Finite Group Databasebeta

← C31×#18304 #21475063

Subgroups — C31×#18304 #21475063

Order 7936.

1054 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3455C211yesmaximal, characteristic, complemented
2C31×#3208C211yesmaximal, characteristic, complemented
3C31×#3205C211yesmaximal, characteristic, complemented
4C31×#2619C211yesmaximal, characteristic, complemented
5C31×#1600C214yesmaximal, complemented
6C31×#1597C214yesmaximal, complemented
7C31×#1518C211yesmaximal, characteristic, complemented
8C31×#1517C211yesmaximal, characteristic, complemented
9C31×#1458C211yesmaximal, characteristic, complemented
10C31×#542—21no—
11C31×#542—21no—
12C31×#541—21no—
13C31×#540—21no—
14C31×#539—21no—
15C31×#539C2212yes—
16C31×#538—21no—
17C31×#538C2212yes—
18C31×#533C2211yescharacteristic
19C31×#532C2211yescharacteristic
20C2×C31×#127C2211yescharacteristic
21C31×#529—22no—
22C31×#529—22no—
23C31×#525—21no—
24C2×C31×#121—21no—
25C4×#41×C31—21no—
26C2×C31×#117C2211yescharacteristic
27C31×#493—21no—
28C31×#492—21no—
29C31×#486—21no—
30C31×#486—21no—
31C31×#460—22no—
32C31×#459—22no—
33C31×#450C2214yes—
34C31×#447C2214yes—
35C31×#431C2211yescharacteristic
36C31×#431C2211yescharacteristic
37C31×#394—21no—
38C31×#394C2212yes—
39C31×#392—21no—
40C31×#392C2212yes—
41C31×#390—21no—
42C31×#382C2211yescharacteristic
43C31×#328C2214yes—
44C31×#327C2214yes—
45C31×#324C2212yes—
46C31×#323C2212yes—
47C2×#41×C31—21no—
48C22×D4×C31—22no—
49C22×D4×C31—21no—
50C22×D4×C31—21no—
51C22×D4×C31C2312yes—
52C23×C4×C31D411yescharacteristic
53C2×SD16×C31—24no—
54C2×D8×C31—24no—
55C31×C4⋊Q8D411yescharacteristic, complemented
56C31×C4⋊Q8D411yescharacteristic, complemented
57C31×C4⋊Q8D411yescharacteristic, complemented
58C31×#127—21no—
59C31×#127—21no—
60C31×#127—21no—
61C31×#127—21no—
62C31×#127D411yescharacteristic, complemented
63C31×#126—22no—
64C31×#126—22no—
65C31×#126—22no—
66C31×#126—22no—
67C31×#125—22no—
68C31×#125—22no—
69C31×#125—21no—
70C31×#124—22no—
71C31×#124—22no—
72C31×#124—21no—
73C31×#123—22no—
74C31×#123—22no—
75C31×#123—22no—
76C31×#123—22no—
77C31×#123—22no—
78C31×C22⋊Q8—21no—
79C31×C22⋊Q8—21no—
80C31×C22⋊Q8—22no—
81C31×C22⋊Q8—22no—
82C31×C22⋊Q8—21no—
83C31×C22⋊Q8C2312yes—
84C31×#121—21no—
85C31×#121—21no—
86C31×#121—22no—
87C31×#121—22no—
88C31×#121—21no—
89C31×#121—44no—
90C31×#121C2312yes—
91C31×#120—22no—
92C4×Q8×C31—21no—
93C4×Q8×C31—21no—
94C4×Q8×C31—21no—
95C4×Q8×C31D412yescomplemented
96C4×D4×C31—21no—
97C4×D4×C31—21no—
98C4×D4×C31—22no—
99C4×D4×C31—22no—
100C4×D4×C31—22no—
101C4×D4×C31—22no—
102C4×D4×C31—21no—
103C4×D4×C31D412yescomplemented
104C31×#117—22no—
105C31×#117—21no—
106C31×#117—21no—
107C31×#117—21no—
108C31×#117—21no—
109C31×#117—21no—
110C2×C4⋊C4×C31—21no—
111C2×C4⋊C4×C31D411yescharacteristic
112C2×C4⋊C4×C31C2311yescharacteristic
