Finite Group Databasebeta

← C31×#13088 #21469847

Subgroups — C31×#13088 #21469847

Order 7936.

1040 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C31×#99C211yesmaximal, characteristic, complemented
2C31×#2433C211yesmaximal, characteristic
3C2×C31×#353C212yesmaximal, complemented
4C31×#1781C211yesmaximal, characteristic, complemented
5C31×#1393C212yesmaximal, complemented
6C23×D4×C31C2211yescharacteristic
7C2×C31×#127C411yescharacteristic, complemented
8C2×C31×#127C2211yescharacteristic
9C2×C31×#121C411yescharacteristic, complemented
10C2×C31×#120—21no—
11C2×C31×#120—21no—
12C22×C22⋊C4×C31—21no—
13C2×C31×#99—21no—
14C2×C31×#99—24no—
15C2×C31×#99—21no—
16C2×C31×#99C2212yes—
17C2×C31×#99C2212yes—
18C31×#392—21no—
19C31×#392—21no—
20C31×#390C411yescharacteristic, complemented
21C31×#390C2211yescharacteristic
22C31×#379—21no—
23C4×C22⋊C4×C31C411yescharacteristic, complemented
24C31×#353—24no—
25C25×C31D411yescharacteristic
26C25×C31D411yescharacteristic
27C22×D4×C31—22no—
28C22×D4×C31—22no—
29C22×D4×C31—21no—
30C22×D4×C31—21no—
31C22×D4×C31—21no—
32C22×D4×C31—21no—
33C22×D4×C31—21no—
34C22×D4×C31—41no—
35C22×D4×C31—41no—
36C22×D4×C31—42no—
37C22×D4×C31C2×C411yescharacteristic
38C22×D4×C31C2×C411yescharacteristic, complemented
39C22×D4×C31D411yescharacteristic
40C22×D4×C31D411yescharacteristic
41C22×D4×C31C2311yescharacteristic
42C23×C4×C31—41no—
43C23×C4×C31C2×C411yescharacteristic
44C31×#127—22no—
45C31×#127—22no—
46C31×#127—42no—
47C31×#121—41no—
48C31×#121—41no—
49C31×#120—44no—
50C31×#120—44no—
51C2×C4⋊C4×C31C2×C411yescharacteristic, complemented
52C2×C22⋊C4×C31—21no—
53C2×C22⋊C4×C31—24no—
54C2×C22⋊C4×C31—22no—
55C2×C22⋊C4×C31—21no—
56C2×C22⋊C4×C31—21no—
57C2×C22⋊C4×C31—21no—
58C2×C22⋊C4×C31—21no—
59C2×C22⋊C4×C31—21no—
60C2×C22⋊C4×C31—22no—
61C2×C22⋊C4×C31—41no—
62C2×C22⋊C4×C31—42no—
63C2×C22⋊C4×C31—41no—
64C2×C22⋊C4×C31—41no—
65C2×C42×C31C2×C411yescharacteristic, complemented
66C2×C42×C31C2×C411yescharacteristic, complemented
67C31×#99—24no—
68C31×#99—22no—
69C31×#99—22no—
70C31×#99—22no—
71C31×#99—42no—
72C31×#99D414yescomplemented
73C31×#95—21no—
74C24×C31—21no—
75C24×C31—22no—
76C24×C31—21no—
77C24×C31—21no—
78C24×C31—21no—
79C24×C31—21no—
80C24×C31—21no—
81C24×C31—22no—
82C24×C31—21no—
83C24×C31—41no—
84C24×C31—42no—
85C24×C31—42no—
86C24×C31—81no—
87C24×C31—81no—
88C24×C31C22⋊C411yescharacteristic
89C24×C31C22⋊C411yescharacteristic
90C24×C31C22⋊C411yescharacteristic
91C24×C31C2×D411yescharacteristic
92C24×C31C2×D411yescharacteristic
93C2×D4×C31—22no—
94C2×D4×C31—22no—
95C2×D4×C31—21no—
96C2×D4×C31—22no—
97C2×D4×C31—21no—
98C2×D4×C31—22no—
99C2×D4×C31—22no—
100C2×D4×C31—41no—
101C2×D4×C31—41no—
