Finite Group Databasebeta

← C31×#18237 #21474996

Subgroups — C31×#18237 #21474996

Order 7936.

976 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3536C211yesmaximal, characteristic
2C31×#3095C211yesmaximal, characteristic, complemented
3C31×#3091C211yesmaximal, characteristic, complemented
4C31×#2624C211yesmaximal, characteristic, complemented
5C31×#1618C212yesmaximal, complemented
6C31×#1614C212yesmaximal, complemented
7C31×#1612C212yesmaximal, complemented
8C31×#1608C212yesmaximal, complemented
9C31×#1520C211yesmaximal, characteristic, complemented
10C31×#1513C211yesmaximal, characteristic, complemented
11C31×#1471C211yesmaximal, characteristic, complemented
12C31×#562—22no—
13C31×#554—21no—
14C31×#548—21no—
15C31×#547—21no—
16C31×#546—22no—
17C31×#546—21no—
18C31×#543C2212yes—
19C31×#542—21no—
20C31×#539—21no—
21C31×#538C2212yes—
22C31×#537—21no—
23C31×#537—21no—
24C31×#525C2211yescharacteristic
25C2×C31×C22⋊Q8C2211yescharacteristic
26C2×C31×#121C2211yescharacteristic
27C2×C31×#117C2211yescharacteristic
28C31×#485—21no—
29C31×#485—22no—
30C31×#484C2211yescharacteristic
31C31×#484C2212yes—
32C31×#483C2211yescharacteristic
33C31×#483C2212yes—
34C31×#482—21no—
35C31×#481C2211yescharacteristic
36C31×#480C2211yescharacteristic
37C31×#428—21no—
38C31×#427—21no—
39C31×#419—21no—
40C31×#416—21no—
41C31×#407C2211yescharacteristic
42C2×C31×C22⋊C8C2211yescharacteristic
43C31×#399C2212yes—
44C31×#397C2212yes—
45C31×#394—22no—
46C31×#390—21no—
47C31×#389—21no—
48C31×#380C2211yescharacteristic
49C31×#328C2212yes—
50C31×#328C2212yes—
51C31×#328C2212yes—
52C31×#327C2212yes—
53C31×#323C2212yes—
54C31×#323C2212yes—
55C2×#41×C31—22no—
56C2×#41×C31—21no—
57C2×#41×C31D411yescharacteristic
58C22×Q8×C31D411yescharacteristic
59C22×D4×C31D411yescharacteristic
60C22×D4×C31D412yes—
61C23×C4×C31C2311yescharacteristic
62C2×M16×C31—21no—
63C22×C8×C31—21no—
64C31×C4⋊Q8—21no—
65C31×#126—22no—
66C31×#125—22no—
67C31×#124—21no—
68C31×#124—22no—
69C31×#124—22no—
70C31×#123—22no—
71C31×#123—22no—
72C31×#123—22no—
73C31×#123—21no—
74C31×#123—22no—
75C31×C22⋊Q8—21no—
76C31×C22⋊Q8—22no—
77C31×C22⋊Q8—22no—
78C31×C22⋊Q8—22no—
79C31×C22⋊Q8—21no—
80C31×C22⋊Q8—21no—
81C31×C22⋊Q8—22no—
82C31×C22⋊Q8—22no—
83C31×C22⋊Q8—22no—
84C31×C22⋊Q8D412yes—
85C31×C22⋊Q8D411yescharacteristic
86C31×C22⋊Q8C2312yes—
87C31×C22⋊Q8C2311yescharacteristic
88C31×#121—21no—
89C31×#121—22no—
90C31×#121—22no—
91C31×#121—22no—
92C31×#121—22no—
93C31×#121—22no—
94C31×#121—21no—
95C31×#121—21no—
