Finite Group Databasebeta

← C31×#18209 #21474968

Subgroups — C31×#18209 #21474968

Order 7936.

938 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3491C211yesmaximal, characteristic
2C31×#3108C212yesmaximal, complemented
3C31×#2599C211yesmaximal, characteristic, complemented
4C31×#1609C212yesmaximal, complemented
5C31×#1608C211yesmaximal, characteristic, complemented
6C31×#1607C211yesmaximal, characteristic
7C31×#1603C212yesmaximal, complemented
8C31×#1602C211yesmaximal, characteristic, complemented
9C31×#1601C211yesmaximal, characteristic
10C31×#1521C212yesmaximal, complemented
11C31×#1458C211yesmaximal, characteristic, complemented
12C31×#552—22no—
13C31×#552—22no—
14C31×#551—22no—
15C31×#551—22no—
16C31×#543C2211yescharacteristic
17C31×#539—21no—
18C31×#539C2212yes—
19C31×#534C2211yescharacteristic
20C31×#529—21no—
21C31×#529C2212yes—
22C2×C31×#123—21no—
23C31×#518—21no—
24C2×C4×Q8×C31C2211yescharacteristic, complemented
25C2×C31×#117C2211yescharacteristic
26C31×#483C2212yescomplemented
27C31×#480C2212yes—
28C31×#479—22no—
29C31×#464—22no—
30C31×#451C2212yescomplemented
31C31×#448C2212yes—
32C4×Q16×C31—22no—
33C4×SD16×C31—22no—
34C31×#431C2212yescomplemented
35C31×#398C2211yescharacteristic, complemented
36C31×#397C2211yescharacteristic
37C31×#395C2211yescharacteristic, complemented
38C31×#393C2211yescharacteristic
39C31×#388—21no—
40C31×#387—21no—
41C31×#387—21no—
42C31×#385—21no—
43C31×#383C2211yescharacteristic, complemented
44C31×#328C2212yescomplemented
45C31×#328C2212yescomplemented
46C31×#327C2212yes—
47C31×#327C2212yes—
48C31×#324C2212yescomplemented
49C31×#323C2211yescharacteristic
50C31×#323C2211yescharacteristic
51C22×Q8×C31C2311yescharacteristic
52C22×D4×C31—21no—
53C23×C4×C31D411yescharacteristic
54C2×Q16×C31—22no—
55C2×SD16×C31—22no—
56C31×#126—22no—
57C31×#126—22no—
58C31×#126—22no—
59C31×#126—22no—
60C31×#125—21no—
61C31×#125—22no—
62C31×#125—22no—
63C31×#125D412yes—
64C31×#124—21no—
65C31×#124D412yes—
66C31×#123—22no—
67C31×#123—22no—
68C31×#123—22no—
69C31×#123—22no—
70C31×#123—22no—
71C31×#123—22no—
72C31×#123—22no—
73C31×C22⋊Q8—22no—
74C31×C22⋊Q8—22no—
75C31×C22⋊Q8—22no—
76C31×C22⋊Q8—21no—
77C31×C22⋊Q8—22no—
78C31×C22⋊Q8C2312yes—
79C31×#121—21no—
80C31×#121C2312yes—
81C31×#120—22no—
82C4×Q8×C31—21no—
83C4×Q8×C31—22no—
84C4×Q8×C31—22no—
85C4×Q8×C31D412yes—
86C4×D4×C31—22no—
87C4×D4×C31—22no—
88C4×D4×C31—21no—
89C4×D4×C31—22no—
90C4×D4×C31—22no—
91C4×D4×C31D412yes—
92C31×#117—22no—
93C31×#117—22no—
94C31×#117—22no—
95C31×#117—22no—
96C31×#117—22no—
97C2×C4⋊C4×C31—21no—
98C2×C4⋊C4×C31—21no—
99C2×C4⋊C4×C31—21no—
