Finite Group Databasebeta

← C31×#20914 #21477673

Subgroups — C31×#20914 #21477673

Order 7936.

1970 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#535C211yesmaximal, characteristic
2C2×C31×#520C211yesmaximal, characteristic, complemented
3C2×C31×#418C211yesmaximal, characteristic, complemented
4C22×C31×#102C211yesmaximal, characteristic, complemented
5C2×C31×#408C211yesmaximal, characteristic, complemented
6C31×#1875C214yesmaximal, complemented
7C31×#1874C214yesmaximal, complemented
8C31×#1736C211yesmaximal, characteristic, complemented
9C31×#1735C211yesmaximal, characteristic, complemented
10C22×#41×C31C2211yescharacteristic
11C23×D4×C31C2211yescharacteristic
12C22×M16×C31C2211yescharacteristic
13C23×C8×C31C2211yescharacteristic
14C31×#535—24no—
15C31×#535C418yescomplemented
16C2×C31×#127—21no—
17C2×C31×#124—21no—
18C2×C31×#121C2214yes—
19C2×C31×#120—22no—
20C31×#520—24no—
21C31×#520—24no—
22C2×C4×Q8×C31—21no—
23C2×C4×D4×C31—21no—
24C2×C4×D4×C31—22no—
25C2×C31×#117C2211yescharacteristic
26C2×C31×#117C2211yescharacteristic
27C2×C31×#117C2212yescomplemented
28C22×C4⋊C4×C31C2211yescharacteristic
29C22×C4⋊C4×C31C2212yescomplemented
30C31×#418—24no—
31C2×C31×#102—24no—
32C2×C31×#102C2214yes—
33C2×C31×#102C2212yes—
34C2×C31×#102C2212yes—
35C31×#408—24no—
36C2×C31×C22⋊C8C2214yescomplemented
37C31×#340C2212yescomplemented
38C31×#340C2214yescomplemented
39C31×#340C2212yescomplemented
40C25×C31—21no—
41C2×#41×C31—22no—
42C2×#41×C31—22no—
43C2×#41×C31—22no—
44C2×#41×C31—22no—
45C2×#41×C31C2×C414yes—
46C2×#41×C31D414yes—
47C22×Q8×C31D411yescharacteristic
48C22×D4×C31—24no—
49C22×D4×C31—22no—
50C22×D4×C31—22no—
51C22×D4×C31—21no—
52C22×D4×C31—22no—
53C22×D4×C31—22no—
54C22×D4×C31—22no—
55C22×D4×C31C2×C414yescomplemented
56C22×D4×C31D412yes—
57C22×D4×C31D411yescharacteristic
58C22×D4×C31C2312yes—
59C23×C4×C31—21no—
60C23×C4×C31—22no—
61C23×C4×C31C2312yes—
62C23×C4×C31C2311yescharacteristic
63C2×M16×C31—22no—
64C2×M16×C31—22no—
65C2×M16×C31—22no—
66C22×C8×C31—22no—
67C22×C8×C31—22no—
68C22×C8×C31—22no—
69C22×C8×C31C2312yes—
70C22×C8×C31C2312yes—
71C31×#127—28no—
72C31×#124—28no—
73C31×#121—28no—
74C31×#121C2×C4116yescomplemented
75C31×#120—216no—
76C4×Q8×C31—28no—
77C4×D4×C31—216no—
78C4×D4×C31—28no—
79C31×#117—22no—
80C31×#117—22no—
81C31×#117—24no—
82C31×#117—24no—
83C31×#117C2×C414yescomplemented
84C31×#117D414yescomplemented
85C2×C4⋊C4×C31—21no—
86C2×C4⋊C4×C31—22no—
87C2×C4⋊C4×C31—22no—
88C2×C4⋊C4×C31—24no—
89C2×C4⋊C4×C31—24no—
90C2×C4⋊C4×C31D414yescomplemented
91C2×C4⋊C4×C31C2312yes—
92C2×C4⋊C4×C31C2312yes—
93C2×C4⋊C4×C31C2311yescharacteristic
94C2×C4⋊C4×C31C2311yescharacteristic
95C2×C22⋊C4×C31—21no—
96C2×C22⋊C4×C31—21no—
97C2×C22⋊C4×C31—22no—
98C2×C22⋊C4×C31—24no—
99C2×C42×C31—21no—
100C2×C42×C31—21no—
101C2×C42×C31—22no—
102C31×#102—28no—
103C31×#102—24no—
