Finite Group Databasebeta

← C31×#25614 #21482373

Subgroups — C31×#25614 #21482373

Order 7936.

1216 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#542C211yesmaximal, characteristic, complemented
2C2×C31×#496C211yesmaximal, characteristic, complemented
3C2×C31×#486C211yesmaximal, characteristic, complemented
4C31×#2619C211yesmaximal, characteristic, complemented
5C31×#2000C212yesmaximal, complemented
6C31×#1999C212yesmaximal, complemented
7C31×#1993C211yesmaximal, characteristic, complemented
8C31×#1958C211yesmaximal, characteristic, complemented
9C31×#1880C212yesmaximal, complemented
10C31×#1879C212yesmaximal, complemented
11C31×#1750C211yesmaximal, characteristic, complemented
12C22×SD16×C31C2212yes—
13C22×D8×C31C2212yes—
14C31×#542—24no—
15C31×#542—24no—
16C2×C31×C4⋊Q8C2211yescharacteristic
17C2×C31×#127C2211yescharacteristic
18C2×C31×#126—21no—
19C2×C31×#124—21no—
20C2×C31×#124C2211yescharacteristic
21C2×C31×#123—21no—
22C2×C31×C22⋊Q8—21no—
23C2×C31×C22⋊Q8C2211yescharacteristic
24C2×C31×#121C2211yescharacteristic
25C2×C31×#121C2212yes—
26C2×C4×Q8×C31C2211yescharacteristic
27C2×C4×D4×C31C2211yescharacteristic
28C2×C31×#117C2211yescharacteristic
29C31×#496—24no—
30C31×#486—24no—
31C2×C31×#103C2212yes—
32C2×C31×#102C2212yes—
33C2×C31×#102C2212yes—
34C2×C31×#102C2212yes—
35C2×C31×C22⋊C8C2212yes—
36C2×C31×#97C2211yescharacteristic
37C2×C4×C8×C31C2211yescharacteristic
38C31×#394—21no—
39C31×#394—21no—
40C31×#392—21no—
41C31×#392—21no—
42C31×#390C2212yes—
43C31×#382C2211yescharacteristic
44C31×#340C2212yes—
45C31×#340C2212yes—
46C31×#336C2212yes—
47C22×Q8×C31—21no—
48C22×Q8×C31C2311yescharacteristic
49C22×D4×C31—21no—
50C22×D4×C31—22no—
51C22×D4×C31—21no—
52C22×D4×C31—21no—
53C22×D4×C31C2312yes—
54C22×D4×C31C2311yescharacteristic
55C23×C4×C31—21no—
56C23×C4×C31C2311yescharacteristic
57C2×SD16×C31—24no—
58C2×SD16×C31—24no—
59C2×SD16×C31—42no—
60C2×D8×C31—24no—
61C2×D8×C31—24no—
62C2×D8×C31—42no—
63C22×C8×C31C2312yes—
64C22×C8×C31C2312yes—
65C31×C4⋊Q8—22no—
66C31×C4⋊Q8D414yescomplemented
67C31×#127—22no—
68C31×#127—22no—
69C31×#126—28no—
70C31×#124—28no—
71C31×#124—22no—
72C31×#124D414yescomplemented
73C31×#123—28no—
74C31×C22⋊Q8—28no—
75C31×C22⋊Q8—22no—
76C31×C22⋊Q8—22no—
77C31×#121—22no—
78C31×#121—22no—
79C31×#121—42no—
80C31×#121—42no—
81C4×Q8×C31—22no—
82C4×Q8×C31—22no—
83C4×D4×C31—22no—
84C4×D4×C31—22no—
85C31×#117—22no—
86C31×#117—22no—
87C2×C4⋊C4×C31—21no—
88C2×C4⋊C4×C31—21no—
89C2×C4⋊C4×C31—21no—
90C2×C4⋊C4×C31D411yescharacteristic
91C2×C4⋊C4×C31D411yescharacteristic
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—21no—
