Finite Group Databasebeta

← C31×#18955 #21475714

Subgroups — C31×#18955 #21475714

Order 7936.

938 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3546C211yesmaximal, characteristic
2C31×#3238C211yesmaximal, characteristic, complemented
3C31×#3227C211yesmaximal, characteristic, complemented
4C31×#3206C211yesmaximal, characteristic, complemented
5C31×#1638C212yesmaximal, complemented
6C31×#1635C212yesmaximal, complemented
7C31×#1631C212yesmaximal, complemented
8C31×#1629C212yesmaximal, complemented
9C31×#1553C211yesmaximal, characteristic, complemented
10C31×#1496C211yesmaximal, characteristic, complemented
11C31×#1482C211yesmaximal, characteristic, complemented
12C31×#560—22no—
13C31×#556C2211yescharacteristic
14C31×#553—21no—
15C31×#550—21no—
16C31×#550C2211yescharacteristic
17C31×#547C2211yescharacteristic
18C31×#546—21no—
19C31×#546—21no—
20D42×C31C2211yescharacteristic
21C31×#542—21no—
22C31×#538—22no—
23C31×#537—21no—
24C31×#535—21no—
25C31×#535—21no—
26C31×#532C2211yescharacteristic
27C31×#532C2211yescharacteristic
28C2×C31×C4⋊Q8C2211yescharacteristic, complemented
29C31×#529C2211yescharacteristic
30C31×#501—21no—
31C31×#498—21no—
32C31×#492C2212yes—
33C31×#489—21no—
34C31×#487C2212yescomplemented
35C31×#486—21no—
36C31×#483—21no—
37C31×#481—21no—
38C31×#481—21no—
39C31×#480—21no—
40C31×#477C2212yescomplemented
41C31×#476C2212yes—
42C31×#476C2212yes—
43C31×#476C2212yescomplemented
44C31×#461—22no—
45C31×#448—22no—
46C31×#432C2211yescharacteristic, complemented
47C31×#431C2211yescharacteristic, complemented
48C31×#431C2211yescharacteristic, complemented
49C31×#332C2212yescomplemented
50C31×#331C2212yes—
51C31×Q8⋊C8C2212yescomplemented
52C31×D4⋊C8C2212yes—
53C31×D4⋊C8C2212yes—
54C31×D4⋊C8C2212yescomplemented
55C2×#41×C31—21no—
56C2×#41×C31—21no—
57C2×#41×C31—21no—
58C2×#41×C31—21no—
59C22×Q8×C31—21no—
60C22×D4×C31—21no—
61C22×D4×C31—21no—
62C2×SD16×C31—24no—
63C31×C4⋊Q8—21no—
64C31×C4⋊Q8—21no—
65C31×C4⋊Q8—21no—
66C31×C4⋊Q8D411yescharacteristic
67C31×C4⋊Q8C2312yes—
68C31×C4⋊Q8C2311yescharacteristic
69C31×#127—22no—
70C31×#127—21no—
71C31×#127D411yescharacteristic
72C31×#127C2311yescharacteristic
73C31×#126—22no—
74C31×#126—22no—
75C31×#126—21no—
76C31×#125D411yescharacteristic
77C31×#124—22no—
78C31×#124—21no—
79C31×#124—21no—
80C31×#124—21no—
81C31×#124D411yescharacteristic
82C31×#123—22no—
83C31×#123—22no—
84C31×#123—21no—
85C31×#123—22no—
86C31×C22⋊Q8—22no—
87C31×C22⋊Q8—21no—
88C31×C22⋊Q8—21no—
89C31×C22⋊Q8D411yescharacteristic
90C31×#121—22no—
91C31×#121—22no—
92C31×#121—22no—
93C31×#121—22no—
94C31×#121—22no—
95C31×#121—21no—
96C31×#121—22no—
97C31×#121—21no—
98C31×#121—21no—
99C31×#121—21no—
100C31×#121—21no—
101C31×#121—21no—
102C31×#121D412yes—
103C31×#121D411yescharacteristic
104C31×#121D411yescharacteristic
105C31×#121D411yescharacteristic
106C31×#120—22no—
107C31×#120—22no—
108C31×#120—22no—
109C31×#120—22no—
110C4×Q8×C31—21no—
