Finite Group Databasebeta

← C31×#9042 #21465801

Subgroups — C31×#9042 #21465801

Order 7936.

966 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#450C211yesmaximal, characteristic
2C31×#2329C211yesmaximal, characteristic, complemented
3C31×#1786C211yesmaximal, characteristic
4C2×C31×#328C211yesmaximal, characteristic, complemented
5C31×#1438C211yesmaximal, characteristic
6C31×#1294C211yesmaximal, characteristic, complemented
7C31×#1280C211yesmaximal, characteristic, complemented
8C23×D4×C31C2211yescharacteristic
9C22×SD16×C31—21no—
10C2×C31×C22⋊Q8C2211yescharacteristic
11C22×C22⋊C4×C31—21no—
12C22×C22⋊C4×C31—21no—
13C31×#450—22no—
14C31×#450C414yescomplemented
15C2×C31×#102—21no—
16C2×C31×#102—21no—
17C2×C31×#102—21no—
18C2×C31×C22⋊C8C2211yescharacteristic
19C2×C31×C22⋊C8C2211yescharacteristic
20C31×#391—21no—
21C31×#389—21no—
22C31×#387—21no—
23C31×#387—21no—
24C31×#385—21no—
25C31×#380C2211yescharacteristic
26C4×C22⋊C4×C31—21no—
27C2×C31×#95C2211yescharacteristic
28C2×C31×#95C2211yescharacteristic
29C31×#328—22no—
30C25×C31—21no—
31C22×Q8×C31D411yescharacteristic
32C22×D4×C31—21no—
33C22×D4×C31—21no—
34C22×D4×C31—21no—
35C22×D4×C31—21no—
36C22×D4×C31—21no—
37C22×D4×C31—42no—
38C22×D4×C31—42no—
39C22×D4×C31C2×C412yes—
40C23×C4×C31—21no—
41C23×C4×C31D411yescharacteristic
42C23×C4×C31D411yescharacteristic
43C23×C4×C31C2311yescharacteristic
44C2×SD16×C31—24no—
45C2×SD16×C31—42no—
46C2×SD16×C31—42no—
47C22×C8×C31—21no—
48C22×C8×C31—21no—
49C31×C22⋊Q8—21no—
50C31×C22⋊Q8—21no—
51C31×C22⋊Q8C2×C412yes—
52C31×C22⋊Q8D412yes—
53C2×C4⋊C4×C31—21no—
54C2×C4⋊C4×C31—21no—
55C2×C4⋊C4×C31—21no—
56C2×C4⋊C4×C31D411yescharacteristic
57C2×C4⋊C4×C31D411yescharacteristic
58C2×C4⋊C4×C31D411yescharacteristic
59C2×C22⋊C4×C31—21no—
60C2×C22⋊C4×C31—21no—
61C2×C22⋊C4×C31—42no—
62C2×C22⋊C4×C31—42no—
63C2×C22⋊C4×C31—42no—
64C2×C22⋊C4×C31—42no—
65C2×C22⋊C4×C31—42no—
66C2×C22⋊C4×C31—42no—
67C2×C42×C31—21no—
68C2×C42×C31—21no—
69C31×#102—24no—
70C31×#102—42no—
71C31×#102—42no—
72C31×C22⋊C8—21no—
73C31×C22⋊C8—21no—
74C31×C22⋊C8—21no—
75C31×C22⋊C8C2×C412yes—
76C31×#95—21no—
77C31×#95—21no—
78C31×#95—21no—
79C31×#95—21no—
80C31×#95—21no—
81C31×#95—21no—
82C31×#95—21no—
83C31×#95—21no—
84C31×#95—21no—
85C24×C31—22no—
86C24×C31—22no—
87C24×C31—22no—
88C24×C31—41no—
89C24×C31—41no—
90C24×C31—41no—
91C24×C31—41no—
92C24×C31—41no—
93C24×C31—41no—
94C24×C31—41no—
95C24×C31—41no—
96C24×C31—41no—
97C24×C31—41no—
98C24×C31—41no—
99C24×C31—41no—
100C24×C31C2×D411yescharacteristic
101C2×Q8×C31—21no—
102C2×Q8×C31—41no—
103C2×Q8×C31—41no—
104C2×Q8×C31C22⋊C412yescomplemented
105C2×D4×C31—22no—
106C2×D4×C31—22no—
107C2×D4×C31—22no—
108C2×D4×C31—22no—
109C2×D4×C31—22no—
110C2×D4×C31—22no—
111C2×D4×C31—41no—
