Finite Group Databasebeta

← C31×#18200 #21474959

Subgroups — C31×#18200 #21474959

Order 7936.

1078 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3490C211yesmaximal, characteristic
2C31×#3091C212yesmaximal, complemented
3C31×#2482C211yesmaximal, characteristic, complemented
4C31×#1618C212yesmaximal, complemented
5C31×#1614C211yesmaximal, characteristic, complemented
6C31×#1613C211yesmaximal, characteristic, complemented
7C31×#1612C212yesmaximal, complemented
8C31×#1608C211yesmaximal, characteristic, complemented
9C31×#1607C211yesmaximal, characteristic, complemented
10C31×#1516C212yesmaximal, complemented
11C2×C31×#323C211yesmaximal, characteristic, complemented
12C31×#551—22no—
13C31×#551—22no—
14C31×#546—22no—
15C31×#546—22no—
16C31×#543C2211yescharacteristic
17C31×#538—21no—
18C31×#538C2212yes—
19C31×#534C2211yescharacteristic
20C31×#525—21no—
21C31×#525C2212yes—
22C2×C31×#120—21no—
23C31×#518—21no—
24C2×C31×#117C2211yescharacteristic, complemented
25C22×C22⋊C4×C31C2211yescharacteristic
26C31×#485—22no—
27C31×#484C2212yescomplemented
28C31×#483C2212yescomplemented
29C31×#482—22no—
30C31×#481C2212yescomplemented
31C31×#480C2212yescomplemented
32C31×#427—22no—
33C31×#416—22no—
34C2×C31×C22⋊C8C2212yescomplemented
35C31×#399C2211yescharacteristic
36C31×#399C2211yescharacteristic, complemented
37C31×#397C2211yescharacteristic
38C31×#397C2211yescharacteristic, complemented
39C31×#388—21no—
40C31×#388—21no—
41C31×#385—21no—
42C31×#381C2211yescharacteristic, complemented
43C31×#379—21no—
44C31×#328C2212yescomplemented
45C31×#328C2212yescomplemented
46C31×#327C2212yescomplemented
47C31×#327C2212yescomplemented
48C31×#323C2211yescharacteristic
49C31×#323C2211yescharacteristic
50C31×#323C2212yescomplemented
51C25×C31D411yescharacteristic
52C2×#41×C31—21no—
53C2×#41×C31D412yes—
54C22×D4×C31—21no—
55C22×D4×C31—21no—
56C22×D4×C31—21no—
57C23×C4×C31D411yescharacteristic
58C23×C4×C31C2311yescharacteristic
59C22×C8×C31—22no—
60C31×#126—22no—
61C31×#126—22no—
62C31×#124—22no—
63C31×#124—22no—
64C31×#123—22no—
65C31×#123—22no—
66C31×#123—22no—
67C31×#123—22no—
68C31×#123—22no—
69C31×C22⋊Q8—21no—
70C31×C22⋊Q8—22no—
71C31×C22⋊Q8—21no—
72C31×C22⋊Q8—22no—
73C31×C22⋊Q8D412yes—
74C31×C22⋊Q8C2312yes—
75C31×#121—21no—
76C31×#121—22no—
77C31×#121—22no—
78C31×#121—21no—
79C31×#121D412yes—
80C31×#121C2312yes—
81C31×#120—22no—
82C31×#120—22no—
83C31×#120—22no—
84C31×#120—22no—
85C31×#120—22no—
86C4×D4×C31—22no—
87C4×D4×C31—22no—
88C4×D4×C31—22no—
89C4×D4×C31—22no—
90C4×D4×C31—22no—
91C4×D4×C31—22no—
92C31×#117—22no—
93C31×#117—21no—