111C4×Q8×C31C2311yescharacteristic
112C4×D4×C31—22no—
113C4×D4×C31—21no—
114C4×D4×C31—21no—
115C4×D4×C31—21no—
116C4×D4×C31—21no—
117C4×D4×C31D412yes—
118C4×D4×C31C2311yescharacteristic
119C4×D4×C31C2311yescharacteristic
120C4×D4×C31C2311yescharacteristic
121C31×#117—21no—
122C31×#117—21no—
123C2×C4⋊C4×C31—21no—
124C2×C4⋊C4×C31—21no—
125C2×C4⋊C4×C31—21no—
126C2×C22⋊C4×C31—22no—
127C2×C42×C31C2311yescharacteristic
128C31×#107—22no—
129C31×#106—22no—
130C31×#106—22no—
131C31×#106—22no—
132C31×C4⋊C8—21no—
133C31×C4⋊C8C2312yes—
134C31×C4⋊C8C2312yes—
135C31×#103—22no—
136C31×#103—24no—
137C31×#102—22no—
138C31×#102—22no—
139C31×#102—22no—
140C31×C22⋊C8—21no—
141C31×C22⋊C8—21no—
142C31×C22⋊C8—21no—
143C31×#97—21no—
144C4×C8×C31—21no—
145C4×C8×C31C2312yes—
146C24×C31—22no—
147#41×C31—82no—
148#41×C31—82no—
149#41×C31—82no—
150#41×C31—82no—
151C2×Q8×C31—21no—
152C2×Q8×C31—21no—
153C2×Q8×C31—22no—
154C2×Q8×C31—41no—
155C2×Q8×C31—82no—
156C2×Q8×C31C2×D411yescharacteristic
157C2×D4×C31—22no—
158C2×D4×C31—22no—
159C2×D4×C31—22no—
160C2×D4×C31—22no—
161C2×D4×C31—22no—
162C2×D4×C31—21no—
163C2×D4×C31—21no—
164C2×D4×C31—22no—
165C2×D4×C31—21no—
166C2×D4×C31—21no—
167C2×D4×C31—21no—
168C2×D4×C31—21no—
169C2×D4×C31—21no—
170C2×D4×C31—82no—
171C2×D4×C31—82no—
172C2×D4×C31C2×D411yescharacteristic
173C2×D4×C31C2×D411yescharacteristic
174C2×D4×C31C2×D411yescharacteristic
175C2×D4×C31C2×D412yes—
176C22×C4×C31—22no—
177C22×C4×C31—22no—
178C22×C4×C31—21no—
179C22×C4×C31—21no—
180C22×C4×C31—21no—
181C22×C4×C31—22no—
182C22×C4×C31—41no—
183C22×C4×C31C2×D411yescharacteristic
184C22×C4×C31C2×D411yescharacteristic
185C22×C4×C31C2×D411yescharacteristic
186SD16×C31—84no—
187C2×C8×C31—22no—
188C2×C8×C31—22no—
189C2×C8×C31—22no—
190C4⋊C4×C31—22no—
191C4⋊C4×C31—22no—
192C4⋊C4×C31—21no—
193C4⋊C4×C31—21no—
194C4⋊C4×C31—21no—
195C4⋊C4×C31—21no—
196C4⋊C4×C31—21no—
197C4⋊C4×C31—21no—
198C4⋊C4×C31—22no—
199C4⋊C4×C31—22no—
200C4⋊C4×C31—41no—
201C4⋊C4×C31—41no—
202C4⋊C4×C31C2×D412yes—
203C4⋊C4×C31C2×D411yescharacteristic
204C4⋊C4×C31C2×D411yescharacteristic
205C4⋊C4×C31C2×D411yescharacteristic
206C4⋊C4×C31C2×D411yescharacteristic
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—21no—
214C22⋊C4×C31—21no—
215C22⋊C4×C31—21no—
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—21no—
223C42×C31—22no—
224C42×C31—21no—
225C42×C31C2×D411yescharacteristic
226C42×C31C2×D411yescharacteristic
227C42×C31C2×D411yescharacteristic
228C42×C31C2411yescharacteristic
229#18955C3111yesmaximal, characteristic, direct product factor, Sylow
230C23×C31—21no—
231C23×C31—21no—
232C23×C31—22no—
233C23×C31—21no—
234C23×C31—82no—
235C23×C31—82no—
236C23×C31—82no—
237C23×C31#12011yescharacteristic
238Q8×C31—22no—
239Q8×C31—44no—
240Q8×C31—42no—
241Q8×C31—81no—
242Q8×C31—82no—
243D4×C31—22no—
244D4×C31—22no—
245D4×C31—22no—
246D4×C31—42no—
247D4×C31—42no—
248D4×C31—42no—
249D4×C31—81no—
250D4×C31—81no—
251D4×C31—81no—
252D4×C31—81no—