98C2×C22⋊C4×C31—21no—
99C2×C22⋊C4×C31—21no—
100C2×C22⋊C4×C31—22no—
101C2×C22⋊C4×C31—22no—
102C2×C22⋊C4×C31—22no—
103C2×C22⋊C4×C31D411yescharacteristic
104C2×C22⋊C4×C31D411yescharacteristic
105C2×C42×C31—21no—
106C2×C42×C31C2311yescharacteristic
107C2×C42×C31C2311yescharacteristic
108C31×#103—42no—
109C31×#102—24no—
110C31×#102—24no—
111C31×#102—42no—
112C31×C22⋊C8—24no—
113C31×#97—22no—
114C4×C8×C31—22no—
115C31×#95—21no—
116C31×#95—21no—
117C24×C31—21no—
118C24×C31—22no—
119C24×C31C2×D411yescharacteristic
120C2×Q8×C31—24no—
121C2×Q8×C31—21no—
122C2×Q8×C31—22no—
123C2×Q8×C31—22no—
124C2×Q8×C31—82no—
125C2×Q8×C31C2×D412yes—
126C2×D4×C31—24no—
127C2×D4×C31—21no—
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—22no—
135C2×D4×C31—22no—
136C2×D4×C31—44no—
137C2×D4×C31—42no—
138C2×D4×C31—42no—
139C2×D4×C31—82no—
140C2×D4×C31—82no—
141C2×D4×C31—82no—
142C2×D4×C31—82no—
143C2×D4×C31—82no—
144C2×D4×C31C2×D412yes—
145C22×C4×C31—21no—
146C22×C4×C31—21no—
147C22×C4×C31—21no—
148C22×C4×C31—24no—
149C22×C4×C31—21no—
150C22×C4×C31—21no—
151C22×C4×C31—22no—
152C22×C4×C31—22no—
153C22×C4×C31—22no—
154C22×C4×C31—82no—
155C22×C4×C31C2×D411yescharacteristic
156C22×C4×C31C2×D411yescharacteristic
157C22×C4×C31C2×D411yescharacteristic
158C22×C4×C31C2×D411yescharacteristic
159C22×C4×C31C2×D411yescharacteristic
160C22×C4×C31C2411yescharacteristic
161SD16×C31—48no—
162D8×C31—48no—
163C2×C8×C31—22no—
164C2×C8×C31—22no—
165C2×C8×C31—22no—
166C2×C8×C31—22no—
167C2×C8×C31—22no—
168C2×C8×C31#4114yes—
169C4⋊C4×C31—24no—
170C4⋊C4×C31—24no—
171C4⋊C4×C31—24no—
172C4⋊C4×C31—22no—
173C4⋊C4×C31—22no—
174C4⋊C4×C31—21no—
175C4⋊C4×C31—21no—
176C4⋊C4×C31—22no—
177C4⋊C4×C31—22no—
178C4⋊C4×C31C2×D412yes—
179C4⋊C4×C31C2×D412yes—
180C22⋊C4×C31—24no—
181C22⋊C4×C31—24no—
182C22⋊C4×C31—24no—
183C22⋊C4×C31—24no—
184C22⋊C4×C31—24no—
185C22⋊C4×C31—22no—
186C22⋊C4×C31—22no—
187C22⋊C4×C31—44no—
188C22⋊C4×C31—82no—
189C22⋊C4×C31—82no—
190C42×C31—24no—
191C42×C31—21no—
192C42×C31—21no—
193C42×C31C2×D412yes—
194C42×C31C2×D412yes—
195#25614C3111yesmaximal, characteristic, direct product factor, Sylow
196C23×C31—21no—
197C23×C31—22no—
198C23×C31—21no—
199C23×C31—21no—
200C23×C31—22no—
201C23×C31—22no—
202C23×C31—41no—
203C23×C31—41no—
204C23×C31—42no—
205C23×C31—42no—
206C23×C31—42no—
207C23×C31—42no—
208C23×C31—42no—
209C23×C31—42no—
210C23×C31—42no—
211C23×C31—42no—
212C23×C31—42no—
213C23×C31C22×D411yescharacteristic
214Q8×C31—24no—
215Q8×C31—42no—
216Q8×C31—84no—
217D4×C31—24no—
218D4×C31—42no—
219D4×C31—42no—
220D4×C31—42no—
221D4×C31—42no—
222D4×C31—42no—
223D4×C31—84no—
224D4×C31—84no—
