Finite Group Databasebeta

← C31×#19049 #21475808

Subgroups — C31×#19049 #21475808

Order 7936.

886 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3561C211yesmaximal, characteristic
2C31×#3215C211yesmaximal, characteristic, complemented
3C31×#3181C211yesmaximal, characteristic, complemented
4C31×#3162C211yesmaximal, characteristic, complemented
5C31×#1664C212yesmaximal, complemented
6C31×#1661C212yesmaximal, complemented
7C31×#1661C212yesmaximal, complemented
8C31×#1660C212yesmaximal, complemented
9C31×#1532C211yesmaximal, characteristic, complemented
10C31×#1531C211yesmaximal, characteristic, complemented
11C2×C31×#329C211yesmaximal, characteristic, complemented
12C31×#563C2212yes—
13C31×#562C2212yes—
14C31×#556—21no—
15C31×#551—22no—
16D4×Q8×C31—21no—
17C31×#548—21no—
18C31×#546—21no—
19C31×#543—21no—
20C31×#542—21no—
21C31×#542—22no—
22C31×#538—21no—
23C31×#536—21no—
24C31×#533C2211yescharacteristic
25C31×#533C2211yescharacteristic
26C31×#529C2211yescharacteristic
27C2×C31×#124C2211yescharacteristic
28C31×#501—21no—
29C31×#501—21no—
30C31×#501C2212yescomplemented
31C31×#501C2212yescomplemented
32C31×#501C2211yescharacteristic, complemented
33C31×#497C2212yes—
34C31×#497C2211yescharacteristic, complemented
35C31×#491C2211yescharacteristic, complemented
36C31×#489C2212yes—
37C31×#489C2211yescharacteristic, complemented
38C31×#467—21no—
39C31×#465—21no—
40C31×#464—22no—
41C31×#451—21no—
42C31×#450—21no—
43C31×#441—21no—
44C31×#440—21no—
45C31×#432C2211yescharacteristic
46C31×#427—22no—
47C2×C31×#97C2211yescharacteristic
48C2×C31×#97C2211yescharacteristic
49C31×#330C2212yescomplemented
50C31×#330C2212yescomplemented
51C31×#330C2212yescomplemented
52C31×#329C2212yes—
53C31×#329C2212yes—
54C31×#329C2212yescomplemented
55C2×#41×C31—21no—
56C22×Q8×C31—21no—
57C22×Q8×C31D411yescharacteristic
58C22×D4×C31D411yescharacteristic
59C2×Q16×C31—21no—
60C2×Q16×C31—22no—
61C2×SD16×C31—21no—
62C2×SD16×C31—22no—
63C22×C8×C31—21no—
64C22×C8×C31—21no—
65C31×C4⋊Q8—21no—
66C31×C4⋊Q8—22no—
67C31×C4⋊Q8D411yescharacteristic
68C31×C4⋊Q8D411yescharacteristic
69C31×C4⋊Q8C2311yescharacteristic
70C31×C4⋊Q8C2311yescharacteristic
71C31×#126—22no—
72C31×#126—22no—
73C31×#126—22no—
74C31×#126—21no—
75C31×#125—21no—
76C31×#125C2311yescharacteristic
77C31×#125C2312yes—
78C31×#124—21no—
79C31×#124—22no—
80C31×#124—22no—
81C31×#124—21no—
82C31×#124—21no—
83C31×#124D412yes—
84C31×#124C2312yes—
85C31×#124C2311yescharacteristic
86C31×#123—22no—
87C31×#123—22no—
88C31×#123—21no—
89C31×#123—22no—
90C31×#123—22no—
91C31×C22⋊Q8—22no—
92C31×C22⋊Q8—22no—
93C31×C22⋊Q8—21no—
94C31×C22⋊Q8—22no—
95C31×C22⋊Q8—22no—
96C31×C22⋊Q8—22no—
97C31×C22⋊Q8—21no—
98C31×C22⋊Q8—21no—
99C31×C22⋊Q8—22no—
100C31×C22⋊Q8—22no—
101C31×C22⋊Q8—21no—
102C31×C22⋊Q8—21no—
103C31×C22⋊Q8D411yescharacteristic
104C31×#121—21no—
105C31×#121—22no—
106C31×#121—22no—
