Finite Group Databasebeta

← C31×#18467 #21475226

Subgroups — C31×#18467 #21475226

Order 7936.

890 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3561C211yesmaximal, characteristic
2C31×#3236C211yesmaximal, characteristic, complemented
3C31×#3232C211yesmaximal, characteristic, complemented
4C31×#2820C211yesmaximal, characteristic, complemented
5C31×#1618C212yesmaximal, complemented
6C31×#1614C212yesmaximal, complemented
7C31×#1612C212yesmaximal, complemented
8C31×#1608C212yesmaximal, complemented
9C31×#1518C211yesmaximal, characteristic, complemented
10C31×#1512C211yesmaximal, characteristic, complemented
11C31×#1469C211yesmaximal, characteristic, complemented
12C31×#563—21no—
13C31×#562—21no—
14C31×#556—21no—
15C31×#551—22no—
16D4×Q8×C31—21no—
17C31×#548—21no—
18C31×#546—21no—
19C31×#543C2212yes—
20C31×#542—21no—
21C31×#542—22no—
22C31×#538C2212yes—
23C31×#536—21no—
24C31×#533C2211yescharacteristic
25C31×#529C2211yescharacteristic
26C2×C31×#125C2211yescharacteristic
27C2×C31×#124C2211yescharacteristic
28C31×#501—21no—
29C31×#499—21no—
30C31×#491—21no—
31C31×#488—21no—
32C31×#484C2211yescharacteristic
33C31×#484C2212yes—
34C31×#483C2211yescharacteristic
35C31×#483C2212yes—
36C31×#481C2211yescharacteristic
37C31×#480C2211yescharacteristic
38C31×#479—21no—
39C31×#479—22no—
40C31×#478—21no—
41C31×#432C2211yescharacteristic
42C31×#431C2211yescharacteristic
43C31×#399C2212yes—
44C31×#397C2212yes—
45C31×#396—22no—
46C31×#395—21no—
47C31×#394—21no—
48C31×#382C2211yescharacteristic
49C31×#328C2212yes—
50C31×#328C2212yes—
51C31×#328C2212yes—
52C31×#327C2212yes—
53C31×#323C2212yes—
54C31×#323C2212yes—
55C2×#41×C31—21no—
56C22×Q8×C31—21no—
57C22×Q8×C31D411yescharacteristic
58C22×D4×C31D411yescharacteristic
59C31×C4⋊Q8—21no—
60C31×C4⋊Q8—22no—
61C31×C4⋊Q8D411yescharacteristic
62C31×C4⋊Q8D411yescharacteristic
63C31×#126—22no—
64C31×#126—22no—
65C31×#126—22no—
66C31×#126—21no—
67C31×#125—21no—
68C31×#125—21no—
69C31×#125—21no—
70C31×#125—21no—
71C31×#125—21no—
72C31×#125D411yescharacteristic
73C31×#124—21no—
74C31×#124—22no—
75C31×#124—22no—
76C31×#124—21no—
77C31×#124—21no—
78C31×#124—21no—
79C31×#124—21no—
80C31×#124D411yescharacteristic
81C31×#123—22no—
82C31×#123—22no—
83C31×#123—22no—
84C31×#123—21no—
85C31×#123—22no—
86C31×C22⋊Q8—22no—
87C31×C22⋊Q8—21no—
88C31×C22⋊Q8—22no—
89C31×C22⋊Q8—22no—
90C31×C22⋊Q8—22no—
91C31×C22⋊Q8—21no—
92C31×C22⋊Q8—22no—
93C31×C22⋊Q8—22no—
94C31×C22⋊Q8—22no—
95C31×C22⋊Q8C2311yescharacteristic
96C31×C22⋊Q8C2312yes—
97C31×#121—21no—
98C31×#121—22no—
99C31×#121—22no—
100C31×#121C2311yescharacteristic
101C31×#120—22no—
102C4×Q8×C31—21no—
103C4×Q8×C31—21no—
104C4×Q8×C31D411yescharacteristic
105C4×Q8×C31D412yes—
106C4×D4×C31—21no—
107C4×D4×C31—22no—
108C4×D4×C31—21no—
