Finite Group Databasebeta

← C31×#23951 #21480710

Subgroups — C31×#23951 #21480710

Order 7936.

1234 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3492C211yesmaximal, characteristic, complemented
2C31×#3005C212yesmaximal, complemented
3C31×#2311C211yesmaximal, characteristic, complemented
4C31×#2028C211yesmaximal, characteristic, complemented
5C31×#2027C211yesmaximal, characteristic, complemented
6C31×#2007C211yesmaximal, characteristic, complemented
7C31×#2006C211yesmaximal, characteristic, complemented
8C31×#1897C212yesmaximal, complemented
9C31×#1883C212yesmaximal, complemented
10C31×#1882C212yesmaximal, complemented
11C31×#1821C211yesmaximal, characteristic, complemented
12C22×#41×C31C2211yescharacteristic
13C2×C31×#137C2212yes—
14C2×C31×AGL(1,ℤ/8)C2212yes—
15C31×#554—22no—
16C31×#553—22no—
17C31×#552—22no—
18C31×#547—22no—
19C31×#544C2211yescharacteristic
20C31×#539—22no—
21C31×#537C2211yescharacteristic
22C31×#525—22no—
23C31×#525C2212yes—
24C31×#520C2212yes—
25C2×C31×#117C2211yescharacteristic
26C22×C4⋊C4×C31C2211yescharacteristic
27C31×#452—22no—
28C31×#452—22no—
29C31×#449—22no—
30C31×#449—22no—
31C2×C31×#104C2212yes—
32C2×C31×#104C2212yes—
33C31×#419C2212yes—
34C31×#418C2212yes—
35C31×#407C2212yes—
36C31×#391C2211yescharacteristic
37C31×#391C2211yescharacteristic
38C31×#390C2211yescharacteristic
39C31×#390C2211yescharacteristic
40C31×#386—22no—
41C31×#380—22no—
42C31×#343C2212yes—
43C31×#343C2211yescharacteristic
44C31×#343C2211yescharacteristic
45C31×#339C2212yes—
46C31×#339C2212yes—
47C2×#41×C31—22no—
48C2×#41×C31—22no—
49C2×#41×C31—22no—
50C2×#41×C31—22no—
51C2×#41×C31—22no—
52C2×#41×C31—42no—
53C2×#41×C31C2312yes—
54C22×Q8×C31C2311yescharacteristic
55C22×D4×C31—21no—
56C22×D4×C31C2311yescharacteristic
57C23×C4×C31—21no—
58C23×C4×C31D412yes—
59C23×C4×C31C2311yescharacteristic
60C31×#137—24no—
61C31×#137—42no—
62C31×AGL(1,ℤ/8)—24no—
63C31×AGL(1,ℤ/8)—42no—
64C2×Q16×C31—22no—
65C2×SD16×C31—22no—
66C2×SD16×C31—22no—
67C2×D8×C31—22no—
68C2×M16×C31C2312yes—
69C2×M16×C31C2312yes—
70C31×C4⋊Q8—22no—
71C31×#126—24no—
72C31×#125—24no—
73C31×#125—22no—
74C31×#123—24no—
75C31×#123—24no—
76C31×#123—22no—
77C31×C22⋊Q8—22no—
78C31×C22⋊Q8—24no—
79C31×C22⋊Q8—22no—
80C31×C22⋊Q8—22no—
81C31×C22⋊Q8D412yes—
82C31×C22⋊Q8D412yes—
83C31×#121—22no—
84C31×#121—22no—
85C31×#121—22no—
86C31×#121D412yes—
87C31×#121D412yes—
88C31×#120—22no—
89C4×Q8×C31—22no—
90C4×D4×C31—24no—
91C4×D4×C31—22no—
92C4×D4×C31—24no—
93C4×D4×C31—22no—
94C4×D4×C31—22no—
95C31×#117—22no—
96C31×#117—22no—
97C31×#117—42no—
98C31×#117C2312yes—
99C2×C4⋊C4×C31—22no—
100C2×C4⋊C4×C31—22no—
101C2×C4⋊C4×C31—22no—