107C31×#121D411yescharacteristic
108C31×#120—22no—
109C4×Q8×C31—21no—
110C4×Q8×C31—21no—
111C4×Q8×C31—21no—
112C4×Q8×C31—21no—
113C4×Q8×C31—21no—
114C4×Q8×C31D411yescharacteristic
115C4×D4×C31—22no—
116C4×D4×C31—21no—
117C4×D4×C31—22no—
118C4×D4×C31—21no—
119C4×D4×C31D411yescharacteristic
120C31×#117—22no—
121C31×#117—21no—
122C31×#117—21no—
123C31×#117D412yes—
124C2×C22⋊C4×C31—22no—
125C2×C22⋊C4×C31—21no—
126C2×C22⋊C4×C31—21no—
127C2×C42×C31C2311yescharacteristic
128C31×#107—22no—
129C31×#107—22no—
130C31×#107—21no—
131C31×#106—22no—
132C31×#106—22no—
133C31×#106—21no—
134C31×C4⋊C8—21no—
135C31×#103—21no—
136C31×#103—22no—
137C31×#103—21no—
138C31×#103—21no—
139C31×#102—21no—
140C31×#102—22no—
141C31×C22⋊C8—21no—
142C31×#97—21no—
143C31×#97—21no—
144C31×#97C2311yescharacteristic
145C31×#97C2311yescharacteristic
146C31×#97C2312yes—
147C31×#97C2312yes—
148C24×C31—21no—
149#41×C31—82no—
150C2×Q8×C31—22no—
151C2×Q8×C31—21no—
152C2×Q8×C31—21no—
153C2×Q8×C31—21no—
154C2×Q8×C31—21no—
155C2×Q8×C31—21no—
156C2×Q8×C31—82no—
157C2×Q8×C31—81no—
158C2×Q8×C31C2×D411yescharacteristic
159C2×Q8×C31C2×D412yes—
160C2×D4×C31—21no—
161C2×D4×C31—22no—
162C2×D4×C31—22no—
163C2×D4×C31—21no—
164C2×D4×C31—81no—
165C2×D4×C31C2×D411yescharacteristic
166C2×D4×C31C2×D412yes—
167C22×C4×C31—22no—
168C22×C4×C31—22no—
169C22×C4×C31—22no—
170C22×C4×C31—21no—
171C22×C4×C31—21no—
172C22×C4×C31C2×D411yescharacteristic
173C22×C4×C31C2×D411yescharacteristic
174C22×C4×C31C2×D411yescharacteristic
175Q16×C31—42no—
176Q16×C31—82no—
177SD16×C31—42no—
178SD16×C31—82no—
179C2×C8×C31—24no—
180C2×C8×C31—22no—
181C2×C8×C31—21no—
182C2×C8×C31—21no—
183C2×C8×C31—22no—
184C2×C8×C31—42no—
185C4⋊C4×C31—22no—
186C4⋊C4×C31—22no—
187C4⋊C4×C31—22no—
188C4⋊C4×C31—22no—
189C4⋊C4×C31—21no—
190C4⋊C4×C31—21no—
191C4⋊C4×C31—21no—
192C4⋊C4×C31—21no—
193C4⋊C4×C31—21no—
194C4⋊C4×C31—21no—
195C4⋊C4×C31—22no—
196C4⋊C4×C31—22no—
197C4⋊C4×C31—21no—
198C4⋊C4×C31—21no—
199C4⋊C4×C31—21no—
200C4⋊C4×C31—21no—
201C4⋊C4×C31—41no—
202C4⋊C4×C31—41no—
203C4⋊C4×C31C2×D411yescharacteristic
204C4⋊C4×C31C2×D411yescharacteristic
205C4⋊C4×C31C2×D412yes—
206C4⋊C4×C31C2×D412yes—
207C22⋊C4×C31—22no—
208C22⋊C4×C31—22no—
209C22⋊C4×C31—22no—
210C22⋊C4×C31—22no—
211C22⋊C4×C31—22no—
212C22⋊C4×C31—21no—
213C22⋊C4×C31—22no—
214C22⋊C4×C31—22no—
215C22⋊C4×C31—22no—
216C22⋊C4×C31—21no—
217C22⋊C4×C31—21no—
218C22⋊C4×C31—22no—
219C22⋊C4×C31—22no—
220C22⋊C4×C31—21no—
221C22⋊C4×C31—21no—
222C22⋊C4×C31—41no—
223C42×C31—22no—
224C42×C31—22no—
225C42×C31—21no—
226C42×C31—41no—
227C42×C31C2×D411yescharacteristic
228C42×C31C2×D411yescharacteristic
229C42×C31C2×D411yescharacteristic
230C42×C31C2411yescharacteristic
231#19049C3111yesmaximal, characteristic, direct product factor, Sylow
232C23×C31—22no—
233C23×C31—21no—
234C23×C31—42no—
235C23×C31—81no—
236C23×C31—81no—
237C23×C31#12011yescharacteristic
