Finite Group Databasebeta

← C31×#9041 #21465800

Subgroups — C31×#9041 #21465800

Order 7936.

1050 conjugacy classes of subgroups.

The subgroup lattice has 720 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#447C211yesmaximal, characteristic
2C31×#2325C211yesmaximal, characteristic, complemented
3C31×#1786C211yesmaximal, characteristic, complemented
4C2×C31×#327C211yesmaximal, characteristic, complemented
5C31×#1438C211yesmaximal, characteristic, complemented
6C31×#1294C211yesmaximal, characteristic, complemented
7C31×#1280C211yesmaximal, characteristic, complemented
8C23×D4×C31C2211yescharacteristic
9C22×D8×C31—21no—
10C2×C31×#121C2211yescharacteristic
11C2×C4×D4×C31—21no—
12C22×C22⋊C4×C31—21no—
13C22×C22⋊C4×C31—21no—
14C31×#447—22no—
15C31×#447C414yescomplemented
16C2×C31×#102—21no—
17C2×C31×#102—21no—
18C2×C31×#102—21no—
19C2×C31×C22⋊C8C2211yescharacteristic
20C2×C31×C22⋊C8C2211yescharacteristic
21C31×#388—21no—
22C31×#387—21no—
23C31×#386—21no—
24C31×#386—21no—
25C31×#385—21no—
26C31×#380C2211yescharacteristic
27C2×C31×#95C2211yescharacteristic
28C2×C31×#95C2211yescharacteristic
29C31×#327—22no—
30C25×C31—21no—
31C22×D4×C31—21no—
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×C412yescomplemented
40C22×D4×C31D411yescharacteristic
41C23×C4×C31—21no—
42C23×C4×C31—21no—
43C23×C4×C31D411yescharacteristic
44C23×C4×C31D411yescharacteristic
45C23×C4×C31C2311yescharacteristic
46C2×D8×C31—24no—
47C2×D8×C31—42no—
48C2×D8×C31—42no—
49C22×C8×C31—21no—
50C22×C8×C31—21no—
51C31×#121—21no—
52C31×#121—21no—
53C31×#121C2×C412yes—
54C31×#121D412yes—
55C4×D4×C31—42no—
56C4×D4×C31—42no—
57C2×C4⋊C4×C31—21no—
58C2×C4⋊C4×C31D411yescharacteristic
59C2×C4⋊C4×C31D411yescharacteristic
60C2×C4⋊C4×C31D411yescharacteristic
61C2×C22⋊C4×C31—21no—
62C2×C22⋊C4×C31—21no—
63C2×C22⋊C4×C31—21no—
64C2×C22⋊C4×C31—21no—
65C2×C22⋊C4×C31—21no—
66C2×C22⋊C4×C31—42no—
67C2×C22⋊C4×C31—42no—
68C2×C22⋊C4×C31—42no—
69C2×C22⋊C4×C31—42no—
70C2×C22⋊C4×C31—42no—
71C2×C22⋊C4×C31—42no—
72C2×C42×C31—21no—
73C31×#102—24no—
74C31×#102—42no—
75C31×#102—42no—
76C31×C22⋊C8—21no—
77C31×C22⋊C8—21no—
78C31×C22⋊C8—21no—
79C31×C22⋊C8C2×C412yescomplemented
80C31×#95—21no—
81C31×#95—21no—
82C31×#95—21no—
83C31×#95—21no—
84C31×#95—21no—
85C31×#95—21no—
86C31×#95—21no—
87C24×C31—22no—
88C24×C31—22no—
89C24×C31—22no—
90C24×C31—21no—
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×C31—41no—
101C24×C31—41no—
102C24×C31—41no—
103C24×C31C2×D411yescharacteristic
104C2×D4×C31—22no—
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—22no—
112C2×D4×C31—21no—
113C2×D4×C31—41no—
114C2×D4×C31—41no—
115C2×D4×C31—41no—
116C2×D4×C31—41no—
117C2×D4×C31—41no—
118C2×D4×C31—41no—
119C2×D4×C31—44no—
120C2×D4×C31—44no—
121C2×D4×C31—44no—
122C2×D4×C31—44no—
123C2×D4×C31—41no—
124C2×D4×C31—41no—
125C2×D4×C31—41no—
126C2×D4×C31—41no—
127C2×D4×C31—41no—
128C2×D4×C31—41no—
129C2×D4×C31C22⋊C412yescomplemented
130C22×C4×C31—21no—
131C22×C4×C31—21no—
132C22×C4×C31—21no—