100C2×C4⋊C4×C31—21no—
101C2×C4⋊C4×C31—21no—
102C2×C4⋊C4×C31D411yescharacteristic
103C2×C4⋊C4×C31C2311yescharacteristic
104C2×C4⋊C4×C31C2311yescharacteristic
105C2×C22⋊C4×C31—22no—
106C2×C22⋊C4×C31—21no—
107C2×C22⋊C4×C31—21no—
108C2×C22⋊C4×C31—21no—
109C2×C22⋊C4×C31—21no—
110C2×C22⋊C4×C31—21no—
111C2×C22⋊C4×C31D411yescharacteristic
112C2×C22⋊C4×C31C2311yescharacteristic
113C2×C42×C31—21no—
114C2×C42×C31D411yescharacteristic
115C2×C42×C31C2311yescharacteristic
116C31×#107—22no—
117C31×#106—22no—
118C31×C4⋊C8—22no—
119C31×#103—22no—
120C31×#103—22no—
121C31×#103—22no—
122C31×#102—22no—
123C31×C22⋊C8C2312yes—
124C31×C22⋊C8C2312yes—
125C4×C8×C31—22no—
126C31×#95—21no—
127C31×#95—21no—
128C31×#95—21no—
129C31×#95—21no—
130C31×#95—21no—
131C31×#95—21no—
132C31×#95C2311yescharacteristic
133C31×#95C2311yescharacteristic
134C24×C31—21no—
135C24×C31C2×D411yescharacteristic
136C2×Q8×C31—21no—
137C2×Q8×C31—22no—
138C2×Q8×C31—42no—
139C2×Q8×C31C2×D412yes—
140C2×D4×C31—22no—
141C2×D4×C31—22no—
142C2×D4×C31—21no—
143C2×D4×C31—42no—
144C2×D4×C31—42no—
145C2×D4×C31C2×D412yes—
146C22×C4×C31—21no—
147C22×C4×C31—22no—
148C22×C4×C31—21no—
149C22×C4×C31—21no—
150C22×C4×C31—21no—
151C22×C4×C31—22no—
152C22×C4×C31—21no—
153C22×C4×C31—21no—
154C22×C4×C31—22no—
155C22×C4×C31—21no—
156C22×C4×C31—21no—
157C22×C4×C31—21no—
158C22×C4×C31—41no—
159C22×C4×C31C2×D411yescharacteristic
160C22×C4×C31C2×D411yescharacteristic
161C22×C4×C31C2×D411yescharacteristic
162C22×C4×C31C2×D411yescharacteristic
163C22×C4×C31C2×D411yescharacteristic
164C22×C4×C31C2411yescharacteristic
165Q16×C31—44no—
166SD16×C31—44no—
167C2×C8×C31—22no—
168C2×C8×C31—22no—
169C4⋊C4×C31—22no—
170C4⋊C4×C31—22no—
171C4⋊C4×C31—22no—
172C4⋊C4×C31—22no—
173C4⋊C4×C31—21no—
174C4⋊C4×C31—21no—
175C4⋊C4×C31—22no—
176C4⋊C4×C31—22no—
177C4⋊C4×C31—22no—
178C4⋊C4×C31—22no—
179C4⋊C4×C31—22no—
180C4⋊C4×C31—22no—
181C4⋊C4×C31—22no—
182C4⋊C4×C31—42no—
183C4⋊C4×C31—42no—
184C4⋊C4×C31—41no—
185C4⋊C4×C31—41no—
186C4⋊C4×C31C2×D412yes—
187C4⋊C4×C31C2×D412yes—
188C22⋊C4×C31—22no—
189C22⋊C4×C31—22no—
190C22⋊C4×C31—22no—
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—22no—
200C22⋊C4×C31—42no—
201C22⋊C4×C31—42no—
202C42×C31—22no—
203C42×C31—21no—
204C42×C31—21no—
205C42×C31—22no—
206C42×C31—42no—
207C42×C31C2×D412yes—
208C42×C31C2×D412yes—
209#18209C3111yesmaximal, characteristic, direct product factor, Sylow
210C23×C31—21no—
211C23×C31—21no—
212C23×C31—22no—
213C23×C31—41no—
214C23×C31—41no—
215C23×C31—42no—
216C23×C31—81no—
217C23×C31—81no—
218C23×C31#13512yes—
219C23×C31C22×D411yescharacteristic
220Q8×C31—24no—
221Q8×C31—42no—
