Finite Group Databasebeta

← C31×#18330 #21475089

Subgroups — C31×#18330 #21475089

Order 7936.

1068 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3460C211yesmaximal, characteristic, complemented
2C31×#3194C212yesmaximal, complemented
3C31×#2543C211yesmaximal, characteristic, complemented
4C31×#1612C212yesmaximal, complemented
5C31×#1611C211yesmaximal, characteristic, complemented
6C31×#1610C211yesmaximal, characteristic, complemented
7C31×#1600C212yesmaximal, complemented
8C31×#1599C211yesmaximal, characteristic, complemented
9C31×#1598C211yesmaximal, characteristic, complemented
10C31×#1504C212yesmaximal, complemented
11C31×#1465C211yesmaximal, characteristic, complemented
12C31×#544C2211yescharacteristic
13C31×#542—22no—
14C31×#540—22no—
15C31×#538—21no—
16C31×#538C2212yes—
17C31×#537C2211yescharacteristic
18C31×#535—21no—
19C31×#532—22no—
20C31×#529—22no—
21C31×#525—21no—
22C31×#525C2212yes—
23C2×C31×#121—21no—
24C2×C31×#121C2211yescharacteristic
25C4×#41×C31—21no—
26C2×C4×D4×C31C2211yescharacteristic
27C31×#483C2212yes—
28C31×#482—22no—
29C31×#480C2212yes—
30C31×#469—22no—
31C31×#468—22no—
32C31×#450C2212yes—
33C31×#449—22no—
34C31×#447C2212yes—
35C31×#407C2212yes—
36C31×#398C2211yescharacteristic
37C31×#397C2211yescharacteristic
38C31×#394C2211yescharacteristic
39C31×#392C2211yescharacteristic
40C31×#388—21no—
41C31×#386—21no—
42C31×#386—21no—
43C2×C31×#95C2211yescharacteristic
44C31×#328C2212yes—
45C31×#328C2212yes—
46C31×#327C2212yes—
47C31×#327C2212yes—
48C31×#324C2211yescharacteristic
49C31×#324C2211yescharacteristic
50C31×#323C2212yes—
51C2×#41×C31—21no—
52C2×#41×C31—21no—
53C2×#41×C31D412yescomplemented
54C22×D4×C31—21no—
55C22×D4×C31—21no—
56C22×D4×C31D411yescharacteristic
57C22×D4×C31C2311yescharacteristic
58C23×C4×C31—21no—
59C23×C4×C31D411yescharacteristic
60C23×C4×C31C2311yescharacteristic
61C2×SD16×C31—22no—
62C2×D8×C31—22no—
63C2×M16×C31—22no—
64C31×C4⋊Q8—22no—
65C31×#127—22no—
66C31×#126—22no—
67C31×#126—22no—
68C31×#125—22no—
69C31×#124—22no—
70C31×#124—22no—
71C31×#124—22no—
72C31×#123—22no—
73C31×#123—22no—
74C31×#123—22no—
75C31×#123—22no—
76C31×C22⋊Q8—21no—
77C31×C22⋊Q8—21no—
78C31×C22⋊Q8—22no—
79C31×C22⋊Q8D412yescomplemented
80C31×C22⋊Q8C2312yes—
81C31×#121—21no—
82C31×#121—22no—
83C31×#121—22no—
84C31×#121—22no—
85C31×#121—22no—
86C31×#121—22no—
87C31×#121—21no—
88C31×#121—22no—
89C31×#121—22no—
90C31×#121—44no—
91C31×#121D412yescomplemented
92C31×#121C2312yes—
93C31×#120—22no—
94C31×#120—22no—
95C4×Q8×C31—21no—
96C4×Q8×C31—21no—
97C4×D4×C31—21no—
98C4×D4×C31—21no—
99C4×D4×C31—22no—
100C4×D4×C31—22no—
101C4×D4×C31—22no—
102C4×D4×C31—22no—
103C4×D4×C31—22no—
104C4×D4×C31—22no—
105C31×#117—22no—
106C31×#117—21no—
107C31×#117—21no—
108C31×#117D412yescomplemented
109C2×C4⋊C4×C31—21no—
110C2×C4⋊C4×C31C2311yescharacteristic
111C2×C4⋊C4×C31C2311yescharacteristic
112C2×C22⋊C4×C31—21no—
