Finite Group Databasebeta

← C31×#24178 #21480937

Subgroups — C31×#24178 #21480937

Order 7936.

1102 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3506C211yesmaximal, characteristic
2C31×#3078C211yesmaximal, characteristic, complemented
3C31×#3037C211yesmaximal, characteristic, complemented
4C4×C31×#124C211yesmaximal, characteristic, complemented
5C31×#2037C212yesmaximal, complemented
6C31×#2004C212yesmaximal, complemented
7C31×#1994C211yesmaximal, characteristic, complemented
8C31×#1962C211yesmaximal, characteristic, complemented
9C31×#1882C212yesmaximal, complemented
10C31×#1860C212yesmaximal, complemented
11C31×#1800C211yesmaximal, characteristic, complemented
12C2×C31×#137C2212yes—
13C31×#559—22no—
14C31×#556—21no—
15C31×#555—21no—
16C31×#553—22no—
17C31×#550—21no—
18C31×#547—21no—
19C31×#538—22no—
20C31×#535C2212yes—
21C31×#533C2211yescharacteristic
22C31×#532C2211yescharacteristic
23C2×C31×#124C2211yescharacteristic
24C31×#525—22no—
25C31×#520C2212yes—
26C2×C4×Q8×C31C2211yescharacteristic
27C2×C4×D4×C31C2211yescharacteristic
28C31×#479—22no—
29C31×#477—21no—
30C31×#475—21no—
31C31×#464—22no—
32C31×#462—21no—
33C31×#460—21no—
34C31×#428C2212yes—
35C31×#423C2211yescharacteristic
36C31×#423C2211yescharacteristic
37C2×C31×#104C2212yes—
38C2×C31×#104C2212yes—
39C31×#419C2212yes—
40C31×#417C2212yes—
41C2×C31×#100C2212yes—
42C31×#391C2211yescharacteristic
43C31×#390C2211yescharacteristic
44C31×#388—22no—
45C31×#382C2211yescharacteristic
46C4×C22⋊C4×C31—22no—
47C43×C31C2211yescharacteristic
48C31×#343C2212yes—
49C31×#339C2212yes—
50C31×#339C2212yes—
51C2×#41×C31—22no—
52C2×#41×C31C2312yes—
53C22×Q8×C31C2311yescharacteristic
54C22×D4×C31C2311yescharacteristic
55C23×C4×C31D412yes—
56C31×#137—24no—
57C31×#137—42no—
58C2×Q16×C31—22no—
59C2×SD16×C31—22no—
60C2×M16×C31C2312yes—
61C2×M16×C31C2312yes—
62C31×C4⋊Q8D411yescharacteristic
63C31×C4⋊Q8D411yescharacteristic
64C31×C4⋊Q8D411yescharacteristic
65C31×#127—22no—
66C31×#127D411yescharacteristic
67C31×#126—24no—
68C31×#125D412yes—
69C31×#124—24no—
70C31×#124—22no—
71C31×#124—22no—
72C31×#124—22no—
73C31×#124D412yes—
74C31×#123—24no—
75C31×#123—24no—
76C31×C22⋊Q8—22no—
77C31×C22⋊Q8—21no—
78C31×C22⋊Q8—21no—
79C31×#121—22no—
80C31×#121—24no—
81C31×#121—21no—
82C31×#121—21no—
83C31×#121—22no—
84C31×#121—22no—
85C31×#120—24no—
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—24no—
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—22no—
103C31×#117—22no—
104C31×#117C2312yes—
105C2×C4⋊C4×C31—22no—
106C2×C4⋊C4×C31—21no—
107C2×C4⋊C4×C31C2311yescharacteristic
108C2×C4⋊C4×C31C2311yescharacteristic
109C2×C22⋊C4×C31—22no—
110C2×C22⋊C4×C31—22no—
111C2×C22⋊C4×C31D412yes—