222Q8×C31—42no—
223Q8×C31—81no—
224D4×C31—42no—
225D4×C31—44no—
226D4×C31—42no—
227D4×C31—42no—
228D4×C31—81no—
229C2×C4×C31—21no—
230C2×C4×C31—21no—
231C2×C4×C31—22no—
232C2×C4×C31—21no—
233C2×C4×C31—21no—
234C2×C4×C31—22no—
235C2×C4×C31—22no—
236C2×C4×C31—22no—
237C2×C4×C31—21no—
238C2×C4×C31—21no—
239C2×C4×C31—21no—
240C2×C4×C31—41no—
241C2×C4×C31—41no—
242C2×C4×C31—42no—
243C2×C4×C31—42no—
244C2×C4×C31—42no—
245C2×C4×C31—41no—
246C2×C4×C31—41no—
247C2×C4×C31—41no—
248C2×C4×C31—41no—
249C2×C4×C31—41no—
250C2×C4×C31—41no—
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—41no—
258C2×C4×C31—82no—
259C2×C4×C31—81no—
260C2×C4×C31—82no—
261C2×C4×C31—81no—
262C2×C4×C31#12012yes—
263C2×C4×C31#12011yescharacteristic
264C2×C4×C31#12011yescharacteristic
265C2×C4×C31C22×D412yes—
266C8×C31—42no—
267C8×C31—42no—
268#3491C2×C3111yescharacteristic
269#3108C2×C3112yescomplemented
270#2599C2×C3111yescharacteristic, complemented
271#1609C2×C3112yescomplemented
272#1608C2×C3111yescharacteristic, complemented
273#1607C2×C3111yescharacteristic
274#1603C2×C3112yescomplemented
275#1602C2×C3111yescharacteristic, complemented
276#1601C2×C3111yescharacteristic
277#1521C2×C3112yescomplemented
278#1458C2×C3111yescharacteristic, complemented
279C22×C31—21no—
280C22×C31—21no—
281C22×C31—22no—
282C22×C31—42no—
283C22×C31—41no—
284C22×C31—41no—
285C22×C31—42no—
286C22×C31—42no—
287C22×C31—42no—
288C22×C31—81no—
289C22×C31—162no—
290C22×C31C2×#12011yescharacteristic
291C22×C31C2×#13511yescharacteristic
292C22×C31#57511yescharacteristic
293C4×C31—21no—
294C4×C31—21no—
295C4×C31—22no—
296C4×C31—22no—
297C4×C31—41no—
298C4×C31—42no—
299C4×C31—41no—
300C4×C31—42no—
301C4×C31—42no—
302C4×C31—41no—
303C4×C31—42no—
304C4×C31—81no—
305C4×C31—81no—
306C4×C31—81no—
307#552—22no—
308#552—22no—
309#551—22no—
310#551—22no—
311#543C22×C3111yescharacteristic
312#539—21no—
313#539C22×C3112yes—
314#534C22×C3111yescharacteristic
315#529—21no—
316#529C22×C3112yes—
317C2×#123—21no—
318#518—21no—
319C2×C4×Q8C22×C3111yescharacteristic, complemented
320C2×#117C22×C3111yescharacteristic
321#483C22×C3112yescomplemented
322#480C22×C3112yes—
323#479—22no—
324#464—22no—
325#451C22×C3112yescomplemented
326#448C22×C3112yes—
327C4×Q16—22no—
328C4×SD16—22no—
329#431C22×C3112yescomplemented
330#398C22×C3111yescharacteristic, complemented
331#397C22×C3111yescharacteristic
332#395C22×C3111yescharacteristic, complemented
333#393C22×C3111yescharacteristic
334#388—21no—
335#387—21no—
336#387—21no—
337#385—21no—
338#383C22×C3111yescharacteristic, complemented
339#328C22×C3112yescomplemented
