Finite Group Databasebeta

← C31×#25626 #21482385

Subgroups — C31×#25626 #21482385

Order 7936.

1076 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3507C211yesmaximal, characteristic, complemented
2C31×#3153C211yesmaximal, characteristic
3C31×#3140C211yesmaximal, characteristic, complemented
4C31×#2545C211yesmaximal, characteristic, complemented
5C31×#2005C212yesmaximal, complemented
6C31×#2003C212yesmaximal, complemented
7C31×#1986C211yesmaximal, characteristic, complemented
8C31×#1980C211yesmaximal, characteristic, complemented
9C31×#1883C212yesmaximal, complemented
10C31×#1882C212yesmaximal, complemented
11C4×C31×#101C211yesmaximal, characteristic, complemented
12C2×C31×#137C2212yes—
13C2×C31×AGL(1,ℤ/8)C2212yes—
14C31×#559—21no—
15C31×#557—21no—
16C31×#556—21no—
17C31×#556—22no—
18C31×#554—21no—
19C31×#552—22no—
20C31×#539—22no—
21C31×#536C2212yes—
22C31×#533C2211yescharacteristic
23C31×#532C2211yescharacteristic
24C2×C31×C4⋊Q8C2211yescharacteristic
25C31×#529—22no—
26C2×C31×#124C2211yescharacteristic
27C2×C31×#124C2211yescharacteristic
28C31×#520C2212yes—
29C2×C4×Q8×C31C2211yescharacteristic
30C31×#497—21no—
31C31×#496—21no—
32C31×#495—22no—
33C31×#489—21no—
34C31×#489—21no—
35C31×#488—22no—
36C2×C31×#104C2212yes—
37C2×C31×#104C2212yes—
38C31×#419C2212yes—
39C31×#418C2212yes—
40C2×C31×#101C2212yes—
41C4×M16×C31C2211yescharacteristic
42C4×M16×C31C2211yescharacteristic
43C31×#396—22no—
44C31×#391C2211yescharacteristic
45C31×#390C2211yescharacteristic
46C31×#388—22no—
47C4×C4⋊C4×C31C2211yescharacteristic
48C31×#343C2212yes—
49C31×#339C2212yes—
50C31×#339C2212yes—
51C2×#41×C31C2312yes—
52C22×Q8×C31C2311yescharacteristic
53C22×Q8×C31C2311yescharacteristic
54C22×D4×C31C2311yescharacteristic
55C31×#137—24no—
56C31×#137—42no—
57C31×AGL(1,ℤ/8)—24no—
58C31×AGL(1,ℤ/8)—42no—
59C2×Q16×C31—22no—
60C2×SD16×C31—22no—
61C2×SD16×C31—22no—
62C2×D8×C31—22no—
63C2×M16×C31C2312yes—
64C2×M16×C31C2312yes—
65C31×C4⋊Q8—22no—
66C31×C4⋊Q8—42no—
67C31×C4⋊Q8D411yescharacteristic
68C31×C4⋊Q8D411yescharacteristic
69C31×C4⋊Q8D411yescharacteristic
70C31×#127D411yescharacteristic
71C31×#126—24no—
72C31×#126—24no—
73C31×#125—22no—
74C31×#125—24no—
75C31×#125D412yes—
76C31×#124—22no—
77C31×#124—22no—
78C31×#124—21no—
79C31×#124—21no—
80C31×#124—21no—
81C31×#124—21no—
82C31×#124—42no—
83C31×#124D412yes—
84C31×#123—24no—
85C31×#123—24no—
86C31×C22⋊Q8—24no—
87C31×C22⋊Q8—21no—
88C31×C22⋊Q8—21no—
89C31×C22⋊Q8—22no—
90C31×C22⋊Q8—22no—
91C31×#121—21no—
92C31×#121—21no—
93C4×Q8×C31—22no—
94C4×Q8×C31—22no—
95C4×Q8×C31—21no—
96C4×Q8×C31—21no—
97C4×Q8×C31—21no—
98C4×Q8×C31—21no—
99C4×Q8×C31—21no—
100C4×Q8×C31—21no—
101C4×D4×C31—22no—
102C4×D4×C31—22no—
103C4×D4×C31—24no—