102C2×D4×C31—41no—
103C2×D4×C31—41no—
104C2×D4×C31—41no—
105C2×D4×C31—41no—
106C2×D4×C31—42no—
107C2×D4×C31—42no—
108C2×D4×C31—42no—
109C2×D4×C31—42no—
110C2×D4×C31—41no—
111C2×D4×C31—41no—
112C2×D4×C31—44no—
113C2×D4×C31—44no—
114C2×D4×C31—44no—
115C2×D4×C31—44no—
116C2×D4×C31—42no—
117C2×D4×C31—42no—
118C2×D4×C31—42no—
119C2×D4×C31—41no—
120C2×D4×C31—41no—
121C2×D4×C31—42no—
122C2×D4×C31—41no—
123C2×D4×C31—42no—
124C2×D4×C31—41no—
125C2×D4×C31—81no—
126C2×D4×C31—81no—
127C2×D4×C31—81no—
128C2×D4×C31—81no—
129C2×D4×C31C2×D412yes—
130C2×D4×C31#4112yes—
131C22×C4×C31—21no—
132C22×C4×C31—21no—
133C22×C4×C31—21no—
134C22×C4×C31—21no—
135C22×C4×C31—22no—
136C22×C4×C31—21no—
137C22×C4×C31—41no—
138C22×C4×C31—44no—
139C22×C4×C31—42no—
140C22×C4×C31—42no—
141C22×C4×C31—82no—
142C22×C4×C31—82no—
143C22×C4×C31C22⋊C411yescharacteristic
144C22×C4×C31C22×C411yescharacteristic
145C4⋊C4×C31—41no—
146C22⋊C4×C31—24no—
147C22⋊C4×C31—24no—
148C22⋊C4×C31—42no—
149C22⋊C4×C31—42no—
150C22⋊C4×C31—42no—
151C22⋊C4×C31—42no—
152C22⋊C4×C31—42no—
153C22⋊C4×C31—44no—
154C22⋊C4×C31—42no—
155C22⋊C4×C31—42no—
156C22⋊C4×C31—42no—
157C22⋊C4×C31—82no—
158C22⋊C4×C31—82no—
159C42×C31—22no—
160C42×C31—41no—
161#13088C3111yesmaximal, characteristic, direct product factor, Sylow
162C23×C31—22no—
163C23×C31—22no—
164C23×C31—22no—
165C23×C31—22no—
166C23×C31—22no—
167C23×C31—21no—
168C23×C31—21no—
169C23×C31—21no—
170C23×C31—21no—
171C23×C31—21no—
172C23×C31—21no—
173C23×C31—21no—
174C23×C31—21no—
175C23×C31—21no—
176C23×C31—21no—
177C23×C31—21no—
178C23×C31—41no—
179C23×C31—42no—
180C23×C31—42no—
181C23×C31—42no—
182C23×C31—41no—
183C23×C31—41no—
184C23×C31—41no—
185C23×C31—41no—
186C23×C31—41no—
187C23×C31—41no—
188C23×C31—41no—
189C23×C31—41no—
190C23×C31—42no—
191C23×C31—41no—
192C23×C31—41no—
193C23×C31—41no—
194C23×C31—41no—
195C23×C31—42no—
196C23×C31—42no—
197C23×C31—41no—
198C23×C31—41no—
199C23×C31—41no—
200C23×C31—81no—
201C23×C31—81no—
202C23×C31—81no—
203C23×C31—81no—
204C23×C31—84no—
205C23×C31—82no—
206C23×C31—82no—
207C23×C31—84no—
208C23×C31—84no—
209C23×C31#9911yescharacteristic
210C23×C31#9911yescharacteristic
211C23×C31#9912yes—
212C23×C31C2×C22⋊C411yescharacteristic
213C23×C31#12011yescharacteristic
214C23×C31#12112yes—
215C23×C31#12311yescharacteristic
216D4×C31—42no—
217D4×C31—42no—
218D4×C31—44no—
219D4×C31—44no—
220D4×C31—42no—
221D4×C31—42no—
222D4×C31—82no—
223D4×C31—84no—
224D4×C31—82no—
225D4×C31—84no—
226D4×C31—82no—
227D4×C31—84no—
228D4×C31—82no—
229D4×C31—82no—
230C2×C4×C31—21no—
231C2×C4×C31—21no—
232C2×C4×C31—22no—
233C2×C4×C31—22no—
234C2×C4×C31—21no—
235C2×C4×C31—21no—
236C2×C4×C31—21no—
237C2×C4×C31—42no—