94C31×#117—22no—
95C31×#117—22no—
96C31×#117D412yes—
97C2×C4⋊C4×C31—21no—
98C2×C4⋊C4×C31—21no—
99C2×C4⋊C4×C31C2311yescharacteristic
100C2×C4⋊C4×C31C2311yescharacteristic
101C2×C22⋊C4×C31—22no—
102C2×C22⋊C4×C31—21no—
103C2×C22⋊C4×C31—21no—
104C2×C22⋊C4×C31—21no—
105C2×C22⋊C4×C31—22no—
106C2×C22⋊C4×C31—22no—
107C2×C22⋊C4×C31—21no—
108C2×C22⋊C4×C31—21no—
109C2×C22⋊C4×C31—21no—
110C2×C22⋊C4×C31—21no—
111C2×C22⋊C4×C31—21no—
112C2×C22⋊C4×C31—21no—
113C2×C22⋊C4×C31D411yescharacteristic
114C2×C22⋊C4×C31D411yescharacteristic
115C2×C22⋊C4×C31C2311yescharacteristic
116C2×C22⋊C4×C31C2311yescharacteristic
117C2×C42×C31—21no—
118C31×#107—22no—
119C31×#107—22no—
120C31×#106—22no—
121C31×#106—22no—
122C31×#103—22no—
123C31×#103—22no—
124C31×#102—22no—
125C31×#102—22no—
126C31×C22⋊C8—22no—
127C31×C22⋊C8C2312yes—
128C31×C22⋊C8C2312yes—
129C31×#95—21no—
130C31×#95—21no—
131C31×#95—21no—
132C31×#95C2311yescharacteristic
133C31×#95C2311yescharacteristic
134C24×C31—21no—
135C24×C31—21no—
136C24×C31—21no—
137C24×C31—22no—
138C24×C31—41no—
139C24×C31—42no—
140C24×C31—81no—
141C24×C31C2×D411yescharacteristic
142C24×C31C2×D411yescharacteristic
143C24×C31C2×D411yescharacteristic
144#41×C31—44no—
145#41×C31—82no—
146C2×Q8×C31—21no—
147C2×Q8×C31C2×D412yes—
148C2×D4×C31—22no—
149C2×D4×C31—22no—
150C2×D4×C31—22no—
151C2×D4×C31—22no—
152C2×D4×C31—21no—
153C2×D4×C31—22no—
154C2×D4×C31—22no—
155C2×D4×C31—42no—
156C2×D4×C31—42no—
157C2×D4×C31—82no—
158C2×D4×C31—82no—
159C2×D4×C31C2×D412yes—
160C22×C4×C31—21no—
161C22×C4×C31—21no—
162C22×C4×C31—21no—
163C22×C4×C31—21no—
164C22×C4×C31—22no—
165C22×C4×C31—21no—
166C22×C4×C31—22no—
167C22×C4×C31—21no—
168C22×C4×C31—21no—
169C22×C4×C31—21no—
170C22×C4×C31—21no—
171C22×C4×C31—42no—
172C22×C4×C31—81no—
173C22×C4×C31C2×D412yes—
174C22×C4×C31C2×D411yescharacteristic
175C22×C4×C31C2×D411yescharacteristic
176C22×C4×C31C2×D411yescharacteristic
177C22×C4×C31C2×D412yes—
178C22×C4×C31C2411yescharacteristic
179C2×C8×C31—22no—
180C2×C8×C31—22no—
181C2×C8×C31—44no—
182C4⋊C4×C31—22no—
183C4⋊C4×C31—21no—
184C4⋊C4×C31—21no—
185C4⋊C4×C31—22no—
186C4⋊C4×C31—22no—
187C4⋊C4×C31—22no—
188C4⋊C4×C31—22no—
189C4⋊C4×C31—22no—
190C4⋊C4×C31—81no—
191C4⋊C4×C31—81no—
192C4⋊C4×C31C2×D412yes—
193C4⋊C4×C31C2×D412yes—
194C22⋊C4×C31—22no—
195C22⋊C4×C31—22no—
196C22⋊C4×C31—22no—
197C22⋊C4×C31—22no—
198C22⋊C4×C31—22no—
199C22⋊C4×C31—22no—
200C22⋊C4×C31—22no—
201C22⋊C4×C31—22no—
202C22⋊C4×C31—22no—
203C22⋊C4×C31—22no—
204C22⋊C4×C31—22no—
205C22⋊C4×C31—22no—
206C22⋊C4×C31—22no—
207C22⋊C4×C31—22no—
208C22⋊C4×C31—42no—
209C22⋊C4×C31—42no—
210C22⋊C4×C31—41no—
