Finite Group Databasebeta

← C31×#24181 #21480940

Subgroups — C31×#24181 #21480940

Order 7936.

1050 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3507C211yesmaximal, characteristic, complemented
2C31×#3077C211yesmaximal, characteristic, complemented
3C31×#3036C211yesmaximal, characteristic
4C31×#2326C211yesmaximal, characteristic, complemented
5C31×#2038C212yesmaximal, complemented
6C31×#2003C212yesmaximal, complemented
7C31×#1995C211yesmaximal, characteristic, complemented
8C31×#1963C211yesmaximal, characteristic, complemented
9C31×#1883C212yesmaximal, complemented
10C31×#1859C212yesmaximal, complemented
11C31×#1801C211yesmaximal, characteristic, complemented
12C2×C31×AGL(1,ℤ/8)C2212yes—
13C31×#559—21no—
14C31×#557—21no—
15C31×#556—21no—
16C31×#556—22no—
17C31×#554—21no—
18C31×#552—22no—
19C31×#539—22no—
20C31×#536C2212yes—
21C31×#533C2211yescharacteristic
22C31×#532C2211yescharacteristic
23C31×#529—22no—
24C2×C31×#124C2211yescharacteristic
25C31×#520C2212yes—
26C2×C4×Q8×C31C2211yescharacteristic
27C2×C4×D4×C31C2211yescharacteristic
28C31×#478—22no—
29C31×#476—21no—
30C31×#474—21no—
31C31×#463—22no—
32C31×#461—21no—
33C31×#459—21no—
34C31×#428C2212yes—
35C31×#423C2211yescharacteristic
36C31×#423C2211yescharacteristic
37C2×C31×#104C2212yes—
38C2×C31×#104C2212yes—
39C31×#418C2212yes—
40C31×#417C2212yes—
41C2×C31×#101C2212yes—
42C31×#391C2211yescharacteristic
43C31×#390C2211yescharacteristic
44C31×#388—22no—
45C31×#385—22no—
46C31×#384C2211yescharacteristic
47C31×#376C2211yescharacteristic
48C31×#343C2212yes—
49C31×#339C2212yes—
50C31×#339C2212yes—
51C2×#41×C31C2312yes—
52C22×Q8×C31C2311yescharacteristic
53C22×D4×C31C2311yescharacteristic
54C23×C4×C31—21no—
55C31×AGL(1,ℤ/8)—24no—
56C31×AGL(1,ℤ/8)—42no—
57C2×SD16×C31—22no—
58C2×D8×C31—22no—
59C2×M16×C31C2312yes—
60C2×M16×C31C2312yes—
61C31×C4⋊Q8—22no—
62C31×C4⋊Q8D411yescharacteristic
63C31×C4⋊Q8D411yescharacteristic
64C31×C4⋊Q8D411yescharacteristic
65C31×#127D411yescharacteristic
66C31×#126—24no—
67C31×#126—24no—
68C31×#125—22no—
69C31×#125—24no—
70C31×#125D412yes—
71C31×#124—22no—
72C31×#124—22no—
73C31×#124—22no—
74C31×#124—22no—
75C31×#124D412yes—
76C31×#123—24no—
77C31×#123—24no—
78C31×C22⋊Q8—24no—
79C31×C22⋊Q8—21no—
80C31×C22⋊Q8—21no—
81C31×C22⋊Q8—22no—
82C31×C22⋊Q8—22no—
83C31×#121—21no—
84C31×#121—21no—
85C4×Q8×C31—22no—
86C4×Q8×C31—22no—
87C4×Q8×C31—21no—
88C4×Q8×C31—21no—
89C4×Q8×C31—21no—
90C4×Q8×C31—21no—
91C4×Q8×C31—21no—
92C4×Q8×C31—21no—
93C4×D4×C31—22no—
94C4×D4×C31—22no—
95C4×D4×C31—24no—
96C4×D4×C31—21no—
97C4×D4×C31—21no—
98C4×D4×C31—21no—
99C4×D4×C31—21no—
100C4×D4×C31—21no—
101C4×D4×C31—21no—
102C31×#117—24no—
103C31×#117—22no—
104C31×#117C2312yes—
105C2×C4⋊C4×C31—22no—