96C31×#121—21no—
97C31×#121—21no—
98C31×#121D411yescharacteristic
99C31×#121C2311yescharacteristic
100C31×#120—21no—
101C31×#120—22no—
102C4×Q8×C31—21no—
103C4×D4×C31—22no—
104C4×D4×C31—21no—
105C4×D4×C31—21no—
106C4×D4×C31—21no—
107C4×D4×C31—22no—
108C31×#117—21no—
109C31×#117—21no—
110C31×#117—21no—
111C31×#117—21no—
112C31×#117D411yescharacteristic
113C2×C4⋊C4×C31—21no—
114C2×C4⋊C4×C31—21no—
115C2×C4⋊C4×C31—21no—
116C2×C4⋊C4×C31—22no—
117C2×C4⋊C4×C31D411yescharacteristic
118C2×C4⋊C4×C31D411yescharacteristic
119C2×C4⋊C4×C31C2311yescharacteristic
120C2×C4⋊C4×C31C2311yescharacteristic
121C2×C22⋊C4×C31—21no—
122C2×C22⋊C4×C31—22no—
123C2×C22⋊C4×C31—21no—
124C2×C22⋊C4×C31—22no—
125C2×C22⋊C4×C31—22no—
126C2×C22⋊C4×C31—21no—
127C2×C22⋊C4×C31C2312yes—
128C2×C42×C31—21no—
129C31×#107—21no—
130C31×#107—21no—
131C31×#107—22no—
132C31×#106—21no—
133C31×#106—21no—
134C31×#106—22no—
135C31×#103—21no—
136C31×#103—21no—
137C31×#103—22no—
138C31×#103—22no—
139C31×#102—21no—
140C31×#102—21no—
141C31×C22⋊C8—21no—
142C31×C22⋊C8—21no—
143C31×C22⋊C8C2311yescharacteristic
144C31×C22⋊C8C2311yescharacteristic
145C31×C22⋊C8C2312yes—
146C31×#95—22no—
147C31×#95C2312yes—
148C24×C31C2×D412yes—
149C24×C31C2×D411yescharacteristic
150#41×C31—42no—
151#41×C31—84no—
152#41×C31—82no—
153C2×Q8×C31—22no—
154C2×Q8×C31—21no—
155C2×Q8×C31—42no—
156C2×Q8×C31C2×D411yescharacteristic
157C2×Q8×C31C2×D412yes—
158C2×D4×C31—21no—
159C2×D4×C31—22no—
160C2×D4×C31—21no—
161C2×D4×C31—22no—
162C2×D4×C31—22no—
163C2×D4×C31—21no—
164C2×D4×C31—22no—
165C2×D4×C31—21no—
166C2×D4×C31—21no—
167C2×D4×C31—42no—
168C2×D4×C31—82no—
169C2×D4×C31C2×D411yescharacteristic
170C22×C4×C31—21no—
171C22×C4×C31—22no—
172C22×C4×C31—22no—
173C22×C4×C31—22no—
174C22×C4×C31—21no—
175C22×C4×C31—21no—
176C22×C4×C31—21no—
177C22×C4×C31—22no—
178C22×C4×C31—21no—
179C22×C4×C31—42no—
180C22×C4×C31C2×D411yescharacteristic
181C22×C4×C31C2×D411yescharacteristic
182C22×C4×C31C2×D411yescharacteristic
183C22×C4×C31C2×D412yes—
184C22×C4×C31C2×D411yescharacteristic
185C22×C4×C31C2×D411yescharacteristic
186C22×C4×C31C2411yescharacteristic
187M16×C31—42no—
188C2×C8×C31—21no—
189C2×C8×C31—21no—
190C2×C8×C31—22no—
191C2×C8×C31—42no—
192C4⋊C4×C31—22no—
193C4⋊C4×C31—22no—
194C4⋊C4×C31—22no—
195C4⋊C4×C31—22no—
196C4⋊C4×C31—21no—
197C4⋊C4×C31—21no—
198C4⋊C4×C31—22no—
199C4⋊C4×C31—21no—
200C4⋊C4×C31—21no—
201C4⋊C4×C31—22no—
202C4⋊C4×C31—21no—
203C4⋊C4×C31—21no—
204C4⋊C4×C31—21no—
205C4⋊C4×C31—21no—
206C4⋊C4×C31—21no—