109C4×D4×C31—22no—
110C4×D4×C31D411yescharacteristic
111C31×#117—22no—
112C31×#117—21no—
113C31×#117—21no—
114C2×C4⋊C4×C31—21no—
115C2×C4⋊C4×C31—21no—
116C2×C4⋊C4×C31—22no—
117C2×C4⋊C4×C31—21no—
118C2×C4⋊C4×C31C2311yescharacteristic
119C2×C4⋊C4×C31C2311yescharacteristic
120C2×C22⋊C4×C31—22no—
121C2×C22⋊C4×C31—22no—
122C2×C22⋊C4×C31—22no—
123C2×C22⋊C4×C31D412yes—
124C2×C22⋊C4×C31C2312yes—
125C2×C42×C31C2311yescharacteristic
126C31×#107—21no—
127C31×#107—21no—
128C31×#107—22no—
129C31×#106—21no—
130C31×#106—21no—
131C31×#106—22no—
132C31×C4⋊C8—21no—
133C31×C4⋊C8—21no—
134C31×#103—21no—
135C31×#103—22no—
136C31×#103—21no—
137C31×#103—22no—
138C31×#102—21no—
139C31×#102—21no—
140C31×C22⋊C8C2311yescharacteristic
141C31×C22⋊C8C2311yescharacteristic
142C31×C22⋊C8C2312yes—
143C31×#97—21no—
144C4×C8×C31—21no—
145C31×#95—22no—
146C31×#95—21no—
147C31×#95C2312yes—
148C24×C31C2×D412yes—
149#41×C31—82no—
150C2×Q8×C31—22no—
151C2×Q8×C31—21no—
152C2×Q8×C31—21no—
153C2×Q8×C31—21no—
154C2×Q8×C31—81no—
155C2×Q8×C31—82no—
156C2×Q8×C31C2×D411yescharacteristic
157C2×Q8×C31C2×D412yes—
158C2×D4×C31—21no—
159C2×D4×C31—22no—
160C2×D4×C31—22no—
161C2×D4×C31—21no—
162C2×D4×C31—21no—
163C2×D4×C31—81no—
164C2×D4×C31C2×D411yescharacteristic
165C22×C4×C31—22no—
166C22×C4×C31—22no—
167C22×C4×C31—22no—
168C22×C4×C31—21no—
169C22×C4×C31—22no—
170C22×C4×C31—21no—
171C22×C4×C31—21no—
172C22×C4×C31C2×D411yescharacteristic
173C22×C4×C31C2×D412yes—
174C22×C4×C31C2×D411yescharacteristic
175C22×C4×C31C2411yescharacteristic
176C2×C8×C31—21no—
177C2×C8×C31—21no—
178C2×C8×C31—22no—
179C4⋊C4×C31—22no—
180C4⋊C4×C31—22no—
181C4⋊C4×C31—22no—
182C4⋊C4×C31—22no—
183C4⋊C4×C31—21no—
184C4⋊C4×C31—21no—
185C4⋊C4×C31—21no—
186C4⋊C4×C31—21no—
187C4⋊C4×C31—21no—
188C4⋊C4×C31—21no—
189C4⋊C4×C31—22no—
190C4⋊C4×C31—21no—
191C4⋊C4×C31—22no—
192C4⋊C4×C31—21no—
193C4⋊C4×C31—21no—
194C4⋊C4×C31—21no—
195C4⋊C4×C31—21no—
196C4⋊C4×C31—41no—
197C4⋊C4×C31—41no—
198C4⋊C4×C31—41no—
199C4⋊C4×C31—41no—
200C4⋊C4×C31—81no—
201C4⋊C4×C31—82no—
202C4⋊C4×C31C2×D411yescharacteristic
203C4⋊C4×C31C2×D411yescharacteristic
204C4⋊C4×C31C2×D412yes—
205C22⋊C4×C31—22no—
206C22⋊C4×C31—22no—
207C22⋊C4×C31—22no—
208C22⋊C4×C31—22no—
209C22⋊C4×C31—22no—
210C22⋊C4×C31—21no—
211C22⋊C4×C31—22no—
212C22⋊C4×C31—22no—
213C22⋊C4×C31—22no—
214C22⋊C4×C31—21no—
215C22⋊C4×C31—21no—
216C22⋊C4×C31—21no—
217C22⋊C4×C31—22no—
218C22⋊C4×C31—22no—
219C22⋊C4×C31—21no—
220C22⋊C4×C31—82no—
221C22⋊C4×C31—82no—
222C42×C31—22no—
223C42×C31—22no—
224C42×C31—21no—
225C42×C31C2×D411yescharacteristic
226C42×C31C2×D411yescharacteristic
227C42×C31C2×D411yescharacteristic
228C42×C31C2×D411yescharacteristic
229#18467C3111yesmaximal, characteristic, direct product factor, Sylow
230C23×C31—22no—
231C23×C31—21no—