104C31×#102—24no—
105C31×C22⋊C8—28no—
106C24×C31—22no—
107C24×C31—24no—
108C24×C31—42no—
109C24×C31—42no—
110C24×C31—82no—
111C24×C31—82no—
112C24×C31C2×D412yes—
113C24×C31C2×D411yescharacteristic
114#41×C31—28no—
115#41×C31—28no—
116#41×C31—28no—
117#41×C31—28no—
118C2×Q8×C31—22no—
119C2×Q8×C31—22no—
120C2×Q8×C31C22⋊C414yes—
121C2×D4×C31—24no—
122C2×D4×C31—24no—
123C2×D4×C31—216no—
124C2×D4×C31—24no—
125C2×D4×C31—28no—
126C2×D4×C31—28no—
127C2×D4×C31—22no—
128C2×D4×C31—22no—
129C2×D4×C31—22no—
130C2×D4×C31—22no—
131C2×D4×C31—22no—
132C2×D4×C31—22no—
133C2×D4×C31—22no—
134C2×D4×C31—42no—
135C2×D4×C31—44no—
136C2×D4×C31—44no—
137C2×D4×C31—42no—
138C2×D4×C31—84no—
139C2×D4×C31—84no—
140C2×D4×C31C22⋊C418yes—
141C2×D4×C31C22⋊C414yes—
142C2×D4×C31C22×C414yes—
143C22×C4×C31—22no—
144C22×C4×C31—22no—
145C22×C4×C31—22no—
146C22×C4×C31—22no—
147C22×C4×C31—21no—
148C22×C4×C31—22no—
149C22×C4×C31—21no—
150C22×C4×C31—22no—
151C22×C4×C31—28no—
152C22×C4×C31—24no—
153C22×C4×C31—82no—
154C22×C4×C31—84no—
155C22×C4×C31C22×C414yes—
156C22×C4×C31C22×C412yes—
157C22×C4×C31C2×D412yes—
158C22×C4×C31C2×D414yes—
159C22×C4×C31C2×D411yescharacteristic
160C22×C4×C31C2×D412yes—
161C22×C4×C31C2×D412yes—
162C22×C4×C31C2×D412yes—
163C22×C4×C31#4112yes—
164C22×C4×C31#4112yes—
165C22×C4×C31C2411yescharacteristic
166M16×C31—44no—
167C2×C8×C31—22no—
168C2×C8×C31—22no—
169C2×C8×C31—22no—
170C2×C8×C31—22no—
171C2×C8×C31—22no—
172C2×C8×C31—22no—
173C2×C8×C31—44no—
174C4⋊C4×C31—28no—
175C4⋊C4×C31—24no—
176C4⋊C4×C31—22no—
177C4⋊C4×C31—21no—
178C4⋊C4×C31—21no—
179C4⋊C4×C31—22no—
180C4⋊C4×C31C22×C414yes—
181C4⋊C4×C31C2×D412yes—
182C4⋊C4×C31C2×D414yes—
183C4⋊C4×C31C2×D412yes—
184C22⋊C4×C31—28no—
185C22⋊C4×C31—24no—
186C22⋊C4×C31—24no—
187C22⋊C4×C31—216no—
188C42×C31—28no—
189C42×C31—24no—
190C42×C31—24no—
191#20914C3111yesmaximal, characteristic, direct product factor, Sylow
192C23×C31—22no—
193C23×C31—22no—
194C23×C31—22no—
195C23×C31—22no—
196C23×C31—24no—
197C23×C31—24no—
198C23×C31—24no—
199C23×C31—42no—
200C23×C31—42no—
201C23×C31—44no—
202C23×C31—44no—
203C23×C31—48no—
204C23×C31—88no—
205C23×C31—88no—
206C23×C31—84no—
207C23×C31—84no—
208C23×C31C2×C22⋊C414yes—
209C23×C31C2×C22⋊C412yes—
210C23×C31#12012yes—
211C23×C31#12312yes—
212C23×C31C22×D411yescharacteristic
213Q8×C31—28no—
214D4×C31—216no—
215D4×C31—24no—
216D4×C31—24no—
217D4×C31—28no—
218D4×C31—42no—
219D4×C31—42no—
220D4×C31—48no—
221D4×C31—816no—
222C2×C4×C31—22no—
223C2×C4×C31—22no—
224C2×C4×C31—24no—
225C2×C4×C31—24no—
226C2×C4×C31—24no—
227C2×C4×C31—24no—
228C2×C4×C31—24no—
229C2×C4×C31—22no—
230C2×C4×C31—24no—
231C2×C4×C31—22no—
232C2×C4×C31—44no—
233C2×C4×C31—42no—
234C2×C4×C31—42no—
235C2×C4×C31—42no—
236C2×C4×C31—42no—
237C2×C4×C31—84no—
238C2×C4×C31—88no—
239C2×C4×C31C2×C22⋊C414yes—
240C2×C4×C31C2×C22⋊C412yes—
241C2×C4×C31C4×D414yes—