207C4⋊C4×C31—42no—
208C4⋊C4×C31—82no—
209C4⋊C4×C31C2×D412yes—
210C4⋊C4×C31C2×D411yescharacteristic
211C4⋊C4×C31C2×D411yescharacteristic
212C22⋊C4×C31—22no—
213C22⋊C4×C31—22no—
214C22⋊C4×C31—22no—
215C22⋊C4×C31—22no—
216C22⋊C4×C31—22no—
217C22⋊C4×C31—22no—
218C22⋊C4×C31—21no—
219C22⋊C4×C31—21no—
220C22⋊C4×C31—22no—
221C22⋊C4×C31—21no—
222C22⋊C4×C31—21no—
223C22⋊C4×C31—22no—
224C22⋊C4×C31—22no—
225C22⋊C4×C31—21no—
226C22⋊C4×C31—41no—
227C22⋊C4×C31—41no—
228C22⋊C4×C31—81no—
229C22⋊C4×C31—82no—
230C22⋊C4×C31—82no—
231C42×C31—22no—
232C42×C31—21no—
233C42×C31—41no—
234C42×C31—41no—
235#18237C3111yesmaximal, characteristic, direct product factor, Sylow
236C23×C31—21no—
237C23×C31—21no—
238C23×C31—22no—
239C23×C31—21no—
240C23×C31—22no—
241C23×C31—22no—
242C23×C31—41no—
243C23×C31—41no—
244C23×C31—42no—
245C23×C31—42no—
246C23×C31#12011yescharacteristic
247C23×C31#12011yescharacteristic
248C23×C31C22×D411yescharacteristic
249Q8×C31—42no—
250Q8×C31—41no—
251Q8×C31—42no—
252Q8×C31—82no—
253D4×C31—41no—
254D4×C31—44no—
255D4×C31—42no—
256D4×C31—41no—
257D4×C31—41no—
258D4×C31—42no—
259D4×C31—81no—
260D4×C31—82no—
261D4×C31—81no—
262D4×C31—82no—
263D4×C31—82no—
264C2×C4×C31—22no—
265C2×C4×C31—22no—
266C2×C4×C31—22no—
267C2×C4×C31—21no—
268C2×C4×C31—22no—
269C2×C4×C31—21no—
270C2×C4×C31—22no—
271C2×C4×C31—21no—
272C2×C4×C31—22no—
273C2×C4×C31—41no—
274C2×C4×C31—41no—
275C2×C4×C31—41no—
276C2×C4×C31—42no—
277C2×C4×C31—42no—
278C2×C4×C31—42no—
279C2×C4×C31—41no—
280C2×C4×C31—42no—
281C2×C4×C31—42no—
282C2×C4×C31—41no—
283C2×C4×C31—41no—
284C2×C4×C31—41no—
285C2×C4×C31—42no—
286C2×C4×C31—41no—
287C2×C4×C31—41no—
288C2×C4×C31—81no—
289C2×C4×C31—82no—
290C2×C4×C31—82no—
291C2×C4×C31—82no—
292C2×C4×C31#12011yescharacteristic
293C2×C4×C31#12011yescharacteristic
294C2×C4×C31C22×D411yescharacteristic
295C2×C4×C31C22×D411yescharacteristic
296C8×C31—41no—
297C8×C31—41no—
298C8×C31—42no—
299#3536C2×C3111yescharacteristic
300#3095C2×C3111yescharacteristic, complemented
301#3091C2×C3111yescharacteristic, complemented
302#2624C2×C3111yescharacteristic, complemented
303#1618C2×C3112yescomplemented
304#1614C2×C3112yescomplemented
305#1612C2×C3112yescomplemented
306#1608C2×C3112yescomplemented
307#1520C2×C3111yescharacteristic, complemented
308#1513C2×C3111yescharacteristic, complemented
309#1471C2×C3111yescharacteristic, complemented
310C22×C31—21no—
311C22×C31—21no—
312C22×C31—21no—
313C22×C31—21no—
314C22×C31—21no—
315C22×C31—21no—
316C22×C31—21no—