112C2×C42×C31—21no—
113C2×C42×C31—21no—
114C2×C42×C31—21no—
115C2×C42×C31C2311yescharacteristic
116C2×C42×C31C2311yescharacteristic
117C2×C42×C31C2311yescharacteristic
118C31×#107—22no—
119C31×#106—22no—
120C31×C4⋊C8—21no—
121C31×C4⋊C8—21no—
122C31×C4⋊C8—21no—
123C31×C4⋊C8—21no—
124C31×#104—24no—
125C31×#104—42no—
126C31×#103—22no—
127C31×#103—22no—
128C31×#103—22no—
129C31×#102—22no—
130C31×#100—42no—
131C31×#95—22no—
132C31×#95—22no—
133C24×C31C2×D412yes—
134#41×C31—22no—
135#41×C31—22no—
136#41×C31—22no—
137#41×C31—22no—
138#41×C31—84no—
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×Q8×C31#4112yescomplemented
148C2×D4×C31—22no—
149C2×D4×C31—24no—
150C2×D4×C31—24no—
151C2×D4×C31—21no—
152C2×D4×C31—21no—
153C2×D4×C31—21no—
154C2×D4×C31—21no—
155C2×D4×C31—21no—
156C2×D4×C31—21no—
157C2×D4×C31—21no—
158C2×D4×C31—21no—
159C2×D4×C31C2×D412yes—
160C22×C4×C31—21no—
161C22×C4×C31—21no—
162C22×C4×C31—21no—
163C22×C4×C31—24no—
164C22×C4×C31—22no—
165C22×C4×C31—22no—
166C22×C4×C31—22no—
167C22×C4×C31—21no—
168C22×C4×C31—44no—
169C22×C4×C31C2×D411yescharacteristic
170C22×C4×C31C2×D412yes—
171C22×C4×C31C2×D411yescharacteristic
172C22×C4×C31C2411yescharacteristic
173Q16×C31—44no—
174SD16×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—21no—
183C4⋊C4×C31—21no—
184C4⋊C4×C31—22no—
185C4⋊C4×C31—21no—
186C4⋊C4×C31—22no—
187C4⋊C4×C31—21no—
188C4⋊C4×C31—21no—
189C4⋊C4×C31—21no—
190C4⋊C4×C31—21no—
191C4⋊C4×C31—21no—
192C4⋊C4×C31—41no—
193C4⋊C4×C31—41no—
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—22no—
204C22⋊C4×C31—22no—
205C22⋊C4×C31—44no—
206C42×C31—24no—
207C42×C31—22no—
208C42×C31—21no—
209C42×C31—21no—
210C42×C31—21no—
211C42×C31—42no—
212C42×C31—42no—
213C42×C31—41no—
214C42×C31—41no—
215C42×C31C2×D411yescharacteristic
216C42×C31C2×D411yescharacteristic
217C42×C31C2×D411yescharacteristic
218C42×C31C2×D411yescharacteristic
219C42×C31#4112yescomplemented
220#24178C3111yesmaximal, characteristic, direct product factor, Sylow
221C23×C31—22no—
222C23×C31—22no—
223C23×C31—22no—
224C23×C31—22no—
225C23×C31—42no—
226C23×C31—42no—
227C23×C31C22×D411yescharacteristic
228Q8×C31—22no—
229Q8×C31—22no—
230Q8×C31—22no—
231Q8×C31—22no—
232Q8×C31—41no—
233Q8×C31—41no—
234Q8×C31—42no—
235Q8×C31—81no—
236D4×C31—22no—
237D4×C31—22no—
238D4×C31—42no—
239D4×C31—42no—
240D4×C31—41no—
241D4×C31—41no—
242D4×C31—41no—
243D4×C31—41no—
244D4×C31—42no—
245D4×C31—81no—
246D4×C31—84no—
247D4×C31—84no—
248C2×C4×C31—22no—
249C2×C4×C31—22no—
250C2×C4×C31—22no—
251C2×C4×C31—22no—
252C2×C4×C31—22no—
253C2×C4×C31—22no—
254C2×C4×C31—22no—
255C2×C4×C31—22no—
256C2×C4×C31—41no—
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—42no—