340#328C22×C3112yescomplemented
341#327C22×C3112yes—
342#327C22×C3112yes—
343#324C22×C3112yescomplemented
344#323C22×C3111yescharacteristic
345#323C22×C3111yescharacteristic
346C2×C31—21no—
347C2×C31—21no—
348C2×C31—42no—
349C2×C31—82no—
350C2×C31#300211yescharacteristic
351C2×C31#302211yescharacteristic
352C2×C31#302511yescharacteristic
353C22×Q8C23×C3111yescharacteristic
354C22×D4—21no—
355C23×C4D4×C3111yescharacteristic
356C2×Q16—22no—
357C2×SD16—22no—
358#126—22no—
359#126—22no—
360#126—22no—
361#126—22no—
362#125—21no—
363#125—22no—
364#125—22no—
365#125D4×C3112yes—
366#124—21no—
367#124D4×C3112yes—
368#123—22no—
369#123—22no—
370#123—22no—
371#123—22no—
372#123—22no—
373#123—22no—
374#123—22no—
375C22⋊Q8—22no—
376C22⋊Q8—22no—
377C22⋊Q8—22no—
378C22⋊Q8—21no—
379C22⋊Q8—22no—
380C22⋊Q8C23×C3112yes—
381#121—21no—
382#121C23×C3112yes—
383#120—22no—
384C4×Q8—21no—
385C4×Q8—22no—
386C4×Q8—22no—
387C4×Q8D4×C3112yes—
388C4×D4—22no—
389C4×D4—22no—
390C4×D4—21no—
391C4×D4—22no—
392C4×D4—22no—
393C4×D4D4×C3112yes—
394#117—22no—
395#117—22no—
396#117—22no—
397#117—22no—
398#117—22no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—21no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4D4×C3111yescharacteristic
405C2×C4⋊C4C23×C3111yescharacteristic
406C2×C4⋊C4C23×C3111yescharacteristic
407C2×C22⋊C4—22no—
408C2×C22⋊C4—21no—
409C2×C22⋊C4—21no—
410C2×C22⋊C4—21no—
411C2×C22⋊C4—21no—
412C2×C22⋊C4—21no—
413C2×C22⋊C4D4×C3111yescharacteristic
414C2×C22⋊C4C23×C3111yescharacteristic
415C2×C42—21no—
416C2×C42D4×C3111yescharacteristic
417C2×C42C23×C3111yescharacteristic
418#107—22no—
419#106—22no—
420C4⋊C8—22no—
421#103—22no—
422#103—22no—
423#103—22no—
424#102—22no—
425C22⋊C8C23×C3112yes—
426C22⋊C8C23×C3112yes—
427C4×C8—22no—
428#95—21no—
429#95—21no—
430#95—21no—
431#95—21no—
432#95—21no—
433#95—21no—
434#95C23×C3111yescharacteristic
435#95C23×C3111yescharacteristic
436C31#1820911yescharacteristic, direct product factor, Sylow
437C24—21no—
438C24C2×D4×C3111yescharacteristic
439C2×Q8—21no—
440C2×Q8—22no—
441C2×Q8—42no—
442C2×Q8C2×D4×C3112yes—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—21no—
446C2×D4—42no—
447C2×D4—42no—
448C2×D4C2×D4×C3112yes—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—41no—
462C22×C4C2×D4×C3111yescharacteristic
463C22×C4C2×D4×C3111yescharacteristic
464C22×C4C2×D4×C3111yescharacteristic
465C22×C4C2×D4×C3111yescharacteristic
466C22×C4C2×D4×C3111yescharacteristic
467C22×C4C24×C3111yescharacteristic
468Q16—44no—
469SD16—44no—
470C2×C8—22no—
471C2×C8—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—41no—
488C4⋊C4—41no—