317C22×C31—21no—
318C22×C31—41no—
319C22×C31—42no—
320C22×C31—41no—
321C22×C31—42no—
322C22×C31—41no—
323C22×C31—42no—
324C22×C31—81no—
325C22×C31—81no—
326C22×C31—82no—
327C22×C31—82no—
328C22×C31C2×#12011yescharacteristic
329C22×C31#57711yescharacteristic
330C22×C31#57811yescharacteristic
331C4×C31—21no—
332C4×C31—21no—
333C4×C31—21no—
334C4×C31—41no—
335C4×C31—42no—
336C4×C31—42no—
337C4×C31—41no—
338C4×C31—41no—
339C4×C31—42no—
340C4×C31—82no—
341C4×C31—81no—
342C4×C31—81no—
343C4×C31#45812yes—
344#562—22no—
345#554—21no—
346#548—21no—
347#547—21no—
348#546—22no—
349#546—21no—
350#543C22×C3112yes—
351#542—21no—
352#539—21no—
353#538C22×C3112yes—
354#537—21no—
355#537—21no—
356#525C22×C3111yescharacteristic
357C2×C22⋊Q8C22×C3111yescharacteristic
358C2×#121C22×C3111yescharacteristic
359C2×#117C22×C3111yescharacteristic
360#485—21no—
361#485—22no—
362#484C22×C3111yescharacteristic
363#484C22×C3112yes—
364#483C22×C3111yescharacteristic
365#483C22×C3112yes—
366#482—21no—
367#481C22×C3111yescharacteristic
368#480C22×C3111yescharacteristic
369#428—21no—
370#427—21no—
371#419—21no—
372#416—21no—
373#407C22×C3111yescharacteristic
374C2×C22⋊C8C22×C3111yescharacteristic
375#399C22×C3112yes—
376#397C22×C3112yes—
377#394—22no—
378#390—21no—
379#389—21no—
380#380C22×C3111yescharacteristic
381#328C22×C3112yes—
382#328C22×C3112yes—
383#328C22×C3112yes—
384#327C22×C3112yes—
385#323C22×C3112yes—
386#323C22×C3112yes—
387C2×C31—21no—
388C2×C31—21no—
389C2×C31—41no—
390C2×C31—41no—
391C2×C31—81no—
392C2×C31—82no—
393C2×C31#301311yescharacteristic
394C2×C31#301911yescharacteristic
395C2×C31C2×#45811yescharacteristic
396C2×#41—22no—
397C2×#41—21no—
398C2×#41D4×C3111yescharacteristic
399C22×Q8D4×C3111yescharacteristic
400C22×D4D4×C3111yescharacteristic
401C22×D4D4×C3112yes—
402C23×C4C23×C3111yescharacteristic
403C2×M16—21no—
404C22×C8—21no—
405C4⋊Q8—21no—
406#126—22no—
407#125—22no—
408#124—21no—
409#124—22no—
410#124—22no—
411#123—22no—
412#123—22no—
413#123—22no—
414#123—21no—
415#123—22no—
416C22⋊Q8—21no—
417C22⋊Q8—22no—
418C22⋊Q8—22no—
419C22⋊Q8—22no—
420C22⋊Q8—21no—
421C22⋊Q8—21no—
422C22⋊Q8—22no—
423C22⋊Q8—22no—
424C22⋊Q8—22no—
425C22⋊Q8D4×C3112yes—
426C22⋊Q8D4×C3111yescharacteristic
427C22⋊Q8C23×C3112yes—
428C22⋊Q8C23×C3111yescharacteristic
429#121—21no—
430#121—22no—
431#121—22no—
432#121—22no—
433#121—22no—
434#121—22no—
435#121—21no—
436#121—21no—
437#121—21no—
438#121—21no—
439#121D4×C3111yescharacteristic
440#121C23×C3111yescharacteristic