232C23×C31—22no—
233C23×C31—22no—
234C23×C31—42no—
235C23×C31—42no—
236C23×C31C22×D411yescharacteristic
237Q8×C31—41no—
238Q8×C31—42no—
239Q8×C31—42no—
240Q8×C31—42no—
241Q8×C31—81no—
242Q8×C31—82no—
243D4×C31—41no—
244D4×C31—42no—
245D4×C31—41no—
246D4×C31—41no—
247D4×C31—81no—
248D4×C31—82no—
249D4×C31—82no—
250C2×C4×C31—21no—
251C2×C4×C31—21no—
252C2×C4×C31—21no—
253C2×C4×C31—22no—
254C2×C4×C31—21no—
255C2×C4×C31—22no—
256C2×C4×C31—22no—
257C2×C4×C31—21no—
258C2×C4×C31—41no—
259C2×C4×C31—41no—
260C2×C4×C31—42no—
261C2×C4×C31—42no—
262C2×C4×C31—42no—
263C2×C4×C31—42no—
264C2×C4×C31—42no—
265C2×C4×C31—41no—
266C2×C4×C31—41no—
267C2×C4×C31—41no—
268C2×C4×C31—42no—
269C2×C4×C31—41no—
270C2×C4×C31—82no—
271C2×C4×C31—82no—
272C2×C4×C31—82no—
273C2×C4×C31#12011yescharacteristic
274C2×C4×C31#12011yescharacteristic
275C2×C4×C31#12011yescharacteristic
276C2×C4×C31#12011yescharacteristic
277C2×C4×C31C22×D411yescharacteristic
278C2×C4×C31C22×D411yescharacteristic
279C8×C31—41no—
280C8×C31—41no—
281C8×C31—42no—
282#3561C2×C3111yescharacteristic
283#3236C2×C3111yescharacteristic, complemented
284#3232C2×C3111yescharacteristic, complemented
285#2820C2×C3111yescharacteristic, complemented
286#1618C2×C3112yescomplemented
287#1614C2×C3112yescomplemented
288#1612C2×C3112yescomplemented
289#1608C2×C3112yescomplemented
290#1518C2×C3111yescharacteristic, complemented
291#1512C2×C3111yescharacteristic, complemented
292#1469C2×C3111yescharacteristic, complemented
293C22×C31—21no—
294C22×C31—21no—
295C22×C31—41no—
296C22×C31—42no—
297C22×C31—41no—
298C22×C31—42no—
299C22×C31—42no—
300C22×C31—41no—
301C22×C31—82no—
302C22×C31—82no—
303C22×C31C2×#12011yescharacteristic
304C22×C31#57711yescharacteristic
305C22×C31#57811yescharacteristic
306C4×C31—21no—
307C4×C31—21no—
308C4×C31—21no—
309C4×C31—21no—
310C4×C31—21no—
311C4×C31—21no—
312C4×C31—41no—
313C4×C31—41no—
314C4×C31—42no—
315C4×C31—42no—
316C4×C31—41no—
317C4×C31—42no—
318C4×C31—82no—
319C4×C31—81no—
320C4×C31—81no—
321#563—21no—
322#562—21no—
323#556—21no—
324#551—22no—
325D4×Q8—21no—
326#548—21no—
327#546—21no—
328#543C22×C3112yes—
329#542—21no—
330#542—22no—
331#538C22×C3112yes—
332#536—21no—
333#533C22×C3111yescharacteristic
334#529C22×C3111yescharacteristic
335C2×#125C22×C3111yescharacteristic
336C2×#124C22×C3111yescharacteristic
337#501—21no—
338#499—21no—
339#491—21no—
340#488—21no—
341#484C22×C3111yescharacteristic
342#484C22×C3112yes—
343#483C22×C3111yescharacteristic
344#483C22×C3112yes—
345#481C22×C3111yescharacteristic
346#480C22×C3111yescharacteristic
347#479—21no—
348#479—22no—
349#478—21no—
350#432C22×C3111yescharacteristic
351#431C22×C3111yescharacteristic
352#399C22×C3112yes—
353#397C22×C3112yes—
354#396—22no—
355#395—21no—
356#394—21no—