112C2×D4×C31—41no—
113C2×D4×C31—41no—
114C2×D4×C31—41no—
115C2×D4×C31—44no—
116C2×D4×C31—44no—
117C2×D4×C31—44no—
118C2×D4×C31—44no—
119C2×D4×C31—41no—
120C2×D4×C31—41no—
121C22×C4×C31—21no—
122C22×C4×C31—21no—
123C22×C4×C31—21no—
124C22×C4×C31—21no—
125C22×C4×C31—22no—
126C22×C4×C31—21no—
127C22×C4×C31—21no—
128C22×C4×C31—21no—
129C22×C4×C31—21no—
130C22×C4×C31—21no—
131C22×C4×C31—41no—
132C22×C4×C31—41no—
133C22×C4×C31—41no—
134C22×C4×C31—41no—
135C22×C4×C31—41no—
136C22×C4×C31—41no—
137C22×C4×C31—41no—
138C22×C4×C31—41no—
139C22×C4×C31—41no—
140C22×C4×C31—41no—
141C22×C4×C31—41no—
142C22×C4×C31—41no—
143C22×C4×C31SD1612yescomplemented
144C22×C4×C31C22×C411yescharacteristic
145C22×C4×C31C2×D411yescharacteristic
146C22×C4×C31C2×D411yescharacteristic
147C22×C4×C31C2×D411yescharacteristic
148C22×C4×C31#4111yescharacteristic
149C22×C4×C31#4111yescharacteristic
150SD16×C31—48no—
151C2×C8×C31—22no—
152C2×C8×C31—41no—
153C2×C8×C31—41no—
154C2×C8×C31—41no—
155C2×C8×C31—41no—
156C2×C8×C31—41no—
157C4⋊C4×C31—22no—
158C4⋊C4×C31—22no—
159C4⋊C4×C31—21no—
160C4⋊C4×C31—41no—
161C4⋊C4×C31—41no—
162C4⋊C4×C31—81no—
163C4⋊C4×C31—81no—
164C4⋊C4×C31C22⋊C412yescomplemented
165C22⋊C4×C31—22no—
166C22⋊C4×C31—22no—
167C22⋊C4×C31—48no—
168C22⋊C4×C31—48no—
169C22⋊C4×C31—41no—
170C22⋊C4×C31—41no—
171C42×C31—81no—
172C42×C31—81no—
173#9042C3111yesmaximal, characteristic, direct product factor, Sylow
174C23×C31—21no—
175C23×C31—21no—
176C23×C31—21no—
177C23×C31—21no—
178C23×C31—24no—
179C23×C31—42no—
180C23×C31—42no—
181C23×C31—42no—
182C23×C31—42no—
183C23×C31—42no—
184C23×C31—42no—
185C23×C31—42no—
186C23×C31—42no—
187C23×C31—42no—
188C23×C31—42no—
189C23×C31—42no—
190C23×C31—42no—
191C23×C31—42no—
192C23×C31—42no—
193C23×C31—42no—
194C23×C31—42no—
195C23×C31—42no—
196C23×C31—42no—
197C23×C31—81no—
198C23×C31—81no—
199C23×C31—81no—
200C23×C31—81no—
201C23×C31—81no—
202C23×C31—81no—
203C23×C31—81no—
204C23×C31—81no—
205C23×C31—81no—
206C23×C31—81no—
207C23×C31—81no—
208C23×C31—81no—
209C23×C31—81no—
210C23×C31—81no—
211C23×C31—81no—
212C23×C31—81no—
213C23×C31C4×D411yescharacteristic
214C23×C31#12011yescharacteristic
215C23×C31#12111yescharacteristic
216C23×C31C2×SD1611yescharacteristic
217C23×C31#13511yescharacteristic
218C23×C31AGL(1,ℤ/8)11yescharacteristic
219C23×C31#13711yescharacteristic
220Q8×C31—42no—
221Q8×C31—81no—
222D4×C31—24no—
223D4×C31—24no—
224D4×C31—42no—
225D4×C31—42no—
226D4×C31—42no—
227D4×C31—42no—
228D4×C31—42no—
229D4×C31—42no—
230C2×C4×C31—21no—
231C2×C4×C31—21no—
232C2×C4×C31—21no—
233C2×C4×C31—21no—
234C2×C4×C31—22no—
235C2×C4×C31—22no—
236C2×C4×C31—22no—
237C2×C4×C31—41no—
238C2×C4×C31—41no—
239C2×C4×C31—41no—
240C2×C4×C31—41no—
241C2×C4×C31—41no—
242C2×C4×C31—41no—