102C2×C4⋊C4×C31—21no—
103C2×C4⋊C4×C31D412yes—
104C2×C4⋊C4×C31C2311yescharacteristic
105C2×C4⋊C4×C31C2311yescharacteristic
106C2×C22⋊C4×C31—21no—
107C2×C22⋊C4×C31—22no—
108C2×C22⋊C4×C31—22no—
109C2×C42×C31C2311yescharacteristic
110C2×C42×C31C2311yescharacteristic
111C31×#104—42no—
112C31×#104—42no—
113C31×#103—22no—
114C31×#103—22no—
115C31×#102—22no—
116C31×#102—22no—
117C31×C22⋊C8—22no—
118C31×C22⋊C8—22no—
119C31×#95—22no—
120C31×#95—22no—
121C24×C31—21no—
122C24×C31C2×D411yescharacteristic
123#41×C31—22no—
124#41×C31—22no—
125#41×C31—22no—
126#41×C31—22no—
127#41×C31—44no—
128#41×C31—44no—
129#41×C31—44no—
130#41×C31—44no—
131#41×C31—84no—
132C2×Q8×C31—22no—
133C2×Q8×C31—21no—
134C2×Q8×C31—22no—
135C2×Q8×C31—22no—
136C2×Q8×C31C2×D412yes—
137C2×Q8×C31#4112yescomplemented
138C2×D4×C31—24no—
139C2×D4×C31—22no—
140C2×D4×C31—22no—
141C2×D4×C31—22no—
142C2×D4×C31—22no—
143C2×D4×C31—22no—
144C2×D4×C31—21no—
145C2×D4×C31—22no—
146C2×D4×C31—42no—
147C2×D4×C31—41no—
148C2×D4×C31—41no—
149C2×D4×C31C2×D412yes—
150C2×D4×C31#4112yescomplemented
151C22×C4×C31—21no—
152C22×C4×C31—21no—
153C22×C4×C31—22no—
154C22×C4×C31—24no—
155C22×C4×C31—22no—
156C22×C4×C31—21no—
157C22×C4×C31—24no—
158C22×C4×C31—22no—
159C22×C4×C31—22no—
160C22×C4×C31—21no—
161C22×C4×C31—21no—
162C22×C4×C31—42no—
163C22×C4×C31—42no—
164C22×C4×C31—82no—
165C22×C4×C31C2×D411yescharacteristic
166C22×C4×C31C2×D412yes—
167C22×C4×C31C2×D412yes—
168C22×C4×C31C2×D412yes—
169C22×C4×C31C2×D412yes—
170C22×C4×C31C2411yescharacteristic
171Q16×C31—44no—
172SD16×C31—44no—
173SD16×C31—44no—
174D8×C31—44no—
175M16×C31—22no—
176M16×C31—22no—
177M16×C31—22no—
178M16×C31—22no—
179C2×C8×C31—22no—
180C2×C8×C31—22no—
181C4⋊C4×C31—24no—
182C4⋊C4×C31—22no—
183C4⋊C4×C31—22no—
184C4⋊C4×C31—24no—
185C4⋊C4×C31—24no—
186C4⋊C4×C31—22no—
187C4⋊C4×C31—22no—
188C4⋊C4×C31—22no—
189C4⋊C4×C31—22no—
190C4⋊C4×C31—42no—
191C4⋊C4×C31C2×D412yes—
192C4⋊C4×C31C2×D412yes—
193C22⋊C4×C31—24no—
194C22⋊C4×C31—24no—
195C22⋊C4×C31—22no—
196C22⋊C4×C31—22no—
197C22⋊C4×C31—22no—
198C22⋊C4×C31—22no—
199C22⋊C4×C31—22no—
200C22⋊C4×C31—42no—
201C22⋊C4×C31—82no—
202C22⋊C4×C31—82no—
203C42×C31—24no—
204C42×C31—22no—
205C42×C31—22no—
206C42×C31—22no—
207#23951C3111yesmaximal, characteristic, direct product factor, Sylow
208C23×C31—21no—
209C23×C31—21no—
210C23×C31—22no—
211C23×C31—22no—
212C23×C31—41no—
213C23×C31—41no—
214C23×C31—42no—
215C23×C31—42no—
216C23×C31—41no—
217C23×C31—41no—
218C23×C31#12011yescharacteristic
219C23×C31#12011yescharacteristic
220C23×C31C22×D411yescharacteristic
221Q8×C31—22no—
222Q8×C31—24no—
223Q8×C31—22no—
224Q8×C31—42no—
225Q8×C31—82no—