113C2×C22⋊C4×C31—21no—
114C2×C22⋊C4×C31—21no—
115C2×C22⋊C4×C31—21no—
116C2×C22⋊C4×C31—21no—
117C2×C22⋊C4×C31D411yescharacteristic
118C2×C22⋊C4×C31C2311yescharacteristic
119C2×C42×C31—22no—
120C2×C42×C31D411yescharacteristic
121C31×#107—22no—
122C31×#106—22no—
123C31×#103—22no—
124C31×#102—22no—
125C31×#102—22no—
126C31×#102—22no—
127C31×C22⋊C8—22no—
128C31×C22⋊C8C2312yes—
129C31×C22⋊C8C2312yes—
130C31×#95—21no—
131C31×#95—21no—
132C31×#95—21no—
133C31×#95C2311yescharacteristic
134C31×#95C2311yescharacteristic
135C24×C31—21no—
136C24×C31C2×D411yescharacteristic
137C24×C31C2×D411yescharacteristic
138#41×C31—42no—
139#41×C31—42no—
140#41×C31—82no—
141#41×C31—82no—
142C2×Q8×C31—21no—
143C2×Q8×C31—21no—
144C2×Q8×C31C2×D412yes—
145C2×D4×C31—21no—
146C2×D4×C31—22no—
147C2×D4×C31—22no—
148C2×D4×C31—22no—
149C2×D4×C31—22no—
150C2×D4×C31—21no—
151C2×D4×C31—22no—
152C2×D4×C31—22no—
153C2×D4×C31—22no—
154C2×D4×C31—22no—
155C2×D4×C31—41no—
156C2×D4×C31—42no—
157C2×D4×C31—42no—
158C2×D4×C31—42no—
159C2×D4×C31—42no—
160C2×D4×C31—42no—
161C2×D4×C31—81no—
162C2×D4×C31—81no—
163C2×D4×C31C2×D412yes—
164C22×C4×C31—22no—
165C22×C4×C31—21no—
166C22×C4×C31—22no—
167C22×C4×C31—22no—
168C22×C4×C31—21no—
169C22×C4×C31—21no—
170C22×C4×C31—21no—
171C22×C4×C31—21no—
172C22×C4×C31—42no—
173C22×C4×C31—42no—
174C22×C4×C31—81no—
175C22×C4×C31—81no—
176C22×C4×C31—81no—
177C22×C4×C31C2×D411yescharacteristic
178C22×C4×C31C2×D411yescharacteristic
179C22×C4×C31C2×D411yescharacteristic
180C22×C4×C31C2×D411yescharacteristic
181C22×C4×C31C2×D412yes—
182C22×C4×C31C2×D412yes—
183C22×C4×C31C2411yescharacteristic
184SD16×C31—82no—
185D8×C31—82no—
186M16×C31—82no—
187C2×C8×C31—22no—
188C2×C8×C31—22no—
189C4⋊C4×C31—21no—
190C4⋊C4×C31—21no—
191C4⋊C4×C31—22no—
192C4⋊C4×C31—21no—
193C4⋊C4×C31—21no—
194C4⋊C4×C31—22no—
195C4⋊C4×C31—22no—
196C4⋊C4×C31—22no—
197C4⋊C4×C31—22no—
198C4⋊C4×C31—22no—
199C4⋊C4×C31—42no—
200C4⋊C4×C31C2×D412yes—
201C4⋊C4×C31C2×D412yes—
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—22no—
209C22⋊C4×C31—22no—
210C22⋊C4×C31—22no—
211C22⋊C4×C31—22no—
212C22⋊C4×C31—22no—
213C22⋊C4×C31—42no—
214C22⋊C4×C31—42no—
215C22⋊C4×C31—41no—
216C22⋊C4×C31—41no—
217C22⋊C4×C31—42no—
218C22⋊C4×C31—81no—
219C42×C31—22no—
220C42×C31—22no—
221C42×C31—21no—
222C42×C31—22no—
223C42×C31—21no—
224#18330C3111yesmaximal, characteristic, direct product factor, Sylow
225C23×C31—21no—
226C23×C31—21no—
227C23×C31—21no—
228C23×C31—22no—
229C23×C31—21no—
230C23×C31—21no—
231C23×C31—41no—
232C23×C31—41no—
233C23×C31—41no—
234C23×C31—41no—
235C23×C31—41no—
236C23×C31—41no—
237C23×C31—41no—
238C23×C31—41no—
239C23×C31—41no—
240C23×C31—41no—
241C23×C31—41no—
242C23×C31#12012yes—
243C23×C31C22×D411yescharacteristic
244Q8×C31—42no—
245Q8×C31—42no—
246Q8×C31—81no—
247D4×C31—44no—
248D4×C31—42no—
249D4×C31—42no—
250D4×C31—42no—
251D4×C31—42no—
252D4×C31—42no—
253D4×C31—42no—