242C2×C4×C31C4×D414yes—
243C2×C4×C31#12012yes—
244C2×C4×C31#12114yes—
245C2×C4×C31#12114yes—
246C2×C4×C31#12312yes—
247C2×C4×C31C23×C411yescharacteristic
248C2×C4×C31C22×D411yescharacteristic
249C2×C4×C31C22×D411yescharacteristic
250C2×C4×C31C22×D411yescharacteristic
251C2×C4×C31C2×#4111yescharacteristic
252C2×C4×C31C2×#4111yescharacteristic
253C8×C31—42no—
254C8×C31—42no—
255C2×#535C2×C3111yescharacteristic
256C2×#520C2×C3111yescharacteristic, complemented
257C2×#418C2×C3111yescharacteristic, complemented
258C22×#102C2×C3111yescharacteristic, complemented
259C2×#408C2×C3111yescharacteristic, complemented
260#1875C2×C3114yescomplemented
261#1874C2×C3114yescomplemented
262#1736C2×C3111yescharacteristic, complemented
263#1735C2×C3111yescharacteristic, complemented
264C22×C31—24no—
265C22×C31—24no—
266C22×C31—22no—
267C22×C31—22no—
268C22×C31—28no—
269C22×C31—24no—
270C22×C31—44no—
271C22×C31—42no—
272C22×C31—42no—
273C22×C31—48no—
274C22×C31—44no—
275C22×C31—816no—
276C22×C31—84no—
277C22×C31—82no—
278C22×C31—82no—
279C22×C31#38614yes—
280C22×C31C22×C22⋊C411yescharacteristic
281C22×C31C2×#12011yescharacteristic
282C22×C31C2×#12311yescharacteristic
283C22×C31#57612yes—
284C22×C31#57712yes—
285C4×C31—24no—
286C4×C31—44no—
287C4×C31—44no—
288C4×C31—42no—
289C4×C31—42no—
290C4×C31#38614yes—
291C4×C31C2×C4×D411yescharacteristic
292C4×C31C2×C4×D411yescharacteristic
293C4×C31C2×#12111yescharacteristic
294C4×C31C2×#12111yescharacteristic
295C22×#41C22×C3111yescharacteristic
296C23×D4C22×C3111yescharacteristic
297C22×M16C22×C3111yescharacteristic
298C23×C8C22×C3111yescharacteristic
299#535—24no—
300#535C4×C3118yescomplemented
301C2×#127—21no—
302C2×#124—21no—
303C2×#121C22×C3114yes—
304C2×#120—22no—
305#520—24no—
306#520—24no—
307C2×C4×Q8—21no—
308C2×C4×D4—21no—
309C2×C4×D4—22no—
310C2×#117C22×C3111yescharacteristic
311C2×#117C22×C3111yescharacteristic
312C2×#117C22×C3112yescomplemented
313C22×C4⋊C4C22×C3111yescharacteristic
314C22×C4⋊C4C22×C3112yescomplemented
315#418—24no—
316C2×#102—24no—
317C2×#102C22×C3114yes—
318C2×#102C22×C3112yes—
319C2×#102C22×C3112yes—
320#408—24no—
321C2×C22⋊C8C22×C3114yescomplemented
322#340C22×C3112yescomplemented
323#340C22×C3114yescomplemented
324#340C22×C3112yescomplemented
325C2×C31—24no—
326C2×C31—44no—
327C2×C31—84no—
328C2×C31C2×#38611yescharacteristic
329C2×C31#289712yes—
330C2×C31#300811yescharacteristic
331C2×C31#301011yescharacteristic
332C2×C31#309612yes—
333C25—21no—
334C2×#41—22no—
335C2×#41—22no—
336C2×#41—22no—
337C2×#41—22no—
338C2×#41C2×C4×C3114yes—
339C2×#41D4×C3114yes—
340C22×Q8D4×C3111yescharacteristic
341C22×D4—24no—
342C22×D4—22no—
343C22×D4—22no—
344C22×D4—21no—
345C22×D4—22no—
346C22×D4—22no—
347C22×D4—22no—
348C22×D4C2×C4×C3114yescomplemented
349C22×D4D4×C3112yes—
350C22×D4D4×C3111yescharacteristic