226D4×C31—22no—
227D4×C31—24no—
228D4×C31—22no—
229D4×C31—44no—
230D4×C31—44no—
231D4×C31—44no—
232D4×C31—44no—
233D4×C31—42no—
234D4×C31—42no—
235D4×C31—42no—
236D4×C31—82no—
237C2×C4×C31—21no—
238C2×C4×C31—21no—
239C2×C4×C31—22no—
240C2×C4×C31—22no—
241C2×C4×C31—22no—
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—48no—
250C2×C4×C31—44no—
251C2×C4×C31—41no—
252C2×C4×C31—41no—
253C2×C4×C31—42no—
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—82no—
264C2×C4×C31—84no—
265C2×C4×C31#12012yes—
266C2×C4×C31C22×D412yes—
267C2×C4×C31C2×#4111yescharacteristic
268C2×C4×C31C2×#4111yescharacteristic
269C2×C4×C31H(5,𝔽2)12yes—
270C8×C31—42no—
271C8×C31—42no—
272#3492C2×C3111yescharacteristic, complemented
273#3005C2×C3112yescomplemented
274#2311C2×C3111yescharacteristic, complemented
275#2028C2×C3111yescharacteristic, complemented
276#2027C2×C3111yescharacteristic, complemented
277#2007C2×C3111yescharacteristic, complemented
278#2006C2×C3111yescharacteristic, complemented
279#1897C2×C3112yescomplemented
280#1883C2×C3112yescomplemented
281#1882C2×C3112yescomplemented
282#1821C2×C3111yescharacteristic, complemented
283C22×C31—21no—
284C22×C31—21no—
285C22×C31—21no—
286C22×C31—21no—
287C22×C31—21no—
288C22×C31—21no—
289C22×C31—21no—
290C22×C31—21no—
291C22×C31—42no—
292C22×C31—42no—
293C22×C31—44no—
294C22×C31—42no—
295C22×C31—42no—
296C22×C31—42no—
297C22×C31—42no—
298C22×C31—82no—
299C22×C31—81no—
300C22×C31—81no—
301C22×C31C2×#12011yescharacteristic
302C22×C31#52511yescharacteristic
303C22×C31#53511yescharacteristic
304C4×C31—22no—
305C4×C31—42no—
306C4×C31—42no—
307C4×C31—42no—
308C4×C31—42no—
309C4×C31—42no—
310C4×C31—42no—
311C4×C31—81no—
312C4×C31—81no—
313C4×C31#54612yes—
314C4×C31#54612yes—
315C22×#41C22×C3111yescharacteristic
316C2×#137C22×C3112yes—
317C2×AGL(1,ℤ/8)C22×C3112yes—
318#554—22no—
319#553—22no—
320#552—22no—
321#547—22no—
322#544C22×C3111yescharacteristic
323#539—22no—
324#537C22×C3111yescharacteristic
325#525—22no—
326#525C22×C3112yes—
327#520C22×C3112yes—
328C2×#117C22×C3111yescharacteristic
329C22×C4⋊C4C22×C3111yescharacteristic
330#452—22no—
331#452—22no—
332#449—22no—
333#449—22no—
334C2×#104C22×C3112yes—
335C2×#104C22×C3112yes—
336#419C22×C3112yes—
337#418C22×C3112yes—
338#407C22×C3112yes—
339#391C22×C3111yescharacteristic
340#391C22×C3111yescharacteristic
341#390C22×C3111yescharacteristic
342#390C22×C3111yescharacteristic
343#386—22no—
344#380—22no—
345#343C22×C3112yes—
346#343C22×C3111yescharacteristic
347#343C22×C3111yescharacteristic
348#339C22×C3112yes—
349#339C22×C3112yes—
350C2×C31—21no—
351C2×C31—21no—
352C2×C31—41no—