357#382C22×C3111yescharacteristic
358#328C22×C3112yes—
359#328C22×C3112yes—
360#328C22×C3112yes—
361#327C22×C3112yes—
362#323C22×C3112yes—
363#323C22×C3112yes—
364C2×C31—21no—
365C2×C31—21no—
366C2×C31—81no—
367C2×C31—82no—
368C2×C31#301311yescharacteristic
369C2×C31#301911yescharacteristic
370C2×C31#302811yescharacteristic
371C2×#41—21no—
372C22×Q8—21no—
373C22×Q8D4×C3111yescharacteristic
374C22×D4D4×C3111yescharacteristic
375C4⋊Q8—21no—
376C4⋊Q8—22no—
377C4⋊Q8D4×C3111yescharacteristic
378C4⋊Q8D4×C3111yescharacteristic
379#126—22no—
380#126—22no—
381#126—22no—
382#126—21no—
383#125—21no—
384#125—21no—
385#125—21no—
386#125—21no—
387#125—21no—
388#125D4×C3111yescharacteristic
389#124—21no—
390#124—22no—
391#124—22no—
392#124—21no—
393#124—21no—
394#124—21no—
395#124—21no—
396#124D4×C3111yescharacteristic
397#123—22no—
398#123—22no—
399#123—22no—
400#123—21no—
401#123—22no—
402C22⋊Q8—22no—
403C22⋊Q8—21no—
404C22⋊Q8—22no—
405C22⋊Q8—22no—
406C22⋊Q8—22no—
407C22⋊Q8—21no—
408C22⋊Q8—22no—
409C22⋊Q8—22no—
410C22⋊Q8—22no—
411C22⋊Q8C23×C3111yescharacteristic
412C22⋊Q8C23×C3112yes—
413#121—21no—
414#121—22no—
415#121—22no—
416#121C23×C3111yescharacteristic
417#120—22no—
418C4×Q8—21no—
419C4×Q8—21no—
420C4×Q8D4×C3111yescharacteristic
421C4×Q8D4×C3112yes—
422C4×D4—21no—
423C4×D4—22no—
424C4×D4—21no—
425C4×D4—22no—
426C4×D4D4×C3111yescharacteristic
427#117—22no—
428#117—21no—
429#117—21no—
430C2×C4⋊C4—21no—
431C2×C4⋊C4—21no—
432C2×C4⋊C4—21no—
433C2×C4⋊C4—22no—
434C2×C4⋊C4C23×C3111yescharacteristic
435C2×C4⋊C4C23×C3111yescharacteristic
436C2×C22⋊C4—22no—
437C2×C22⋊C4—22no—
438C2×C22⋊C4—22no—
439C2×C22⋊C4D4×C3112yes—
440C2×C22⋊C4C23×C3112yes—
441C2×C42C23×C3111yescharacteristic
442#107—21no—
443#107—21no—
444#107—22no—
445#106—21no—
446#106—21no—
447#106—22no—
448C4⋊C8—21no—
449C4⋊C8—21no—
450#103—21no—
451#103—22no—
452#103—21no—
453#103—22no—
454#102—21no—
455#102—21no—
456C22⋊C8C23×C3112yes—
457C22⋊C8C23×C3111yescharacteristic
458C22⋊C8C23×C3111yescharacteristic
459#97—21no—
460C4×C8—21no—
461#95—21no—
462#95—22no—
463#95C23×C3112yes—
464C31#1846711yescharacteristic, direct product factor, Sylow
465C24C2×D4×C3112yes—
466#41—82no—
467C2×Q8—22no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—81no—
472C2×Q8—82no—
473C2×Q8C2×D4×C3111yescharacteristic
474C2×Q8C2×D4×C3112yes—
475C2×D4—21no—
476C2×D4—22no—
477C2×D4—22no—
478C2×D4—21no—
479C2×D4—21no—
480C2×D4—81no—
481C2×D4C2×D4×C3111yescharacteristic
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—21no—
486C22×C4—22no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4C2×D4×C3111yescharacteristic
490C22×C4C2×D4×C3112yes—
491C22×C4C2×D4×C3111yescharacteristic
492C22×C4C24×C3111yescharacteristic
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—22no—