106C2×C4⋊C4×C31—22no—
107C2×C4⋊C4×C31D412yes—
108C2×C4⋊C4×C31C2311yescharacteristic
109C2×C4⋊C4×C31C2311yescharacteristic
110C2×C22⋊C4×C31—22no—
111C2×C22⋊C4×C31—21no—
112C2×C22⋊C4×C31—22no—
113C2×C42×C31D412yes—
114C2×C42×C31C2311yescharacteristic
115C2×C42×C31C2311yescharacteristic
116C2×C42×C31C2311yescharacteristic
117C31×#107—22no—
118C31×#106—22no—
119C31×C4⋊C8—21no—
120C31×C4⋊C8—21no—
121C31×C4⋊C8—21no—
122C31×C4⋊C8—21no—
123C31×#104—24no—
124C31×#104—42no—
125C31×#103—22no—
126C31×#102—22no—
127C31×#102—22no—
128C31×#102—22no—
129C31×#101—42no—
130C31×#95—21no—
131C31×#95—21no—
132C31×#95—22no—
133C24×C31—21no—
134#41×C31—22no—
135#41×C31—22no—
136#41×C31—22no—
137#41×C31—22no—
138C2×Q8×C31—21no—
139C2×Q8×C31—21no—
140C2×Q8×C31—21no—
141C2×Q8×C31—21no—
142C2×Q8×C31—21no—
143C2×Q8×C31—21no—
144C2×Q8×C31—21no—
145C2×Q8×C31—21no—
146C2×Q8×C31C2×D412yes—
147C2×D4×C31—22no—
148C2×D4×C31—21no—
149C2×D4×C31—21no—
150C2×D4×C31—21no—
151C2×D4×C31—21no—
152C2×D4×C31—21no—
153C2×D4×C31—21no—
154C2×D4×C31—21no—
155C2×D4×C31C2×D412yes—
156C2×D4×C31#4112yes—
157C22×C4×C31—21no—
158C22×C4×C31—24no—
159C22×C4×C31—22no—
160C22×C4×C31—21no—
161C22×C4×C31—21no—
162C22×C4×C31—41no—
163C22×C4×C31—41no—
164C22×C4×C31—82no—
165C22×C4×C31C2×D412yes—
166C22×C4×C31C2×D412yes—
167C22×C4×C31C2×D411yescharacteristic
168C22×C4×C31C2×D411yescharacteristic
169C22×C4×C31C2411yescharacteristic
170SD16×C31—44no—
171D8×C31—44no—
172M16×C31—22no—
173M16×C31—22no—
174M16×C31—22no—
175M16×C31—22no—
176C2×C8×C31—22no—
177C2×C8×C31—22no—
178C4⋊C4×C31—24no—
179C4⋊C4×C31—24no—
180C4⋊C4×C31—24no—
181C4⋊C4×C31—22no—
182C4⋊C4×C31—22no—
183C4⋊C4×C31—22no—
184C4⋊C4×C31—21no—
185C4⋊C4×C31—21no—
186C4⋊C4×C31—21no—
187C4⋊C4×C31—22no—
188C4⋊C4×C31—21no—
189C4⋊C4×C31—21no—
190C4⋊C4×C31—21no—
191C4⋊C4×C31—21no—
192C4⋊C4×C31—21no—
193C4⋊C4×C31—82no—
194C4⋊C4×C31C2×D412yes—
195C4⋊C4×C31C2×D412yes—
196C22⋊C4×C31—28no—
197C22⋊C4×C31—24no—
198C22⋊C4×C31—24no—
199C22⋊C4×C31—24no—
200C22⋊C4×C31—22no—
201C22⋊C4×C31—22no—
202C22⋊C4×C31—22no—
203C22⋊C4×C31—41no—
204C22⋊C4×C31—41no—
205C22⋊C4×C31—82no—
206C42×C31—24no—
207C42×C31—22no—
208C42×C31—22no—
209C42×C31—22no—
210C42×C31—21no—
211C42×C31—21no—
212C42×C31—21no—
213C42×C31C2×D411yescharacteristic
214C42×C31C2×D411yescharacteristic
215C42×C31C2×D411yescharacteristic
216C42×C31C2×D411yescharacteristic
217C42×C31#4112yes—
218#24181C3111yesmaximal, characteristic, direct product factor, Sylow
219C23×C31—22no—
220C23×C31—22no—
221C23×C31—42no—
222C23×C31—42no—
223C23×C31—41no—
224C23×C31—41no—
225C23×C31C22×D411yescharacteristic
226Q8×C31—22no—
227Q8×C31—22no—
228Q8×C31—42no—
229Q8×C31—41no—
230Q8×C31—41no—
231Q8×C31—41no—
232Q8×C31—41no—
233Q8×C31—81no—