264C2×C4×C31—42no—
265C2×C4×C31—44no—
266C2×C4×C31—42no—
267C2×C4×C31—42no—
268C2×C4×C31—41no—
269C2×C4×C31—82no—
270C2×C4×C31—84no—
271C2×C4×C31#12011yescharacteristic
272C2×C4×C31#12011yescharacteristic
273C2×C4×C31#12011yescharacteristic
274C2×C4×C31#12011yescharacteristic
275C2×C4×C31C22×D411yescharacteristic
276C2×C4×C31C22×D411yescharacteristic
277C2×C4×C31C2×#4111yescharacteristic
278C2×C4×C31C2×#4111yescharacteristic
279C2×C4×C31H(5,𝔽2)11yescharacteristic
280C2×C4×C312−1+411yescharacteristic
281C8×C31—42no—
282C8×C31—42no—
283#3506C2×C3111yescharacteristic
284#3078C2×C3111yescharacteristic, complemented
285#3037C2×C3111yescharacteristic, complemented
286C4×#124C2×C3111yescharacteristic, complemented
287#2037C2×C3112yescomplemented
288#2004C2×C3112yescomplemented
289#1994C2×C3111yescharacteristic, complemented
290#1962C2×C3111yescharacteristic, complemented
291#1882C2×C3112yescomplemented
292#1860C2×C3112yescomplemented
293#1800C2×C3111yescharacteristic, complemented
294C22×C31—21no—
295C22×C31—21no—
296C22×C31—44no—
297C22×C31—42no—
298C22×C31—42no—
299C22×C31—42no—
300C22×C31—42no—
301C22×C31—42no—
302C22×C31—42no—
303C22×C31—82no—
304C22×C31C2×#12011yescharacteristic
305C22×C31#52511yescharacteristic
306C22×C31#53711yescharacteristic
307C4×C31—21no—
308C4×C31—21no—
309C4×C31—21no—
310C4×C31—42no—
311C4×C31—42no—
312C4×C31—42no—
313C4×C31—42no—
314C4×C31—42no—
315C4×C31—42no—
316C4×C31—42no—
317C4×C31—81no—
318C4×C31#45512yes—
319C4×C31#54611yescharacteristic
320C4×C31#54711yescharacteristic
321C4×C31#54811yescharacteristic
322C4×C31#55011yescharacteristic
323C2×#137C22×C3112yes—
324#559—22no—
325#556—21no—
326#555—21no—
327#553—22no—
328#550—21no—
329#547—21no—
330#538—22no—
331#535C22×C3112yes—
332#533C22×C3111yescharacteristic
333#532C22×C3111yescharacteristic
334C2×#124C22×C3111yescharacteristic
335#525—22no—
336#520C22×C3112yes—
337C2×C4×Q8C22×C3111yescharacteristic
338C2×C4×D4C22×C3111yescharacteristic
339#479—22no—
340#477—21no—
341#475—21no—
342#464—22no—
343#462—21no—
344#460—21no—
345#428C22×C3112yes—
346#423C22×C3111yescharacteristic
347#423C22×C3111yescharacteristic
348C2×#104C22×C3112yes—
349C2×#104C22×C3112yes—
350#419C22×C3112yes—
351#417C22×C3112yes—
352C2×#100C22×C3112yes—
353#391C22×C3111yescharacteristic
354#390C22×C3111yescharacteristic
355#388—22no—
356#382C22×C3111yescharacteristic
357C4×C22⋊C4—22no—
358C43C22×C3111yescharacteristic
359#343C22×C3112yes—
360#339C22×C3112yes—
361#339C22×C3112yes—
362C2×C31—21no—
363C2×C31—21no—
364C2×C31—82no—
365C2×C31—82no—
366C2×C31#240711yescharacteristic
367C2×C31C2×#45511yescharacteristic
368C2×C31#301811yescharacteristic
369C2×#41—22no—
370C2×#41C23×C3112yes—