113C2×C22⋊C4×C31—22no—
114C2×C22⋊C4×C31—22no—
115C2×C22⋊C4×C31—22no—
116C2×C22⋊C4×C31D411yescharacteristic
117C2×C22⋊C4×C31C2311yescharacteristic
118C2×C42×C31—22no—
119C2×C42×C31D411yescharacteristic
120C2×C42×C31C2311yescharacteristic
121C31×C4⋊C8—21no—
122C31×C4⋊C8—21no—
123C31×#102—24no—
124C31×#102—24no—
125C31×C22⋊C8C2312yes—
126C31×C22⋊C8C2312yes—
127C4×C8×C31—21no—
128C4×C8×C31—21no—
129C31×#95—21no—
130C31×#95C2312yes—
131C24×C31—22no—
132C24×C31C2×D411yescharacteristic
133#41×C31—42no—
134#41×C31—42no—
135C2×Q8×C31—21no—
136C2×Q8×C31—21no—
137C2×Q8×C31C2×D412yes—
138C2×D4×C31—22no—
139C2×D4×C31—22no—
140C2×D4×C31—22no—
141C2×D4×C31—21no—
142C2×D4×C31—21no—
143C2×D4×C31—22no—
144C2×D4×C31—22no—
145C2×D4×C31—44no—
146C2×D4×C31—41no—
147C2×D4×C31—41no—
148C2×D4×C31—41no—
149C2×D4×C31—41no—
150C2×D4×C31—42no—
151C2×D4×C31—42no—
152C2×D4×C31—82no—
153C2×D4×C31—82no—
154C2×D4×C31—82no—
155C2×D4×C31—82no—
156C2×D4×C31C2×D412yes—
157C22×C4×C31—22no—
158C22×C4×C31—22no—
159C22×C4×C31—22no—
160C22×C4×C31—21no—
161C22×C4×C31—21no—
162C22×C4×C31—22no—
163C22×C4×C31—22no—
164C22×C4×C31—22no—
165C22×C4×C31C2×D411yescharacteristic
166C22×C4×C31C2×D411yescharacteristic
167C22×C4×C31C2×D411yescharacteristic
168C22×C4×C31C2×D411yescharacteristic
169C22×C4×C31C2×D411yescharacteristic
170C22×C4×C31C2411yescharacteristic
171SD16×C31—84no—
172D8×C31—84no—
173C2×C8×C31—22no—
174C2×C8×C31—22no—
175C4⋊C4×C31—22no—
176C4⋊C4×C31—22no—
177C4⋊C4×C31—22no—
178C4⋊C4×C31—22no—
179C4⋊C4×C31—21no—
180C4⋊C4×C31—21no—
181C4⋊C4×C31—21no—
182C4⋊C4×C31—21no—
183C4⋊C4×C31—21no—
184C4⋊C4×C31—21no—
185C4⋊C4×C31—22no—
186C4⋊C4×C31—21no—
187C4⋊C4×C31—21no—
188C4⋊C4×C31—22no—
189C4⋊C4×C31—42no—
190C4⋊C4×C31C2×D412yes—
191C4⋊C4×C31C2×D412yes—
192C22⋊C4×C31—22no—
193C22⋊C4×C31—22no—
194C22⋊C4×C31—22no—
195C22⋊C4×C31—22no—
196C22⋊C4×C31—22no—
197C22⋊C4×C31—22no—
198C22⋊C4×C31—22no—
199C22⋊C4×C31—21no—
200C22⋊C4×C31—21no—
201C22⋊C4×C31—22no—
202C22⋊C4×C31—21no—
203C22⋊C4×C31—21no—
204C22⋊C4×C31—22no—
205C22⋊C4×C31—44no—
206C22⋊C4×C31—82no—
207C22⋊C4×C31—82no—
208C42×C31—22no—
209C42×C31—22no—
210C42×C31—21no—
211C42×C31—22no—
212C42×C31—21no—
213C42×C31C2×D411yescharacteristic
214C42×C31C2×D411yescharacteristic
215C42×C31C2×D411yescharacteristic
216C42×C31C2×D411yescharacteristic
217#18304C3111yesmaximal, characteristic, direct product factor, Sylow
218C23×C31—21no—
219C23×C31—22no—
220C23×C31—21no—
221C23×C31—21no—
222C23×C31—41no—
223C23×C31—41no—
224C23×C31—42no—
225C23×C31—42no—
226C23×C31—42no—
227C23×C31—42no—
228C23×C31—42no—
229C23×C31—42no—
230C23×C31—42no—
231C23×C31C22×D411yescharacteristic
232Q8×C31—42no—
233Q8×C31—42no—
234Q8×C31—81no—
235D4×C31—42no—
236D4×C31—44no—
237D4×C31—44no—
238D4×C31—42no—
239D4×C31—42no—
240D4×C31—42no—
241D4×C31—81no—
242D4×C31—82no—