225D4×C31—84no—
226D4×C31—88no—
227C2×C4×C31—22no—
228C2×C4×C31—22no—
229C2×C4×C31—21no—
230C2×C4×C31—22no—
231C2×C4×C31—21no—
232C2×C4×C31—22no—
233C2×C4×C31—21no—
234C2×C4×C31—22no—
235C2×C4×C31—21no—
236C2×C4×C31—21no—
237C2×C4×C31—22no—
238C2×C4×C31—22no—
239C2×C4×C31—41no—
240C2×C4×C31—41no—
241C2×C4×C31—42no—
242C2×C4×C31—42no—
243C2×C4×C31—42no—
244C2×C4×C31—42no—
245C2×C4×C31—42no—
246C2×C4×C31—42no—
247C2×C4×C31—42no—
248C2×C4×C31—84no—
249C2×C4×C31#12012yes—
250C2×C4×C31#12012yes—
251C2×C4×C31C22×D411yescharacteristic
252C2×C4×C31C22×D411yescharacteristic
253C2×C4×C31C2×#4112yes—
254C2×C4×C31H(5,𝔽2)12yes—
255C8×C31—24no—
256C8×C31—42no—
257C2×#542C2×C3111yescharacteristic, complemented
258C2×#496C2×C3111yescharacteristic, complemented
259C2×#486C2×C3111yescharacteristic, complemented
260#2619C2×C3111yescharacteristic, complemented
261#2000C2×C3112yescomplemented
262#1999C2×C3112yescomplemented
263#1993C2×C3111yescharacteristic, complemented
264#1958C2×C3111yescharacteristic, complemented
265#1880C2×C3112yescomplemented
266#1879C2×C3112yescomplemented
267#1750C2×C3111yescharacteristic, complemented
268C22×C31—42no—
269C22×C31—41no—
270C22×C31—42no—
271C22×C31—42no—
272C22×C31—42no—
273C22×C31—41no—
274C22×C31—41no—
275C22×C31—42no—
276C22×C31—82no—
277C22×C31—82no—
278C22×C31—82no—
279C22×C31—82no—
280C22×C31—82no—
281C22×C31—82no—
282C22×C31—82no—
283C22×C31—82no—
284C22×C31—82no—
285C22×C31C2×#12011yescharacteristic
286C22×C31#53211yescharacteristic
287C22×C31#53511yescharacteristic
288C22×C31#57612yes—
289C22×C31#57712yes—
290C4×C31—22no—
291C4×C31—22no—
292C4×C31—41no—
293C4×C31—42no—
294C4×C31—41no—
295C4×C31—42no—
296C4×C31—42no—
297C4×C31—41no—
298C4×C31—82no—
299C4×C31#54612yes—
300C4×C31#55012yes—
301C22×SD16C22×C3112yes—
302C22×D8C22×C3112yes—
303#542—24no—
304#542—24no—
305C2×C4⋊Q8C22×C3111yescharacteristic
306C2×#127C22×C3111yescharacteristic
307C2×#126—21no—
308C2×#124—21no—
309C2×#124C22×C3111yescharacteristic
310C2×#123—21no—
311C2×C22⋊Q8—21no—
312C2×C22⋊Q8C22×C3111yescharacteristic
313C2×#121C22×C3111yescharacteristic
314C2×#121C22×C3112yes—
315C2×C4×Q8C22×C3111yescharacteristic
316C2×C4×D4C22×C3111yescharacteristic
317C2×#117C22×C3111yescharacteristic
318#496—24no—
319#486—24no—
320C2×#103C22×C3112yes—
321C2×#102C22×C3112yes—
322C2×#102C22×C3112yes—
323C2×#102C22×C3112yes—
324C2×C22⋊C8C22×C3112yes—
325C2×#97C22×C3111yescharacteristic
326C2×C4×C8C22×C3111yescharacteristic
327#394—21no—
328#394—21no—
329#392—21no—
330#392—21no—
331#390C22×C3112yes—
332#382C22×C3111yescharacteristic
333#340C22×C3112yes—