243C2×C4×C31—41no—
244C2×C4×C31—41no—
245C2×C4×C31—41no—
246C2×C4×C31—41no—
247C2×C4×C31—42no—
248C2×C4×C31—42no—
249C2×C4×C31—42no—
250C2×C4×C31—42no—
251C2×C4×C31—42no—
252C2×C4×C31—41no—
253C2×C4×C31—41no—
254C2×C4×C31—41no—
255C2×C4×C31—41no—
256C2×C4×C31—41no—
257C2×C4×C31—41no—
258C2×C4×C31—41no—
259C2×C4×C31—41no—
260C2×C4×C31—41no—
261C2×C4×C31—81no—
262C2×C4×C31—81no—
263C2×C4×C31—81no—
264C2×C4×C31—81no—
265C2×C4×C31—81no—
266C2×C4×C31—81no—
267C2×C4×C31C2×C22⋊C411yescharacteristic
268C2×C4×C31C4×D411yescharacteristic
269C2×C4×C31#12111yescharacteristic
270C2×C4×C31#12311yescharacteristic
271C8×C31—42no—
272C8×C31—81no—
273C2×#450C2×C3111yescharacteristic
274#2329C2×C3111yescharacteristic, complemented
275#1786C2×C3111yescharacteristic
276C2×#328C2×C3111yescharacteristic, complemented
277#1438C2×C3111yescharacteristic
278#1294C2×C3111yescharacteristic, complemented
279#1280C2×C3111yescharacteristic, complemented
280C22×C31—21no—
281C22×C31—21no—
282C22×C31—21no—
283C22×C31—21no—
284C22×C31—21no—
285C22×C31—21no—
286C22×C31—21no—
287C22×C31—21no—
288C22×C31—21no—
289C22×C31—21no—
290C22×C31—21no—
291C22×C31—21no—
292C22×C31—44no—
293C22×C31—44no—
294C22×C31—44no—
295C22×C31—82no—
296C22×C31—82no—
297C22×C31—82no—
298C22×C31—82no—
299C22×C31—82no—
300C22×C31—82no—
301C22×C31—82no—
302C22×C31—82no—
303C22×C31—82no—
304C22×C31—82no—
305C22×C31—82no—
306C22×C31—82no—
307C22×C31#38611yescharacteristic
308C22×C31C4×SD1611yescharacteristic
309C22×C31#44011yescharacteristic
310C22×C31#44811yescharacteristic
311C22×C31#44911yescharacteristic
312C22×C31#45311yescharacteristic
313C22×C31#45411yescharacteristic
314C22×C31B(4,𝔽2)11yescharacteristic
315C22×C31B(4,𝔽2)11yescharacteristic
316C22×C31#46011yescharacteristic
317C22×C31#46411yescharacteristic
318C4×C31—21no—
319C4×C31—21no—
320C4×C31—21no—
321C4×C31—21no—
322C4×C31—42no—
323C4×C31—42no—
324C4×C31—42no—
325C4×C31—82no—
326C4×C31—81no—
327C4×C31—81no—
328C4×C31—81no—
329C23×D4C22×C3111yescharacteristic
330C22×SD16—21no—
331C2×C22⋊Q8C22×C3111yescharacteristic
332C22×C22⋊C4—21no—
333C22×C22⋊C4—21no—
334#450—22no—
335#450C4×C3114yescomplemented
336C2×#102—21no—
337C2×#102—21no—
338C2×#102—21no—
339C2×C22⋊C8C22×C3111yescharacteristic
340C2×C22⋊C8C22×C3111yescharacteristic
341#391—21no—
342#389—21no—
343#387—21no—
344#387—21no—
345#385—21no—
346#380C22×C3111yescharacteristic
347C4×C22⋊C4—21no—
348C2×#95C22×C3111yescharacteristic
349C2×#95C22×C3111yescharacteristic
350#328—22no—
351C2×C31—21no—
352C2×C31—21no—
353C2×C31—21no—
354C2×C31—21no—
355C2×C31—84no—
356C2×C31#159611yescharacteristic
357C2×C31#159711yescharacteristic
358C2×C31#159911yescharacteristic
359C2×C31#160011yescharacteristic
360C2×C31#188011yescharacteristic