351C22×D4C23×C3112yes—
352C23×C4—21no—
353C23×C4—22no—
354C23×C4C23×C3112yes—
355C23×C4C23×C3111yescharacteristic
356C2×M16—22no—
357C2×M16—22no—
358C2×M16—22no—
359C22×C8—22no—
360C22×C8—22no—
361C22×C8—22no—
362C22×C8C23×C3112yes—
363C22×C8C23×C3112yes—
364#127—28no—
365#124—28no—
366#121—28no—
367#121C2×C4×C31116yescomplemented
368#120—216no—
369C4×Q8—28no—
370C4×D4—216no—
371C4×D4—28no—
372#117—22no—
373#117—22no—
374#117—24no—
375#117—24no—
376#117C2×C4×C3114yescomplemented
377#117D4×C3114yescomplemented
378C2×C4⋊C4—21no—
379C2×C4⋊C4—22no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—24no—
382C2×C4⋊C4—24no—
383C2×C4⋊C4D4×C3114yescomplemented
384C2×C4⋊C4C23×C3112yes—
385C2×C4⋊C4C23×C3112yes—
386C2×C4⋊C4C23×C3111yescharacteristic
387C2×C4⋊C4C23×C3111yescharacteristic
388C2×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—22no—
391C2×C22⋊C4—24no—
392C2×C42—21no—
393C2×C42—21no—
394C2×C42—22no—
395#102—28no—
396#102—24no—
397#102—24no—
398C22⋊C8—28no—
399C31#2091411yescharacteristic, direct product factor, Sylow
400C24—22no—
401C24—24no—
402C24—42no—
403C24—42no—
404C24—82no—
405C24—82no—
406C24C2×D4×C3112yes—
407C24C2×D4×C3111yescharacteristic
408#41—28no—
409#41—28no—
410#41—28no—
411#41—28no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8C22⋊C4×C3114yes—
415C2×D4—24no—
416C2×D4—24no—
417C2×D4—216no—
418C2×D4—24no—
419C2×D4—28no—
420C2×D4—28no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—42no—
429C2×D4—44no—
430C2×D4—44no—
431C2×D4—42no—
432C2×D4—84no—
433C2×D4—84no—
434C2×D4C22⋊C4×C3118yes—
435C2×D4C22⋊C4×C3114yes—
436C2×D4C22×C4×C3114yes—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—28no—
446C22×C4—24no—
447C22×C4—82no—
448C22×C4—84no—
449C22×C4C22×C4×C3114yes—
450C22×C4C22×C4×C3112yes—
451C22×C4C2×D4×C3112yes—
452C22×C4C2×D4×C3114yes—
453C22×C4C2×D4×C3111yescharacteristic
454C22×C4C2×D4×C3112yes—
455C22×C4C2×D4×C3112yes—
456C22×C4C2×D4×C3112yes—
457C22×C4#41×C3112yes—
458C22×C4#41×C3112yes—
459C22×C4C24×C3111yescharacteristic
460M16—44no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—44no—
468C4⋊C4—28no—
469C4⋊C4—24no—
470C4⋊C4—22no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—22no—
474C4⋊C4C22×C4×C3114yes—
475C4⋊C4C2×D4×C3112yes—
476C4⋊C4C2×D4×C3114yes—
477C4⋊C4C2×D4×C3112yes—
478C22⋊C4—28no—
479C22⋊C4—24no—
480C22⋊C4—24no—
481C22⋊C4—216no—
482C42—28no—
483C42—24no—
484C42—24no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—42no—
493C23—42no—
494C23—44no—
495C23—44no—
496C23—48no—
497C23—88no—
498C23—88no—
499C23—84no—
500C23—84no—
501C23C2×C22⋊C4×C3114yes—
502C23C2×C22⋊C4×C3112yes—
503C23C31×#12012yes—
504C23C31×#12312yes—
505C23C22×D4×C3111yescharacteristic
506Q8—28no—
507D4—216no—
508D4—24no—
509D4—24no—
510D4—28no—
511D4—42no—
512D4—42no—
513D4—48no—
514D4—816no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—84no—