104C4×D4×C31—21no—
105C4×D4×C31—21no—
106C31×#117—24no—
107C31×#117—22no—
108C31×#117C2312yes—
109C2×C4⋊C4×C31—22no—
110C2×C4⋊C4×C31—21no—
111C2×C4⋊C4×C31—22no—
112C2×C4⋊C4×C31D412yes—
113C2×C4⋊C4×C31C2311yescharacteristic
114C2×C22⋊C4×C31—21no—
115C2×C22⋊C4×C31—22no—
116C2×C22⋊C4×C31—22no—
117C2×C22⋊C4×C31—21no—
118C2×C42×C31D412yes—
119C2×C42×C31C2311yescharacteristic
120C2×C42×C31C2311yescharacteristic
121C2×C42×C31C2311yescharacteristic
122C31×#104—42no—
123C31×#104—42no—
124C31×#103—22no—
125C31×#103—22no—
126C31×#102—22no—
127C31×#102—22no—
128C31×#101—24no—
129C31×#97—21no—
130C31×#97—21no—
131C4×C8×C31—21no—
132C4×C8×C31—21no—
133C31×#95—22no—
134C31×#95—21no—
135C24×C31—21no—
136#41×C31—22no—
137#41×C31—22no—
138#41×C31—22no—
139#41×C31—22no—
140C2×Q8×C31—21no—
141C2×Q8×C31—21no—
142C2×Q8×C31—21no—
143C2×Q8×C31—21no—
144C2×Q8×C31—22no—
145C2×Q8×C31—22no—
146C2×Q8×C31—21no—
147C2×Q8×C31—21no—
148C2×Q8×C31—22no—
149C2×Q8×C31—22no—
150C2×Q8×C31C2×D412yes—
151C2×D4×C31—22no—
152C2×D4×C31—21no—
153C2×D4×C31—21no—
154C2×D4×C31—21no—
155C2×D4×C31—21no—
156C2×D4×C31—22no—
157C2×D4×C31—22no—
158C2×D4×C31C2×D412yes—
159C22×C4×C31—21no—
160C22×C4×C31—24no—
161C22×C4×C31—22no—
162C22×C4×C31—21no—
163C22×C4×C31—21no—
164C22×C4×C31C2×D412yes—
165C22×C4×C31C2×D412yes—
166C22×C4×C31C2×D411yescharacteristic
167C22×C4×C31C2×D411yescharacteristic
168C22×C4×C31C2411yescharacteristic
169Q16×C31—44no—
170SD16×C31—44no—
171SD16×C31—44no—
172D8×C31—44no—
173M16×C31—22no—
174M16×C31—22no—
175M16×C31—22no—
176M16×C31#4114yes—
177C2×C8×C31—22no—
178C2×C8×C31—22no—
179C4⋊C4×C31—24no—
180C4⋊C4×C31—24no—
181C4⋊C4×C31—24no—
182C4⋊C4×C31—22no—
183C4⋊C4×C31—22no—
184C4⋊C4×C31—22no—
185C4⋊C4×C31—21no—
186C4⋊C4×C31—21no—
187C4⋊C4×C31—21no—
188C4⋊C4×C31—22no—
189C4⋊C4×C31—21no—
190C4⋊C4×C31—21no—
191C4⋊C4×C31—21no—
192C4⋊C4×C31—42no—
193C4⋊C4×C31—44no—
194C4⋊C4×C31C2×D412yes—
195C4⋊C4×C31C2×D412yes—
196C22⋊C4×C31—24no—
197C22⋊C4×C31—24no—
198C22⋊C4×C31—24no—
199C22⋊C4×C31—22no—
200C22⋊C4×C31—22no—
201C22⋊C4×C31—22no—
202C22⋊C4×C31—44no—
203C22⋊C4×C31—44no—
204C22⋊C4×C31—41no—
205C22⋊C4×C31—41no—
206C22⋊C4×C31—41no—
207C22⋊C4×C31—41no—
208C42×C31—24no—
209C42×C31—22no—
210C42×C31—22no—
211C42×C31—22no—
212C42×C31—22no—
213C42×C31—22no—
214C42×C31C2×D411yescharacteristic
215C42×C31C2×D411yescharacteristic
216C42×C31C2×D411yescharacteristic
217C42×C31C2×D411yescharacteristic
218#25626C3111yesmaximal, characteristic, direct product factor, Sylow
219C23×C31—22no—
220C23×C31—42no—
221C23×C31—42no—
222C23×C31—41no—
223C23×C31—41no—
224C23×C31—41no—
225C23×C31C22×D411yescharacteristic
226Q8×C31—22no—
227Q8×C31—22no—
228Q8×C31—42no—
229Q8×C31—42no—
230Q8×C31—42no—
231Q8×C31—42no—