238C2×C4×C31—42no—
239C2×C4×C31—41no—
240C2×C4×C31—41no—
241C2×C4×C31—41no—
242C2×C4×C31—41no—
243C2×C4×C31—44no—
244C2×C4×C31—41no—
245C2×C4×C31—41no—
246C2×C4×C31—41no—
247C2×C4×C31—88no—
248C2×C4×C31—82no—
249C2×C4×C31—82no—
250C2×C4×C31—82no—
251C2×C4×C31—82no—
252C2×C4×C31—82no—
253C2×C4×C31C4×D412yes—
254C22×#99C2×C3111yescharacteristic, complemented
255#2433C2×C3111yescharacteristic
256C2×#353C2×C3112yescomplemented
257#1781C2×C3111yescharacteristic, complemented
258#1393C2×C3112yescomplemented
259C22×C31—21no—
260C22×C31—21no—
261C22×C31—21no—
262C22×C31—21no—
263C22×C31—21no—
264C22×C31—21no—
265C22×C31—21no—
266C22×C31—21no—
267C22×C31—22no—
268C22×C31—22no—
269C22×C31—21no—
270C22×C31—42no—
271C22×C31—42no—
272C22×C31—42no—
273C22×C31—41no—
274C22×C31—42no—
275C22×C31—41no—
276C22×C31—44no—
277C22×C31—44no—
278C22×C31—42no—
279C22×C31—41no—
280C22×C31—41no—
281C22×C31—41no—
282C22×C31—41no—
283C22×C31—41no—
284C22×C31—41no—
285C22×C31—41no—
286C22×C31—41no—
287C22×C31—41no—
288C22×C31—81no—
289C22×C31—81no—
290C22×C31—81no—
291C22×C31—84no—
292C22×C31—81no—
293C22×C31—81no—
294C22×C31—81no—
295C22×C31—81no—
296C22×C31—81no—
297C22×C31—82no—
298C22×C31—82no—
299C22×C31—84no—
300C22×C31—81no—
301C22×C31—81no—
302C22×C31—81no—
303C22×C31#35312yescomplemented
304C22×C31#38611yescharacteristic
305C22×C31C2×#9911yescharacteristic
306C22×C31C2×#9911yescharacteristic
307C4×C31—22no—
308C4×C31—42no—
309C4×C31—42no—
310C4×C31—41no—
311C4×C31—84no—
312C4×C31—81no—
313C4×C31—162no—
314C23×D4C22×C3111yescharacteristic
315C2×#127C4×C3111yescharacteristic, complemented
316C2×#127C22×C3111yescharacteristic
317C2×#121C4×C3111yescharacteristic, complemented
318C2×#120—21no—
319C2×#120—21no—
320C22×C22⋊C4—21no—
321C2×#99—21no—
322C2×#99—24no—
323C2×#99—21no—
324C2×#99C22×C3112yes—
325C2×#99C22×C3112yes—
326#392—21no—
327#392—21no—
328#390C4×C3111yescharacteristic, complemented
329#390C22×C3111yescharacteristic
330#379—21no—
331C4×C22⋊C4C4×C3111yescharacteristic, complemented
332#353—24no—
333C2×C31—21no—
334C2×C31—21no—
335C2×C31—22no—
336C2×C31—41no—
337C2×C31—42no—
338C2×C31—41no—
339C2×C31—42no—
340C2×C31—41no—
341C2×C31—81no—
342C2×C31—161no—
343C2×C31#189811yescharacteristic
344C2×C31C2×#35311yescharacteristic
345C2×C31#212711yescharacteristic
346C25D4×C3111yescharacteristic
347C25D4×C3111yescharacteristic
348C22×D4—22no—
349C22×D4—22no—
350C22×D4—21no—
351C22×D4—21no—
352C22×D4—21no—
353C22×D4—21no—
354C22×D4—21no—
355C22×D4—41no—
356C22×D4—41no—
357C22×D4—42no—
358C22×D4C2×C4×C3111yescharacteristic
359C22×D4C2×C4×C3111yescharacteristic, complemented
360C22×D4D4×C3111yescharacteristic
361C22×D4D4×C3111yescharacteristic