211C22⋊C4×C31—41no—
212C22⋊C4×C31—42no—
213C22⋊C4×C31—81no—
214C22⋊C4×C31—81no—
215C42×C31—22no—
216C42×C31—22no—
217C42×C31—42no—
218#18200C3111yesmaximal, characteristic, direct product factor, Sylow
219C23×C31—21no—
220C23×C31—21no—
221C23×C31—21no—
222C23×C31—22no—
223C23×C31—21no—
224C23×C31—21no—
225C23×C31—22no—
226C23×C31—21no—
227C23×C31—21no—
228C23×C31—41no—
229C23×C31—42no—
230C23×C31—42no—
231C23×C31—41no—
232C23×C31—41no—
233C23×C31—41no—
234C23×C31—41no—
235C23×C31—42no—
236C23×C31—42no—
237C23×C31—42no—
238C23×C31—81no—
239C23×C31—81no—
240C23×C31—81no—
241C23×C31—81no—
242C23×C31—84no—
243C23×C31—84no—
244C23×C31#12011yescharacteristic
245C23×C31#12011yescharacteristic
246C23×C31#13512yes—
247C23×C31C22×D411yescharacteristic
248Q8×C31—42no—
249Q8×C31—81no—
250D4×C31—42no—
251D4×C31—44no—
252D4×C31—44no—
253D4×C31—42no—
254D4×C31—42no—
255D4×C31—44no—
256D4×C31—82no—
257D4×C31—82no—
258D4×C31—81no—
259C2×C4×C31—24no—
260C2×C4×C31—22no—
261C2×C4×C31—21no—
262C2×C4×C31—22no—
263C2×C4×C31—22no—
264C2×C4×C31—21no—
265C2×C4×C31—42no—
266C2×C4×C31—41no—
267C2×C4×C31—41no—
268C2×C4×C31—42no—
269C2×C4×C31—42no—
270C2×C4×C31—41no—
271C2×C4×C31—41no—
272C2×C4×C31—41no—
273C2×C4×C31—41no—
274C2×C4×C31—41no—
275C2×C4×C31—41no—
276C2×C4×C31—41no—
277C2×C4×C31—41no—
278C2×C4×C31—41no—
279C2×C4×C31—41no—
280C2×C4×C31—41no—
281C2×C4×C31—41no—
282C2×C4×C31—41no—
283C2×C4×C31—41no—
284C2×C4×C31—82no—
285C2×C4×C31—81no—
286C2×C4×C31—81no—
287C2×C4×C31—81no—
288C2×C4×C31—82no—
289C2×C4×C31#12012yes—
290C2×C4×C31C22×D412yes—
291C8×C31—42no—
292C8×C31—42no—
293#3490C2×C3111yescharacteristic
294#3091C2×C3112yescomplemented
295#2482C2×C3111yescharacteristic, complemented
296#1618C2×C3112yescomplemented
297#1614C2×C3111yescharacteristic, complemented
298#1613C2×C3111yescharacteristic, complemented
299#1612C2×C3112yescomplemented
300#1608C2×C3111yescharacteristic, complemented
301#1607C2×C3111yescharacteristic, complemented
302#1516C2×C3112yescomplemented
303C2×#323C2×C3111yescharacteristic, complemented
304C22×C31—21no—
305C22×C31—21no—
306C22×C31—21no—
307C22×C31—21no—
308C22×C31—22no—
309C22×C31—21no—
310C22×C31—21no—
311C22×C31—21no—
312C22×C31—42no—
313C22×C31—42no—
314C22×C31—42no—
315C22×C31—42no—
316C22×C31—41no—
317C22×C31—41no—
318C22×C31—42no—
319C22×C31—41no—
320C22×C31—42no—
321C22×C31—41no—
322C22×C31—42no—
323C22×C31—41no—
324C22×C31—81no—
325C22×C31—84no—
326C22×C31—81no—
327C22×C31—84no—
328C22×C31—162no—
329C22×C31#45812yes—
330C22×C31C2×#12011yescharacteristic
331C22×C31C2×#13511yescharacteristic