238Q8×C31—41no—
239Q8×C31—42no—
240Q8×C31—42no—
241Q8×C31—42no—
242Q8×C31—41no—
243Q8×C31—42no—
244Q8×C31—41no—
245Q8×C31—81no—
246Q8×C31—82no—
247D4×C31—41no—
248D4×C31—42no—
249D4×C31—42no—
250D4×C31—42no—
251D4×C31—82no—
252D4×C31—82no—
253C2×C4×C31—22no—
254C2×C4×C31—21no—
255C2×C4×C31—22no—
256C2×C4×C31—21no—
257C2×C4×C31—22no—
258C2×C4×C31—22no—
259C2×C4×C31—21no—
260C2×C4×C31—41no—
261C2×C4×C31—41no—
262C2×C4×C31—42no—
263C2×C4×C31—41no—
264C2×C4×C31—41no—
265C2×C4×C31—81no—
266C2×C4×C31—82no—
267C2×C4×C31—82no—
268C2×C4×C31—82no—
269C2×C4×C31#12011yescharacteristic
270C2×C4×C31#12011yescharacteristic
271C2×C4×C31#12011yescharacteristic
272C2×C4×C31C22×D411yescharacteristic
273C2×C4×C31C22×D411yescharacteristic
274C2×C4×C31C22×D411yescharacteristic
275C8×C31—24no—
276C8×C31—22no—
277C8×C31—41no—
278C8×C31—42no—
279#3561C2×C3111yescharacteristic
280#3215C2×C3111yescharacteristic, complemented
281#3181C2×C3111yescharacteristic, complemented
282#3162C2×C3111yescharacteristic, complemented
283#1664C2×C3112yescomplemented
284#1661C2×C3112yescomplemented
285#1661C2×C3112yescomplemented
286#1660C2×C3112yescomplemented
287#1532C2×C3111yescharacteristic, complemented
288#1531C2×C3111yescharacteristic, complemented
289C2×#329C2×C3111yescharacteristic, complemented
290C22×C31—21no—
291C22×C31—21no—
292C22×C31—41no—
293C22×C31—42no—
294C22×C31—42no—
295C22×C31—41no—
296C22×C31—41no—
297C22×C31—42no—
298C22×C31—162no—
299C22×C31#45612yes—
300C22×C31C2×#12011yescharacteristic
301C4×C31—21no—
302C4×C31—21no—
303C4×C31—21no—
304C4×C31—21no—
305C4×C31—21no—
306C4×C31—21no—
307C4×C31—41no—
308C4×C31—41no—
309C4×C31—42no—
310C4×C31—41no—
311C4×C31—42no—
312C4×C31—42no—
313C4×C31—81no—
314C4×C31—81no—
315#563C22×C3112yes—
316#562C22×C3112yes—
317#556—21no—
318#551—22no—
319D4×Q8—21no—
320#548—21no—
321#546—21no—
322#543—21no—
323#542—21no—
324#542—22no—
325#538—21no—
326#536—21no—
327#533C22×C3111yescharacteristic
328#533C22×C3111yescharacteristic
329#529C22×C3111yescharacteristic
330C2×#124C22×C3111yescharacteristic
331#501—21no—
332#501—21no—
333#501C22×C3112yescomplemented
334#501C22×C3112yescomplemented
335#501C22×C3111yescharacteristic, complemented
336#497C22×C3112yes—
337#497C22×C3111yescharacteristic, complemented
338#491C22×C3111yescharacteristic, complemented
339#489C22×C3112yes—
340#489C22×C3111yescharacteristic, complemented
341#467—21no—
342#465—21no—
343#464—22no—
344#451—21no—
345#450—21no—
346#441—21no—
347#440—21no—
348#432C22×C3111yescharacteristic
349#427—22no—
350C2×#97C22×C3111yescharacteristic
351C2×#97C22×C3111yescharacteristic
352#330C22×C3112yescomplemented
353#330C22×C3112yescomplemented
354#330C22×C3112yescomplemented
355#329C22×C3112yes—
356#329C22×C3112yes—
357#329C22×C3112yescomplemented
358C2×C31—22no—
359C2×C31—81no—
360C2×C31—82no—
361C2×C31C2×#45611yescharacteristic