254D4×C31—82no—
255D4×C31—82no—
256D4×C31—82no—
257D4×C31—81no—
258D4×C31—82no—
259D4×C31—84no—
260D4×C31—84no—
261C2×C4×C31—21no—
262C2×C4×C31—21no—
263C2×C4×C31—22no—
264C2×C4×C31—21no—
265C2×C4×C31—22no—
266C2×C4×C31—22no—
267C2×C4×C31—22no—
268C2×C4×C31—21no—
269C2×C4×C31—42no—
270C2×C4×C31—41no—
271C2×C4×C31—42no—
272C2×C4×C31—41no—
273C2×C4×C31—42no—
274C2×C4×C31—44no—
275C2×C4×C31—44no—
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—41no—
285C2×C4×C31—41no—
286C2×C4×C31—81no—
287C2×C4×C31—82no—
288C2×C4×C31—84no—
289C2×C4×C31#12011yescharacteristic
290C2×C4×C31#12011yescharacteristic
291C2×C4×C31AGL(1,ℤ/8)12yes—
292C2×C4×C31C22×D412yes—
293C8×C31—42no—
294C8×C31—42no—
295#3460C2×C3111yescharacteristic, complemented
296#3194C2×C3112yescomplemented
297#2543C2×C3111yescharacteristic, complemented
298#1612C2×C3112yescomplemented
299#1611C2×C3111yescharacteristic, complemented
300#1610C2×C3111yescharacteristic, complemented
301#1600C2×C3112yescomplemented
302#1599C2×C3111yescharacteristic, complemented
303#1598C2×C3111yescharacteristic, complemented
304#1504C2×C3112yescomplemented
305#1465C2×C3111yescharacteristic, complemented
306C22×C31—21no—
307C22×C31—21no—
308C22×C31—22no—
309C22×C31—22no—
310C22×C31—22no—
311C22×C31—42no—
312C22×C31—42no—
313C22×C31—42no—
314C22×C31—42no—
315C22×C31—41no—
316C22×C31—42no—
317C22×C31—41no—
318C22×C31—41no—
319C22×C31—42no—
320C22×C31—81no—
321C22×C31—81no—
322C22×C31—81no—
323C22×C31—81no—
324C22×C31—81no—
325C22×C31—81no—
326C22×C31—81no—
327C22×C31—81no—
328C22×C31C2×#12011yescharacteristic
329C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
330C22×C31#57511yescharacteristic
331C4×C31—22no—
332C4×C31—22no—
333C4×C31—21no—
334C4×C31—21no—
335C4×C31—42no—
336C4×C31—42no—
337C4×C31—41no—
338C4×C31—42no—
339C4×C31—41no—
340C4×C31—42no—
341C4×C31—81no—
342C4×C31—81no—
343#544C22×C3111yescharacteristic
344#542—22no—
345#540—22no—
346#538—21no—
347#538C22×C3112yes—
348#537C22×C3111yescharacteristic
349#535—21no—
350#532—22no—
351#529—22no—
352#525—21no—
353#525C22×C3112yes—
354C2×#121—21no—
355C2×#121C22×C3111yescharacteristic
356C4×#41—21no—
357C2×C4×D4C22×C3111yescharacteristic
358#483C22×C3112yes—
359#482—22no—
360#480C22×C3112yes—
361#469—22no—
362#468—22no—
363#450C22×C3112yes—
364#449—22no—
365#447C22×C3112yes—
366#407C22×C3112yes—
367#398C22×C3111yescharacteristic
368#397C22×C3111yescharacteristic
369#394C22×C3111yescharacteristic
370#392C22×C3111yescharacteristic
371#388—21no—
372#386—21no—
373#386—21no—
374C2×#95C22×C3111yescharacteristic
375#328C22×C3112yes—
376#328C22×C3112yes—
377#327C22×C3112yes—
378#327C22×C3112yes—
379#324C22×C3111yescharacteristic
380#324C22×C3111yescharacteristic
381#323C22×C3112yes—
382C2×C31—21no—
383C2×C31—21no—
384C2×C31—42no—
385C2×C31—82no—
386C2×C31—81no—
387C2×C31—161no—
388C2×C31#300611yescharacteristic
