Finite Group Databasebeta

← C31×#18169 #21474928

Subgroups — C31×#18169 #21474928

Order 7936.

1144 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3490C211yesmaximal, characteristic
2C31×#3092C212yesmaximal, complemented
3C2×C31×#397C211yesmaximal, characteristic, complemented
4C31×#1612C214yesmaximal, complemented
5C31×#1608C212yesmaximal, complemented
6C31×#1607C212yesmaximal, complemented
7C31×#1504C212yesmaximal, complemented
8C31×#1462C211yesmaximal, characteristic, complemented
9C31×#551—22no—
10C31×#551—22no—
11C31×#546—22no—
12C31×#546—22no—
13C31×#543C2211yescharacteristic
14C31×#538—21no—
15C31×#538C2212yes—
16C31×#534C2211yescharacteristic
17C31×#525—21no—
18C31×#525C2212yes—
19C2×C31×#120—21no—
20C31×#518—21no—
21C22×C4⋊C4×C31C2211yescharacteristic, complemented
22C22×C22⋊C4×C31—21no—
23C22×C22⋊C4×C31C2211yescharacteristic
24C31×#484—22no—
25C31×#483C2214yescomplemented
26C31×#481—22no—
27C31×#480C2214yescomplemented
28C31×#428—22no—
29C31×#417—22no—
30C31×#407C2212yescomplemented
31C31×#397—21no—
32C31×#397—21no—
33C31×#397C2212yes—
34C31×#397C2212yescomplemented
35C2×C31×#95—21no—
36C2×C31×#95C2211yescharacteristic, complemented
37C31×#328C2214yescomplemented
38C31×#327C2214yescomplemented
39C31×#323C2212yes—
40C31×#323C2212yescomplemented
41C25×C31D411yescharacteristic
42C2×#41×C31—21no—
43C2×#41×C31D412yes—
44C22×D4×C31—21no—
45C22×D4×C31—21no—
46C22×D4×C31—21no—
47C23×C4×C31—21no—
48C23×C4×C31—21no—
49C23×C4×C31D411yescharacteristic
50C23×C4×C31C2311yescharacteristic
51C2×M16×C31—22no—
52C31×#126—22no—
53C31×#126—22no—
54C31×#124—22no—
55C31×#124—22no—
56C31×#123—22no—
57C31×#123—22no—
58C31×#123—22no—
59C31×#123—22no—
60C31×#123—22no—
61C31×C22⋊Q8—21no—
62C31×C22⋊Q8—22no—
63C31×C22⋊Q8—21no—
64C31×C22⋊Q8—22no—
65C31×C22⋊Q8D412yes—
66C31×C22⋊Q8C2312yes—
67C31×#121—21no—
68C31×#121—22no—
69C31×#121—22no—
70C31×#121—21no—
71C31×#121D412yes—
72C31×#121C2312yes—
73C31×#120—22no—
74C31×#120—22no—
75C31×#120—22no—
76C31×#120—22no—
77C31×#120—22no—
78C4×D4×C31—22no—
79C4×D4×C31—22no—
80C4×D4×C31—22no—
81C4×D4×C31—22no—
82C4×D4×C31—22no—
83C4×D4×C31—22no—
84C31×#117—22no—
85C31×#117—21no—
86C31×#117D412yes—
87C2×C4⋊C4×C31—22no—
88C2×C4⋊C4×C31—22no—
89C2×C4⋊C4×C31—21no—
90C2×C4⋊C4×C31C2312yes—
91C2×C22⋊C4×C31—22no—
92C2×C22⋊C4×C31—21no—
93C2×C22⋊C4×C31—21no—
94C2×C22⋊C4×C31—21no—
95C2×C22⋊C4×C31—22no—
96C2×C22⋊C4×C31—22no—
97C2×C22⋊C4×C31—22no—
98C2×C22⋊C4×C31—22no—
99C2×C22⋊C4×C31—44no—
100C2×C22⋊C4×C31D411yescharacteristic
101C2×C22⋊C4×C31D411yescharacteristic
102C2×C22⋊C4×C31C2311yescharacteristic
103C2×C22⋊C4×C31C2311yescharacteristic
104C31×#107—24no—
105C31×#106—24no—