243D4×C31—82no—
244D4×C31—82no—
245D4×C31—82no—
246D4×C31—88no—
247D4×C31—82no—
248D4×C31—82no—
249C2×C4×C31—21no—
250C2×C4×C31—21no—
251C2×C4×C31—22no—
252C2×C4×C31—22no—
253C2×C4×C31—22no—
254C2×C4×C31—21no—
255C2×C4×C31—21no—
256C2×C4×C31—21no—
257C2×C4×C31—41no—
258C2×C4×C31—41no—
259C2×C4×C31—42no—
260C2×C4×C31—42no—
261C2×C4×C31—44no—
262C2×C4×C31—41no—
263C2×C4×C31—41no—
264C2×C4×C31—42no—
265C2×C4×C31—42no—
266C2×C4×C31—42no—
267C2×C4×C31—82no—
268C2×C4×C31—82no—
269C2×C4×C31#12011yescharacteristic
270C2×C4×C31#12011yescharacteristic
271C2×C4×C31#12011yescharacteristic
272C2×C4×C31#12011yescharacteristic
273C2×C4×C31#13512yes—
274C2×C4×C31AGL(1,ℤ/8)12yes—
275C2×C4×C31C22×D411yescharacteristic
276C2×C4×C31C22×D411yescharacteristic
277C8×C31—42no—
278C8×C31—42no—
279#3455C2×C3111yescharacteristic, complemented
280#3208C2×C3111yescharacteristic, complemented
281#3205C2×C3111yescharacteristic, complemented
282#2619C2×C3111yescharacteristic, complemented
283#1600C2×C3114yescomplemented
284#1597C2×C3114yescomplemented
285#1518C2×C3111yescharacteristic, complemented
286#1517C2×C3111yescharacteristic, complemented
287#1458C2×C3111yescharacteristic, complemented
288C22×C31—21no—
289C22×C31—21no—
290C22×C31—42no—
291C22×C31—41no—
292C22×C31—42no—
293C22×C31—42no—
294C22×C31—41no—
295C22×C31—42no—
296C22×C31—41no—
297C22×C31—81no—
298C22×C31—82no—
299C22×C31—82no—
300C22×C31—82no—
301C22×C31—82no—
302C22×C31—82no—
303C22×C31—82no—
304C22×C31—82no—
305C22×C31C2×#12011yescharacteristic
306C22×C31C2×#13511yescharacteristic
307C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
308C4×C31—21no—
309C4×C31—21no—
310C4×C31—21no—
311C4×C31—21no—
312C4×C31—41no—
313C4×C31—42no—
314C4×C31—41no—
315C4×C31—42no—
316C4×C31—41no—
317C4×C31—42no—
318C4×C31—82no—
319C4×C31#44912yes—
320C4×C31#44912yes—
321#542—21no—
322#542—21no—
323#541—21no—
324#540—21no—
325#539—21no—
326#539C22×C3112yes—
327#538—21no—
328#538C22×C3112yes—
329#533C22×C3111yescharacteristic
330#532C22×C3111yescharacteristic
331C2×#127C22×C3111yescharacteristic
332#529—22no—
333#529—22no—
334#525—21no—
335C2×#121—21no—
336C4×#41—21no—
337C2×#117C22×C3111yescharacteristic
338#493—21no—
339#492—21no—
340#486—21no—
341#486—21no—
342#460—22no—
343#459—22no—
344#450C22×C3114yes—
345#447C22×C3114yes—
346#431C22×C3111yescharacteristic
347#431C22×C3111yescharacteristic
348#394—21no—
349#394C22×C3112yes—
350#392—21no—
351#392C22×C3112yes—
352#390—21no—
353#382C22×C3111yescharacteristic
354#328C22×C3114yes—
355#327C22×C3114yes—
356#324C22×C3112yes—
357#323C22×C3112yes—
358C2×C31—21no—
359C2×C31—21no—
360C2×C31—82no—
361C2×C31—81no—
362C2×C31—162no—
363C2×C31C2×#44911yescharacteristic
364C2×C31#302111yescharacteristic
365C2×C31#302711yescharacteristic
366C2×#41—21no—
367C22×D4—22no—