362C22×D4C23×C3111yescharacteristic
363C23×C4—41no—
364C23×C4C2×C4×C3111yescharacteristic
365#127—22no—
366#127—22no—
367#127—42no—
368#121—41no—
369#121—41no—
370#120—44no—
371#120—44no—
372C2×C4⋊C4C2×C4×C3111yescharacteristic, complemented
373C2×C22⋊C4—21no—
374C2×C22⋊C4—24no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—21no—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4—41no—
383C2×C22⋊C4—42no—
384C2×C22⋊C4—41no—
385C2×C22⋊C4—41no—
386C2×C42C2×C4×C3111yescharacteristic, complemented
387C2×C42C2×C4×C3111yescharacteristic, complemented
388#99—24no—
389#99—22no—
390#99—22no—
391#99—22no—
392#99—42no—
393#99D4×C3114yescomplemented
394#95—21no—
395C31#1308811yescharacteristic, direct product factor, Sylow
396C24—21no—
397C24—22no—
398C24—21no—
399C24—21no—
400C24—21no—
401C24—21no—
402C24—21no—
403C24—22no—
404C24—21no—
405C24—41no—
406C24—42no—
407C24—42no—
408C24—81no—
409C24—81no—
410C24C22⋊C4×C3111yescharacteristic
411C24C22⋊C4×C3111yescharacteristic
412C24C22⋊C4×C3111yescharacteristic
413C24C2×D4×C3111yescharacteristic
414C24C2×D4×C3111yescharacteristic
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—21no—
418C2×D4—22no—
419C2×D4—21no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—41no—
423C2×D4—41no—
424C2×D4—41no—
425C2×D4—41no—
426C2×D4—41no—
427C2×D4—41no—
428C2×D4—41no—
429C2×D4—41no—
430C2×D4—42no—
431C2×D4—42no—
432C2×D4—42no—
433C2×D4—42no—
434C2×D4—44no—
435C2×D4—44no—
436C2×D4—44no—
437C2×D4—44no—
438C2×D4—42no—
439C2×D4—42no—
440C2×D4—42no—
441C2×D4—41no—
442C2×D4—41no—
443C2×D4—42no—
444C2×D4—41no—
445C2×D4—42no—
446C2×D4—41no—
447C2×D4—81no—
448C2×D4—81no—
449C2×D4—81no—
450C2×D4—81no—
451C2×D4C2×D4×C3112yes—
452C2×D4#41×C3112yes—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—41no—
460C22×C4—44no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—82no—
464C22×C4—82no—
465C22×C4C22⋊C4×C3111yescharacteristic
466C22×C4C22×C4×C3111yescharacteristic
467C4⋊C4—41no—
468C22⋊C4—24no—
469C22⋊C4—24no—
470C22⋊C4—42no—
471C22⋊C4—42no—
472C22⋊C4—42no—
473C22⋊C4—42no—
474C22⋊C4—42no—
475C22⋊C4—44no—
476C22⋊C4—42no—
477C22⋊C4—42no—
478C22⋊C4—42no—
479C22⋊C4—82no—
480C22⋊C4—82no—
481C42—22no—
482C42—41no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—22no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—41no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—42no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—42no—
517C23—42no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—81no—
522C23—81no—
523C23—81no—
524C23—81no—
525C23—84no—
526C23—82no—
527C23—82no—
528C23—84no—
529C23—84no—
530C23C31×#9911yescharacteristic
531C23C31×#9911yescharacteristic