106C31×#103—24no—
107C31×#102—24no—
108C31×C22⋊C8—22no—
109C31×C22⋊C8C2314yes—
110C31×#95—21no—
111C31×#95—22no—
112C31×#95—22no—
113C31×#95C2312yes—
114C24×C31—21no—
115C24×C31—22no—
116C24×C31—21no—
117C24×C31—21no—
118C24×C31—41no—
119C24×C31—42no—
120C24×C31—42no—
121C24×C31C2×D411yescharacteristic
122C24×C31C2×D411yescharacteristic
123C24×C31C2×D411yescharacteristic
124#41×C31—44no—
125#41×C31—82no—
126C2×Q8×C31—21no—
127C2×Q8×C31C2×D412yes—
128C2×D4×C31—22no—
129C2×D4×C31—22no—
130C2×D4×C31—22no—
131C2×D4×C31—22no—
132C2×D4×C31—21no—
133C2×D4×C31—22no—
134C2×D4×C31—22no—
135C2×D4×C31—42no—
136C2×D4×C31—42no—
137C2×D4×C31—82no—
138C2×D4×C31—82no—
139C2×D4×C31C2×D412yes—
140C22×C4×C31—21no—
141C22×C4×C31—21no—
142C22×C4×C31—21no—
143C22×C4×C31—21no—
144C22×C4×C31—22no—
145C22×C4×C31—21no—
146C22×C4×C31—22no—
147C22×C4×C31—22no—
148C22×C4×C31—22no—
149C22×C4×C31—42no—
150C22×C4×C31—42no—
151C22×C4×C31—42no—
152C22×C4×C31—42no—
153C22×C4×C31—42no—
154C22×C4×C31—42no—
155C22×C4×C31—42no—
156C22×C4×C31—41no—
157C22×C4×C31—41no—
158C22×C4×C31C2×D412yes—
159C22×C4×C31C2×D411yescharacteristic
160C22×C4×C31C2×D411yescharacteristic
161C22×C4×C31C2×D411yescharacteristic
162C22×C4×C31C2×D412yes—
163C22×C4×C31C2411yescharacteristic
164M16×C31—44no—
165C2×C8×C31—24no—
166C4⋊C4×C31—22no—
167C4⋊C4×C31—21no—
168C4⋊C4×C31—21no—
169C4⋊C4×C31—22no—
170C4⋊C4×C31—22no—
171C4⋊C4×C31—22no—
172C4⋊C4×C31—24no—
173C4⋊C4×C31—42no—
174C4⋊C4×C31C2×D412yes—
175C4⋊C4×C31C2×D412yes—
176C22⋊C4×C31—22no—
177C22⋊C4×C31—22no—
178C22⋊C4×C31—22no—
179C22⋊C4×C31—22no—
180C22⋊C4×C31—22no—
181C22⋊C4×C31—22no—
182C22⋊C4×C31—22no—
183C22⋊C4×C31—22no—
184C22⋊C4×C31—22no—
185C22⋊C4×C31—22no—
186C22⋊C4×C31—22no—
187C22⋊C4×C31—22no—
188C22⋊C4×C31—22no—
189C22⋊C4×C31—22no—
190C22⋊C4×C31—44no—
191C22⋊C4×C31—44no—
192C42×C31—22no—
193C42×C31—22no—
194#18169C3111yesmaximal, characteristic, direct product factor, Sylow
195C23×C31—21no—
196C23×C31—21no—
197C23×C31—22no—
198C23×C31—22no—
199C23×C31—21no—
200C23×C31—21no—
201C23×C31—21no—
202C23×C31—22no—
203C23×C31—21no—
204C23×C31—21no—
205C23×C31—42no—
206C23×C31—42no—
207C23×C31—41no—
208C23×C31—41no—
209C23×C31—41no—
210C23×C31—41no—
211C23×C31—48no—
212C23×C31—42no—
213C23×C31—42no—
214C23×C31—42no—
215C23×C31—81no—
216C23×C31—81no—
217C23×C31—84no—
218C23×C31—81no—
219C23×C31—81no—
220C23×C31#12011yescharacteristic
221C23×C31#12011yescharacteristic
222C23×C31AGL(1,ℤ/8)12yes—
223C23×C31C22×D411yescharacteristic
224Q8×C31—42no—
225Q8×C31—81no—
226D4×C31—42no—
227D4×C31—44no—
228D4×C31—42no—
229D4×C31—42no—
230D4×C31—44no—
231D4×C31—44no—
232D4×C31—82no—
233D4×C31—82no—
