Finite Group Databasebeta

← C31×#18328 #21475087

Subgroups — C31×#18328 #21475087

Order 7936.

1192 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#538C211yesmaximal, characteristic, complemented
2C31×#3193C211yesmaximal, characteristic, complemented
3C31×#3192C211yesmaximal, characteristic, complemented
4C31×#2539C211yesmaximal, characteristic, complemented
5C31×#1618C214yesmaximal, complemented
6C31×#1600C214yesmaximal, complemented
7C31×#1508C211yesmaximal, characteristic, complemented
8C31×#1507C211yesmaximal, characteristic, complemented
9C31×#1462C211yesmaximal, characteristic, complemented
10C31×#538—22no—
11C31×#538—22no—
12C31×#538—22no—
13C31×#538C2214yes—
14C2×C31×#126—21no—
15C2×C31×#124—21no—
16C2×C31×#123—21no—
17C2×C31×#123—21no—
18C2×C31×C22⋊Q8C2211yescharacteristic
19C2×C31×#121—21no—
20C2×C31×#121C2211yescharacteristic
21C2×C31×#121C2211yescharacteristic
22C2×C31×#120—21no—
23C2×C31×#120—21no—
24C2×C4×D4×C31—21no—
25C22×C22⋊C4×C31C2211yescharacteristic
26C31×#484C2212yescomplemented
27C31×#481—21no—
28C31×#481C2212yescomplemented
29C31×#480—21no—
30C31×#469—22no—
31C31×#468—22no—
32C31×#450C2212yescomplemented
33C31×#448—21no—
34C31×#447—21no—
35C31×#447C2212yescomplemented
36C31×#407C2211yescharacteristic
37C31×#407C2211yescharacteristic
38C31×#399C2212yescomplemented
39C31×#392C2212yescomplemented
40C31×#390—22no—
41C4×C22⋊C4×C31C2211yescharacteristic
42C31×#328C2214yescomplemented
43C31×#327C2214yescomplemented
44C31×#323C2214yescomplemented
45C25×C31D411yescharacteristic
46C22×Q8×C31D411yescharacteristic, complemented
47C22×D4×C31—21no—
48C22×D4×C31—21no—
49C22×D4×C31—21no—
50C22×D4×C31—21no—
51C22×D4×C31—22no—
52C22×D4×C31D411yescharacteristic, complemented
53C22×D4×C31C2311yescharacteristic
54C23×C4×C31—21no—
55C23×C4×C31D411yescharacteristic
56C23×C4×C31C2311yescharacteristic
57C2×SD16×C31—22no—
58C2×D8×C31—22no—
59C2×M16×C31—21no—
60C2×M16×C31—21no—
61C31×#126—24no—
62C31×#126—24no—
63C31×#124—24no—
64C31×#124—24no—
65C31×#123—24no—
66C31×#123—24no—
67C31×#123—44no—
68C31×C22⋊Q8—21no—
69C31×C22⋊Q8—21no—
70C31×C22⋊Q8D412yescomplemented
71C31×C22⋊Q8C2312yes—
72C31×#121—24no—
73C31×#121—24no—
74C31×#121—21no—
75C31×#121—21no—
76C31×#121—21no—
77C31×#121—21no—
78C31×#121—21no—
79C31×#121—21no—
80C31×#121D412yescomplemented
81C31×#121C2312yes—
82C31×#120—24no—
83C31×#120—24no—
84C31×#120—44no—
85C4×D4×C31—24no—
86C4×D4×C31—24no—
87C2×C4⋊C4×C31—21no—
88C2×C4⋊C4×C31—21no—
89C2×C4⋊C4×C31—22no—
90C2×C4⋊C4×C31D411yescharacteristic, complemented
91C2×C4⋊C4×C31D411yescharacteristic, complemented
92C2×C4⋊C4×C31C2311yescharacteristic
93C2×C22⋊C4×C31—21no—
94C2×C22⋊C4×C31—21no—
95C2×C22⋊C4×C31—21no—
96C2×C22⋊C4×C31—21no—
97C2×C22⋊C4×C31—21no—
98C2×C22⋊C4×C31—21no—
99C2×C22⋊C4×C31—21no—
100C2×C22⋊C4×C31—21no—
101C2×C22⋊C4×C31—21no—
102C2×C22⋊C4×C31—21no—
103C2×C22⋊C4×C31—22no—
104C2×C22⋊C4×C31—21no—
105C2×C22⋊C4×C31—22no—