133C22×C4×C31—22no—
134C22×C4×C31—21no—
135C22×C4×C31—21no—
136C22×C4×C31—21no—
137C22×C4×C31—21no—
138C22×C4×C31—21no—
139C22×C4×C31—41no—
140C22×C4×C31—41no—
141C22×C4×C31—42no—
142C22×C4×C31—41no—
143C22×C4×C31—41no—
144C22×C4×C31—41no—
145C22×C4×C31—41no—
146C22×C4×C31—41no—
147C22×C4×C31—41no—
148C22×C4×C31—41no—
149C22×C4×C31—41no—
150C22×C4×C31—41no—
151C22×C4×C31—41no—
152C22×C4×C31—81no—
153C22×C4×C31—81no—
154C22×C4×C31D812yescomplemented
155C22×C4×C31C22×C411yescharacteristic
156C22×C4×C31C2×D411yescharacteristic
157C22×C4×C31C2×D411yescharacteristic
158C22×C4×C31C2×D411yescharacteristic
159C22×C4×C31#4111yescharacteristic
160C22×C4×C31#4111yescharacteristic
161D8×C31—48no—
162C2×C8×C31—22no—
163C2×C8×C31—41no—
164C2×C8×C31—41no—
165C2×C8×C31—41no—
166C2×C8×C31—41no—
167C2×C8×C31—41no—
168C4⋊C4×C31—21no—
169C4⋊C4×C31—42no—
170C4⋊C4×C31—41no—
171C4⋊C4×C31—41no—
172C4⋊C4×C31C22⋊C412yescomplemented
173C22⋊C4×C31—22no—
174C22⋊C4×C31—22no—
175C22⋊C4×C31—42no—
176C22⋊C4×C31—48no—
177C22⋊C4×C31—48no—
178C22⋊C4×C31—41no—
179C22⋊C4×C31—41no—
180C22⋊C4×C31—81no—
181C22⋊C4×C31—81no—
182C42×C31—42no—
183#9041C3111yesmaximal, characteristic, direct product factor, Sylow
184C23×C31—21no—
185C23×C31—21no—
186C23×C31—21no—
187C23×C31—21no—
188C23×C31—24no—
189C23×C31—22no—
190C23×C31—42no—
191C23×C31—42no—
192C23×C31—42no—
193C23×C31—42no—
194C23×C31—42no—
195C23×C31—42no—
196C23×C31—42no—
197C23×C31—42no—
198C23×C31—42no—
199C23×C31—42no—
200C23×C31—42no—
201C23×C31—42no—
202C23×C31—42no—
203C23×C31—42no—
204C23×C31—42no—
205C23×C31—42no—
206C23×C31—41no—
207C23×C31—41no—
208C23×C31—41no—
209C23×C31—41no—
210C23×C31—41no—
211C23×C31—41no—
212C23×C31—42no—
213C23×C31—42no—
214C23×C31—81no—
215C23×C31—81no—
216C23×C31—81no—
217C23×C31—81no—
218C23×C31—81no—
219C23×C31—81no—
220C23×C31—81no—
221C23×C31—81no—
222C23×C31—81no—
223C23×C31—81no—
224C23×C31—81no—
225C23×C31—81no—
226C23×C31—81no—
227C23×C31—81no—
228C23×C31—81no—
229C23×C31—81no—
230C23×C31C4×D411yescharacteristic
231C23×C31#12011yescharacteristic
232C23×C31#12111yescharacteristic
233C23×C31C2×D811yescharacteristic
234C23×C31#13511yescharacteristic
235C23×C31AGL(1,ℤ/8)11yescharacteristic
236C23×C31AGL(1,ℤ/8)11yescharacteristic
237D4×C31—24no—
238D4×C31—24no—
239D4×C31—42no—
240D4×C31—42no—
241D4×C31—42no—
242D4×C31—42no—
243D4×C31—42no—
244D4×C31—42no—
245D4×C31—42no—
246D4×C31—42no—
247D4×C31—42no—
248D4×C31—42no—
249D4×C31—42no—
250D4×C31—81no—
251C2×C4×C31—21no—
252C2×C4×C31—21no—
253C2×C4×C31—21no—
254C2×C4×C31—21no—
255C2×C4×C31—22no—
256C2×C4×C31—22no—
257C2×C4×C31—41no—
258C2×C4×C31—41no—
259C2×C4×C31—41no—
260C2×C4×C31—41no—
261C2×C4×C31—41no—
262C2×C4×C31—41no—
263C2×C4×C31—41no—
264C2×C4×C31—41no—
265C2×C4×C31—42no—
266C2×C4×C31—42no—
267C2×C4×C31—42no—
268C2×C4×C31—42no—
269C2×C4×C31—42no—
270C2×C4×C31—41no—
271C2×C4×C31—41no—
272C2×C4×C31—41no—
273C2×C4×C31—41no—
274C2×C4×C31—41no—
275C2×C4×C31—41no—
276C2×C4×C31—41no—
277C2×C4×C31—41no—