234D4×C31—22no—
235D4×C31—22no—
236D4×C31—22no—
237D4×C31—22no—
238D4×C31—42no—
239D4×C31—42no—
240D4×C31—42no—
241D4×C31—41no—
242D4×C31—41no—
243D4×C31—81no—
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—22no—
251C2×C4×C31—22no—
252C2×C4×C31—41no—
253C2×C4×C31—41no—
254C2×C4×C31—41no—
255C2×C4×C31—41no—
256C2×C4×C31—42no—
257C2×C4×C31—42no—
258C2×C4×C31—42no—
259C2×C4×C31—41no—
260C2×C4×C31—42no—
261C2×C4×C31—41no—
262C2×C4×C31—41no—
263C2×C4×C31—41no—
264C2×C4×C31—82no—
265C2×C4×C31—82no—
266C2×C4×C31—84no—
267C2×C4×C31#12011yescharacteristic
268C2×C4×C31#12011yescharacteristic
269C2×C4×C31#12011yescharacteristic
270C2×C4×C31#12011yescharacteristic
271C2×C4×C31C22×D411yescharacteristic
272C2×C4×C31C22×D411yescharacteristic
273C2×C4×C31C2×#4111yescharacteristic
274C2×C4×C31C2×#4111yescharacteristic
275C2×C4×C31H(5,𝔽2)11yescharacteristic
276C2×C4×C312−1+411yescharacteristic
277C8×C31—42no—
278C8×C31—42no—
279#3507C2×C3111yescharacteristic, complemented
280#3077C2×C3111yescharacteristic, complemented
281#3036C2×C3111yescharacteristic
282#2326C2×C3111yescharacteristic, complemented
283#2038C2×C3112yescomplemented
284#2003C2×C3112yescomplemented
285#1995C2×C3111yescharacteristic, complemented
286#1963C2×C3111yescharacteristic, complemented
287#1883C2×C3112yescomplemented
288#1859C2×C3112yescomplemented
289#1801C2×C3111yescharacteristic, complemented
290C22×C31—21no—
291C22×C31—21no—
292C22×C31—44no—
293C22×C31—42no—
294C22×C31—42no—
295C22×C31—42no—
296C22×C31—42no—
297C22×C31—82no—
298C22×C31—81no—
299C22×C31—81no—
300C22×C31C2×#12011yescharacteristic
301C22×C31#52511yescharacteristic
302C22×C31#53711yescharacteristic
303C4×C31—21no—
304C4×C31—21no—
305C4×C31—21no—
306C4×C31—21no—
307C4×C31—42no—
308C4×C31—42no—
309C4×C31—42no—
310C4×C31—42no—
311C4×C31—42no—
312C4×C31—42no—
313C4×C31—81no—
314C4×C31—81no—
315C4×C31#54611yescharacteristic
316C4×C31#54711yescharacteristic
317C4×C31#54811yescharacteristic
318C4×C31#55011yescharacteristic
319C2×AGL(1,ℤ/8)C22×C3112yes—
320#559—21no—
321#557—21no—
322#556—21no—
323#556—22no—
324#554—21no—
325#552—22no—
326#539—22no—
327#536C22×C3112yes—
328#533C22×C3111yescharacteristic
329#532C22×C3111yescharacteristic
330#529—22no—
331C2×#124C22×C3111yescharacteristic
332#520C22×C3112yes—
333C2×C4×Q8C22×C3111yescharacteristic
334C2×C4×D4C22×C3111yescharacteristic
335#478—22no—
336#476—21no—
337#474—21no—
338#463—22no—
339#461—21no—
340#459—21no—
341#428C22×C3112yes—
342#423C22×C3111yescharacteristic
343#423C22×C3111yescharacteristic
344C2×#104C22×C3112yes—
345C2×#104C22×C3112yes—
346#418C22×C3112yes—
347#417C22×C3112yes—
348C2×#101C22×C3112yes—
349#391C22×C3111yescharacteristic
350#390C22×C3111yescharacteristic
