Finite Group Databasebeta

← C31×#18127 #21474886

Subgroups — C31×#18127 #21474886

Order 7936.

950 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3455C211yesmaximal, characteristic
2C31×#3104C211yesmaximal, characteristic, complemented
3C31×#3103C211yesmaximal, characteristic, complemented
4C31×#2396C211yesmaximal, characteristic, complemented
5C31×#1618C212yesmaximal, complemented
6C31×#1615C212yesmaximal, complemented
7C31×#1606C212yesmaximal, complemented
8C31×#1603C212yesmaximal, complemented
9C31×#1503C211yesmaximal, characteristic, complemented
10C31×#1502C211yesmaximal, characteristic, complemented
11C31×#1458C211yesmaximal, characteristic, complemented
12C31×#542—21no—
13C31×#542—21no—
14C31×#541—21no—
15C31×#540—21no—
16C31×#539—21no—
17C31×#539C2212yes—
18C31×#538—21no—
19C31×#538C2212yes—
20C31×#533C2211yescharacteristic
21C31×#532C2211yescharacteristic
22C31×#529—22no—
23C31×#529—22no—
24C31×#525—21no—
25C4×#41×C31—21no—
26C2×C4×Q8×C31C2211yescharacteristic
27C2×C31×#117C2211yescharacteristic
28C31×#484C2212yes—
29C31×#481C2212yes—
30C31×#477—21no—
31C31×#475—21no—
32C31×#462—21no—
33C31×#460—21no—
34C31×#451C2212yes—
35C31×#448C2212yes—
36C4×Q16×C31—22no—
37C4×SD16×C31—22no—
38C31×#431C2211yescharacteristic
39C31×#431C2211yescharacteristic
40C31×#400C2211yescharacteristic
41C31×#399C2211yescharacteristic
42C31×#398—21no—
43C31×#396—21no—
44C31×#395C2211yescharacteristic
45C31×#393C2211yescharacteristic
46C4×C4⋊C4×C31—21no—
47C4×C4⋊C4×C31C2211yescharacteristic
48C4×C22⋊C4×C31—21no—
49C31×#328C2212yes—
50C31×#328C2212yes—
51C31×#327C2212yes—
52C31×#327C2212yes—
53C31×#324C2212yes—
54C31×#323C2212yes—
55C2×#41×C31—21no—
56C22×Q8×C31C2311yescharacteristic
57C23×C4×C31D411yescharacteristic
58C2×Q16×C31—22no—
59C2×SD16×C31—22no—
60C31×C4⋊Q8D411yescharacteristic
61C31×C4⋊Q8D411yescharacteristic
62C31×C4⋊Q8D411yescharacteristic
63C31×#127D411yescharacteristic
64C31×#126—22no—
65C31×#126—22no—
66C31×#126—22no—
67C31×#126—22no—
68C31×#125—22no—
69C31×#125—22no—
70C31×#125—21no—
71C31×#124—22no—
72C31×#124—22no—
73C31×#124—21no—
74C31×#123—22no—
75C31×#123—22no—
76C31×#123—22no—
77C31×#123—22no—
78C31×#123—22no—
79C31×C22⋊Q8—21no—
80C31×C22⋊Q8—21no—
81C31×C22⋊Q8—22no—
82C31×C22⋊Q8—22no—
83C31×C22⋊Q8—21no—
84C31×C22⋊Q8C2312yes—
85C31×#121—21no—
86C31×#121—21no—
87C31×#121—22no—
88C31×#121—22no—
89C31×#121—21no—
90C31×#121C2312yes—
91C31×#120—22no—
92C4×Q8×C31—21no—
93C4×Q8×C31—21no—
94C4×Q8×C31—21no—
95C4×Q8×C31—21no—
96C4×Q8×C31—21no—
97C4×Q8×C31—21no—
98C4×Q8×C31—21no—
99C4×Q8×C31D412yes—
100C4×D4×C31—21no—
101C4×D4×C31—21no—
102C4×D4×C31—22no—
103C4×D4×C31—22no—
104C4×D4×C31—22no—
105C4×D4×C31—22no—
106C4×D4×C31—21no—
107C4×D4×C31D412yes—
108C31×#117—22no—
109C31×#117—21no—
110C31×#117—21no—
111C31×#117—21no—
112C31×#117—21no—