489C4⋊C4C2×D4×C3112yes—
490C4⋊C4C2×D4×C3112yes—
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—42no—
504C22⋊C4—42no—
505C42—22no—
506C42—21no—
507C42—21no—
508C42—22no—
509C42—42no—
510C42C2×D4×C3112yes—
511C42C2×D4×C3112yes—
512C23—21no—
513C23—21no—
514C23—22no—
515C23—41no—
516C23—41no—
517C23—42no—
518C23—81no—
519C23—81no—
520C23C31×#13512yes—
521C23C22×D4×C3111yescharacteristic
522Q8—24no—
523Q8—42no—
524Q8—42no—
525Q8—81no—
526D4—42no—
527D4—44no—
528D4—42no—
529D4—42no—
530D4—81no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—82no—
561C2×C4—81no—
562C2×C4—82no—
563C2×C4—81no—
564C2×C4C31×#12012yes—
565C2×C4C31×#12011yescharacteristic
566C2×C4C31×#12011yescharacteristic
567C2×C4C22×D4×C3112yes—
568C8—42no—
569C8—42no—
570C22—21no—
571C22—21no—
572C22—22no—
573C22—42no—
574C22—41no—
575C22—41no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—81no—
580C22—162no—
581C22C2×C31×#12011yescharacteristic
582C22C2×C31×#13511yescharacteristic
583C22C31×#57511yescharacteristic
584C4—21no—
585C4—21no—
586C4—22no—
587C4—22no—
588C4—41no—
589C4—42no—
590C4—41no—
591C4—42no—
592C4—42no—
593C4—41no—
594C4—42no—
595C4—81no—
596C4—81no—
597C4—81no—
598C2—21no—
599C2—21no—
600C2—42no—
601C2—82no—
602C2C31×#300211yescharacteristic
603C2C31×#302211yescharacteristic
604C2C31×#302511yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#534C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#3108C2semidirect
3. C31×#2599C2semidirect
4. C31×#1609C2semidirect
5. C31×#1608C2semidirect
7. C31×#1603C2semidirect
8. C31×#1602C2semidirect
10. C31×#1521C2semidirect
11. C31×#1458C2semidirect
24. C2×C4×Q8×C31C22semidirect
26. C31×#483C22semidirect
30. C31×#451C22semidirect
34. C31×#431C22semidirect
35. C31×#398C22semidirect
37. C31×#395C22semidirect
43. C31×#383C22semidirect
44. C31×#328C22semidirect
45. C31×#328C22semidirect
48. C31×#324C22semidirect
209. #18209C31direct (N × H)
269. #3108C2×C31semidirect
270. #2599C2×C31semidirect
271. #1609C2×C31semidirect
272. #1608C2×C31semidirect
274. #1603C2×C31semidirect
275. #1602C2×C31semidirect
277. #1521C2×C31semidirect
278. #1458C2×C31semidirect
319. C2×C4×Q8C22×C31semidirect
321. #483C22×C31semidirect
325. #451C22×C31semidirect
329. #431C22×C31semidirect
330. #398C22×C31semidirect
332. #395C22×C31semidirect
338. #383C22×C31semidirect
339. #328C22×C31semidirect
340. #328C22×C31semidirect
343. #324C22×C31semidirect
436. C31#18209direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#34912
C31×#31082
C31×#25992
C31×#16092
C31×#16082
C31×#16072
C31×#16032
C31×#16022
C31×#16012
C31×#15212
C31×#14582
#1820931