253D4×C31—81no—
254D4×C31—82no—
255D4×C31—82no—
256D4×C31—82no—
257D4×C31—82no—
258D4×C31—82no—
259D4×C31—82no—
260C2×C4×C31—22no—
261C2×C4×C31—22no—
262C2×C4×C31—21no—
263C2×C4×C31—21no—
264C2×C4×C31—21no—
265C2×C4×C31—22no—
266C2×C4×C31—21no—
267C2×C4×C31—21no—
268C2×C4×C31—41no—
269C2×C4×C31—42no—
270C2×C4×C31—42no—
271C2×C4×C31—42no—
272C2×C4×C31—42no—
273C2×C4×C31—82no—
274C2×C4×C31—82no—
275C2×C4×C31—82no—
276C2×C4×C31—82no—
277C2×C4×C31#12011yescharacteristic
278C2×C4×C31#12011yescharacteristic
279C2×C4×C31#12011yescharacteristic
280C2×C4×C31C22×D411yescharacteristic
281C2×C4×C31C22×D411yescharacteristic
282C2×C4×C31C22×D411yescharacteristic
283C8×C31—42no—
284C8×C31—42no—
285C8×C31—42no—
286#3546C2×C3111yescharacteristic
287#3238C2×C3111yescharacteristic, complemented
288#3227C2×C3111yescharacteristic, complemented
289#3206C2×C3111yescharacteristic, complemented
290#1638C2×C3112yescomplemented
291#1635C2×C3112yescomplemented
292#1631C2×C3112yescomplemented
293#1629C2×C3112yescomplemented
294#1553C2×C3111yescharacteristic, complemented
295#1496C2×C3111yescharacteristic, complemented
296#1482C2×C3111yescharacteristic, complemented
297C22×C31—21no—
298C22×C31—21no—
299C22×C31—21no—
300C22×C31—41no—
301C22×C31—41no—
302C22×C31—41no—
303C22×C31—41no—
304C22×C31—42no—
305C22×C31—42no—
306C22×C31—41no—
307C22×C31—41no—
308C22×C31—42no—
309C22×C31—41no—
310C22×C31—41no—
311C22×C31—41no—
312C22×C31—164no—
313C22×C31C2×#12011yescharacteristic
314C4×C31—21no—
315C4×C31—21no—
316C4×C31—21no—
317C4×C31—21no—
318C4×C31—41no—
319C4×C31—41no—
320C4×C31—41no—
321C4×C31—41no—
322C4×C31—42no—
323C4×C31—41no—
324C4×C31—82no—
325C4×C31#57712yes—
326C4×C31#57712yes—
327#560—22no—
328#556C22×C3111yescharacteristic
329#553—21no—
330#550—21no—
331#550C22×C3111yescharacteristic
332#547C22×C3111yescharacteristic
333#546—21no—
334#546—21no—
335D42C22×C3111yescharacteristic
336#542—21no—
337#538—22no—
338#537—21no—
339#535—21no—
340#535—21no—
341#532C22×C3111yescharacteristic
342#532C22×C3111yescharacteristic
343C2×C4⋊Q8C22×C3111yescharacteristic, complemented
344#529C22×C3111yescharacteristic
345#501—21no—
346#498—21no—
347#492C22×C3112yes—
348#489—21no—
349#487C22×C3112yescomplemented
350#486—21no—
351#483—21no—
352#481—21no—
353#481—21no—
354#480—21no—
355#477C22×C3112yescomplemented
356#476C22×C3112yes—
357#476C22×C3112yes—
358#476C22×C3112yescomplemented
359#461—22no—
360#448—22no—
361#432C22×C3111yescharacteristic, complemented
362#431C22×C3111yescharacteristic, complemented
363#431C22×C3111yescharacteristic, complemented
364#332C22×C3112yescomplemented
365#331C22×C3112yes—
366Q8⋊C8C22×C3112yescomplemented
367D4⋊C8C22×C3112yes—
368D4⋊C8C22×C3112yes—
369D4⋊C8C22×C3112yescomplemented
370C2×C31—41no—
371C2×C31—81no—
372C2×C31—82no—
373C2×C31—81no—
374C2×C31—81no—
375C2×C31#300911yescharacteristic
376C2×C31#300911yescharacteristic
377C2×C31#302111yescharacteristic
378C2×#41—21no—
379C2×#41—21no—