361C2×C31#188311yescharacteristic
362C2×C31#188911yescharacteristic
363C25—21no—
364C22×Q8D4×C3111yescharacteristic
365C22×D4—21no—
366C22×D4—21no—
367C22×D4—21no—
368C22×D4—21no—
369C22×D4—21no—
370C22×D4—42no—
371C22×D4—42no—
372C22×D4C2×C4×C3112yes—
373C23×C4—21no—
374C23×C4D4×C3111yescharacteristic
375C23×C4D4×C3111yescharacteristic
376C23×C4C23×C3111yescharacteristic
377C2×SD16—24no—
378C2×SD16—42no—
379C2×SD16—42no—
380C22×C8—21no—
381C22×C8—21no—
382C22⋊Q8—21no—
383C22⋊Q8—21no—
384C22⋊Q8C2×C4×C3112yes—
385C22⋊Q8D4×C3112yes—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4D4×C3111yescharacteristic
390C2×C4⋊C4D4×C3111yescharacteristic
391C2×C4⋊C4D4×C3111yescharacteristic
392C2×C22⋊C4—21no—
393C2×C22⋊C4—21no—
394C2×C22⋊C4—42no—
395C2×C22⋊C4—42no—
396C2×C22⋊C4—42no—
397C2×C22⋊C4—42no—
398C2×C22⋊C4—42no—
399C2×C22⋊C4—42no—
400C2×C42—21no—
401C2×C42—21no—
402#102—24no—
403#102—42no—
404#102—42no—
405C22⋊C8—21no—
406C22⋊C8—21no—
407C22⋊C8—21no—
408C22⋊C8C2×C4×C3112yes—
409#95—21no—
410#95—21no—
411#95—21no—
412#95—21no—
413#95—21no—
414#95—21no—
415#95—21no—
416#95—21no—
417#95—21no—
418C31#904211yescharacteristic, direct product factor, Sylow
419C24—22no—
420C24—22no—
421C24—22no—
422C24—41no—
423C24—41no—
424C24—41no—
425C24—41no—
426C24—41no—
427C24—41no—
428C24—41no—
429C24—41no—
430C24—41no—
431C24—41no—
432C24—41no—
433C24—41no—
434C24C2×D4×C3111yescharacteristic
435C2×Q8—21no—
436C2×Q8—41no—
437C2×Q8—41no—
438C2×Q8C22⋊C4×C3112yescomplemented
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—41no—
446C2×D4—41no—
447C2×D4—41no—
448C2×D4—41no—
449C2×D4—44no—
450C2×D4—44no—
451C2×D4—44no—
452C2×D4—44no—
453C2×D4—41no—
454C2×D4—41no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—41no—
466C22×C4—41no—
467C22×C4—41no—
468C22×C4—41no—
469C22×C4—41no—
470C22×C4—41no—
471C22×C4—41no—
472C22×C4—41no—
473C22×C4—41no—
474C22×C4—41no—
475C22×C4—41no—
476C22×C4—41no—
477C22×C4SD16×C3112yescomplemented
478C22×C4C22×C4×C3111yescharacteristic
479C22×C4C2×D4×C3111yescharacteristic
480C22×C4C2×D4×C3111yescharacteristic
481C22×C4C2×D4×C3111yescharacteristic
482C22×C4#41×C3111yescharacteristic
483C22×C4#41×C3111yescharacteristic
484SD16—48no—
485C2×C8—22no—
486C2×C8—41no—
487C2×C8—41no—
488C2×C8—41no—
489C2×C8—41no—
490C2×C8—41no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—21no—
494C4⋊C4—41no—
495C4⋊C4—41no—
496C4⋊C4—81no—
497C4⋊C4—81no—
498C4⋊C4C22⋊C4×C3112yescomplemented
499C22⋊C4—22no—
500C22⋊C4—22no—
501C22⋊C4—48no—
502C22⋊C4—48no—
503C22⋊C4—41no—
504C22⋊C4—41no—
505C42—81no—
506C42—81no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—24no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—81no—
531C23—81no—
532C23—81no—
533C23—81no—
534C23—81no—