234D4×C31—81no—
235C2×C4×C31—24no—
236C2×C4×C31—22no—
237C2×C4×C31—21no—
238C2×C4×C31—22no—
239C2×C4×C31—22no—
240C2×C4×C31—21no—
241C2×C4×C31—42no—
242C2×C4×C31—41no—
243C2×C4×C31—41no—
244C2×C4×C31—42no—
245C2×C4×C31—42no—
246C2×C4×C31—41no—
247C2×C4×C31—41no—
248C2×C4×C31—42no—
249C2×C4×C31—42no—
250C2×C4×C31—42no—
251C2×C4×C31—42no—
252C2×C4×C31—42no—
253C2×C4×C31—42no—
254C2×C4×C31—82no—
255C2×C4×C31—81no—
256C2×C4×C31—81no—
257C2×C4×C31—81no—
258C2×C4×C31—82no—
259C2×C4×C31—84no—
260C2×C4×C31—82no—
261C2×C4×C31—82no—
262C2×C4×C31#12012yes—
263C2×C4×C31C22×D412yes—
264C8×C31—44no—
265#3490C2×C3111yescharacteristic
266#3092C2×C3112yescomplemented
267C2×#397C2×C3111yescharacteristic, complemented
268#1612C2×C3114yescomplemented
269#1608C2×C3112yescomplemented
270#1607C2×C3112yescomplemented
271#1504C2×C3112yescomplemented
272#1462C2×C3111yescharacteristic, complemented
273C22×C31—24no—
274C22×C31—24no—
275C22×C31—21no—
276C22×C31—21no—
277C22×C31—21no—
278C22×C31—21no—
279C22×C31—21no—
280C22×C31—21no—
281C22×C31—22no—
282C22×C31—42no—
283C22×C31—42no—
284C22×C31—41no—
285C22×C31—41no—
286C22×C31—42no—
287C22×C31—41no—
288C22×C31—42no—
289C22×C31—41no—
290C22×C31—42no—
291C22×C31—41no—
292C22×C31—81no—
293C22×C31—84no—
294C22×C31—84no—
295C22×C31—81no—
296C22×C31—162no—
297C22×C31#45512yes—
298C22×C31#45812yes—
299C22×C31C2×#12011yescharacteristic
300C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
301C22×C31#57511yescharacteristic
302C4×C31—22no—
303C4×C31—22no—
304C4×C31—42no—
305C4×C31—41no—
306C4×C31—41no—
307C4×C31—42no—
308C4×C31—42no—
309C4×C31—82no—
310C4×C31—82no—
311#551—22no—
312#551—22no—
313#546—22no—
314#546—22no—
315#543C22×C3111yescharacteristic
316#538—21no—
317#538C22×C3112yes—
318#534C22×C3111yescharacteristic
319#525—21no—
320#525C22×C3112yes—
321C2×#120—21no—
322#518—21no—
323C22×C4⋊C4C22×C3111yescharacteristic, complemented
324C22×C22⋊C4—21no—
325C22×C22⋊C4C22×C3111yescharacteristic
326#484—22no—
327#483C22×C3114yescomplemented
328#481—22no—
329#480C22×C3114yescomplemented
330#428—22no—
331#417—22no—
332#407C22×C3112yescomplemented
333#397—21no—
334#397—21no—
335#397C22×C3112yes—
336#397C22×C3112yescomplemented
337C2×#95—21no—
338C2×#95C22×C3111yescharacteristic, complemented
339#328C22×C3114yescomplemented
340#327C22×C3114yescomplemented
341#323C22×C3112yes—
342#323C22×C3112yescomplemented
343C2×C31—21no—
344C2×C31—21no—
345C2×C31—22no—
346C2×C31—22no—
347C2×C31—42no—
348C2×C31—81no—
349C2×C31—82no—
350C2×C31#300611yescharacteristic
351C2×C31C2×#45511yescharacteristic
352C2×C31C2×#45811yescharacteristic
353C25D4×C3111yescharacteristic
354C2×#41—21no—