380C2×#41—21no—
381C2×#41—21no—
382C22×Q8—21no—
383C22×D4—21no—
384C22×D4—21no—
385C2×SD16—24no—
386C4⋊Q8—21no—
387C4⋊Q8—21no—
388C4⋊Q8—21no—
389C4⋊Q8D4×C3111yescharacteristic
390C4⋊Q8C23×C3112yes—
391C4⋊Q8C23×C3111yescharacteristic
392#127—22no—
393#127—21no—
394#127D4×C3111yescharacteristic
395#127C23×C3111yescharacteristic
396#126—22no—
397#126—22no—
398#126—21no—
399#125D4×C3111yescharacteristic
400#124—22no—
401#124—21no—
402#124—21no—
403#124—21no—
404#124D4×C3111yescharacteristic
405#123—22no—
406#123—22no—
407#123—21no—
408#123—22no—
409C22⋊Q8—22no—
410C22⋊Q8—21no—
411C22⋊Q8—21no—
412C22⋊Q8D4×C3111yescharacteristic
413#121—22no—
414#121—22no—
415#121—22no—
416#121—22no—
417#121—22no—
418#121—21no—
419#121—22no—
420#121—21no—
421#121—21no—
422#121—21no—
423#121—21no—
424#121—21no—
425#121D4×C3111yescharacteristic
426#121D4×C3111yescharacteristic
427#121D4×C3111yescharacteristic
428#121D4×C3112yes—
429#120—22no—
430#120—22no—
431#120—22no—
432#120—22no—
433C4×Q8—21no—
434C4×Q8C23×C3111yescharacteristic
435C4×D4—22no—
436C4×D4—21no—
437C4×D4—21no—
438C4×D4—21no—
439C4×D4—21no—
440C4×D4D4×C3112yes—
441C4×D4C23×C3111yescharacteristic
442C4×D4C23×C3111yescharacteristic
443C4×D4C23×C3111yescharacteristic
444#117—21no—
445#117—21no—
446C2×C4⋊C4—21no—
447C2×C4⋊C4—21no—
448C2×C4⋊C4—21no—
449C2×C22⋊C4—22no—
450C2×C42C23×C3111yescharacteristic
451#107—22no—
452#106—22no—
453#106—22no—
454#106—22no—
455C4⋊C8—21no—
456C4⋊C8C23×C3112yes—
457C4⋊C8C23×C3112yes—
458#103—22no—
459#103—24no—
460#102—22no—
461#102—22no—
462#102—22no—
463C22⋊C8—21no—
464C22⋊C8—21no—
465C22⋊C8—21no—
466#97—21no—
467C4×C8—21no—
468C4×C8C23×C3112yes—
469C31#1895511yescharacteristic, direct product factor, Sylow
470C24—22no—
471#41—82no—
472#41—82no—
473#41—82no—
474#41—82no—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—22no—
478C2×Q8—41no—
479C2×Q8—82no—
480C2×Q8C2×D4×C3111yescharacteristic
481C2×D4—22no—
482C2×D4—22no—
483C2×D4—22no—
484C2×D4—22no—
485C2×D4—22no—
486C2×D4—21no—
487C2×D4—21no—
488C2×D4—22no—
489C2×D4—21no—
490C2×D4—21no—
491C2×D4—21no—
492C2×D4—21no—
493C2×D4—21no—
494C2×D4—82no—
495C2×D4—82no—
496C2×D4C2×D4×C3111yescharacteristic
497C2×D4C2×D4×C3111yescharacteristic
498C2×D4C2×D4×C3111yescharacteristic
499C2×D4C2×D4×C3112yes—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—22no—
506C22×C4—41no—
507C22×C4C2×D4×C3111yescharacteristic
508C22×C4C2×D4×C3111yescharacteristic
509C22×C4C2×D4×C3111yescharacteristic
510SD16—84no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—41no—
525C4⋊C4—41no—
526C4⋊C4C2×D4×C3112yes—
527C4⋊C4C2×D4×C3111yescharacteristic
528C4⋊C4C2×D4×C3111yescharacteristic
529C4⋊C4C2×D4×C3111yescharacteristic
530C4⋊C4C2×D4×C3111yescharacteristic
531C22⋊C4—22no—
532C22⋊C4—22no—
533C22⋊C4—22no—
534C22⋊C4—22no—
535C22⋊C4—22no—
536C22⋊C4—21no—
537C22⋊C4—21no—
538C22⋊C4—21no—
539C22⋊C4—21no—
540C22⋊C4—21no—
541C22⋊C4—21no—
542C22⋊C4—22no—
543C22⋊C4—22no—