106C2×C22⋊C4×C31D412yes—
107C2×C22⋊C4×C31C2312yes—
108C2×C42×C31—21no—
109C2×C42×C31—21no—
110C31×#107—22no—
111C31×#106—22no—
112C31×#103—22no—
113C31×#102—22no—
114C31×#102—22no—
115C31×#102—22no—
116C31×C22⋊C8—21no—
117C31×C22⋊C8—21no—
118C31×C22⋊C8C2312yes—
119C31×C22⋊C8C2312yes—
120C31×#95C2312yes—
121C24×C31—21no—
122C24×C31—21no—
123C24×C31—21no—
124C24×C31—21no—
125C24×C31—42no—
126C24×C31—42no—
127C24×C31—81no—
128C24×C31C2×D411yescharacteristic
129C24×C31C2×D412yes—
130C2×Q8×C31—21no—
131C2×Q8×C31—81no—
132C2×Q8×C31C2×D412yes—
133C2×D4×C31—22no—
134C2×D4×C31—22no—
135C2×D4×C31—24no—
136C2×D4×C31—24no—
137C2×D4×C31—21no—
138C2×D4×C31—21no—
139C2×D4×C31—21no—
140C2×D4×C31—44no—
141C2×D4×C31—44no—
142C2×D4×C31—42no—
143C2×D4×C31—42no—
144C2×D4×C31—44no—
145C2×D4×C31—41no—
146C2×D4×C31—41no—
147C2×D4×C31—81no—
148C2×D4×C31—81no—
149C2×D4×C31—82no—
150C2×D4×C31—82no—
151C2×D4×C31—82no—
152C2×D4×C31C2×D412yes—
153C22×C4×C31—21no—
154C22×C4×C31—21no—
155C22×C4×C31—21no—
156C22×C4×C31—21no—
157C22×C4×C31—22no—
158C22×C4×C31—24no—
159C22×C4×C31—21no—
160C22×C4×C31—21no—
161C22×C4×C31—42no—
162C22×C4×C31—81no—
163C22×C4×C31—82no—
164C22×C4×C31C2×D412yes—
165C22×C4×C31C2×D411yescharacteristic
166C22×C4×C31C2×D411yescharacteristic
167C22×C4×C31C2×D411yescharacteristic
168C22×C4×C31C2×D411yescharacteristic
169C22×C4×C31C2×D411yescharacteristic
170C22×C4×C31C2411yescharacteristic
171SD16×C31—82no—
172D8×C31—82no—
173M16×C31—81no—
174M16×C31—81no—
175C2×C8×C31—22no—
176C2×C8×C31—22no—
177C4⋊C4×C31—24no—
178C4⋊C4×C31—22no—
179C4⋊C4×C31—21no—
180C4⋊C4×C31—21no—
181C4⋊C4×C31—22no—
182C4⋊C4×C31—21no—
183C4⋊C4×C31—21no—
184C4⋊C4×C31—44no—
185C4⋊C4×C31C2×D412yes—
186C4⋊C4×C31C2×D412yes—
187C22⋊C4×C31—24no—
188C22⋊C4×C31—24no—
189C22⋊C4×C31—24no—
190C22⋊C4×C31—24no—
191C22⋊C4×C31—22no—
192C22⋊C4×C31—22no—
193C22⋊C4×C31—22no—
194C22⋊C4×C31—22no—
195C22⋊C4×C31—22no—
196C22⋊C4×C31—22no—
197C22⋊C4×C31—22no—
198C22⋊C4×C31—22no—
199C22⋊C4×C31—44no—
200C22⋊C4×C31—44no—
201C22⋊C4×C31—41no—
202C22⋊C4×C31—41no—
203C42×C31—24no—
204C42×C31—81no—
205#18328C3111yesmaximal, characteristic, direct product factor, Sylow
206C23×C31—22no—
207C23×C31—22no—
208C23×C31—22no—
209C23×C31—22no—
210C23×C31—22no—
211C23×C31—22no—
212C23×C31—41no—
213C23×C31—42no—
214C23×C31—42no—
215C23×C31—42no—
216C23×C31—42no—
217C23×C31—42no—
218C23×C31—42no—
219C23×C31—42no—
220C23×C31—41no—
221C23×C31—41no—
222C23×C31—42no—
223C23×C31—42no—
224C23×C31—42no—
225C23×C31—41no—
226C23×C31—42no—
227C23×C31—81no—
228C23×C31—81no—
229C23×C31—88no—
230C23×C31#12011yescharacteristic
231C23×C31#12011yescharacteristic
232C23×C31C22×D411yescharacteristic
233Q8×C31—42no—
234Q8×C31—81no—
235D4×C31—42no—
236D4×C31—42no—
237D4×C31—42no—
238D4×C31—44no—
239D4×C31—44no—
240D4×C31—81no—
241D4×C31—84no—
242D4×C31—84no—
243D4×C31—88no—
244D4×C31—82no—
245D4×C31—82no—
246C2×C4×C31—24no—
247C2×C4×C31—22no—
248C2×C4×C31—22no—