353C2×C31—41no—
354C2×C31—82no—
355C2×C31—82no—
356C2×C31#240411yescharacteristic
357C2×C31#301911yescharacteristic
358C2×C31#302011yescharacteristic
359C2×#41—22no—
360C2×#41—22no—
361C2×#41—22no—
362C2×#41—22no—
363C2×#41—22no—
364C2×#41—42no—
365C2×#41C23×C3112yes—
366C22×Q8C23×C3111yescharacteristic
367C22×D4—21no—
368C22×D4C23×C3111yescharacteristic
369C23×C4—21no—
370C23×C4D4×C3112yes—
371C23×C4C23×C3111yescharacteristic
372#137—24no—
373#137—42no—
374AGL(1,ℤ/8)—24no—
375AGL(1,ℤ/8)—42no—
376C2×Q16—22no—
377C2×SD16—22no—
378C2×SD16—22no—
379C2×D8—22no—
380C2×M16C23×C3112yes—
381C2×M16C23×C3112yes—
382C4⋊Q8—22no—
383#126—24no—
384#125—24no—
385#125—22no—
386#123—24no—
387#123—24no—
388#123—22no—
389C22⋊Q8—22no—
390C22⋊Q8—24no—
391C22⋊Q8—22no—
392C22⋊Q8—22no—
393C22⋊Q8D4×C3112yes—
394C22⋊Q8D4×C3112yes—
395#121—22no—
396#121—22no—
397#121—22no—
398#121D4×C3112yes—
399#121D4×C3112yes—
400#120—22no—
401C4×Q8—22no—
402C4×D4—24no—
403C4×D4—22no—
404C4×D4—24no—
405C4×D4—22no—
406C4×D4—22no—
407#117—22no—
408#117—22no—
409#117—42no—
410#117C23×C3112yes—
411C2×C4⋊C4—22no—
412C2×C4⋊C4—22no—
413C2×C4⋊C4—22no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4D4×C3112yes—
416C2×C4⋊C4C23×C3111yescharacteristic
417C2×C4⋊C4C23×C3111yescharacteristic
418C2×C22⋊C4—21no—
419C2×C22⋊C4—22no—
420C2×C22⋊C4—22no—
421C2×C42C23×C3111yescharacteristic
422C2×C42C23×C3111yescharacteristic
423#104—42no—
424#104—42no—
425#103—22no—
426#103—22no—
427#102—22no—
428#102—22no—
429C22⋊C8—22no—
430C22⋊C8—22no—
431#95—22no—
432#95—22no—
433C31#2395111yescharacteristic, direct product factor, Sylow
434C24—21no—
435C24C2×D4×C3111yescharacteristic
436#41—22no—
437#41—22no—
438#41—22no—
439#41—22no—
440#41—44no—
441#41—44no—
442#41—44no—
443#41—44no—
444#41—84no—
445C2×Q8—22no—
446C2×Q8—21no—
447C2×Q8—22no—
448C2×Q8—22no—
449C2×Q8C2×D4×C3112yes—
450C2×Q8#41×C3112yescomplemented
451C2×D4—24no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—22no—
456C2×D4—22no—
457C2×D4—21no—
458C2×D4—22no—
459C2×D4—42no—
460C2×D4—41no—
461C2×D4—41no—
462C2×D4C2×D4×C3112yes—
463C2×D4#41×C3112yescomplemented
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—21no—
470C22×C4—24no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—82no—
478C22×C4C2×D4×C3111yescharacteristic
479C22×C4C2×D4×C3112yes—
480C22×C4C2×D4×C3112yes—
481C22×C4C2×D4×C3112yes—
482C22×C4C2×D4×C3112yes—
483C22×C4C24×C3111yescharacteristic
484Q16—44no—
485SD16—44no—
486SD16—44no—
487D8—44no—
488M16—22no—
489M16—22no—
490M16—22no—
491M16—22no—
492C2×C8—22no—
493C2×C8—22no—
494C4⋊C4—24no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—42no—