389C2×C31#301811yescharacteristic
390C2×C31#302611yescharacteristic
391C2×#41—21no—
392C2×#41—21no—
393C2×#41D4×C3112yescomplemented
394C22×D4—21no—
395C22×D4—21no—
396C22×D4D4×C3111yescharacteristic
397C22×D4C23×C3111yescharacteristic
398C23×C4—21no—
399C23×C4D4×C3111yescharacteristic
400C23×C4C23×C3111yescharacteristic
401C2×SD16—22no—
402C2×D8—22no—
403C2×M16—22no—
404C4⋊Q8—22no—
405#127—22no—
406#126—22no—
407#126—22no—
408#125—22no—
409#124—22no—
410#124—22no—
411#124—22no—
412#123—22no—
413#123—22no—
414#123—22no—
415#123—22no—
416C22⋊Q8—21no—
417C22⋊Q8—21no—
418C22⋊Q8—22no—
419C22⋊Q8D4×C3112yescomplemented
420C22⋊Q8C23×C3112yes—
421#121—21no—
422#121—22no—
423#121—22no—
424#121—22no—
425#121—22no—
426#121—22no—
427#121—21no—
428#121—22no—
429#121—22no—
430#121—44no—
431#121D4×C3112yescomplemented
432#121C23×C3112yes—
433#120—22no—
434#120—22no—
435C4×Q8—21no—
436C4×Q8—21no—
437C4×D4—21no—
438C4×D4—21no—
439C4×D4—22no—
440C4×D4—22no—
441C4×D4—22no—
442C4×D4—22no—
443C4×D4—22no—
444C4×D4—22no—
445#117—22no—
446#117—21no—
447#117—21no—
448#117D4×C3112yescomplemented
449C2×C4⋊C4—21no—
450C2×C4⋊C4C23×C3111yescharacteristic
451C2×C4⋊C4C23×C3111yescharacteristic
452C2×C22⋊C4—21no—
453C2×C22⋊C4—21no—
454C2×C22⋊C4—21no—
455C2×C22⋊C4—21no—
456C2×C22⋊C4—21no—
457C2×C22⋊C4D4×C3111yescharacteristic
458C2×C22⋊C4C23×C3111yescharacteristic
459C2×C42—22no—
460C2×C42D4×C3111yescharacteristic
461#107—22no—
462#106—22no—
463#103—22no—
464#102—22no—
465#102—22no—
466#102—22no—
467C22⋊C8—22no—
468C22⋊C8C23×C3112yes—
469C22⋊C8C23×C3112yes—
470#95—21no—
471#95—21no—
472#95—21no—
473#95C23×C3111yescharacteristic
474#95C23×C3111yescharacteristic
475C31#1833011yescharacteristic, direct product factor, Sylow
476C24—21no—
477C24C2×D4×C3111yescharacteristic
478C24C2×D4×C3111yescharacteristic
479#41—42no—
480#41—42no—
481#41—82no—
482#41—82no—
483C2×Q8—21no—
484C2×Q8—21no—
485C2×Q8C2×D4×C3112yes—
486C2×D4—21no—
487C2×D4—22no—
488C2×D4—22no—
489C2×D4—22no—
490C2×D4—22no—
491C2×D4—21no—
492C2×D4—22no—
493C2×D4—22no—
494C2×D4—22no—
495C2×D4—22no—
496C2×D4—41no—
497C2×D4—42no—
498C2×D4—42no—
499C2×D4—42no—
500C2×D4—42no—
501C2×D4—42no—
502C2×D4—81no—
503C2×D4—81no—
504C2×D4C2×D4×C3112yes—
505C22×C4—22no—
506C22×C4—21no—
507C22×C4—22no—
508C22×C4—22no—
509C22×C4—21no—
510C22×C4—21no—
511C22×C4—21no—
512C22×C4—21no—
513C22×C4—42no—
514C22×C4—42no—
515C22×C4—81no—
516C22×C4—81no—
517C22×C4—81no—
518C22×C4C2×D4×C3111yescharacteristic
519C22×C4C2×D4×C3111yescharacteristic
520C22×C4C2×D4×C3111yescharacteristic
521C22×C4C2×D4×C3111yescharacteristic
522C22×C4C2×D4×C3112yes—
523C22×C4C2×D4×C3112yes—
524C22×C4C24×C3111yescharacteristic
525SD16—82no—
526D8—82no—
527M16—82no—
528C2×C8—22no—
529C2×C8—22no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—22no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—42no—
541C4⋊C4C2×D4×C3112yes—
542C4⋊C4C2×D4×C3112yes—