113C31×#117—21no—
114C2×C4⋊C4×C31—21no—
115C2×C4⋊C4×C31—21no—
116C2×C4⋊C4×C31—21no—
117C2×C4⋊C4×C31—21no—
118C2×C4⋊C4×C31D411yescharacteristic
119C2×C4⋊C4×C31C2311yescharacteristic
120C2×C4⋊C4×C31C2311yescharacteristic
121C2×C22⋊C4×C31—21no—
122C2×C22⋊C4×C31—21no—
123C2×C22⋊C4×C31D411yescharacteristic
124C2×C22⋊C4×C31C2311yescharacteristic
125C2×C42×C31—22no—
126C2×C42×C31—21no—
127C2×C42×C31—21no—
128C2×C42×C31D411yescharacteristic
129C2×C42×C31C2311yescharacteristic
130C31×#107—22no—
131C31×#106—22no—
132C31×C4⋊C8—21no—
133C31×C4⋊C8—21no—
134C31×#103—22no—
135C31×#103—22no—
136C31×#103—22no—
137C31×#102—22no—
138C31×C22⋊C8C2312yes—
139C31×C22⋊C8C2312yes—
140C4×C8×C31—21no—
141C4×C8×C31—21no—
142C31×#95—21no—
143C31×#95—21no—
144C31×#95—21no—
145C31×#95—21no—
146C31×#95C2311yescharacteristic
147C31×#95C2311yescharacteristic
148C24×C31C2×D411yescharacteristic
149#41×C31—42no—
150#41×C31—42no—
151C2×Q8×C31—21no—
152C2×Q8×C31—21no—
153C2×Q8×C31—21no—
154C2×Q8×C31—21no—
155C2×Q8×C31—42no—
156C2×Q8×C31C2×D412yes—
157C2×D4×C31—22no—
158C2×D4×C31—22no—
159C2×D4×C31—22no—
160C2×D4×C31—21no—
161C2×D4×C31—21no—
162C2×D4×C31C2×D412yes—
163C22×C4×C31—21no—
164C22×C4×C31—21no—
165C22×C4×C31—22no—
166C22×C4×C31—22no—
167C22×C4×C31—21no—
168C22×C4×C31—22no—
169C22×C4×C31—21no—
170C22×C4×C31—22no—
171C22×C4×C31—22no—
172C22×C4×C31—21no—
173C22×C4×C31—21no—
174C22×C4×C31C2×D411yescharacteristic
175C22×C4×C31C2×D411yescharacteristic
176C22×C4×C31C2×D411yescharacteristic
177C22×C4×C31C2×D411yescharacteristic
178C22×C4×C31C2×D411yescharacteristic
179C22×C4×C31C2411yescharacteristic
180Q16×C31—44no—
181SD16×C31—44no—
182C2×C8×C31—22no—
183C2×C8×C31—22no—
184C4⋊C4×C31—22no—
185C4⋊C4×C31—22no—
186C4⋊C4×C31—22no—
187C4⋊C4×C31—22no—
188C4⋊C4×C31—21no—
189C4⋊C4×C31—21no—
190C4⋊C4×C31—21no—
191C4⋊C4×C31—21no—
192C4⋊C4×C31—21no—
193C4⋊C4×C31—21no—
194C4⋊C4×C31—22no—
195C4⋊C4×C31—21no—
196C4⋊C4×C31—21no—
197C4⋊C4×C31—22no—
198C4⋊C4×C31—21no—
199C4⋊C4×C31—21no—
200C4⋊C4×C31—42no—
201C4⋊C4×C31—42no—
202C4⋊C4×C31—42no—
203C4⋊C4×C31—41no—
204C4⋊C4×C31—41no—
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—22no—
213C22⋊C4×C31—22no—
214C22⋊C4×C31—21no—
215C22⋊C4×C31—21no—
216C22⋊C4×C31—22no—
217C22⋊C4×C31—21no—
218C22⋊C4×C31—21no—
219C22⋊C4×C31—22no—
220C22⋊C4×C31—42no—
221C22⋊C4×C31—42no—
222C42×C31—22no—
223C42×C31—22no—
224C42×C31—21no—
225C42×C31—21no—
226C42×C31—22no—
227C42×C31—41no—
228C42×C31—41no—
229C42×C31—42no—
230C42×C31C2×D411yescharacteristic
231C42×C31C2×D411yescharacteristic
232C42×C31C2×D411yescharacteristic
233C42×C31C2×D411yescharacteristic
234#18127C3111yesmaximal, characteristic, direct product factor, Sylow
235C23×C31—21no—
236C23×C31—22no—
237C23×C31—21no—
238C23×C31—21no—
239C23×C31—41no—
240C23×C31—41no—
241C23×C31C22×D411yescharacteristic