441#120—21no—
442#120—22no—
443C4×Q8—21no—
444C4×D4—22no—
445C4×D4—21no—
446C4×D4—21no—
447C4×D4—21no—
448C4×D4—22no—
449#117—21no—
450#117—21no—
451#117—21no—
452#117—21no—
453#117D4×C3111yescharacteristic
454C2×C4⋊C4—21no—
455C2×C4⋊C4—21no—
456C2×C4⋊C4—21no—
457C2×C4⋊C4—22no—
458C2×C4⋊C4D4×C3111yescharacteristic
459C2×C4⋊C4D4×C3111yescharacteristic
460C2×C4⋊C4C23×C3111yescharacteristic
461C2×C4⋊C4C23×C3111yescharacteristic
462C2×C22⋊C4—21no—
463C2×C22⋊C4—22no—
464C2×C22⋊C4—21no—
465C2×C22⋊C4—21no—
466C2×C22⋊C4—22no—
467C2×C22⋊C4—22no—
468C2×C22⋊C4C23×C3112yes—
469C2×C42—21no—
470#107—21no—
471#107—21no—
472#107—22no—
473#106—21no—
474#106—21no—
475#106—22no—
476#103—21no—
477#103—21no—
478#103—22no—
479#103—22no—
480#102—21no—
481#102—21no—
482C22⋊C8—21no—
483C22⋊C8—21no—
484C22⋊C8C23×C3112yes—
485C22⋊C8C23×C3111yescharacteristic
486C22⋊C8C23×C3111yescharacteristic
487#95—22no—
488#95C23×C3112yes—
489C31#1823711yescharacteristic, direct product factor, Sylow
490C24C2×D4×C3112yes—
491C24C2×D4×C3111yescharacteristic
492#41—42no—
493#41—84no—
494#41—82no—
495C2×Q8—22no—
496C2×Q8—21no—
497C2×Q8—42no—
498C2×Q8C2×D4×C3111yescharacteristic
499C2×Q8C2×D4×C3112yes—
500C2×D4—21no—
501C2×D4—22no—
502C2×D4—21no—
503C2×D4—22no—
504C2×D4—22no—
505C2×D4—21no—
506C2×D4—22no—
507C2×D4—21no—
508C2×D4—21no—
509C2×D4—42no—
510C2×D4—82no—
511C2×D4C2×D4×C3111yescharacteristic
512C22×C4—21no—
513C22×C4—22no—
514C22×C4—22no—
515C22×C4—22no—
516C22×C4—21no—
517C22×C4—21no—
518C22×C4—21no—
519C22×C4—22no—
520C22×C4—21no—
521C22×C4—42no—
522C22×C4C2×D4×C3111yescharacteristic
523C22×C4C2×D4×C3111yescharacteristic
524C22×C4C2×D4×C3111yescharacteristic
525C22×C4C2×D4×C3112yes—
526C22×C4C2×D4×C3111yescharacteristic
527C22×C4C2×D4×C3111yescharacteristic
528C22×C4C24×C3111yescharacteristic
529M16—42no—
530C2×C8—21no—
531C2×C8—21no—
532C2×C8—22no—
533C2×C8—42no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—22no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—22no—
544C4⋊C4—21no—
545C4⋊C4—21no—
546C4⋊C4—21no—
547C4⋊C4—21no—
548C4⋊C4—21no—
549C4⋊C4—42no—
550C4⋊C4—82no—
551C4⋊C4C2×D4×C3112yes—
552C4⋊C4C2×D4×C3111yescharacteristic
553C4⋊C4C2×D4×C3111yescharacteristic
554C22⋊C4—22no—
555C22⋊C4—22no—
556C22⋊C4—22no—
557C22⋊C4—22no—
558C22⋊C4—22no—
559C22⋊C4—22no—
560C22⋊C4—21no—
561C22⋊C4—21no—
562C22⋊C4—22no—
563C22⋊C4—21no—
564C22⋊C4—21no—
565C22⋊C4—22no—
566C22⋊C4—22no—
567C22⋊C4—21no—
568C22⋊C4—41no—
569C22⋊C4—41no—
570C22⋊C4—81no—
571C22⋊C4—82no—