532C23C31×#9912yes—
533C23C2×C22⋊C4×C3111yescharacteristic
534C23C31×#12011yescharacteristic
535C23C31×#12112yes—
536C23C31×#12311yescharacteristic
537D4—42no—
538D4—42no—
539D4—44no—
540D4—44no—
541D4—42no—
542D4—42no—
543D4—82no—
544D4—84no—
545D4—82no—
546D4—84no—
547D4—82no—
548D4—84no—
549D4—82no—
550D4—82no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—44no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—88no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4C4×D4×C3112yes—
575C22—21no—
576C22—21no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—22no—
584C22—22no—
585C22—21no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—41no—
590C22—42no—
591C22—41no—
592C22—44no—
593C22—44no—
594C22—42no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—81no—
605C22—81no—
606C22—81no—
607C22—84no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22—82no—
614C22—82no—
615C22—84no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22C31×#35312yescomplemented
620C22C31×#38611yescharacteristic
621C22C2×C31×#9911yescharacteristic
622C22C2×C31×#9911yescharacteristic
623C4—22no—
624C4—42no—
625C4—42no—
626C4—41no—
627C4—84no—
628C4—81no—
629C4—162no—
630C2—21no—
631C2—21no—
632C2—22no—
633C2—41no—
634C2—42no—
635C2—41no—
636C2—42no—
637C2—41no—
638C2—81no—
639C2—161no—
640C2C31×#189811yescharacteristic
641C2C2×C31×#35311yescharacteristic
642C2C31×#212711yescharacteristic
subgroupG/H
center, socleC22×C31C2×#99
commutatorC22×C4C22×C4×C31
Frattini, cosocleC22×D4C23×C31
prime radicalC31×#2433C2
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C24×C31 ◁ C23×D4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×C31×#99C2semidirect
3. C2×C31×#353C2semidirect
4. C31×#1781C2semidirect
5. C31×#1393C2semidirect
7. C2×C31×#127C4semidirect
9. C2×C31×#121C4semidirect
20. C31×#390C4semidirect
23. C4×C22⋊C4×C31C4semidirect
38. C22×D4×C31C2×C4semidirect
51. C2×C4⋊C4×C31C2×C4semidirect
65. C2×C42×C31C2×C4semidirect
66. C2×C42×C31C2×C4semidirect
72. C31×#99D4semidirect
161. #13088C31direct (N × H)
254. C22×#99C2×C31semidirect
256. C2×#353C2×C31semidirect
257. #1781C2×C31semidirect
258. #1393C2×C31semidirect
303. C22×C31#353semidirect
315. C2×#127C4×C31semidirect
317. C2×#121C4×C31semidirect
328. #390C4×C31semidirect
331. C4×C22⋊C4C4×C31semidirect
359. C22×D4C2×C4×C31semidirect
372. C2×C4⋊C4C2×C4×C31semidirect
386. C2×C42C2×C4×C31semidirect
387. C2×C42C2×C4×C31semidirect
393. #99D4×C31semidirect
395. C31#13088direct (N × H)
619. C22C31×#353semidirect

Maximal subgroups

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

subgroupindex
C22×C31×#992
C31×#24332
C2×C31×#3532
C31×#17812
C31×#13932
#1308831