535C23—81no—
536C23—81no—
537C23—81no—
538C23—81no—
539C23—81no—
540C23—81no—
541C23—81no—
542C23—81no—
543C23—81no—
544C23—81no—
545C23—81no—
546C23C4×D4×C3111yescharacteristic
547C23C31×#12011yescharacteristic
548C23C31×#12111yescharacteristic
549C23C2×SD16×C3111yescharacteristic
550C23C31×#13511yescharacteristic
551C23C31×AGL(1,ℤ/8)11yescharacteristic
552C23C31×#13711yescharacteristic
553Q8—42no—
554Q8—81no—
555D4—24no—
556D4—24no—
557D4—42no—
558D4—42no—
559D4—42no—
560D4—42no—
561D4—42no—
562D4—42no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—81no—
595C2×C4—81no—
596C2×C4—81no—
597C2×C4—81no—
598C2×C4—81no—
599C2×C4—81no—
600C2×C4C2×C22⋊C4×C3111yescharacteristic
601C2×C4C4×D4×C3111yescharacteristic
602C2×C4C31×#12111yescharacteristic
603C2×C4C31×#12311yescharacteristic
604C8—42no—
605C8—81no—
606C22—21no—
607C22—21no—
608C22—21no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—21no—
614C22—21no—
615C22—21no—
616C22—21no—
617C22—21no—
618C22—44no—
619C22—44no—
620C22—44no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22C31×#38611yescharacteristic
634C22C4×SD16×C3111yescharacteristic
635C22C31×#44011yescharacteristic
636C22C31×#44811yescharacteristic
637C22C31×#44911yescharacteristic
638C22C31×#45311yescharacteristic
639C22C31×#45411yescharacteristic
640C22C31×B(4,𝔽2)11yescharacteristic
641C22C31×B(4,𝔽2)11yescharacteristic
642C22C31×#46011yescharacteristic
643C22C31×#46411yescharacteristic
644C4—21no—
645C4—21no—
646C4—21no—
647C4—21no—
648C4—42no—
649C4—42no—
650C4—42no—
651C4—82no—
652C4—81no—
653C4—81no—
654C4—81no—
655C2—21no—
656C2—21no—
657C2—21no—
658C2—21no—
659C2—84no—
660C2C31×#159611yescharacteristic
661C2C31×#159711yescharacteristic
662C2C31×#159911yescharacteristic
663C2C31×#160011yescharacteristic
664C2C31×#188011yescharacteristic
665C2C31×#188311yescharacteristic
666C2C31×#188911yescharacteristic
subgroupG/H
center, socleC23×C31#120
commutatorC22×C4C22×C4×C31
Frattini, cosocleC23×C4C23×C31
prime radicalC23×D4×C31C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#2329C2semidirect
4. C2×C31×#328C2semidirect
6. C31×#1294C2semidirect
7. C31×#1280C2semidirect
14. C31×#450C4semidirect
104. C2×Q8×C31C22⋊C4semidirect
143. C22×C4×C31SD16semidirect
164. C4⋊C4×C31C22⋊C4semidirect
173. #9042C31direct (N × H)
274. #2329C2×C31semidirect
276. C2×#328C2×C31semidirect
278. #1294C2×C31semidirect
279. #1280C2×C31semidirect
335. #450C4×C31semidirect
418. C31#9042direct (N × H)
438. C2×Q8C22⋊C4×C31semidirect
477. C22×C4SD16×C31semidirect
498. C4⋊C4C22⋊C4×C31semidirect

Maximal subgroups

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

subgroupindex
C2×C31×#4502
C31×#23292
C31×#17862
C2×C31×#3282
C31×#14382
C31×#12942
C31×#12802
#904231