544C22⋊C4—21no—
545C22⋊C4—21no—
546C22⋊C4—21no—
547C42—22no—
548C42—21no—
549C42C2×D4×C3111yescharacteristic
550C42C2×D4×C3111yescharacteristic
551C42C2×D4×C3111yescharacteristic
552C42C24×C3111yescharacteristic
553C23—21no—
554C23—21no—
555C23—22no—
556C23—21no—
557C23—82no—
558C23—82no—
559C23—82no—
560C23C31×#12011yescharacteristic
561Q8—22no—
562Q8—44no—
563Q8—42no—
564Q8—81no—
565Q8—82no—
566D4—22no—
567D4—22no—
568D4—22no—
569D4—42no—
570D4—42no—
571D4—42no—
572D4—81no—
573D4—81no—
574D4—81no—
575D4—81no—
576D4—81no—
577D4—82no—
578D4—82no—
579D4—82no—
580D4—82no—
581D4—82no—
582D4—82no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—82no—
597C2×C4—82no—
598C2×C4—82no—
599C2×C4—82no—
600C2×C4C31×#12011yescharacteristic
601C2×C4C31×#12011yescharacteristic
602C2×C4C31×#12011yescharacteristic
603C2×C4C22×D4×C3111yescharacteristic
604C2×C4C22×D4×C3111yescharacteristic
605C2×C4C22×D4×C3111yescharacteristic
606C8—42no—
607C8—42no—
608C8—42no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—42no—
617C22—42no—
618C22—41no—
619C22—41no—
620C22—42no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22—164no—
625C22C2×C31×#12011yescharacteristic
626C4—21no—
627C4—21no—
628C4—21no—
629C4—21no—
630C4—41no—
631C4—41no—
632C4—41no—
633C4—41no—
634C4—42no—
635C4—41no—
636C4—82no—
637C4C31×#57712yes—
638C4C31×#57712yes—
639C2—41no—
640C2—81no—
641C2—81no—
642C2—82no—
643C2—81no—
644C2C31×#300911yescharacteristic
645C2C31×#300911yescharacteristic
646C2C31×#302111yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC42C24×C31
prime radicalC31×#3546C2
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×#3238C2semidirect
3. C31×#3227C2semidirect
4. C31×#3206C2semidirect
5. C31×#1638C2semidirect
6. C31×#1635C2semidirect
7. C31×#1631C2semidirect
8. C31×#1629C2semidirect
9. C31×#1553C2semidirect
10. C31×#1496C2semidirect
11. C31×#1482C2semidirect
28. C2×C31×C4⋊Q8C22semidirect
34. C31×#487C22semidirect
40. C31×#477C22semidirect
43. C31×#476C22semidirect
46. C31×#432C22semidirect
47. C31×#431C22semidirect
48. C31×#431C22semidirect
49. C31×#332C22semidirect
51. C31×Q8⋊C8C22semidirect
54. C31×D4⋊C8C22semidirect
229. #18955C31direct (N × H)
287. #3238C2×C31semidirect
288. #3227C2×C31semidirect
289. #3206C2×C31semidirect
290. #1638C2×C31semidirect
291. #1635C2×C31semidirect
292. #1631C2×C31semidirect
293. #1629C2×C31semidirect
294. #1553C2×C31semidirect
295. #1496C2×C31semidirect
296. #1482C2×C31semidirect
343. C2×C4⋊Q8C22×C31semidirect
349. #487C22×C31semidirect
355. #477C22×C31semidirect
358. #476C22×C31semidirect
361. #432C22×C31semidirect
362. #431C22×C31semidirect
363. #431C22×C31semidirect
364. #332C22×C31semidirect
366. Q8⋊C8C22×C31semidirect
369. D4⋊C8C22×C31semidirect
469. C31#18955direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35462
C31×#32382
C31×#32272
C31×#32062
C31×#16382
C31×#16352
C31×#16312
C31×#16292
C31×#15532
C31×#14962
C31×#14822
#1895531