368C22×D4—21no—
369C22×D4—21no—
370C22×D4C23×C3112yes—
371C23×C4D4×C3111yescharacteristic
372C2×SD16—24no—
373C2×D8—24no—
374C4⋊Q8D4×C3111yescharacteristic, complemented
375C4⋊Q8D4×C3111yescharacteristic, complemented
376C4⋊Q8D4×C3111yescharacteristic, complemented
377#127—21no—
378#127—21no—
379#127—21no—
380#127—21no—
381#127D4×C3111yescharacteristic, complemented
382#126—22no—
383#126—22no—
384#126—22no—
385#126—22no—
386#125—22no—
387#125—22no—
388#125—21no—
389#124—22no—
390#124—22no—
391#124—21no—
392#123—22no—
393#123—22no—
394#123—22no—
395#123—22no—
396#123—22no—
397C22⋊Q8—21no—
398C22⋊Q8—21no—
399C22⋊Q8—22no—
400C22⋊Q8—22no—
401C22⋊Q8—21no—
402C22⋊Q8C23×C3112yes—
403#121—21no—
404#121—21no—
405#121—22no—
406#121—22no—
407#121—21no—
408#121—44no—
409#121C23×C3112yes—
410#120—22no—
411C4×Q8—21no—
412C4×Q8—21no—
413C4×Q8—21no—
414C4×Q8D4×C3112yescomplemented
415C4×D4—21no—
416C4×D4—21no—
417C4×D4—22no—
418C4×D4—22no—
419C4×D4—22no—
420C4×D4—22no—
421C4×D4—21no—
422C4×D4D4×C3112yescomplemented
423#117—22no—
424#117—21no—
425#117—21no—
426#117—21no—
427#117—21no—
428#117—21no—
429C2×C4⋊C4—21no—
430C2×C4⋊C4D4×C3111yescharacteristic
431C2×C4⋊C4C23×C3111yescharacteristic
432C2×C22⋊C4—22no—
433C2×C22⋊C4—22no—
434C2×C22⋊C4—22no—
435C2×C22⋊C4D4×C3111yescharacteristic
436C2×C22⋊C4C23×C3111yescharacteristic
437C2×C42—22no—
438C2×C42D4×C3111yescharacteristic
439C2×C42C23×C3111yescharacteristic
440C4⋊C8—21no—
441C4⋊C8—21no—
442#102—24no—
443#102—24no—
444C22⋊C8C23×C3112yes—
445C22⋊C8C23×C3112yes—
446C4×C8—21no—
447C4×C8—21no—
448#95—21no—
449#95C23×C3112yes—
450C31#1830411yescharacteristic, direct product factor, Sylow
451C24—22no—
452C24C2×D4×C3111yescharacteristic
453#41—42no—
454#41—42no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8C2×D4×C3112yes—
458C2×D4—22no—
459C2×D4—22no—
460C2×D4—22no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—22no—
464C2×D4—22no—
465C2×D4—44no—
466C2×D4—41no—
467C2×D4—41no—
468C2×D4—41no—
469C2×D4—41no—
470C2×D4—42no—
471C2×D4—42no—
472C2×D4—82no—
473C2×D4—82no—
474C2×D4—82no—
475C2×D4—82no—
476C2×D4C2×D4×C3112yes—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4C2×D4×C3111yescharacteristic
486C22×C4C2×D4×C3111yescharacteristic
487C22×C4C2×D4×C3111yescharacteristic
488C22×C4C2×D4×C3111yescharacteristic
489C22×C4C2×D4×C3111yescharacteristic
490C22×C4C24×C3111yescharacteristic
491SD16—84no—
492D8—84no—
493C2×C8—22no—
494C2×C8—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—22no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—22no—
509C4⋊C4—42no—
510C4⋊C4C2×D4×C3112yes—
511C4⋊C4C2×D4×C3112yes—
512C22⋊C4—22no—
513C22⋊C4—22no—
514C22⋊C4—22no—
515C22⋊C4—22no—
516C22⋊C4—22no—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—21no—
520C22⋊C4—21no—
521C22⋊C4—22no—
522C22⋊C4—21no—