249C2×C4×C31—22no—
250C2×C4×C31—22no—
251C2×C4×C31—41no—
252C2×C4×C31—41no—
253C2×C4×C31—41no—
254C2×C4×C31—41no—
255C2×C4×C31—41no—
256C2×C4×C31—41no—
257C2×C4×C31—42no—
258C2×C4×C31—42no—
259C2×C4×C31—42no—
260C2×C4×C31—42no—
261C2×C4×C31—42no—
262C2×C4×C31—42no—
263C2×C4×C31—42no—
264C2×C4×C31—41no—
265C2×C4×C31—41no—
266C2×C4×C31—41no—
267C2×C4×C31—81no—
268C2×C4×C31—84no—
269C2×C4×C31#12011yescharacteristic
270C2×C4×C31#12011yescharacteristic
271C2×C4×C31C22×D411yescharacteristic
272C2×C4×C31C22×D411yescharacteristic
273C8×C31—42no—
274C8×C31—42no—
275C2×#538C2×C3111yescharacteristic, complemented
276#3193C2×C3111yescharacteristic, complemented
277#3192C2×C3111yescharacteristic, complemented
278#2539C2×C3111yescharacteristic, complemented
279#1618C2×C3114yescomplemented
280#1600C2×C3114yescomplemented
281#1508C2×C3111yescharacteristic, complemented
282#1507C2×C3111yescharacteristic, complemented
283#1462C2×C3111yescharacteristic, complemented
284C22×C31—21no—
285C22×C31—21no—
286C22×C31—21no—
287C22×C31—21no—
288C22×C31—21no—
289C22×C31—21no—
290C22×C31—21no—
291C22×C31—44no—
292C22×C31—44no—
293C22×C31—42no—
294C22×C31—42no—
295C22×C31—42no—
296C22×C31—42no—
297C22×C31—42no—
298C22×C31—42no—
299C22×C31—82no—
300C22×C31—84no—
301C22×C31—84no—
302C22×C31—82no—
303C22×C31—82no—
304C22×C31—84no—
305C22×C31—81no—
306C22×C31—81no—
307C22×C31—81no—
308C22×C31#45312yes—
309C22×C31C2×#12011yescharacteristic
310C22×C31#57512yes—
311C4×C31—21no—
312C4×C31—21no—
313C4×C31—21no—
314C4×C31—21no—
315C4×C31—44no—
316C4×C31—42no—
317C4×C31—42no—
318C4×C31—42no—
319C4×C31—42no—
320C4×C31—81no—
321#538—22no—
322#538—22no—
323#538—22no—
324#538C22×C3114yes—
325C2×#126—21no—
326C2×#124—21no—
327C2×#123—21no—
328C2×#123—21no—
329C2×C22⋊Q8C22×C3111yescharacteristic
330C2×#121—21no—
331C2×#121C22×C3111yescharacteristic
332C2×#121C22×C3111yescharacteristic
333C2×#120—21no—
334C2×#120—21no—
335C2×C4×D4—21no—
336C22×C22⋊C4C22×C3111yescharacteristic
337#484C22×C3112yescomplemented
338#481—21no—
339#481C22×C3112yescomplemented
340#480—21no—
341#469—22no—
342#468—22no—
343#450C22×C3112yescomplemented
344#448—21no—
345#447—21no—
346#447C22×C3112yescomplemented
347#407C22×C3111yescharacteristic
348#407C22×C3111yescharacteristic
349#399C22×C3112yescomplemented
350#392C22×C3112yescomplemented
351#390—22no—
352C4×C22⋊C4C22×C3111yescharacteristic
353#328C22×C3114yescomplemented
354#327C22×C3114yescomplemented
355#323C22×C3114yescomplemented
356C2×C31—22no—
357C2×C31—22no—
358C2×C31—41no—
359C2×C31—82no—
360C2×C31—82no—
361C2×C31—161no—
362C2×C31#300511yescharacteristic
363C2×C31C2×#45311yescharacteristic
364C2×C31#302611yescharacteristic
365C25D4×C3111yescharacteristic
366C22×Q8D4×C3111yescharacteristic, complemented
367C22×D4—21no—
368C22×D4—21no—
369C22×D4—21no—
370C22×D4—21no—
371C22×D4—22no—
372C22×D4D4×C3111yescharacteristic, complemented
373C22×D4C23×C3111yescharacteristic