362C2×C31#301811yescharacteristic
363C2×C31#301911yescharacteristic
364C2×#41—21no—
365C22×Q8—21no—
366C22×Q8D4×C3111yescharacteristic
367C22×D4D4×C3111yescharacteristic
368C2×Q16—21no—
369C2×Q16—22no—
370C2×SD16—21no—
371C2×SD16—22no—
372C22×C8—21no—
373C22×C8—21no—
374C4⋊Q8—21no—
375C4⋊Q8—22no—
376C4⋊Q8D4×C3111yescharacteristic
377C4⋊Q8D4×C3111yescharacteristic
378C4⋊Q8C23×C3111yescharacteristic
379C4⋊Q8C23×C3111yescharacteristic
380#126—22no—
381#126—22no—
382#126—22no—
383#126—21no—
384#125—21no—
385#125C23×C3111yescharacteristic
386#125C23×C3112yes—
387#124—21no—
388#124—22no—
389#124—22no—
390#124—21no—
391#124—21no—
392#124D4×C3112yes—
393#124C23×C3112yes—
394#124C23×C3111yescharacteristic
395#123—22no—
396#123—22no—
397#123—21no—
398#123—22no—
399#123—22no—
400C22⋊Q8—22no—
401C22⋊Q8—22no—
402C22⋊Q8—21no—
403C22⋊Q8—22no—
404C22⋊Q8—22no—
405C22⋊Q8—22no—
406C22⋊Q8—21no—
407C22⋊Q8—21no—
408C22⋊Q8—22no—
409C22⋊Q8—22no—
410C22⋊Q8—21no—
411C22⋊Q8—21no—
412C22⋊Q8D4×C3111yescharacteristic
413#121—21no—
414#121—22no—
415#121—22no—
416#121D4×C3111yescharacteristic
417#120—22no—
418C4×Q8—21no—
419C4×Q8—21no—
420C4×Q8—21no—
421C4×Q8—21no—
422C4×Q8—21no—
423C4×Q8D4×C3111yescharacteristic
424C4×D4—22no—
425C4×D4—21no—
426C4×D4—22no—
427C4×D4—21no—
428C4×D4D4×C3111yescharacteristic
429#117—22no—
430#117—21no—
431#117—21no—
432#117D4×C3112yes—
433C2×C22⋊C4—22no—
434C2×C22⋊C4—21no—
435C2×C22⋊C4—21no—
436C2×C42C23×C3111yescharacteristic
437#107—22no—
438#107—22no—
439#107—21no—
440#106—22no—
441#106—22no—
442#106—21no—
443C4⋊C8—21no—
444#103—21no—
445#103—22no—
446#103—21no—
447#103—21no—
448#102—21no—
449#102—22no—
450C22⋊C8—21no—
451#97—21no—
452#97—21no—
453#97C23×C3112yes—
454#97C23×C3111yescharacteristic
455#97C23×C3111yescharacteristic
456#97C23×C3112yes—
457C31#1904911yescharacteristic, direct product factor, Sylow
458C24—21no—
459#41—82no—
460C2×Q8—22no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—82no—
467C2×Q8—81no—
468C2×Q8C2×D4×C3111yescharacteristic
469C2×Q8C2×D4×C3112yes—
470C2×D4—21no—
471C2×D4—22no—
472C2×D4—22no—
473C2×D4—21no—
474C2×D4—81no—
475C2×D4C2×D4×C3111yescharacteristic
476C2×D4C2×D4×C3112yes—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4C2×D4×C3111yescharacteristic
483C22×C4C2×D4×C3111yescharacteristic
484C22×C4C2×D4×C3111yescharacteristic
485Q16—42no—
486Q16—82no—
487SD16—42no—
488SD16—82no—
489C2×C8—24no—
490C2×C8—22no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—22no—
494C2×C8—42no—
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—22no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—41no—
512C4⋊C4—41no—
513C4⋊C4C2×D4×C3111yescharacteristic
514C4⋊C4C2×D4×C3111yescharacteristic
515C4⋊C4C2×D4×C3112yes—
516C4⋊C4C2×D4×C3112yes—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—22no—
522C22⋊C4—21no—
523C22⋊C4—22no—
524C22⋊C4—22no—
525C22⋊C4—22no—