278C2×C4×C31—41no—
279C2×C4×C31—81no—
280C2×C4×C31—84no—
281C2×C4×C31—81no—
282C2×C4×C31—81no—
283C2×C4×C31—81no—
284C2×C4×C31—81no—
285C2×C4×C31—81no—
286C2×C4×C31C2×C22⋊C411yescharacteristic
287C2×C4×C31C4×D411yescharacteristic
288C2×C4×C31#12111yescharacteristic
289C2×C4×C31#12311yescharacteristic
290C8×C31—42no—
291C8×C31—81no—
292C2×#447C2×C3111yescharacteristic
293#2325C2×C3111yescharacteristic, complemented
294#1786C2×C3111yescharacteristic, complemented
295C2×#327C2×C3111yescharacteristic, complemented
296#1438C2×C3111yescharacteristic, complemented
297#1294C2×C3111yescharacteristic, complemented
298#1280C2×C3111yescharacteristic, complemented
299C22×C31—21no—
300C22×C31—21no—
301C22×C31—21no—
302C22×C31—21no—
303C22×C31—21no—
304C22×C31—21no—
305C22×C31—21no—
306C22×C31—21no—
307C22×C31—21no—
308C22×C31—21no—
309C22×C31—21no—
310C22×C31—21no—
311C22×C31—44no—
312C22×C31—44no—
313C22×C31—42no—
314C22×C31—42no—
315C22×C31—44no—
316C22×C31—42no—
317C22×C31—82no—
318C22×C31—82no—
319C22×C31—82no—
320C22×C31—82no—
321C22×C31—82no—
322C22×C31—82no—
323C22×C31—81no—
324C22×C31—82no—
325C22×C31—81no—
326C22×C31—82no—
327C22×C31—81no—
328C22×C31—82no—
329C22×C31—81no—
330C22×C31—82no—
331C22×C31—82no—
332C22×C31—82no—
333C22×C31#38611yescharacteristic
334C22×C31C4×D811yescharacteristic
335C22×C31#44211yescharacteristic
336C22×C31#44711yescharacteristic
337C22×C31#44911yescharacteristic
338C22×C31#45311yescharacteristic
339C22×C31#45411yescharacteristic
340C22×C31B(4,𝔽2)11yescharacteristic
341C22×C31B(4,𝔽2)11yescharacteristic
342C22×C31#45911yescharacteristic
343C22×C31#46311yescharacteristic
344C4×C31—21no—
345C4×C31—21no—
346C4×C31—21no—
347C4×C31—21no—
348C4×C31—42no—
349C4×C31—42no—
350C4×C31—82no—
351C4×C31—81no—
352C4×C31—81no—
353C4×C31—81no—
354C23×D4C22×C3111yescharacteristic
355C22×D8—21no—
356C2×#121C22×C3111yescharacteristic
357C2×C4×D4—21no—
358C22×C22⋊C4—21no—
359C22×C22⋊C4—21no—
360#447—22no—
361#447C4×C3114yescomplemented
362C2×#102—21no—
363C2×#102—21no—
364C2×#102—21no—
365C2×C22⋊C8C22×C3111yescharacteristic
366C2×C22⋊C8C22×C3111yescharacteristic
367#388—21no—
368#387—21no—
369#386—21no—
370#386—21no—
371#385—21no—
372#380C22×C3111yescharacteristic
373C2×#95C22×C3111yescharacteristic
374C2×#95C22×C3111yescharacteristic
375#327—22no—
376C2×C31—21no—
377C2×C31—21no—
378C2×C31—21no—
379C2×C31—21no—
380C2×C31—82no—
381C2×C31—84no—
382C2×C31#159511yescharacteristic
383C2×C31#159711yescharacteristic
384C2×C31#159811yescharacteristic
385C2×C31#160011yescharacteristic
386C2×C31#187911yescharacteristic
387C2×C31#188311yescharacteristic
388C2×C31#188911yescharacteristic
389C25—21no—
390C22×D4—21no—
391C22×D4—21no—
392C22×D4—21no—
393C22×D4—21no—
394C22×D4—21no—
395C22×D4—21no—
396C22×D4—42no—
397C22×D4—42no—
398C22×D4C2×C4×C3112yescomplemented
399C22×D4D4×C3111yescharacteristic
400C23×C4—21no—
401C23×C4—21no—
402C23×C4D4×C3111yescharacteristic
403C23×C4D4×C3111yescharacteristic
404C23×C4C23×C3111yescharacteristic