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—22no—
552C22⋊C4—22no—
553C22⋊C4—22no—
554C22⋊C4—42no—
555C22⋊C4—42no—
556C22⋊C4—41no—
557C22⋊C4—41no—
558C22⋊C4—42no—
559C22⋊C4—81no—
560C42—22no—
561C42—22no—
562C42—21no—
563C42—22no—
564C42—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—22no—
569C23—21no—
570C23—21no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23C31×#12012yes—
583C23C22×D4×C3111yescharacteristic
584Q8—42no—
585Q8—42no—
586Q8—81no—
587D4—44no—
588D4—42no—
589D4—42no—
590D4—42no—
591D4—42no—
592D4—42no—
593D4—42no—
594D4—82no—
595D4—82no—
596D4—82no—
597D4—81no—
598D4—82no—
599D4—84no—
600D4—84no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—22no—
604C2×C4—21no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—21no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—44no—
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—81no—
627C2×C4—82no—
628C2×C4—84no—
629C2×C4C31×#12011yescharacteristic
630C2×C4C31×#12011yescharacteristic
631C2×C4C31×AGL(1,ℤ/8)12yes—
632C2×C4C22×D4×C3112yes—
633C8—42no—
634C8—42no—
635C22—21no—
636C22—21no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—41no—
645C22—42no—
646C22—41no—
647C22—41no—
648C22—42no—
649C22—81no—
650C22—81no—
651C22—81no—
652C22—81no—
653C22—81no—
654C22—81no—
655C22—81no—
656C22—81no—
657C22C2×C31×#12011yescharacteristic
658C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
659C22C31×#57511yescharacteristic
660C4—22no—
661C4—22no—
662C4—21no—
663C4—21no—
664C4—42no—
665C4—42no—
666C4—41no—
667C4—42no—
668C4—41no—
669C4—42no—
670C4—81no—
671C4—81no—
672C2—21no—
673C2—21no—
674C2—42no—
675C2—81no—
676C2—82no—
677C2—161no—
678C2C31×#300611yescharacteristic
679C2C31×#301811yescharacteristic
680C2C31×#302611yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
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
1. C31×#3460C2semidirect
2. C31×#3194C2semidirect
3. C31×#2543C2semidirect
4. C31×#1612C2semidirect
5. C31×#1611C2semidirect
6. C31×#1610C2semidirect
7. C31×#1600C2semidirect
8. C31×#1599C2semidirect
9. C31×#1598C2semidirect
10. C31×#1504C2semidirect
11. C31×#1465C2semidirect
53. C2×#41×C31D4semidirect
79. C31×C22⋊Q8D4semidirect
91. C31×#121D4semidirect
108. C31×#117D4semidirect
224. #18330C31direct (N × H)
295. #3460C2×C31semidirect
296. #3194C2×C31semidirect
297. #2543C2×C31semidirect
298. #1612C2×C31semidirect
299. #1611C2×C31semidirect
300. #1610C2×C31semidirect
301. #1600C2×C31semidirect
302. #1599C2×C31semidirect
303. #1598C2×C31semidirect
304. #1504C2×C31semidirect
305. #1465C2×C31semidirect
393. C2×#41D4×C31semidirect
419. C22⋊Q8D4×C31semidirect
431. #121D4×C31semidirect
448. #117D4×C31semidirect
475. C31#18330direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#34602
C31×#31942
C31×#25432
C31×#16122
C31×#16112
C31×#16102
C31×#16002
C31×#15992
C31×#15982
C31×#15042
C31×#14652
#1833031