332C22×C31#57511yescharacteristic
333C4×C31—22no—
334C4×C31—22no—
335C4×C31—42no—
336C4×C31—41no—
337C4×C31—41no—
338C4×C31—42no—
339C4×C31—42no—
340C4×C31—81no—
341C4×C31—81no—
342C4×C31—81no—
343C4×C31—81no—
344#551—22no—
345#551—22no—
346#546—22no—
347#546—22no—
348#543C22×C3111yescharacteristic
349#538—21no—
350#538C22×C3112yes—
351#534C22×C3111yescharacteristic
352#525—21no—
353#525C22×C3112yes—
354C2×#120—21no—
355#518—21no—
356C2×#117C22×C3111yescharacteristic, complemented
357C22×C22⋊C4C22×C3111yescharacteristic
358#485—22no—
359#484C22×C3112yescomplemented
360#483C22×C3112yescomplemented
361#482—22no—
362#481C22×C3112yescomplemented
363#480C22×C3112yescomplemented
364#427—22no—
365#416—22no—
366C2×C22⋊C8C22×C3112yescomplemented
367#399C22×C3111yescharacteristic
368#399C22×C3111yescharacteristic, complemented
369#397C22×C3111yescharacteristic
370#397C22×C3111yescharacteristic, complemented
371#388—21no—
372#388—21no—
373#385—21no—
374#381C22×C3111yescharacteristic, complemented
375#379—21no—
376#328C22×C3112yescomplemented
377#328C22×C3112yescomplemented
378#327C22×C3112yescomplemented
379#327C22×C3112yescomplemented
380#323C22×C3111yescharacteristic
381#323C22×C3111yescharacteristic
382#323C22×C3112yescomplemented
383C2×C31—21no—
384C2×C31—21no—
385C2×C31—22no—
386C2×C31—41no—
387C2×C31—42no—
388C2×C31—81no—
389C2×C31—82no—
390C2×C31#300211yescharacteristic
391C2×C31#301911yescharacteristic
392C2×C31C2×#45811yescharacteristic
393C25D4×C3111yescharacteristic
394C2×#41—21no—
395C2×#41D4×C3112yes—
396C22×D4—21no—
397C22×D4—21no—
398C22×D4—21no—
399C23×C4D4×C3111yescharacteristic
400C23×C4C23×C3111yescharacteristic
401C22×C8—22no—
402#126—22no—
403#126—22no—
404#124—22no—
405#124—22no—
406#123—22no—
407#123—22no—
408#123—22no—
409#123—22no—
410#123—22no—
411C22⋊Q8—21no—
412C22⋊Q8—22no—
413C22⋊Q8—21no—
414C22⋊Q8—22no—
415C22⋊Q8D4×C3112yes—
416C22⋊Q8C23×C3112yes—
417#121—21no—
418#121—22no—
419#121—22no—
420#121—21no—
421#121D4×C3112yes—
422#121C23×C3112yes—
423#120—22no—
424#120—22no—
425#120—22no—
426#120—22no—
427#120—22no—
428C4×D4—22no—
429C4×D4—22no—
430C4×D4—22no—
431C4×D4—22no—
432C4×D4—22no—
433C4×D4—22no—
434#117—22no—
435#117—21no—
436#117—22no—
437#117—22no—
438#117D4×C3112yes—
439C2×C4⋊C4—21no—
440C2×C4⋊C4—21no—
441C2×C4⋊C4C23×C3111yescharacteristic
442C2×C4⋊C4C23×C3111yescharacteristic
443C2×C22⋊C4—22no—
444C2×C22⋊C4—21no—
445C2×C22⋊C4—21no—
446C2×C22⋊C4—21no—
447C2×C22⋊C4—22no—
448C2×C22⋊C4—22no—
449C2×C22⋊C4—21no—
450C2×C22⋊C4—21no—
451C2×C22⋊C4—21no—
452C2×C22⋊C4—21no—
453C2×C22⋊C4—21no—
454C2×C22⋊C4—21no—