371C22×Q8C23×C3111yescharacteristic
372C22×D4C23×C3111yescharacteristic
373C23×C4D4×C3112yes—
374#137—24no—
375#137—42no—
376C2×Q16—22no—
377C2×SD16—22no—
378C2×M16C23×C3112yes—
379C2×M16C23×C3112yes—
380C4⋊Q8D4×C3111yescharacteristic
381C4⋊Q8D4×C3111yescharacteristic
382C4⋊Q8D4×C3111yescharacteristic
383#127—22no—
384#127D4×C3111yescharacteristic
385#126—24no—
386#125D4×C3112yes—
387#124—24no—
388#124—22no—
389#124—22no—
390#124—22no—
391#124D4×C3112yes—
392#123—24no—
393#123—24no—
394C22⋊Q8—22no—
395C22⋊Q8—21no—
396C22⋊Q8—21no—
397#121—22no—
398#121—24no—
399#121—21no—
400#121—21no—
401#121—22no—
402#121—22no—
403#120—24no—
404C4×Q8—22no—
405C4×Q8—21no—
406C4×Q8—21no—
407C4×Q8—21no—
408C4×Q8—21no—
409C4×Q8—21no—
410C4×Q8—21no—
411C4×D4—24no—
412C4×D4—22no—
413C4×D4—24no—
414C4×D4—21no—
415C4×D4—21no—
416C4×D4—21no—
417C4×D4—21no—
418C4×D4—21no—
419C4×D4—21no—
420#117—22no—
421#117—22no—
422#117C23×C3112yes—
423C2×C4⋊C4—22no—
424C2×C4⋊C4—21no—
425C2×C4⋊C4C23×C3111yescharacteristic
426C2×C4⋊C4C23×C3111yescharacteristic
427C2×C22⋊C4—22no—
428C2×C22⋊C4—22no—
429C2×C22⋊C4D4×C3112yes—
430C2×C42—21no—
431C2×C42—21no—
432C2×C42—21no—
433C2×C42C23×C3111yescharacteristic
434C2×C42C23×C3111yescharacteristic
435C2×C42C23×C3111yescharacteristic
436#107—22no—
437#106—22no—
438C4⋊C8—21no—
439C4⋊C8—21no—
440C4⋊C8—21no—
441C4⋊C8—21no—
442#104—24no—
443#104—42no—
444#103—22no—
445#103—22no—
446#103—22no—
447#102—22no—
448#100—42no—
449#95—22no—
450#95—22no—
451C31#2417811yescharacteristic, direct product factor, Sylow
452C24C2×D4×C3112yes—
453#41—22no—
454#41—22no—
455#41—22no—
456#41—22no—
457#41—84no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8C2×D4×C3112yes—
466C2×Q8#41×C3112yescomplemented
467C2×D4—22no—
468C2×D4—24no—
469C2×D4—24no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4C2×D4×C3112yes—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—24no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—21no—
487C22×C4—44no—
488C22×C4C2×D4×C3111yescharacteristic
489C22×C4C2×D4×C3112yes—
490C22×C4C2×D4×C3111yescharacteristic
491C22×C4C24×C3111yescharacteristic
492Q16—44no—
493SD16—44no—
494M16—22no—
495M16—22no—
496M16—22no—
497M16—22no—
498C2×C8—22no—
499C2×C8—22no—
500C4⋊C4—24no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—22no—
504C4⋊C4—21no—
505C4⋊C4—22no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—41no—
512C4⋊C4—41no—
513C4⋊C4C2×D4×C3112yes—
514C4⋊C4C2×D4×C3112yes—
515C22⋊C4—28no—
516C22⋊C4—24no—
517C22⋊C4—24no—
518C22⋊C4—24no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—22no—
522C22⋊C4—22no—
523C22⋊C4—22no—
524C22⋊C4—44no—
525C42—24no—
526C42—22no—
527C42—21no—
528C42—21no—