355C2×#41D4×C3112yes—
356C22×D4—21no—
357C22×D4—21no—
358C22×D4—21no—
359C23×C4—21no—
360C23×C4—21no—
361C23×C4D4×C3111yescharacteristic
362C23×C4C23×C3111yescharacteristic
363C2×M16—22no—
364#126—22no—
365#126—22no—
366#124—22no—
367#124—22no—
368#123—22no—
369#123—22no—
370#123—22no—
371#123—22no—
372#123—22no—
373C22⋊Q8—21no—
374C22⋊Q8—22no—
375C22⋊Q8—21no—
376C22⋊Q8—22no—
377C22⋊Q8D4×C3112yes—
378C22⋊Q8C23×C3112yes—
379#121—21no—
380#121—22no—
381#121—22no—
382#121—21no—
383#121D4×C3112yes—
384#121C23×C3112yes—
385#120—22no—
386#120—22no—
387#120—22no—
388#120—22no—
389#120—22no—
390C4×D4—22no—
391C4×D4—22no—
392C4×D4—22no—
393C4×D4—22no—
394C4×D4—22no—
395C4×D4—22no—
396#117—22no—
397#117—21no—
398#117D4×C3112yes—
399C2×C4⋊C4—22no—
400C2×C4⋊C4—22no—
401C2×C4⋊C4—21no—
402C2×C4⋊C4C23×C3112yes—
403C2×C22⋊C4—22no—
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—21no—
407C2×C22⋊C4—22no—
408C2×C22⋊C4—22no—
409C2×C22⋊C4—22no—
410C2×C22⋊C4—22no—
411C2×C22⋊C4—44no—
412C2×C22⋊C4D4×C3111yescharacteristic
413C2×C22⋊C4D4×C3111yescharacteristic
414C2×C22⋊C4C23×C3111yescharacteristic
415C2×C22⋊C4C23×C3111yescharacteristic
416#107—24no—
417#106—24no—
418#103—24no—
419#102—24no—
420C22⋊C8—22no—
421C22⋊C8C23×C3114yes—
422#95—22no—
423#95—22no—
424#95—21no—
425#95C23×C3112yes—
426C31#1816911yescharacteristic, direct product factor, Sylow
427C24—21no—
428C24—22no—
429C24—21no—
430C24—21no—
431C24—41no—
432C24—42no—
433C24—42no—
434C24C2×D4×C3111yescharacteristic
435C24C2×D4×C3111yescharacteristic
436C24C2×D4×C3111yescharacteristic
437#41—44no—
438#41—82no—
439C2×Q8—21no—
440C2×Q8C2×D4×C3112yes—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—21no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—42no—
449C2×D4—42no—
450C2×D4—82no—
451C2×D4—82no—
452C2×D4C2×D4×C3112yes—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—41no—
470C22×C4—41no—
471C22×C4C2×D4×C3112yes—
472C22×C4C2×D4×C3111yescharacteristic
473C22×C4C2×D4×C3111yescharacteristic
474C22×C4C2×D4×C3111yescharacteristic
475C22×C4C2×D4×C3112yes—
476C22×C4C24×C3111yescharacteristic
477M16—44no—
478C2×C8—24no—
479C4⋊C4—22no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—24no—
486C4⋊C4—42no—
487C4⋊C4C2×D4×C3112yes—
488C4⋊C4C2×D4×C3112yes—
489C22⋊C4—22no—
490C22⋊C4—22no—
491C22⋊C4—22no—
492C22⋊C4—22no—
493C22⋊C4—22no—
494C22⋊C4—22no—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—22no—
498C22⋊C4—22no—
499C22⋊C4—22no—
500C22⋊C4—22no—
501C22⋊C4—22no—
502C22⋊C4—22no—
503C22⋊C4—44no—
504C22⋊C4—44no—
505C42—22no—
506C42—22no—
507C23—22no—
508C23—22no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—22no—
515C23—21no—