374C23×C4—21no—
375C23×C4D4×C3111yescharacteristic
376C23×C4C23×C3111yescharacteristic
377C2×SD16—22no—
378C2×D8—22no—
379C2×M16—21no—
380C2×M16—21no—
381#126—24no—
382#126—24no—
383#124—24no—
384#124—24no—
385#123—24no—
386#123—24no—
387#123—44no—
388C22⋊Q8—21no—
389C22⋊Q8—21no—
390C22⋊Q8D4×C3112yescomplemented
391C22⋊Q8C23×C3112yes—
392#121—24no—
393#121—24no—
394#121—21no—
395#121—21no—
396#121—21no—
397#121—21no—
398#121—21no—
399#121—21no—
400#121D4×C3112yescomplemented
401#121C23×C3112yes—
402#120—24no—
403#120—24no—
404#120—44no—
405C4×D4—24no—
406C4×D4—24no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—22no—
410C2×C4⋊C4D4×C3111yescharacteristic, complemented
411C2×C4⋊C4D4×C3111yescharacteristic, complemented
412C2×C4⋊C4C23×C3111yescharacteristic
413C2×C22⋊C4—21no—
414C2×C22⋊C4—21no—
415C2×C22⋊C4—21no—
416C2×C22⋊C4—21no—
417C2×C22⋊C4—21no—
418C2×C22⋊C4—21no—
419C2×C22⋊C4—21no—
420C2×C22⋊C4—21no—
421C2×C22⋊C4—21no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—22no—
424C2×C22⋊C4—21no—
425C2×C22⋊C4—22no—
426C2×C22⋊C4D4×C3112yes—
427C2×C22⋊C4C23×C3112yes—
428C2×C42—21no—
429C2×C42—21no—
430#107—22no—
431#106—22no—
432#103—22no—
433#102—22no—
434#102—22no—
435#102—22no—
436C22⋊C8—21no—
437C22⋊C8—21no—
438C22⋊C8C23×C3112yes—
439C22⋊C8C23×C3112yes—
440#95C23×C3112yes—
441C31#1832811yescharacteristic, direct product factor, Sylow
442C24—21no—
443C24—21no—
444C24—21no—
445C24—21no—
446C24—42no—
447C24—42no—
448C24—81no—
449C24C2×D4×C3111yescharacteristic
450C24C2×D4×C3112yes—
451C2×Q8—21no—
452C2×Q8—81no—
453C2×Q8C2×D4×C3112yes—
454C2×D4—22no—
455C2×D4—22no—
456C2×D4—24no—
457C2×D4—24no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—44no—
462C2×D4—44no—
463C2×D4—42no—
464C2×D4—42no—
465C2×D4—44no—
466C2×D4—41no—
467C2×D4—41no—
468C2×D4—81no—
469C2×D4—81no—
470C2×D4—82no—
471C2×D4—82no—
472C2×D4—82no—
473C2×D4C2×D4×C3112yes—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—22no—
479C22×C4—24no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—42no—
483C22×C4—81no—
484C22×C4—82no—
485C22×C4C2×D4×C3112yes—
486C22×C4C2×D4×C3111yescharacteristic
487C22×C4C2×D4×C3111yescharacteristic
488C22×C4C2×D4×C3111yescharacteristic
489C22×C4C2×D4×C3111yescharacteristic
490C22×C4C2×D4×C3111yescharacteristic
491C22×C4C24×C3111yescharacteristic
492SD16—82no—
493D8—82no—
494M16—81no—
495M16—81no—
496C2×C8—22no—
497C2×C8—22no—
498C4⋊C4—24no—
499C4⋊C4—22no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—22no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—44no—
506C4⋊C4C2×D4×C3112yes—
507C4⋊C4C2×D4×C3112yes—
508C22⋊C4—24no—
509C22⋊C4—24no—
510C22⋊C4—24no—
511C22⋊C4—24no—
512C22⋊C4—22no—
513C22⋊C4—22no—
514C22⋊C4—22no—
515C22⋊C4—22no—
516C22⋊C4—22no—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—22no—
520C22⋊C4—44no—
521C22⋊C4—44no—
522C22⋊C4—41no—
523C22⋊C4—41no—
524C42—24no—