242Q8×C31—22no—
243Q8×C31—22no—
244Q8×C31—42no—
245Q8×C31—41no—
246Q8×C31—41no—
247Q8×C31—42no—
248Q8×C31—81no—
249D4×C31—42no—
250D4×C31—42no—
251D4×C31—42no—
252D4×C31—42no—
253D4×C31—81no—
254D4×C31—82no—
255D4×C31—82no—
256C2×C4×C31—22no—
257C2×C4×C31—22no—
258C2×C4×C31—21no—
259C2×C4×C31—21no—
260C2×C4×C31—22no—
261C2×C4×C31—22no—
262C2×C4×C31—22no—
263C2×C4×C31—21no—
264C2×C4×C31—21no—
265C2×C4×C31—21no—
266C2×C4×C31—41no—
267C2×C4×C31—41no—
268C2×C4×C31—41no—
269C2×C4×C31—41no—
270C2×C4×C31—41no—
271C2×C4×C31—41no—
272C2×C4×C31—42no—
273C2×C4×C31—42no—
274C2×C4×C31—44no—
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—82no—
284C2×C4×C31—82no—
285C2×C4×C31#12011yescharacteristic
286C2×C4×C31#12011yescharacteristic
287C2×C4×C31#12011yescharacteristic
288C2×C4×C31#12011yescharacteristic
289C2×C4×C31#13512yes—
290C2×C4×C31#13712yes—
291C2×C4×C31C22×D411yescharacteristic
292C2×C4×C31C22×D411yescharacteristic
293C8×C31—42no—
294C8×C31—42no—
295#3455C2×C3111yescharacteristic
296#3104C2×C3111yescharacteristic, complemented
297#3103C2×C3111yescharacteristic, complemented
298#2396C2×C3111yescharacteristic, complemented
299#1618C2×C3112yescomplemented
300#1615C2×C3112yescomplemented
301#1606C2×C3112yescomplemented
302#1603C2×C3112yescomplemented
303#1503C2×C3111yescharacteristic, complemented
304#1502C2×C3111yescharacteristic, complemented
305#1458C2×C3111yescharacteristic, complemented
306C22×C31—21no—
307C22×C31—21no—
308C22×C31—42no—
309C22×C31—41no—
310C22×C31—42no—
311C22×C31—42no—
312C22×C31—41no—
313C22×C31—42no—
314C22×C31—41no—
315C22×C31—81no—
316C22×C31C2×#12011yescharacteristic
317C22×C31C2×#13511yescharacteristic
318C22×C31C2×#13711yescharacteristic
319C4×C31—21no—
320C4×C31—21no—
321C4×C31—21no—
322C4×C31—21no—
323C4×C31—42no—
324C4×C31—41no—
325C4×C31—42no—
326C4×C31—41no—
327C4×C31—42no—
328C4×C31—42no—
329C4×C31—41no—
330C4×C31—42no—
331C4×C31—81no—
332C4×C31—81no—
333C4×C31#45212yes—
334C4×C31#45212yes—
335#542—21no—
336#542—21no—
337#541—21no—
338#540—21no—
339#539—21no—
340#539C22×C3112yes—
341#538—21no—
342#538C22×C3112yes—
343#533C22×C3111yescharacteristic
344#532C22×C3111yescharacteristic
345#529—22no—
346#529—22no—
347#525—21no—
348C4×#41—21no—
349C2×C4×Q8C22×C3111yescharacteristic
350C2×#117C22×C3111yescharacteristic
351#484C22×C3112yes—
352#481C22×C3112yes—
353#477—21no—
354#475—21no—
355#462—21no—
356#460—21no—
357#451C22×C3112yes—
358#448C22×C3112yes—
359C4×Q16—22no—
360C4×SD16—22no—
361#431C22×C3111yescharacteristic
362#431C22×C3111yescharacteristic
363#400C22×C3111yescharacteristic
364#399C22×C3111yescharacteristic
365#398—21no—
366#396—21no—
367#395C22×C3111yescharacteristic
368#393C22×C3111yescharacteristic
369C4×C4⋊C4—21no—
370C4×C4⋊C4C22×C3111yescharacteristic
371C4×C22⋊C4—21no—
372#328C22×C3112yes—
373#328C22×C3112yes—