523C22⋊C4—21no—
524C22⋊C4—22no—
525C22⋊C4—44no—
526C22⋊C4—82no—
527C22⋊C4—82no—
528C42—22no—
529C42—22no—
530C42—21no—
531C42—22no—
532C42—21no—
533C42C2×D4×C3111yescharacteristic
534C42C2×D4×C3111yescharacteristic
535C42C2×D4×C3111yescharacteristic
536C42C2×D4×C3111yescharacteristic
537C23—21no—
538C23—22no—
539C23—21no—
540C23—21no—
541C23—41no—
542C23—41no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23C22×D4×C3111yescharacteristic
551Q8—42no—
552Q8—42no—
553Q8—81no—
554D4—42no—
555D4—44no—
556D4—44no—
557D4—42no—
558D4—42no—
559D4—42no—
560D4—81no—
561D4—82no—
562D4—82no—
563D4—82no—
564D4—82no—
565D4—88no—
566D4—82no—
567D4—82no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—82no—
587C2×C4—82no—
588C2×C4C31×#12011yescharacteristic
589C2×C4C31×#12011yescharacteristic
590C2×C4C31×#12011yescharacteristic
591C2×C4C31×#12011yescharacteristic
592C2×C4C31×#13512yes—
593C2×C4C31×AGL(1,ℤ/8)12yes—
594C2×C4C22×D4×C3111yescharacteristic
595C2×C4C22×D4×C3111yescharacteristic
596C8—42no—
597C8—42no—
598C22—21no—
599C22—21no—
600C22—42no—
601C22—41no—
602C22—42no—
603C22—42no—
604C22—41no—
605C22—42no—
606C22—41no—
607C22—81no—
608C22—82no—
609C22—82no—
610C22—82no—
611C22—82no—
612C22—82no—
613C22—82no—
614C22—82no—
615C22C2×C31×#12011yescharacteristic
616C22C2×C31×#13511yescharacteristic
617C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
618C4—21no—
619C4—21no—
620C4—21no—
621C4—21no—
622C4—41no—
623C4—42no—
624C4—41no—
625C4—42no—
626C4—41no—
627C4—42no—
628C4—82no—
629C4C31×#44912yes—
630C4C31×#44912yes—
631C2—21no—
632C2—21no—
633C2—81no—
634C2—82no—
635C2—162no—
636C2C2×C31×#44911yescharacteristic
637C2C31×#302111yescharacteristic
638C2C31×#302711yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
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
1. C31×#3455C2semidirect
2. C31×#3208C2semidirect
3. C31×#3205C2semidirect
4. C31×#2619C2semidirect
5. C31×#1600C2semidirect
6. C31×#1597C2semidirect
7. C31×#1518C2semidirect
8. C31×#1517C2semidirect
9. C31×#1458C2semidirect
55. C31×C4⋊Q8D4semidirect
56. C31×C4⋊Q8D4semidirect
57. C31×C4⋊Q8D4semidirect
62. C31×#127D4semidirect
95. C4×Q8×C31D4semidirect
103. C4×D4×C31D4semidirect
217. #18304C31direct (N × H)
279. #3455C2×C31semidirect
280. #3208C2×C31semidirect
281. #3205C2×C31semidirect
282. #2619C2×C31semidirect
283. #1600C2×C31semidirect
284. #1597C2×C31semidirect
285. #1518C2×C31semidirect
286. #1517C2×C31semidirect
287. #1458C2×C31semidirect
374. C4⋊Q8D4×C31semidirect
375. C4⋊Q8D4×C31semidirect
376. C4⋊Q8D4×C31semidirect
381. #127D4×C31semidirect
414. C4×Q8D4×C31semidirect
422. C4×D4D4×C31semidirect
450. C31#18304direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#34552
C31×#32082
C31×#32052
C31×#26192
C31×#16002
C31×#15972
C31×#15182
C31×#15172
C31×#14582
#1830431