232Q8×C31—42no—
233Q8×C31—81no—
234D4×C31—22no—
235D4×C31—22no—
236D4×C31—42no—
237D4×C31—42no—
238D4×C31—42no—
239D4×C31—42no—
240D4×C31—42no—
241D4×C31—81no—
242C2×C4×C31—22no—
243C2×C4×C31—22no—
244C2×C4×C31—22no—
245C2×C4×C31—22no—
246C2×C4×C31—22no—
247C2×C4×C31—22no—
248C2×C4×C31—22no—
249C2×C4×C31—22no—
250C2×C4×C31—41no—
251C2×C4×C31—41no—
252C2×C4×C31—41no—
253C2×C4×C31—41no—
254C2×C4×C31—42no—
255C2×C4×C31—42no—
256C2×C4×C31—42no—
257C2×C4×C31—41no—
258C2×C4×C31—41no—
259C2×C4×C31—41no—
260C2×C4×C31—41no—
261C2×C4×C31—41no—
262C2×C4×C31—41no—
263C2×C4×C31—84no—
264C2×C4×C31#12011yescharacteristic
265C2×C4×C31#12011yescharacteristic
266C2×C4×C31#12011yescharacteristic
267C2×C4×C31#12011yescharacteristic
268C2×C4×C31C22×D411yescharacteristic
269C2×C4×C31C22×D411yescharacteristic
270C2×C4×C31C2×#4112yes—
271C2×C4×C31H(5,𝔽2)12yes—
272C8×C31—24no—
273C8×C31—42no—
274#3507C2×C3111yescharacteristic, complemented
275#3153C2×C3111yescharacteristic
276#3140C2×C3111yescharacteristic, complemented
277#2545C2×C3111yescharacteristic, complemented
278#2005C2×C3112yescomplemented
279#2003C2×C3112yescomplemented
280#1986C2×C3111yescharacteristic, complemented
281#1980C2×C3111yescharacteristic, complemented
282#1883C2×C3112yescomplemented
283#1882C2×C3112yescomplemented
284C4×#101C2×C3111yescharacteristic, complemented
285C22×C31—21no—
286C22×C31—21no—
287C22×C31—42no—
288C22×C31—42no—
289C22×C31—42no—
290C22×C31—82no—
291C22×C31—82no—
292C22×C31—81no—
293C22×C31—81no—
294C22×C31—81no—
295C22×C31C2×#12011yescharacteristic
296C22×C31#53211yescharacteristic
297C22×C31#53511yescharacteristic
298C4×C31—21no—
299C4×C31—21no—
300C4×C31—21no—
301C4×C31—21no—
302C4×C31—42no—
303C4×C31—42no—
304C4×C31—42no—
305C4×C31—42no—
306C4×C31—42no—
307C4×C31—42no—
308C4×C31—81no—
309C4×C31—81no—
310C4×C31#54612yes—
311C4×C31#55012yes—
312C2×#137C22×C3112yes—
313C2×AGL(1,ℤ/8)C22×C3112yes—
314#559—21no—
315#557—21no—
316#556—21no—
317#556—22no—
318#554—21no—
319#552—22no—
320#539—22no—
321#536C22×C3112yes—
322#533C22×C3111yescharacteristic
323#532C22×C3111yescharacteristic
324C2×C4⋊Q8C22×C3111yescharacteristic
325#529—22no—
326C2×#124C22×C3111yescharacteristic
327C2×#124C22×C3111yescharacteristic
328#520C22×C3112yes—
329C2×C4×Q8C22×C3111yescharacteristic
330#497—21no—
331#496—21no—
332#495—22no—
333#489—21no—
334#489—21no—
335#488—22no—
336C2×#104C22×C3112yes—
337C2×#104C22×C3112yes—
338#419C22×C3112yes—
339#418C22×C3112yes—
340C2×#101C22×C3112yes—
341C4×M16C22×C3111yescharacteristic
342C4×M16C22×C3111yescharacteristic
343#396—22no—
344#391C22×C3111yescharacteristic
345#390C22×C3111yescharacteristic
346#388—22no—
347C4×C4⋊C4C22×C3111yescharacteristic
348#343C22×C3112yes—