572C22⋊C4—82no—
573C42—22no—
574C42—21no—
575C42—41no—
576C42—41no—
577C23—21no—
578C23—21no—
579C23—22no—
580C23—21no—
581C23—22no—
582C23—22no—
583C23—41no—
584C23—41no—
585C23—42no—
586C23—42no—
587C23C31×#12011yescharacteristic
588C23C31×#12011yescharacteristic
589C23C22×D4×C3111yescharacteristic
590Q8—42no—
591Q8—41no—
592Q8—42no—
593Q8—82no—
594D4—41no—
595D4—44no—
596D4—42no—
597D4—41no—
598D4—41no—
599D4—42no—
600D4—81no—
601D4—82no—
602D4—81no—
603D4—82no—
604D4—82no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—21no—
609C2×C4—22no—
610C2×C4—21no—
611C2×C4—22no—
612C2×C4—21no—
613C2×C4—22no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—42no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—81no—
630C2×C4—82no—
631C2×C4—82no—
632C2×C4—82no—
633C2×C4C31×#12011yescharacteristic
634C2×C4C31×#12011yescharacteristic
635C2×C4C22×D4×C3111yescharacteristic
636C2×C4C22×D4×C3111yescharacteristic
637C8—41no—
638C8—41no—
639C8—42no—
640C22—21no—
641C22—21no—
642C22—21no—
643C22—21no—
644C22—21no—
645C22—21no—
646C22—21no—
647C22—21no—
648C22—41no—
649C22—42no—
650C22—41no—
651C22—42no—
652C22—41no—
653C22—42no—
654C22—81no—
655C22—81no—
656C22—82no—
657C22—82no—
658C22C2×C31×#12011yescharacteristic
659C22C31×#57711yescharacteristic
660C22C31×#57811yescharacteristic
661C4—21no—
662C4—21no—
663C4—21no—
664C4—41no—
665C4—42no—
666C4—42no—
667C4—41no—
668C4—41no—
669C4—42no—
670C4—82no—
671C4—81no—
672C4—81no—
673C4C31×#45812yes—
674C2—21no—
675C2—21no—
676C2—41no—
677C2—41no—
678C2—82no—
679C2—81no—
680C2C31×#301311yescharacteristic
681C2C31×#301911yescharacteristic
682C2C2×C31×#45811yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#3536C2
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×#3095C2semidirect
3. C31×#3091C2semidirect
4. C31×#2624C2semidirect
5. C31×#1618C2semidirect
6. C31×#1614C2semidirect
7. C31×#1612C2semidirect
8. C31×#1608C2semidirect
9. C31×#1520C2semidirect
10. C31×#1513C2semidirect
11. C31×#1471C2semidirect
235. #18237C31direct (N × H)
300. #3095C2×C31semidirect
301. #3091C2×C31semidirect
302. #2624C2×C31semidirect
303. #1618C2×C31semidirect
304. #1614C2×C31semidirect
305. #1612C2×C31semidirect
306. #1608C2×C31semidirect
307. #1520C2×C31semidirect
308. #1513C2×C31semidirect
309. #1471C2×C31semidirect
489. C31#18237direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35362
C31×#30952
C31×#30912
C31×#26242
C31×#16182
C31×#16142
C31×#16122
C31×#16082
C31×#15202
C31×#15132
C31×#14712
#1823731