351#388—22no—
352#385—22no—
353#384C22×C3111yescharacteristic
354#376C22×C3111yescharacteristic
355#343C22×C3112yes—
356#339C22×C3112yes—
357#339C22×C3112yes—
358C2×C31—21no—
359C2×C31—21no—
360C2×C31—82no—
361C2×C31—82no—
362C2×C31#240711yescharacteristic
363C2×C31#302011yescharacteristic
364C2×C31#302111yescharacteristic
365C2×#41C23×C3112yes—
366C22×Q8C23×C3111yescharacteristic
367C22×D4C23×C3111yescharacteristic
368C23×C4—21no—
369AGL(1,ℤ/8)—24no—
370AGL(1,ℤ/8)—42no—
371C2×SD16—22no—
372C2×D8—22no—
373C2×M16C23×C3112yes—
374C2×M16C23×C3112yes—
375C4⋊Q8—22no—
376C4⋊Q8D4×C3111yescharacteristic
377C4⋊Q8D4×C3111yescharacteristic
378C4⋊Q8D4×C3111yescharacteristic
379#127D4×C3111yescharacteristic
380#126—24no—
381#126—24no—
382#125—22no—
383#125—24no—
384#125D4×C3112yes—
385#124—22no—
386#124—22no—
387#124—22no—
388#124—22no—
389#124D4×C3112yes—
390#123—24no—
391#123—24no—
392C22⋊Q8—24no—
393C22⋊Q8—21no—
394C22⋊Q8—21no—
395C22⋊Q8—22no—
396C22⋊Q8—22no—
397#121—21no—
398#121—21no—
399C4×Q8—22no—
400C4×Q8—22no—
401C4×Q8—21no—
402C4×Q8—21no—
403C4×Q8—21no—
404C4×Q8—21no—
405C4×Q8—21no—
406C4×Q8—21no—
407C4×D4—22no—
408C4×D4—22no—
409C4×D4—24no—
410C4×D4—21no—
411C4×D4—21no—
412C4×D4—21no—
413C4×D4—21no—
414C4×D4—21no—
415C4×D4—21no—
416#117—24no—
417#117—22no—
418#117C23×C3112yes—
419C2×C4⋊C4—22no—
420C2×C4⋊C4—22no—
421C2×C4⋊C4D4×C3112yes—
422C2×C4⋊C4C23×C3111yescharacteristic
423C2×C4⋊C4C23×C3111yescharacteristic
424C2×C22⋊C4—22no—
425C2×C22⋊C4—21no—
426C2×C22⋊C4—22no—
427C2×C42D4×C3112yes—
428C2×C42C23×C3111yescharacteristic
429C2×C42C23×C3111yescharacteristic
430C2×C42C23×C3111yescharacteristic
431#107—22no—
432#106—22no—
433C4⋊C8—21no—
434C4⋊C8—21no—
435C4⋊C8—21no—
436C4⋊C8—21no—
437#104—24no—
438#104—42no—
439#103—22no—
440#102—22no—
441#102—22no—
442#102—22no—
443#101—42no—
444#95—21no—
445#95—21no—
446#95—22no—
447C31#2418111yescharacteristic, direct product factor, Sylow
448C24—21no—
449#41—22no—
450#41—22no—
451#41—22no—
452#41—22no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8C2×D4×C3112yes—
462C2×D4—22no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4C2×D4×C3112yes—
471C2×D4#41×C3112yes—
472C22×C4—21no—
473C22×C4—24no—
474C22×C4—22no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—41no—
478C22×C4—41no—
479C22×C4—82no—
480C22×C4C2×D4×C3112yes—
481C22×C4C2×D4×C3112yes—
482C22×C4C2×D4×C3111yescharacteristic
483C22×C4C2×D4×C3111yescharacteristic
484C22×C4C24×C3111yescharacteristic
485SD16—44no—
486D8—44no—
487M16—22no—
488M16—22no—
489M16—22no—
490M16—22no—
491C2×C8—22no—
492C2×C8—22no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—22no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—82no—
509C4⋊C4C2×D4×C3112yes—