455C2×C22⋊C4D4×C3111yescharacteristic
456C2×C22⋊C4D4×C3111yescharacteristic
457C2×C22⋊C4C23×C3111yescharacteristic
458C2×C22⋊C4C23×C3111yescharacteristic
459C2×C42—21no—
460#107—22no—
461#107—22no—
462#106—22no—
463#106—22no—
464#103—22no—
465#103—22no—
466#102—22no—
467#102—22no—
468C22⋊C8—22no—
469C22⋊C8C23×C3112yes—
470C22⋊C8C23×C3112yes—
471#95—21no—
472#95—21no—
473#95—21no—
474#95C23×C3111yescharacteristic
475#95C23×C3111yescharacteristic
476C31#1820011yescharacteristic, direct product factor, Sylow
477C24—21no—
478C24—21no—
479C24—21no—
480C24—22no—
481C24—41no—
482C24—42no—
483C24—81no—
484C24C2×D4×C3111yescharacteristic
485C24C2×D4×C3111yescharacteristic
486C24C2×D4×C3111yescharacteristic
487#41—44no—
488#41—82no—
489C2×Q8—21no—
490C2×Q8C2×D4×C3112yes—
491C2×D4—22no—
492C2×D4—22no—
493C2×D4—22no—
494C2×D4—22no—
495C2×D4—21no—
496C2×D4—22no—
497C2×D4—22no—
498C2×D4—42no—
499C2×D4—42no—
500C2×D4—82no—
501C2×D4—82no—
502C2×D4C2×D4×C3112yes—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—22no—
508C22×C4—21no—
509C22×C4—22no—
510C22×C4—21no—
511C22×C4—21no—
512C22×C4—21no—
513C22×C4—21no—
514C22×C4—42no—
515C22×C4—81no—
516C22×C4C2×D4×C3112yes—
517C22×C4C2×D4×C3111yescharacteristic
518C22×C4C2×D4×C3111yescharacteristic
519C22×C4C2×D4×C3111yescharacteristic
520C22×C4C2×D4×C3112yes—
521C22×C4C24×C3111yescharacteristic
522C2×C8—22no—
523C2×C8—22no—
524C2×C8—44no—
525C4⋊C4—22no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—81no—
534C4⋊C4—81no—
535C4⋊C4C2×D4×C3112yes—
536C4⋊C4C2×D4×C3112yes—
537C22⋊C4—22no—
538C22⋊C4—22no—
539C22⋊C4—22no—
540C22⋊C4—22no—
541C22⋊C4—22no—
542C22⋊C4—22no—
543C22⋊C4—22no—
544C22⋊C4—22no—
545C22⋊C4—22no—
546C22⋊C4—22no—
547C22⋊C4—22no—
548C22⋊C4—22no—
549C22⋊C4—22no—
550C22⋊C4—22no—
551C22⋊C4—42no—
552C22⋊C4—42no—
553C22⋊C4—41no—
554C22⋊C4—41no—
555C22⋊C4—42no—
556C22⋊C4—81no—
557C22⋊C4—81no—
558C42—22no—
559C42—22no—
560C42—42no—
561C23—22no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—22no—
568C23—21no—
569C23—21no—
570C23—41no—
571C23—42no—
572C23—42no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23—41no—
577C23—42no—
578C23—42no—
579C23—42no—
580C23—81no—
581C23—81no—
582C23—81no—
583C23—81no—
584C23—84no—
585C23—84no—
586C23C31×#12011yescharacteristic
587C23C31×#12011yescharacteristic
588C23C31×#13512yes—
589C23C22×D4×C3111yescharacteristic
590Q8—42no—
591Q8—81no—
592D4—42no—
593D4—44no—
594D4—44no—
595D4—44no—
596D4—42no—
597D4—42no—
598D4—82no—
599D4—82no—
600D4—81no—
601C2×C4—24no—
602C2×C4—22no—
603C2×C4—21no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—21no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—82no—