529C42—21no—
530C42—42no—
531C42—42no—
532C42—41no—
533C42—41no—
534C42C2×D4×C3111yescharacteristic
535C42C2×D4×C3111yescharacteristic
536C42C2×D4×C3111yescharacteristic
537C42C2×D4×C3111yescharacteristic
538C42#41×C3112yescomplemented
539C23—22no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—42no—
544C23—42no—
545C23C22×D4×C3111yescharacteristic
546Q8—22no—
547Q8—22no—
548Q8—22no—
549Q8—22no—
550Q8—42no—
551Q8—41no—
552Q8—41no—
553Q8—81no—
554D4—22no—
555D4—22no—
556D4—42no—
557D4—42no—
558D4—42no—
559D4—41no—
560D4—41no—
561D4—41no—
562D4—41no—
563D4—81no—
564D4—84no—
565D4—84no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—82no—
588C2×C4—84no—
589C2×C4C31×#12011yescharacteristic
590C2×C4C31×#12011yescharacteristic
591C2×C4C31×#12011yescharacteristic
592C2×C4C31×#12011yescharacteristic
593C2×C4C22×D4×C3111yescharacteristic
594C2×C4C22×D4×C3111yescharacteristic
595C2×C4C2×#41×C3111yescharacteristic
596C2×C4C2×#41×C3111yescharacteristic
597C2×C4C31×H(5,𝔽2)11yescharacteristic
598C2×C4C31×2−1+411yescharacteristic
599C8—42no—
600C8—42no—
601C22—21no—
602C22—21no—
603C22—44no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—82no—
611C22C2×C31×#12011yescharacteristic
612C22C31×#52511yescharacteristic
613C22C31×#53711yescharacteristic
614C4—21no—
615C4—21no—
616C4—21no—
617C4—42no—
618C4—42no—
619C4—42no—
620C4—42no—
621C4—42no—
622C4—42no—
623C4—42no—
624C4—81no—
625C4C31×#45512yes—
626C4C31×#54611yescharacteristic
627C4C31×#54711yescharacteristic
628C4C31×#54811yescharacteristic
629C4C31×#55011yescharacteristic
630C2—21no—
631C2—21no—
632C2—82no—
633C2—82no—
634C2C31×#240711yescharacteristic
635C2C2×C31×#45511yescharacteristic
636C2C31×#301811yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#3506C2
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
2. C31×#3078C2semidirect
3. C31×#3037C2semidirect
4. C4×C31×#124C2semidirect
5. C31×#2037C2semidirect
6. C31×#2004C2semidirect
7. C31×#1994C2semidirect
8. C31×#1962C2semidirect
9. C31×#1882C2semidirect
10. C31×#1860C2semidirect
11. C31×#1800C2semidirect
147. C2×Q8×C31#41semidirect
219. C42×C31#41semidirect
220. #24178C31direct (N × H)
284. #3078C2×C31semidirect
285. #3037C2×C31semidirect
286. C4×#124C2×C31semidirect
287. #2037C2×C31semidirect
288. #2004C2×C31semidirect
289. #1994C2×C31semidirect
290. #1962C2×C31semidirect
291. #1882C2×C31semidirect
292. #1860C2×C31semidirect
293. #1800C2×C31semidirect
451. C31#24178direct (N × H)
466. C2×Q8#41×C31semidirect
538. C42#41×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#35062
C31×#30782
C31×#30372
C4×C31×#1242
C31×#20372
C31×#20042
C31×#19942
C31×#19622
C31×#18822
C31×#18602
C31×#18002
#2417831