C22×C4C2×D4×C3112yes—
468C22×C4C2×D4×C3112yes—
469C22×C4C2×D4×C3112yes—
470C22×C4C2×D4×C3112yes—
471C22×C4C2×D4×C3111yescharacteristic
472C22×C4C2×D4×C3112yes—
473C22×C4#41×C3112yes—
474C22×C4#41×C3112yes—
475C22×C4C24×C3111yescharacteristic
476SD16—216no—
477D8—216no—
478M16—28no—
479M16—44no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8C22×C4×C3114yescomplemented
486C4⋊C4—24no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4C2×D4×C3114yes—
491C22⋊C4—216no—
492C22⋊C4—44no—
493C22⋊C4—42no—
494C22⋊C4—44no—
495C42—24no—
496C42—22no—
497C42—22no—
498C42—42no—
499C23—22no—
500C23—22no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—216no—
505C23—22no—
506C23—48no—
507C23—416no—
508C23—48no—
509C23—416no—
510C23—44no—
511C23—44no—
512C23—41no—
513C23—41no—
514C23—42no—
515C23—42no—
516C23C4×D4×C3112yes—
517C23C31×#12012yes—
518C23C31×#12112yes—
519C23C22×D4×C3111yescharacteristic
520Q8—22no—
521Q8—22no—
522Q8—41no—
523Q8—41no—
524D4—22no—
525D4—24no—
526D4—22no—
527D4—24no—
528D4—24no—
529D4—216no—
530D4—48no—
531D4—41no—
532D4—41no—
533D4C2×C22⋊C4×C3118yes—
534C2×C4—28no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—432no—
543C2×C4—44no—
544C2×C4—48no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—44no—
548C2×C4—48no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4C2×C22⋊C4×C3114yes—
553C2×C4C4×D4×C3114yes—
554C2×C4C4×D4×C3112yes—
555C2×C4C31×#12012yes—
556C2×C4C31×#12114yes—
557C2×C4C31×#12112yes—
558C2×C4C31×#12314yes—
559C2×C4C31×AGL(1,ℤ/8)14yescomplemented
560C2×C4C23×C4×C3111yescharacteristic
561C2×C4C22×D4×C3111yescharacteristic
562C2×C4C22×D4×C3111yescharacteristic
563C2×C4C22×D4×C3111yescharacteristic
564C2×C4C2×#41×C3111yescharacteristic
565C2×C4C2×#41×C3111yescharacteristic
566C8—24no—
567C8—42no—
568C22—24no—
569C22—22no—
570C22—22no—
571C22—24no—
572C22—28no—
573C22—432no—
574C22—48no—
575C22—44no—
576C22—44no—
577C22—48no—
578C22—44no—
579C22—42no—
580C22—81no—
581C22—81no—
582C22—82no—
583C22C31×#38614yes—
584C22C2×C4×D4×C3111yescharacteristic
585C22C2×C31×#12011yescharacteristic
586C22C2×C31×#12111yescharacteristic
587C22C2×C31×AGL(1,ℤ/8)12yes—
588C22C31×#57512yes—
589C4—24no—
590C4—44no—
591C4—42no—
592C4—42no—
593C4—82no—
594C4C31×#38614yes—
595C4C22×C22⋊C4×C3111yescharacteristic
596C4C2×C4×D4×C3111yescharacteristic
597C4C2×C31×#12111yescharacteristic
598C4C2×C31×#12311yescharacteristic
599C2—24no—
600C2—48no—
601C2—82no—
602C2C2×C31×#38611yescharacteristic
603C2C4×C31×AGL(1,ℤ/8)12yes—
604C2C31×#300311yescharacteristic
605C2C31×#300511yescharacteristic
606C2C31×#303612yes—
subgroupG/H
center, socleC23×C31C22×D4
commutatorC2×C4C23×C4×C31
Frattini, cosocleC22×C4C24×C31
prime radicalC22×C31×AGL(1,ℤ/8)C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C22×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C22×C4×D4×C31C2semidirect
3. C2×C31×#417C2semidirect
4. C2×C31×#418C2semidirect
5. C31×#2297C2semidirect
6. C31×#1879C2semidirect
7. C31×#1877C2semidirect
8. C31×#1736C2semidirect
9. C31×#1728C2semidirect
15. C2×C31×AGL(1,ℤ/8)C4semidirect
30. C2×C31×#103C22semidirect
32. C2×C31×#102C22semidirect
33. C31×#391C22semidirect
34. C31×#390C22semidirect
38. C31×#340C22semidirect
39. C31×#336C22semidirect
46. C2×#41×C31C2×C4semidirect
69. C2×SD16×C31C2×C4semidirect
72. C2×D8×C31C2×C4semidirect
78. C2×M16×C31C2×C4semidirect
123. C2×Q8×C31C22×C4semidirect
141. C2×D4×C31C22⋊C4semidirect
143. C2×D4×C31C22×C4semidirect
181. C2×C8×C31C22×C4semidirect
195. #20925C31direct (N × H)
256. C2×C4×C31AGL(1,ℤ/8)semidirect
266. C22×C4×D4C2×C31semidirect
267. C2×#417C2×C31semidirect
268. C2×#418C2×C31semidirect
269. #2297C2×C31semidirect
270. #1879C2×C31semidirect
271. #1877C2×C31semidirect
272. #1736C2×C31semidirect
273. #1728C2×C31semidirect
310. C2×AGL(1,ℤ/8)C4×C31semidirect
325. C2×#103C22×C31semidirect
327. C2×#102C22×C31semidirect
328. #391C22×C31semidirect
329. #390C22×C31semidirect
333. #340C22×C31semidirect
334. #336C22×C31semidirect
349. C2×#41C2×C4×C31semidirect
372. C2×SD16C2×C4×C31semidirect
375. C2×D8C2×C4×C31semidirect
381. C2×M16C2×C4×C31semidirect
409. C31#20925direct (N × H)
427. C2×Q8C22×C4×C31semidirect
445. C2×D4C22⋊C4×C31semidirect
447. C2×D4C22×C4×C31semidirect
485. C2×C8C22×C4×C31semidirect
559. C2×C4C31×AGL(1,ℤ/8)semidirect

Maximal subgroups

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

subgroupindex
C22×C31×AGL(1,ℤ/8)2
C22×C4×D4×C312
C2×C31×#4172
C2×C31×#4182
C31×#22972
C31×#18792
C31×#18772
C31×#17362
C31×#17282
#2092531