405C2×D8—24no—
406C2×D8—42no—
407C2×D8—42no—
408C22×C8—21no—
409C22×C8—21no—
410#121—21no—
411#121—21no—
412#121C2×C4×C3112yes—
413#121D4×C3112yes—
414C4×D4—42no—
415C4×D4—42no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4D4×C3111yescharacteristic
418C2×C4⋊C4D4×C3111yescharacteristic
419C2×C4⋊C4D4×C3111yescharacteristic
420C2×C22⋊C4—21no—
421C2×C22⋊C4—21no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—21no—
424C2×C22⋊C4—21no—
425C2×C22⋊C4—42no—
426C2×C22⋊C4—42no—
427C2×C22⋊C4—42no—
428C2×C22⋊C4—42no—
429C2×C22⋊C4—42no—
430C2×C22⋊C4—42no—
431C2×C42—21no—
432#102—24no—
433#102—42no—
434#102—42no—
435C22⋊C8—21no—
436C22⋊C8—21no—
437C22⋊C8—21no—
438C22⋊C8C2×C4×C3112yescomplemented
439#95—21no—
440#95—21no—
441#95—21no—
442#95—21no—
443#95—21no—
444#95—21no—
445#95—21no—
446C31#904111yescharacteristic, direct product factor, Sylow
447C24—22no—
448C24—22no—
449C24—22no—
450C24—21no—
451C24—41no—
452C24—41no—
453C24—41no—
454C24—41no—
455C24—41no—
456C24—41no—
457C24—41no—
458C24—41no—
459C24—41no—
460C24—41no—
461C24—41no—
462C24—41no—
463C24C2×D4×C3111yescharacteristic
464C2×D4—22no—
465C2×D4—22no—
466C2×D4—22no—
467C2×D4—22no—
468C2×D4—22no—
469C2×D4—22no—
470C2×D4—22no—
471C2×D4—22no—
472C2×D4—21no—
473C2×D4—41no—
474C2×D4—41no—
475C2×D4—41no—
476C2×D4—41no—
477C2×D4—41no—
478C2×D4—41no—
479C2×D4—44no—
480C2×D4—44no—
481C2×D4—44no—
482C2×D4—44no—
483C2×D4—41no—
484C2×D4—41no—
485C2×D4—41no—
486C2×D4—41no—
487C2×D4—41no—
488C2×D4—41no—
489C2×D4C22⋊C4×C3112yescomplemented
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—22no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—41no—
500C22×C4—41no—
501C22×C4—42no—
502C22×C4—41no—
503C22×C4—41no—
504C22×C4—41no—
505C22×C4—41no—
506C22×C4—41no—
507C22×C4—41no—
508C22×C4—41no—
509C22×C4—41no—
510C22×C4—41no—
511C22×C4—41no—
512C22×C4—81no—
513C22×C4—81no—
514C22×C4D8×C3112yescomplemented
515C22×C4C22×C4×C3111yescharacteristic
516C22×C4C2×D4×C3111yescharacteristic
517C22×C4C2×D4×C3111yescharacteristic
518C22×C4C2×D4×C3111yescharacteristic
519C22×C4#41×C3111yescharacteristic
520C22×C4#41×C3111yescharacteristic
521D8—48no—
522C2×C8—22no—
523C2×C8—41no—
524C2×C8—41no—
525C2×C8—41no—
526C2×C8—41no—
527C2×C8—41no—
528C4⋊C4—21no—
529C4⋊C4—41no—
530C4⋊C4—41no—
531C4⋊C4—42no—
532C4⋊C4C22⋊C4×C3112yescomplemented
533C22⋊C4—22no—
534C22⋊C4—22no—
535C22⋊C4—42no—
536C22⋊C4—48no—
537C22⋊C4—48no—
538C22⋊C4—41no—
539C22⋊C4—41no—
540C22⋊C4—81no—
541C22⋊C4—81no—
542C42—42no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—24no—
548C23—22no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—81no—
574C23—81no—
575C23—81no—
576C23—81no—
577C23—81no—
578C23—81no—
579C23—81no—
580C23—81no—
581C23—81no—
582C23—81no—
583C23—81no—
584C23—81no—
585C23—81no—
586C23—81no—
587C23—81no—
588C23—81no—
589C23C4×D4×C3111yescharacteristic
590C23C31×#12011yescharacteristic
591C23C31×#12111yescharacteristic
592C23C2×D8×C3111yescharacteristic
593C23C31×#13511yescharacteristic