334#340C22×C3112yes—
335#336C22×C3112yes—
336C2×C31—81no—
337C2×C31—82no—
338C2×C31—162no—
339C2×C31#243111yescharacteristic
340C2×C31#300811yescharacteristic
341C2×C31#301011yescharacteristic
342C2×C31#317912yes—
343C2×C31#322412yes—
344C22×Q8—21no—
345C22×Q8C23×C3111yescharacteristic
346C22×D4—21no—
347C22×D4—22no—
348C22×D4—21no—
349C22×D4—21no—
350C22×D4C23×C3112yes—
351C22×D4C23×C3111yescharacteristic
352C23×C4—21no—
353C23×C4C23×C3111yescharacteristic
354C2×SD16—24no—
355C2×SD16—24no—
356C2×SD16—42no—
357C2×D8—24no—
358C2×D8—24no—
359C2×D8—42no—
360C22×C8C23×C3112yes—
361C22×C8C23×C3112yes—
362C4⋊Q8—22no—
363C4⋊Q8D4×C3114yescomplemented
364#127—22no—
365#127—22no—
366#126—28no—
367#124—28no—
368#124—22no—
369#124D4×C3114yescomplemented
370#123—28no—
371C22⋊Q8—28no—
372C22⋊Q8—22no—
373C22⋊Q8—22no—
374#121—22no—
375#121—22no—
376#121—42no—
377#121—42no—
378C4×Q8—22no—
379C4×Q8—22no—
380C4×D4—22no—
381C4×D4—22no—
382#117—22no—
383#117—22no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4D4×C3111yescharacteristic
388C2×C4⋊C4D4×C3111yescharacteristic
389C2×C4⋊C4C23×C3112yes—
390C2×C4⋊C4C23×C3111yescharacteristic
391C2×C4⋊C4C23×C3111yescharacteristic
392C2×C22⋊C4—21no—
393C2×C22⋊C4—21no—
394C2×C22⋊C4—21no—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—22no—
398C2×C22⋊C4—22no—
399C2×C22⋊C4—22no—
400C2×C22⋊C4D4×C3111yescharacteristic
401C2×C22⋊C4D4×C3111yescharacteristic
402C2×C42—21no—
403C2×C42C23×C3111yescharacteristic
404C2×C42C23×C3111yescharacteristic
405#103—42no—
406#102—24no—
407#102—24no—
408#102—42no—
409C22⋊C8—24no—
410#97—22no—
411C4×C8—22no—
412#95—21no—
413#95—21no—
414C31#2561411yescharacteristic, direct product factor, Sylow
415C24—21no—
416C24—22no—
417C24C2×D4×C3111yescharacteristic
418C2×Q8—24no—
419C2×Q8—21no—
420C2×Q8—22no—
421C2×Q8—22no—
422C2×Q8—82no—
423C2×Q8C2×D4×C3112yes—
424C2×D4—24no—
425C2×D4—21no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—44no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4—82no—
438C2×D4—82no—
439C2×D4—82no—
440C2×D4—82no—
441C2×D4—82no—
442C2×D4C2×D4×C3112yes—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—24no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—82no—
453C22×C4C2×D4×C3111yescharacteristic
454C22×C4C2×D4×C3111yescharacteristic
455C22×C4C2×D4×C3111yescharacteristic
456C22×C4C2×D4×C3111yescharacteristic
457C22×C4C2×D4×C3111yescharacteristic
458C22×C4C24×C3111yescharacteristic
459SD16—48no—
460D8—48no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8#41×C3114yes—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4C2×D4×C3112yes—
477C4⋊C4C2×D4×C3112yes—