516C23—21no—
517C23—42no—
518C23—42no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—48no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—81no—
528C23—81no—
529C23—84no—
530C23—81no—
531C23—81no—
532C23C31×#12011yescharacteristic
533C23C31×#12011yescharacteristic
534C23C31×AGL(1,ℤ/8)12yes—
535C23C22×D4×C3111yescharacteristic
536Q8—42no—
537Q8—81no—
538D4—42no—
539D4—44no—
540D4—44no—
541D4—44no—
542D4—42no—
543D4—42no—
544D4—82no—
545D4—82no—
546D4—81no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—82no—
567C2×C4—81no—
568C2×C4—81no—
569C2×C4—81no—
570C2×C4—82no—
571C2×C4—84no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4C31×#12012yes—
575C2×C4C22×D4×C3112yes—
576C8—44no—
577C22—24no—
578C22—24no—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—21no—
584C22—21no—
585C22—22no—
586C22—42no—
587C22—42no—
588C22—41no—
589C22—41no—
590C22—42no—
591C22—41no—
592C22—42no—
593C22—41no—
594C22—42no—
595C22—41no—
596C22—81no—
597C22—84no—
598C22—84no—
599C22—81no—
600C22—162no—
601C22C31×#45512yes—
602C22C31×#45812yes—
603C22C2×C31×#12011yescharacteristic
604C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
605C22C31×#57511yescharacteristic
606C4—22no—
607C4—22no—
608C4—42no—
609C4—41no—
610C4—41no—
611C4—42no—
612C4—42no—
613C4—82no—
614C4—82no—
615C2—21no—
616C2—22no—
617C2—22no—
618C2—21no—
619C2—42no—
620C2—81no—
621C2—82no—
622C2C31×#300611yescharacteristic
623C2C2×C31×#45511yescharacteristic
624C2C2×C31×#45811yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#3490C2
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
2. C31×#3092C2semidirect
3. C2×C31×#397C2semidirect
4. C31×#1612C2semidirect
5. C31×#1608C2semidirect
6. C31×#1607C2semidirect
7. C31×#1504C2semidirect
8. C31×#1462C2semidirect
21. C22×C4⋊C4×C31C22semidirect
25. C31×#483C22semidirect
27. C31×#480C22semidirect
30. C31×#407C22semidirect
34. C31×#397C22semidirect
36. C2×C31×#95C22semidirect
37. C31×#328C22semidirect
38. C31×#327C22semidirect
40. C31×#323C22semidirect
194. #18169C31direct (N × H)
266. #3092C2×C31semidirect
267. C2×#397C2×C31semidirect
268. #1612C2×C31semidirect
269. #1608C2×C31semidirect
270. #1607C2×C31semidirect
271. #1504C2×C31semidirect
272. #1462C2×C31semidirect
323. C22×C4⋊C4C22×C31semidirect
327. #483C22×C31semidirect
329. #480C22×C31semidirect
332. #407C22×C31semidirect
336. #397C22×C31semidirect
338. C2×#95C22×C31semidirect
339. #328C22×C31semidirect
340. #327C22×C31semidirect
342. #323C22×C31semidirect
426. C31#18169direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#34902
C31×#30922
C2×C31×#3972
C31×#16122
C31×#16082
C31×#16072
C31×#15042
C31×#14622
#1816931