349#339C22×C3112yes—
350#339C22×C3112yes—
351C2×C31—21no—
352C2×C31—21no—
353C2×C31—82no—
354C2×C31—161no—
355C2×C31#243111yescharacteristic
356C2×C31#301811yescharacteristic
357C2×C31#302011yescharacteristic
358C2×#41C23×C3112yes—
359C22×Q8C23×C3111yescharacteristic
360C22×Q8C23×C3111yescharacteristic
361C22×D4C23×C3111yescharacteristic
362#137—24no—
363#137—42no—
364AGL(1,ℤ/8)—24no—
365AGL(1,ℤ/8)—42no—
366C2×Q16—22no—
367C2×SD16—22no—
368C2×SD16—22no—
369C2×D8—22no—
370C2×M16C23×C3112yes—
371C2×M16C23×C3112yes—
372C4⋊Q8—22no—
373C4⋊Q8—42no—
374C4⋊Q8D4×C3111yescharacteristic
375C4⋊Q8D4×C3111yescharacteristic
376C4⋊Q8D4×C3111yescharacteristic
377#127D4×C3111yescharacteristic
378#126—24no—
379#126—24no—
380#125—22no—
381#125—24no—
382#125D4×C3112yes—
383#124—22no—
384#124—22no—
385#124—21no—
386#124—21no—
387#124—21no—
388#124—21no—
389#124—42no—
390#124D4×C3112yes—
391#123—24no—
392#123—24no—
393C22⋊Q8—24no—
394C22⋊Q8—21no—
395C22⋊Q8—21no—
396C22⋊Q8—22no—
397C22⋊Q8—22no—
398#121—21no—
399#121—21no—
400C4×Q8—22no—
401C4×Q8—22no—
402C4×Q8—21no—
403C4×Q8—21no—
404C4×Q8—21no—
405C4×Q8—21no—
406C4×Q8—21no—
407C4×Q8—21no—
408C4×D4—22no—
409C4×D4—22no—
410C4×D4—24no—
411C4×D4—21no—
412C4×D4—21no—
413#117—24no—
414#117—22no—
415#117C23×C3112yes—
416C2×C4⋊C4—22no—
417C2×C4⋊C4—22no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4D4×C3112yes—
420C2×C4⋊C4C23×C3111yescharacteristic
421C2×C22⋊C4—21no—
422C2×C22⋊C4—22no—
423C2×C22⋊C4—21no—
424C2×C22⋊C4—22no—
425C2×C42D4×C3112yes—
426C2×C42C23×C3111yescharacteristic
427C2×C42C23×C3111yescharacteristic
428C2×C42C23×C3111yescharacteristic
429#104—42no—
430#104—42no—
431#103—22no—
432#103—22no—
433#102—22no—
434#102—22no—
435#101—24no—
436#97—21no—
437#97—21no—
438C4×C8—21no—
439C4×C8—21no—
440#95—21no—
441#95—22no—
442C31#2562611yescharacteristic, direct product factor, Sylow
443C24—21no—
444#41—22no—
445#41—22no—
446#41—22no—
447#41—22no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—22no—
453C2×Q8—22no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—22no—
457C2×Q8—22no—
458C2×Q8C2×D4×C3112yes—
459C2×D4—22no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—22no—
465C2×D4—22no—
466C2×D4C2×D4×C3112yes—
467C22×C4—21no—
468C22×C4—24no—
469C22×C4—22no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4C2×D4×C3112yes—
473C22×C4C2×D4×C3112yes—
474C22×C4C2×D4×C3111yescharacteristic
475C22×C4C2×D4×C3111yescharacteristic
476C22×C4C24×C3111yescharacteristic
477Q16—44no—
478SD16—44no—
479SD16—44no—
480D8—44no—
481M16—22no—
482M16—22no—
483M16—22no—
484M16#41×C3114yes—
485C2×C8—22no—
486C2×C8—22no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—22no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—42no—