504C4⋊C4C2×D4×C3112yes—
505C4⋊C4C2×D4×C3112yes—
506C22⋊C4—24no—
507C22⋊C4—24no—
508C22⋊C4—22no—
509C22⋊C4—22no—
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—22no—
513C22⋊C4—42no—
514C22⋊C4—82no—
515C22⋊C4—82no—
516C42—24no—
517C42—22no—
518C42—22no—
519C42—22no—
520C23—21no—
521C23—21no—
522C23—22no—
523C23—22no—
524C23—41no—
525C23—41no—
526C23—42no—
527C23—42no—
528C23—41no—
529C23—41no—
530C23C31×#12011yescharacteristic
531C23C31×#12011yescharacteristic
532C23C22×D4×C3111yescharacteristic
533Q8—22no—
534Q8—24no—
535Q8—22no—
536Q8—42no—
537Q8—82no—
538D4—22no—
539D4—24no—
540D4—22no—
541D4—44no—
542D4—44no—
543D4—44no—
544D4—44no—
545D4—42no—
546D4—42no—
547D4—42no—
548D4—82no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—48no—
562C2×C4—44no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—82no—
576C2×C4—84no—
577C2×C4C31×#12012yes—
578C2×C4C22×D4×C3112yes—
579C2×C4C2×#41×C3111yescharacteristic
580C2×C4C2×#41×C3111yescharacteristic
581C2×C4C31×H(5,𝔽2)12yes—
582C8—42no—
583C8—42no—
584C22—21no—
585C22—21no—
586C22—21no—
587C22—21no—
588C22—21no—
589C22—21no—
590C22—21no—
591C22—21no—
592C22—42no—
593C22—42no—
594C22—44no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22—82no—
600C22—81no—
601C22—81no—
602C22C2×C31×#12011yescharacteristic
603C22C31×#52511yescharacteristic
604C22C31×#53511yescharacteristic
605C4—22no—
606C4—42no—
607C4—42no—
608C4—42no—
609C4—42no—
610C4—42no—
611C4—42no—
612C4—81no—
613C4—81no—
614C4C31×#54612yes—
615C4C31×#54612yes—
616C2—21no—
617C2—21no—
618C2—41no—
619C2—41no—
620C2—82no—
621C2—82no—
622C2C31×#240411yescharacteristic
623C2C31×#301911yescharacteristic
624C2C31×#302011yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#3492C2semidirect
2. C31×#3005C2semidirect
3. C31×#2311C2semidirect
4. C31×#2028C2semidirect
5. C31×#2027C2semidirect
6. C31×#2007C2semidirect
7. C31×#2006C2semidirect
8. C31×#1897C2semidirect
9. C31×#1883C2semidirect
10. C31×#1882C2semidirect
11. C31×#1821C2semidirect
137. C2×Q8×C31#41semidirect
150. C2×D4×C31#41semidirect
207. #23951C31direct (N × H)
272. #3492C2×C31semidirect
273. #3005C2×C31semidirect
274. #2311C2×C31semidirect
275. #2028C2×C31semidirect
276. #2027C2×C31semidirect
277. #2007C2×C31semidirect
278. #2006C2×C31semidirect
279. #1897C2×C31semidirect
280. #1883C2×C31semidirect
281. #1882C2×C31semidirect
282. #1821C2×C31semidirect
433. C31#23951direct (N × H)
450. C2×Q8#41×C31semidirect
463. C2×D4#41×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#34922
C31×#30052
C31×#23112
C31×#20282
C31×#20272
C31×#20072
C31×#20062
C31×#18972
C31×#18832
C31×#18822
C31×#18212
#2395131