627C2×C4—81no—
628C2×C4—81no—
629C2×C4—81no—
630C2×C4—82no—
631C2×C4C31×#12012yes—
632C2×C4C22×D4×C3112yes—
633C8—42no—
634C8—42no—
635C22—21no—
636C22—21no—
637C22—21no—
638C22—21no—
639C22—22no—
640C22—21no—
641C22—21no—
642C22—21no—
643C22—42no—
644C22—42no—
645C22—42no—
646C22—42no—
647C22—41no—
648C22—41no—
649C22—42no—
650C22—41no—
651C22—42no—
652C22—41no—
653C22—42no—
654C22—41no—
655C22—81no—
656C22—84no—
657C22—81no—
658C22—84no—
659C22—162no—
660C22C31×#45812yes—
661C22C2×C31×#12011yescharacteristic
662C22C2×C31×#13511yescharacteristic
663C22C31×#57511yescharacteristic
664C4—22no—
665C4—22no—
666C4—42no—
667C4—41no—
668C4—41no—
669C4—42no—
670C4—42no—
671C4—81no—
672C4—81no—
673C4—81no—
674C4—81no—
675C2—21no—
676C2—21no—
677C2—22no—
678C2—41no—
679C2—42no—
680C2—81no—
681C2—82no—
682C2C31×#300211yescharacteristic
683C2C31×#301911yescharacteristic
684C2C2×C31×#45811yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#3490C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#3091C2semidirect
3. C31×#2482C2semidirect
4. C31×#1618C2semidirect
5. C31×#1614C2semidirect
6. C31×#1613C2semidirect
7. C31×#1612C2semidirect
8. C31×#1608C2semidirect
9. C31×#1607C2semidirect
10. C31×#1516C2semidirect
11. C2×C31×#323C2semidirect
24. C2×C31×#117C22semidirect
27. C31×#484C22semidirect
28. C31×#483C22semidirect
30. C31×#481C22semidirect
31. C31×#480C22semidirect
34. C2×C31×C22⋊C8C22semidirect
36. C31×#399C22semidirect
38. C31×#397C22semidirect
42. C31×#381C22semidirect
44. C31×#328C22semidirect
45. C31×#328C22semidirect
46. C31×#327C22semidirect
47. C31×#327C22semidirect
50. C31×#323C22semidirect
218. #18200C31direct (N × H)
294. #3091C2×C31semidirect
295. #2482C2×C31semidirect
296. #1618C2×C31semidirect
297. #1614C2×C31semidirect
298. #1613C2×C31semidirect
299. #1612C2×C31semidirect
300. #1608C2×C31semidirect
301. #1607C2×C31semidirect
302. #1516C2×C31semidirect
303. C2×#323C2×C31semidirect
356. C2×#117C22×C31semidirect
359. #484C22×C31semidirect
360. #483C22×C31semidirect
362. #481C22×C31semidirect
363. #480C22×C31semidirect
366. C2×C22⋊C8C22×C31semidirect
368. #399C22×C31semidirect
370. #397C22×C31semidirect
374. #381C22×C31semidirect
376. #328C22×C31semidirect
377. #328C22×C31semidirect
378. #327C22×C31semidirect
379. #327C22×C31semidirect
382. #323C22×C31semidirect
476. C31#18200direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#34902
C31×#30912
C31×#24822
C31×#16182
C31×#16142
C31×#16132
C31×#16122
C31×#16082
C31×#16072
C31×#15162
C2×C31×#3232
#1820031