510C4⋊C4C2×D4×C3112yes—
511C22⋊C4—28no—
512C22⋊C4—24no—
513C22⋊C4—24no—
514C22⋊C4—24no—
515C22⋊C4—22no—
516C22⋊C4—22no—
517C22⋊C4—22no—
518C22⋊C4—41no—
519C22⋊C4—41no—
520C22⋊C4—82no—
521C42—24no—
522C42—22no—
523C42—22no—
524C42—22no—
525C42—21no—
526C42—21no—
527C42—21no—
528C42C2×D4×C3111yescharacteristic
529C42C2×D4×C3111yescharacteristic
530C42C2×D4×C3111yescharacteristic
531C42C2×D4×C3111yescharacteristic
532C42#41×C3112yes—
533C23—22no—
534C23—22no—
535C23—42no—
536C23—42no—
537C23—41no—
538C23—41no—
539C23C22×D4×C3111yescharacteristic
540Q8—22no—
541Q8—22no—
542Q8—42no—
543Q8—41no—
544Q8—41no—
545Q8—41no—
546Q8—41no—
547Q8—81no—
548D4—22no—
549D4—22no—
550D4—22no—
551D4—22no—
552D4—42no—
553D4—42no—
554D4—42no—
555D4—41no—
556D4—41no—
557D4—81no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4—84no—
581C2×C4C31×#12011yescharacteristic
582C2×C4C31×#12011yescharacteristic
583C2×C4C31×#12011yescharacteristic
584C2×C4C31×#12011yescharacteristic
585C2×C4C22×D4×C3111yescharacteristic
586C2×C4C22×D4×C3111yescharacteristic
587C2×C4C2×#41×C3111yescharacteristic
588C2×C4C2×#41×C3111yescharacteristic
589C2×C4C31×H(5,𝔽2)11yescharacteristic
590C2×C4C31×2−1+411yescharacteristic
591C8—42no—
592C8—42no—
593C22—21no—
594C22—21no—
595C22—44no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22—42no—
600C22—82no—
601C22—81no—
602C22—81no—
603C22C2×C31×#12011yescharacteristic
604C22C31×#52511yescharacteristic
605C22C31×#53711yescharacteristic
606C4—21no—
607C4—21no—
608C4—21no—
609C4—21no—
610C4—42no—
611C4—42no—
612C4—42no—
613C4—42no—
614C4—42no—
615C4—42no—
616C4—81no—
617C4—81no—
618C4C31×#54611yescharacteristic
619C4C31×#54711yescharacteristic
620C4C31×#54811yescharacteristic
621C4C31×#55011yescharacteristic
622C2—21no—
623C2—21no—
624C2—82no—
625C2—82no—
626C2C31×#240711yescharacteristic
627C2C31×#302011yescharacteristic
628C2C31×#302111yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#3036C2
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
2. C31×#3077C2semidirect
4. C31×#2326C2semidirect
5. C31×#2038C2semidirect
6. C31×#2003C2semidirect
7. C31×#1995C2semidirect
8. C31×#1963C2semidirect
9. C31×#1883C2semidirect
10. C31×#1859C2semidirect
11. C31×#1801C2semidirect
218. #24181C31direct (N × H)
279. #3507C2×C31semidirect
280. #3077C2×C31semidirect
282. #2326C2×C31semidirect
283. #2038C2×C31semidirect
284. #2003C2×C31semidirect
285. #1995C2×C31semidirect
286. #1963C2×C31semidirect
287. #1883C2×C31semidirect
288. #1859C2×C31semidirect
289. #1801C2×C31semidirect
447. C31#24181direct (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×#30772
C31×#30362
C31×#23262
C31×#20382
C31×#20032
C31×#19952
C31×#19632
C31×#18832
C31×#18592
C31×#18012
#2418131