507C4⋊C4—21no—
508C4⋊C4—22no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—41no—
514C4⋊C4—41no—
515C4⋊C4—41no—
516C4⋊C4—41no—
517C4⋊C4—81no—
518C4⋊C4—82no—
519C4⋊C4C2×D4×C3111yescharacteristic
520C4⋊C4C2×D4×C3111yescharacteristic
521C4⋊C4C2×D4×C3112yes—
522C22⋊C4—22no—
523C22⋊C4—22no—
524C22⋊C4—22no—
525C22⋊C4—22no—
526C22⋊C4—22no—
527C22⋊C4—21no—
528C22⋊C4—22no—
529C22⋊C4—22no—
530C22⋊C4—22no—
531C22⋊C4—21no—
532C22⋊C4—21no—
533C22⋊C4—21no—
534C22⋊C4—22no—
535C22⋊C4—22no—
536C22⋊C4—21no—
537C22⋊C4—82no—
538C22⋊C4—82no—
539C42—22no—
540C42—22no—
541C42—21no—
542C42C2×D4×C3111yescharacteristic
543C42C2×D4×C3111yescharacteristic
544C42C2×D4×C3111yescharacteristic
545C42C2×D4×C3111yescharacteristic
546C23—22no—
547C23—21no—
548C23—22no—
549C23—22no—
550C23—42no—
551C23—42no—
552C23C22×D4×C3111yescharacteristic
553Q8—41no—
554Q8—42no—
555Q8—42no—
556Q8—42no—
557Q8—81no—
558Q8—82no—
559D4—41no—
560D4—42no—
561D4—41no—
562D4—41no—
563D4—81no—
564D4—82no—
565D4—82no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—82no—
587C2×C4—82no—
588C2×C4—82no—
589C2×C4C31×#12011yescharacteristic
590C2×C4C31×#12011yescharacteristic
591C2×C4C31×#12011yescharacteristic
592C2×C4C31×#12011yescharacteristic
593C2×C4C22×D4×C3111yescharacteristic
594C2×C4C22×D4×C3111yescharacteristic
595C8—41no—
596C8—41no—
597C8—42no—
598C22—21no—
599C22—21no—
600C22—41no—
601C22—42no—
602C22—41no—
603C22—42no—
604C22—42no—
605C22—41no—
606C22—82no—
607C22—82no—
608C22C2×C31×#12011yescharacteristic
609C22C31×#57711yescharacteristic
610C22C31×#57811yescharacteristic
611C4—21no—
612C4—21no—
613C4—21no—
614C4—21no—
615C4—21no—
616C4—21no—
617C4—41no—
618C4—41no—
619C4—42no—
620C4—42no—
621C4—41no—
622C4—42no—
623C4—82no—
624C4—81no—
625C4—81no—
626C2—21no—
627C2—21no—
628C2—82no—
629C2—81no—
630C2C31×#301311yescharacteristic
631C2C31×#301911yescharacteristic
632C2C31×#302811yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#3561C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#3236C2semidirect
3. C31×#3232C2semidirect
4. C31×#2820C2semidirect
5. C31×#1618C2semidirect
6. C31×#1614C2semidirect
7. C31×#1612C2semidirect
8. C31×#1608C2semidirect
9. C31×#1518C2semidirect
10. C31×#1512C2semidirect
11. C31×#1469C2semidirect
229. #18467C31direct (N × H)
283. #3236C2×C31semidirect
284. #3232C2×C31semidirect
285. #2820C2×C31semidirect
286. #1618C2×C31semidirect
287. #1614C2×C31semidirect
288. #1612C2×C31semidirect
289. #1608C2×C31semidirect
290. #1518C2×C31semidirect
291. #1512C2×C31semidirect
292. #1469C2×C31semidirect
464. C31#18467direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35612
C31×#32362
C31×#32322
C31×#28202
C31×#16182
C31×#16142
C31×#16122
C31×#16082
C31×#15182
C31×#15122
C31×#14692
#1846731