531C2×C4—88no—
532C2×C4C2×C22⋊C4×C3114yes—
533C2×C4C2×C22⋊C4×C3112yes—
534C2×C4C4×D4×C3114yes—
535C2×C4C4×D4×C3114yes—
536C2×C4C31×#12012yes—
537C2×C4C31×#12114yes—
538C2×C4C31×#12114yes—
539C2×C4C31×#12312yes—
540C2×C4C23×C4×C3111yescharacteristic
541C2×C4C22×D4×C3111yescharacteristic
542C2×C4C22×D4×C3111yescharacteristic
543C2×C4C22×D4×C3111yescharacteristic
544C2×C4C2×#41×C3111yescharacteristic
545C2×C4C2×#41×C3111yescharacteristic
546C8—42no—
547C8—42no—
548C22—24no—
549C22—24no—
550C22—22no—
551C22—22no—
552C22—28no—
553C22—24no—
554C22—44no—
555C22—42no—
556C22—42no—
557C22—48no—
558C22—44no—
559C22—816no—
560C22—84no—
561C22—82no—
562C22—82no—
563C22C31×#38614yes—
564C22C22×C22⋊C4×C3111yescharacteristic
565C22C2×C31×#12011yescharacteristic
566C22C2×C31×#12311yescharacteristic
567C22C31×#57612yes—
568C22C31×#57712yes—
569C4—24no—
570C4—44no—
571C4—44no—
572C4—42no—
573C4—42no—
574C4C31×#38614yes—
575C4C2×C4×D4×C3111yescharacteristic
576C4C2×C4×D4×C3111yescharacteristic
577C4C2×C31×#12111yescharacteristic
578C4C2×C31×#12111yescharacteristic
579C2—24no—
580C2—44no—
581C2—84no—
582C2C2×C31×#38611yescharacteristic
583C2C31×#289712yes—
584C2C31×#300811yescharacteristic
585C2C31×#301011yescharacteristic
586C2C31×#309612yes—
subgroupG/H
center, socleC23×C31C22×D4
commutatorC2×C4C23×C4×C31
Frattini, cosocleC22×C4C24×C31
prime radicalC2×C31×#535C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C22×#41×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×#520C2semidirect
3. C2×C31×#418C2semidirect
4. C22×C31×#102C2semidirect
5. C2×C31×#408C2semidirect
6. C31×#1875C2semidirect
7. C31×#1874C2semidirect
8. C31×#1736C2semidirect
9. C31×#1735C2semidirect
15. C31×#535C4semidirect
27. C2×C31×#117C22semidirect
29. C22×C4⋊C4×C31C22semidirect
36. C2×C31×C22⋊C8C22semidirect
37. C31×#340C22semidirect
38. C31×#340C22semidirect
39. C31×#340C22semidirect
55. C22×D4×C31C2×C4semidirect
74. C31×#121C2×C4semidirect
83. C31×#117C2×C4semidirect
84. C31×#117D4semidirect
90. C2×C4⋊C4×C31D4semidirect
191. #20914C31direct (N × H)
256. C2×#520C2×C31semidirect
257. C2×#418C2×C31semidirect
258. C22×#102C2×C31semidirect
259. C2×#408C2×C31semidirect
260. #1875C2×C31semidirect
261. #1874C2×C31semidirect
262. #1736C2×C31semidirect
263. #1735C2×C31semidirect
300. #535C4×C31semidirect
312. C2×#117C22×C31semidirect
314. C22×C4⋊C4C22×C31semidirect
321. C2×C22⋊C8C22×C31semidirect
322. #340C22×C31semidirect
323. #340C22×C31semidirect
324. #340C22×C31semidirect
348. C22×D4C2×C4×C31semidirect
367. #121C2×C4×C31semidirect
376. #117C2×C4×C31semidirect
377. #117D4×C31semidirect
383. C2×C4⋊C4D4×C31semidirect
399. C31#20914direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5352
C2×C31×#5202
C2×C31×#4182
C22×C31×#1022
C2×C31×#4082
C31×#18752
C31×#18742
C31×#17362
C31×#17352
#2091431