594C23C31×AGL(1,ℤ/8)11yescharacteristic
595C23C31×AGL(1,ℤ/8)11yescharacteristic
596D4—24no—
597D4—24no—
598D4—42no—
599D4—42no—
600D4—42no—
601D4—42no—
602D4—42no—
603D4—42no—
604D4—42no—
605D4—42no—
606D4—42no—
607D4—42no—
608D4—42no—
609D4—81no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—22no—
615C2×C4—22no—
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—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—81no—
639C2×C4—84no—
640C2×C4—81no—
641C2×C4—81no—
642C2×C4—81no—
643C2×C4—81no—
644C2×C4—81no—
645C2×C4C2×C22⋊C4×C3111yescharacteristic
646C2×C4C4×D4×C3111yescharacteristic
647C2×C4C31×#12111yescharacteristic
648C2×C4C31×#12311yescharacteristic
649C8—42no—
650C8—81no—
651C22—21no—
652C22—21no—
653C22—21no—
654C22—21no—
655C22—21no—
656C22—21no—
657C22—21no—
658C22—21no—
659C22—21no—
660C22—21no—
661C22—21no—
662C22—21no—
663C22—44no—
664C22—44no—
665C22—42no—
666C22—42no—
667C22—44no—
668C22—42no—
669C22—82no—
670C22—82no—
671C22—82no—
672C22—82no—
673C22—82no—
674C22—82no—
675C22—81no—
676C22—82no—
677C22—82no—
678C22—81no—
679C22—82no—
680C22—81no—
681C22—82no—
682C22—81no—
683C22—82no—
684C22—82no—
685C22C31×#38611yescharacteristic
686C22C4×D8×C3111yescharacteristic
687C22C31×#44211yescharacteristic
688C22C31×#44711yescharacteristic
689C22C31×#44911yescharacteristic
690C22C31×#45311yescharacteristic
691C22C31×#45411yescharacteristic
692C22C31×B(4,𝔽2)11yescharacteristic
693C22C31×B(4,𝔽2)11yescharacteristic
694C22C31×#45911yescharacteristic
695C22C31×#46311yescharacteristic
696C4—21no—
697C4—21no—
698C4—21no—
699C4—21no—
700C4—42no—
701C4—42no—
702C4—82no—
703C4—81no—
704C4—81no—
705C4—81no—
706C2—21no—
707C2—21no—
708C2—21no—
709C2—21no—
710C2—84no—
711C2—82no—
712C2C31×#159511yescharacteristic
713C2C31×#159711yescharacteristic
714C2C31×#159811yescharacteristic
715C2C31×#160011yescharacteristic
716C2C31×#187911yescharacteristic
717C2C31×#188311yescharacteristic
718C2C31×#188911yescharacteristic
subgroupG/H
center, socleC23×C31#120
commutatorC22×C4C22×C4×C31
Frattini, cosocleC23×C4C23×C31
prime radicalC2×C31×#447C2
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×#2325C2semidirect
3. C31×#1786C2semidirect
4. C2×C31×#327C2semidirect
5. C31×#1438C2semidirect
6. C31×#1294C2semidirect
7. C31×#1280C2semidirect
15. C31×#447C4semidirect
39. C22×D4×C31C2×C4semidirect
79. C31×C22⋊C8C2×C4semidirect
129. C2×D4×C31C22⋊C4semidirect
154. C22×C4×C31D8semidirect
172. C4⋊C4×C31C22⋊C4semidirect
183. #9041C31direct (N × H)
293. #2325C2×C31semidirect
294. #1786C2×C31semidirect
295. C2×#327C2×C31semidirect
296. #1438C2×C31semidirect
297. #1294C2×C31semidirect
298. #1280C2×C31semidirect
361. #447C4×C31semidirect
398. C22×D4C2×C4×C31semidirect
438. C22⋊C8C2×C4×C31semidirect
446. C31#9041direct (N × H)
489. C2×D4C22⋊C4×C31semidirect
514. C22×C4D8×C31semidirect
532. 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×#4472
C31×#23252
C31×#17862
C2×C31×#3272
C31×#14382
C31×#12942
C31×#12802
#904131