374#327C22×C3112yes—
375#327C22×C3112yes—
376#324C22×C3112yes—
377#323C22×C3112yes—
378C2×C31—21no—
379C2×C31—21no—
380C2×C31—82no—
381C2×C31—81no—
382C2×C31C2×#45211yescharacteristic
383C2×C31#301811yescharacteristic
384C2×C31#302511yescharacteristic
385C2×#41—21no—
386C22×Q8C23×C3111yescharacteristic
387C23×C4D4×C3111yescharacteristic
388C2×Q16—22no—
389C2×SD16—22no—
390C4⋊Q8D4×C3111yescharacteristic
391C4⋊Q8D4×C3111yescharacteristic
392C4⋊Q8D4×C3111yescharacteristic
393#127D4×C3111yescharacteristic
394#126—22no—
395#126—22no—
396#126—22no—
397#126—22no—
398#125—22no—
399#125—22no—
400#125—21no—
401#124—22no—
402#124—22no—
403#124—21no—
404#123—22no—
405#123—22no—
406#123—22no—
407#123—22no—
408#123—22no—
409C22⋊Q8—21no—
410C22⋊Q8—21no—
411C22⋊Q8—22no—
412C22⋊Q8—22no—
413C22⋊Q8—21no—
414C22⋊Q8C23×C3112yes—
415#121—21no—
416#121—21no—
417#121—22no—
418#121—22no—
419#121—21no—
420#121C23×C3112yes—
421#120—22no—
422C4×Q8—21no—
423C4×Q8—21no—
424C4×Q8—21no—
425C4×Q8—21no—
426C4×Q8—21no—
427C4×Q8—21no—
428C4×Q8—21no—
429C4×Q8D4×C3112yes—
430C4×D4—21no—
431C4×D4—21no—
432C4×D4—22no—
433C4×D4—22no—
434C4×D4—22no—
435C4×D4—22no—
436C4×D4—21no—
437C4×D4D4×C3112yes—
438#117—22no—
439#117—21no—
440#117—21no—
441#117—21no—
442#117—21no—
443#117—21no—
444C2×C4⋊C4—21no—
445C2×C4⋊C4—21no—
446C2×C4⋊C4—21no—
447C2×C4⋊C4—21no—
448C2×C4⋊C4D4×C3111yescharacteristic
449C2×C4⋊C4C23×C3111yescharacteristic
450C2×C4⋊C4C23×C3111yescharacteristic
451C2×C22⋊C4—21no—
452C2×C22⋊C4—21no—
453C2×C22⋊C4D4×C3111yescharacteristic
454C2×C22⋊C4C23×C3111yescharacteristic
455C2×C42—22no—
456C2×C42—21no—
457C2×C42—21no—
458C2×C42D4×C3111yescharacteristic
459C2×C42C23×C3111yescharacteristic
460#107—22no—
461#106—22no—
462C4⋊C8—21no—
463C4⋊C8—21no—
464#103—22no—
465#103—22no—
466#103—22no—
467#102—22no—
468C22⋊C8C23×C3112yes—
469C22⋊C8C23×C3112yes—
470C4×C8—21no—
471C4×C8—21no—
472#95—21no—
473#95—21no—
474#95—21no—
475#95—21no—
476#95C23×C3111yescharacteristic
477#95C23×C3111yescharacteristic
478C31#1812711yescharacteristic, direct product factor, Sylow
479C24C2×D4×C3111yescharacteristic
480#41—42no—
481#41—42no—
482C2×Q8—21no—
483C2×Q8—21no—
484C2×Q8—21no—
485C2×Q8—21no—
486C2×Q8—42no—
487C2×Q8C2×D4×C3112yes—
488C2×D4—22no—
489C2×D4—22no—
490C2×D4—22no—
491C2×D4—21no—
492C2×D4—21no—
493C2×D4C2×D4×C3112yes—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—22no—
502C22×C4—22no—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4C2×D4×C3111yescharacteristic
506C22×C4C2×D4×C3111yescharacteristic
507C22×C4C2×D4×C3111yescharacteristic
508C22×C4C2×D4×C3111yescharacteristic
509C22×C4C2×D4×C3111yescharacteristic
510C22×C4C24×C3111yescharacteristic
511Q16—44no—
512SD16—44no—
513C2×C8—22no—