501C4⋊C4—44no—
502C4⋊C4C2×D4×C3112yes—
503C4⋊C4C2×D4×C3112yes—
504C22⋊C4—24no—
505C22⋊C4—24no—
506C22⋊C4—24no—
507C22⋊C4—22no—
508C22⋊C4—22no—
509C22⋊C4—22no—
510C22⋊C4—44no—
511C22⋊C4—44no—
512C22⋊C4—41no—
513C22⋊C4—41no—
514C22⋊C4—41no—
515C22⋊C4—41no—
516C42—24no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—22no—
521C42—22no—
522C42C2×D4×C3111yescharacteristic
523C42C2×D4×C3111yescharacteristic
524C42C2×D4×C3111yescharacteristic
525C42C2×D4×C3111yescharacteristic
526C23—22no—
527C23—42no—
528C23—42no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23C22×D4×C3111yescharacteristic
533Q8—22no—
534Q8—22no—
535Q8—42no—
536Q8—42no—
537Q8—42no—
538Q8—42no—
539Q8—42no—
540Q8—81no—
541D4—22no—
542D4—22no—
543D4—42no—
544D4—42no—
545D4—42no—
546D4—42no—
547D4—42no—
548D4—81no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
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—42no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—84no—
571C2×C4C31×#12011yescharacteristic
572C2×C4C31×#12011yescharacteristic
573C2×C4C31×#12011yescharacteristic
574C2×C4C31×#12011yescharacteristic
575C2×C4C22×D4×C3111yescharacteristic
576C2×C4C22×D4×C3111yescharacteristic
577C2×C4C2×#41×C3112yes—
578C2×C4C31×H(5,𝔽2)12yes—
579C8—24no—
580C8—42no—
581C22—21no—
582C22—21no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—82no—
587C22—82no—
588C22—81no—
589C22—81no—
590C22—81no—
591C22C2×C31×#12011yescharacteristic
592C22C31×#53211yescharacteristic
593C22C31×#53511yescharacteristic
594C4—21no—
595C4—21no—
596C4—21no—
597C4—21no—
598C4—42no—
599C4—42no—
600C4—42no—
601C4—42no—
602C4—42no—
603C4—42no—
604C4—81no—
605C4—81no—
606C4C31×#54612yes—
607C4C31×#55012yes—
608C2—21no—
609C2—21no—
610C2—82no—
611C2—161no—
612C2C31×#243111yescharacteristic
613C2C31×#301811yescharacteristic
614C2C31×#302011yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#3153C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#3507C2semidirect
3. C31×#3140C2semidirect
4. C31×#2545C2semidirect
5. C31×#2005C2semidirect
6. C31×#2003C2semidirect
7. C31×#1986C2semidirect
8. C31×#1980C2semidirect
9. C31×#1883C2semidirect
10. C31×#1882C2semidirect
11. C4×C31×#101C2semidirect
218. #25626C31direct (N × H)
274. #3507C2×C31semidirect
276. #3140C2×C31semidirect
277. #2545C2×C31semidirect
278. #2005C2×C31semidirect
279. #2003C2×C31semidirect
280. #1986C2×C31semidirect
281. #1980C2×C31semidirect
282. #1883C2×C31semidirect
283. #1882C2×C31semidirect
284. C4×#101C2×C31semidirect
442. C31#25626direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35072
C31×#31532
C31×#31402
C31×#25452
C31×#20052
C31×#20032
C31×#19862
C31×#19802
C31×#18832
C31×#18822
C4×C31×#1012
#2562631