526C22⋊C4—21no—
527C22⋊C4—21no—
528C22⋊C4—22no—
529C22⋊C4—22no—
530C22⋊C4—21no—
531C22⋊C4—21no—
532C22⋊C4—41no—
533C42—22no—
534C42—22no—
535C42—21no—
536C42—41no—
537C42C2×D4×C3111yescharacteristic
538C42C2×D4×C3111yescharacteristic
539C42C2×D4×C3111yescharacteristic
540C42C24×C3111yescharacteristic
541C23—22no—
542C23—21no—
543C23—42no—
544C23—81no—
545C23—81no—
546C23C31×#12011yescharacteristic
547Q8—41no—
548Q8—42no—
549Q8—42no—
550Q8—42no—
551Q8—41no—
552Q8—42no—
553Q8—41no—
554Q8—81no—
555Q8—82no—
556D4—41no—
557D4—42no—
558D4—42no—
559D4—42no—
560D4—82no—
561D4—82no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—81no—
575C2×C4—82no—
576C2×C4—82no—
577C2×C4—82no—
578C2×C4C31×#12011yescharacteristic
579C2×C4C31×#12011yescharacteristic
580C2×C4C31×#12011yescharacteristic
581C2×C4C22×D4×C3111yescharacteristic
582C2×C4C22×D4×C3111yescharacteristic
583C2×C4C22×D4×C3111yescharacteristic
584C8—24no—
585C8—22no—
586C8—41no—
587C8—42no—
588C22—21no—
589C22—21no—
590C22—41no—
591C22—42no—
592C22—42no—
593C22—41no—
594C22—41no—
595C22—42no—
596C22—162no—
597C22C31×#45612yes—
598C22C2×C31×#12011yescharacteristic
599C4—21no—
600C4—21no—
601C4—21no—
602C4—21no—
603C4—21no—
604C4—21no—
605C4—41no—
606C4—41no—
607C4—42no—
608C4—41no—
609C4—42no—
610C4—42no—
611C4—81no—
612C4—81no—
613C2—22no—
614C2—82no—
615C2—81no—
616C2C2×C31×#45611yescharacteristic
617C2C31×#301811yescharacteristic
618C2C31×#301911yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC42C24×C31
prime radicalC31×#3561C2
LCSG ▷ C42 ▷ 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×#3215C2semidirect
3. C31×#3181C2semidirect
4. C31×#3162C2semidirect
5. C31×#1664C2semidirect
6. C31×#1661C2semidirect
7. C31×#1661C2semidirect
8. C31×#1660C2semidirect
9. C31×#1532C2semidirect
10. C31×#1531C2semidirect
11. C2×C31×#329C2semidirect
30. C31×#501C22semidirect
31. C31×#501C22semidirect
32. C31×#501C22semidirect
34. C31×#497C22semidirect
35. C31×#491C22semidirect
37. C31×#489C22semidirect
49. C31×#330C22semidirect
50. C31×#330C22semidirect
51. C31×#330C22semidirect
54. C31×#329C22semidirect
231. #19049C31direct (N × H)
280. #3215C2×C31semidirect
281. #3181C2×C31semidirect
282. #3162C2×C31semidirect
283. #1664C2×C31semidirect
284. #1661C2×C31semidirect
285. #1661C2×C31semidirect
286. #1660C2×C31semidirect
287. #1532C2×C31semidirect
288. #1531C2×C31semidirect
289. C2×#329C2×C31semidirect
333. #501C22×C31semidirect
334. #501C22×C31semidirect
335. #501C22×C31semidirect
337. #497C22×C31semidirect
338. #491C22×C31semidirect
340. #489C22×C31semidirect
352. #330C22×C31semidirect
353. #330C22×C31semidirect
354. #330C22×C31semidirect
357. #329C22×C31semidirect
457. C31#19049direct (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×#32152
C31×#31812
C31×#31622
C31×#16642
C31×#16612
C31×#16602
C31×#15322
C31×#15312
C2×C31×#3292
#1904931