478C22⋊C4—24no—
479C22⋊C4—24no—
480C22⋊C4—24no—
481C22⋊C4—24no—
482C22⋊C4—24no—
483C22⋊C4—22no—
484C22⋊C4—22no—
485C22⋊C4—44no—
486C22⋊C4—82no—
487C22⋊C4—82no—
488C42—24no—
489C42—21no—
490C42—21no—
491C42C2×D4×C3112yes—
492C42C2×D4×C3112yes—
493C23—21no—
494C23—22no—
495C23—21no—
496C23—21no—
497C23—22no—
498C23—22no—
499C23—41no—
500C23—41no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23C22×D4×C3111yescharacteristic
511Q8—24no—
512Q8—42no—
513Q8—84no—
514D4—24no—
515D4—42no—
516D4—42no—
517D4—42no—
518D4—42no—
519D4—42no—
520D4—84no—
521D4—84no—
522D4—84no—
523D4—88no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—84no—
546C2×C4C31×#12012yes—
547C2×C4C31×#12012yes—
548C2×C4C22×D4×C3111yescharacteristic
549C2×C4C22×D4×C3111yescharacteristic
550C2×C4C2×#41×C3112yes—
551C2×C4C31×H(5,𝔽2)12yes—
552C8—24no—
553C8—42no—
554C22—42no—
555C22—41no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—41no—
560C22—41no—
561C22—42no—
562C22—82no—
563C22—82no—
564C22—82no—
565C22—82no—
566C22—82no—
567C22—82no—
568C22—82no—
569C22—82no—
570C22—82no—
571C22C2×C31×#12011yescharacteristic
572C22C31×#53211yescharacteristic
573C22C31×#53511yescharacteristic
574C22C31×#57612yes—
575C22C31×#57712yes—
576C4—22no—
577C4—22no—
578C4—41no—
579C4—42no—
580C4—41no—
581C4—42no—
582C4—42no—
583C4—41no—
584C4—82no—
585C4C31×#54612yes—
586C4C31×#55012yes—
587C2—81no—
588C2—82no—
589C2—162no—
590C2C31×#243111yescharacteristic
591C2C31×#300811yescharacteristic
592C2C31×#301011yescharacteristic
593C2C31×#317912yes—
594C2C31×#322412yes—
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C2×C31×#117 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C31×#542C2semidirect
2. C2×C31×#496C2semidirect
3. C2×C31×#486C2semidirect
4. C31×#2619C2semidirect
5. C31×#2000C2semidirect
6. C31×#1999C2semidirect
7. C31×#1993C2semidirect
8. C31×#1958C2semidirect
9. C31×#1880C2semidirect
10. C31×#1879C2semidirect
11. C31×#1750C2semidirect
66. C31×C4⋊Q8D4semidirect
72. C31×#124D4semidirect
195. #25614C31direct (N × H)
257. C2×#542C2×C31semidirect
258. C2×#496C2×C31semidirect
259. C2×#486C2×C31semidirect
260. #2619C2×C31semidirect
261. #2000C2×C31semidirect
262. #1999C2×C31semidirect
263. #1993C2×C31semidirect
264. #1958C2×C31semidirect
265. #1880C2×C31semidirect
266. #1879C2×C31semidirect
267. #1750C2×C31semidirect
363. C4⋊Q8D4×C31semidirect
369. #124D4×C31semidirect
414. C31#25614direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5422
C2×C31×#4962
C2×C31×#4862
C31×#26192
C31×#20002
C31×#19992
C31×#19932
C31×#19582
C31×#18802
C31×#18792
C31×#17502
#2561431