525C42—81no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—41no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—41no—
541C23—41no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—41no—
546C23—42no—
547C23—81no—
548C23—81no—
549C23—88no—
550C23C31×#12011yescharacteristic
551C23C31×#12011yescharacteristic
552C23C22×D4×C3111yescharacteristic
553Q8—42no—
554Q8—81no—
555D4—42no—
556D4—42no—
557D4—42no—
558D4—44no—
559D4—44no—
560D4—81no—
561D4—84no—
562D4—84no—
563D4—88no—
564D4—82no—
565D4—82no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—81no—
588C2×C4—84no—
589C2×C4C31×#12011yescharacteristic
590C2×C4C31×#12011yescharacteristic
591C2×C4C22×D4×C3111yescharacteristic
592C2×C4C22×D4×C3111yescharacteristic
593C8—42no—
594C8—42no—
595C22—21no—
596C22—21no—
597C22—21no—
598C22—21no—
599C22—21no—
600C22—21no—
601C22—21no—
602C22—44no—
603C22—44no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—82no—
611C22—84no—
612C22—84no—
613C22—82no—
614C22—82no—
615C22—84no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22C31×#45312yes—
620C22C2×C31×#12011yescharacteristic
621C22C31×#57512yes—
622C4—21no—
623C4—21no—
624C4—21no—
625C4—21no—
626C4—44no—
627C4—42no—
628C4—42no—
629C4—42no—
630C4—42no—
631C4—81no—
632C2—22no—
633C2—22no—
634C2—41no—
635C2—82no—
636C2—82no—
637C2—161no—
638C2C31×#300511yescharacteristic
639C2C2×C31×#45311yescharacteristic
640C2C31×#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. C2×C31×#538C2semidirect
2. C31×#3193C2semidirect
3. C31×#3192C2semidirect
4. C31×#2539C2semidirect
5. C31×#1618C2semidirect
6. C31×#1600C2semidirect
7. C31×#1508C2semidirect
8. C31×#1507C2semidirect
9. C31×#1462C2semidirect
26. C31×#484C22semidirect
28. C31×#481C22semidirect
32. C31×#450C22semidirect
35. C31×#447C22semidirect
38. C31×#399C22semidirect
39. C31×#392C22semidirect
42. C31×#328C22semidirect
43. C31×#327C22semidirect
44. C31×#323C22semidirect
46. C22×Q8×C31D4semidirect
52. C22×D4×C31D4semidirect
70. C31×C22⋊Q8D4semidirect
80. C31×#121D4semidirect
90. C2×C4⋊C4×C31D4semidirect
91. C2×C4⋊C4×C31D4semidirect
205. #18328C31direct (N × H)
275. C2×#538C2×C31semidirect
276. #3193C2×C31semidirect
277. #3192C2×C31semidirect
278. #2539C2×C31semidirect
279. #1618C2×C31semidirect
280. #1600C2×C31semidirect
281. #1508C2×C31semidirect
282. #1507C2×C31semidirect
283. #1462C2×C31semidirect
337. #484C22×C31semidirect
339. #481C22×C31semidirect
343. #450C22×C31semidirect
346. #447C22×C31semidirect
349. #399C22×C31semidirect
350. #392C22×C31semidirect
353. #328C22×C31semidirect
354. #327C22×C31semidirect
355. #323C22×C31semidirect
366. C22×Q8D4×C31semidirect
372. C22×D4D4×C31semidirect
390. C22⋊Q8D4×C31semidirect
400. #121D4×C31semidirect
410. C2×C4⋊C4D4×C31semidirect
411. C2×C4⋊C4D4×C31semidirect
441. C31#18328direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5382
C31×#31932
C31×#31922
C31×#25392
C31×#16182
C31×#16002
C31×#15082
C31×#15072
C31×#14622
#1832831