514C2×C8—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—22no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—22no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—41no—
535C4⋊C4—41no—
536C4⋊C4C2×D4×C3112yes—
537C4⋊C4C2×D4×C3112yes—
538C22⋊C4—22no—
539C22⋊C4—22no—
540C22⋊C4—22no—
541C22⋊C4—22no—
542C22⋊C4—22no—
543C22⋊C4—22no—
544C22⋊C4—22no—
545C22⋊C4—21no—
546C22⋊C4—21no—
547C22⋊C4—22no—
548C22⋊C4—21no—
549C22⋊C4—21no—
550C22⋊C4—22no—
551C22⋊C4—42no—
552C22⋊C4—42no—
553C42—22no—
554C42—22no—
555C42—21no—
556C42—22no—
557C42—21no—
558C42—41no—
559C42—41no—
560C42—42no—
561C42C2×D4×C3111yescharacteristic
562C42C2×D4×C3111yescharacteristic
563C42C2×D4×C3111yescharacteristic
564C42C2×D4×C3111yescharacteristic
565C23—21no—
566C23—22no—
567C23—21no—
568C23—21no—
569C23—41no—
570C23—41no—
571C23C22×D4×C3111yescharacteristic
572Q8—22no—
573Q8—22no—
574Q8—42no—
575Q8—41no—
576Q8—41no—
577Q8—42no—
578Q8—81no—
579D4—42no—
580D4—42no—
581D4—42no—
582D4—42no—
583D4—81no—
584D4—82no—
585D4—82no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—82no—
614C2×C4—82no—
615C2×C4C31×#12011yescharacteristic
616C2×C4C31×#12011yescharacteristic
617C2×C4C31×#12011yescharacteristic
618C2×C4C31×#12011yescharacteristic
619C2×C4C31×#13512yes—
620C2×C4C31×#13712yes—
621C2×C4C22×D4×C3111yescharacteristic
622C2×C4C22×D4×C3111yescharacteristic
623C8—42no—
624C8—42no—
625C22—21no—
626C22—21no—
627C22—42no—
628C22—41no—
629C22—42no—
630C22—42no—
631C22—41no—
632C22—42no—
633C22—41no—
634C22—81no—
635C22C2×C31×#12011yescharacteristic
636C22C2×C31×#13511yescharacteristic
637C22C2×C31×#13711yescharacteristic
638C4—21no—
639C4—21no—
640C4—21no—
641C4—21no—
642C4—42no—
643C4—41no—
644C4—42no—
645C4—41no—
646C4—42no—
647C4—42no—
648C4—41no—
649C4—42no—
650C4—81no—
651C4—81no—
652C4C31×#45212yes—
653C4C31×#45212yes—
654C2—21no—
655C2—21no—
656C2—81no—
657C2—82no—
658C2C2×C31×#45211yescharacteristic
659C2C31×#301811yescharacteristic
660C2C31×#302511yescharacteristic
subgroupG/H
centerC2×C4×C31#120
commutator, Frattini, cosocleC22×C4C24×C31
socleC22×C31C2×#120
prime radicalC31×#3455C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#3104C2semidirect
3. C31×#3103C2semidirect
4. C31×#2396C2semidirect
5. C31×#1618C2semidirect
6. C31×#1615C2semidirect
7. C31×#1606C2semidirect
8. C31×#1603C2semidirect
9. C31×#1503C2semidirect
10. C31×#1502C2semidirect
11. C31×#1458C2semidirect
234. #18127C31direct (N × H)
296. #3104C2×C31semidirect
297. #3103C2×C31semidirect
298. #2396C2×C31semidirect
299. #1618C2×C31semidirect
300. #1615C2×C31semidirect
301. #1606C2×C31semidirect
302. #1603C2×C31semidirect
303. #1503C2×C31semidirect
304. #1502C2×C31semidirect
305. #1458C2×C31semidirect
478. C31#18127direct (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×#31042
C31×#31032
C31×#23962
C31×#16182
C31×#16152
C31×#16062
C31×#16032
C31×#15032
C31×#15022
C31×#14582
#1812731