Finite Group Databasebeta

← C31×#18437 #21475196

Subgroups — C31×#18437 #21475196

Order 7936.

1028 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3558C211yesmaximal, characteristic
2C31×#3239C211yesmaximal, characteristic, complemented
3C31×#3227C211yesmaximal, characteristic, complemented
4C31×#2658C211yesmaximal, characteristic, complemented
5C31×#1618C214yesmaximal, complemented
6C31×#1613C214yesmaximal, complemented
7C31×#1505C211yesmaximal, characteristic, complemented
8C31×#1500C211yesmaximal, characteristic, complemented
9C31×#1469C211yesmaximal, characteristic, complemented
10C31×#562—21no—
11C31×H(3,𝔽4)—21no—
12C31×#559—21no—
13C31×#548—21no—
14C31×#546—21no—
15C31×#546—22no—
16D42×C31—21no—
17C31×#542—21no—
18C31×#538—22no—
19C31×#538C2212yes—
20C31×#535—21no—
21C31×#534C2212yes—
22C31×#532C2211yescharacteristic
23C2×C31×C4⋊Q8—21no—
24C2×C31×C4⋊Q8C2211yescharacteristic, complemented
25C31×#529C2211yescharacteristic
26C2×C31×#124—21no—
27C2×C31×#124C2211yescharacteristic
28C31×#501—21no—
29C31×#498—21no—
30C31×#489—21no—
31C31×#486—21no—
32C31×#484C2212yescomplemented
33C31×#481C2212yes—
34C31×#481C2214yescomplemented
35C31×#476—22no—
36C31×#475—21no—
37C31×#474—21no—
38C31×#432C2211yescharacteristic, complemented
39C31×#431C2211yescharacteristic, complemented
40C31×#399—22no—
41C31×#399C2214yescomplemented
42C31×#376C2211yescharacteristic, complemented
43C31×#328C2212yescomplemented
44C31×#327C2212yes—
45C31×#327C2214yescomplemented
46C31×#323C2214yescomplemented
47C2×#41×C31—21no—
48C2×#41×C31—21no—
49C22×Q8×C31—21no—
50C22×Q8×C31D411yescharacteristic
51C22×D4×C31—21no—
52C22×D4×C31—22no—
53C22×D4×C31D411yescharacteristic
54C31×C4⋊Q8—21no—
55C31×C4⋊Q8—21no—
56C31×C4⋊Q8—21no—
57C31×C4⋊Q8—21no—
58C31×C4⋊Q8—44no—
59C31×C4⋊Q8D411yescharacteristic
60C31×#127—21no—
61C31×#127D411yescharacteristic
62C31×#126—22no—
63C31×#126—21no—
64C31×#125D411yescharacteristic
65C31×#124—21no—
66C31×#124—22no—
67C31×#124—22no—
68C31×#124—22no—
69C31×#124—21no—
70C31×#124—21no—
71C31×#124—21no—
72C31×#124—21no—
73C31×#124—21no—
74C31×#124—44no—
75C31×#124D411yescharacteristic
76C31×#123—22no—
77C31×#123—21no—
78C31×#123—22no—
79C31×#123—22no—
80C31×#123—22no—
81C31×C22⋊Q8—22no—
82C31×C22⋊Q8—21no—
83C31×C22⋊Q8—22no—
84C31×C22⋊Q8C2311yescharacteristic
85C31×#121—22no—
86C31×#121—22no—
87C31×#121—22no—
88C31×#121—21no—
89C31×#121—22no—
90C31×#121—21no—
91C31×#121—22no—
92C31×#121C2311yescharacteristic
93C31×#121C2312yes—
94C31×#120—22no—
95C31×#120—22no—
96C31×#120—22no—
97C31×#120—22no—
98C31×#120—22no—
99C4×Q8×C31—21no—
100C4×Q8×C31D411yescharacteristic
101C4×D4×C31—22no—
102C4×D4×C31—21no—
103C4×D4×C31—21no—
104C4×D4×C31—22no—
105C4×D4×C31—21no—
106C4×D4×C31D411yescharacteristic
107C4×D4×C31D412yes—
108C31×#117—21no—
109C2×C4⋊C4×C31—21no—
110C2×C4⋊C4×C31—24no—
111C2×C4⋊C4×C31C2312yes—
112C2×C22⋊C4×C31—22no—
113C2×C22⋊C4×C31—24no—
114C2×C22⋊C4×C31D412yes—
115C2×C22⋊C4×C31C2312yes—
116C2×C42×C31—21no—
117C2×C42×C31C2311yescharacteristic
118C31×#107—22no—
119C31×#106—22no—
120C31×#106—24no—
121C31×C4⋊C8—21no—
122C31×C4⋊C8—21no—
123C31×#103—22no—
124C31×#102—22no—
125C31×#102—24no—
126C31×C22⋊C8C2312yes—
127C31×C22⋊C8C2312yes—
128C31×#97—21no—
129C4×C8×C31—21no—
130C31×#95—21no—
131C31×#95C2312yes—
132C24×C31—22no—
133C24×C31C2×D412yes—
134#41×C31—82no—
135#41×C31—82no—
136C2×Q8×C31—21no—
137C2×Q8×C31—21no—
138C2×Q8×C31—21no—
139C2×Q8×C31—42no—
140C2×Q8×C31—42no—
141C2×Q8×C31—42no—
142C2×Q8×C31—41no—
143C2×Q8×C31—41no—
144C2×Q8×C31C2×D411yescharacteristic
145C2×D4×C31—22no—
146C2×D4×C31—21no—
147C2×D4×C31—22no—
148C2×D4×C31—22no—
149C2×D4×C31—22no—
150C2×D4×C31—22no—
151C2×D4×C31—22no—
152C2×D4×C31—21no—
153C2×D4×C31—21no—
154C2×D4×C31—21no—
155C2×D4×C31—21no—
156C2×D4×C31—42no—
157C2×D4×C31—82no—
158C2×D4×C31—84no—
159C2×D4×C31C2×D411yescharacteristic
160C2×D4×C31C2×D412yes—
161C22×C4×C31—22no—
162C22×C4×C31—22no—
163C22×C4×C31—22no—
164C22×C4×C31—21no—
165C22×C4×C31—22no—
166C22×C4×C31—22no—
167C22×C4×C31C2×D411yescharacteristic
168C22×C4×C31C2×D411yescharacteristic
169C22×C4×C31C2×D412yes—
170C22×C4×C31C2411yescharacteristic
171C2×C8×C31—22no—
172C2×C8×C31—22no—
173C4⋊C4×C31—22no—
174C4⋊C4×C31—22no—
175C4⋊C4×C31—21no—
176C4⋊C4×C31—21no—
177C4⋊C4×C31—21no—
178C4⋊C4×C31—21no—
179C4⋊C4×C31—21no—
180C4⋊C4×C31—22no—
181C4⋊C4×C31—22no—
182C4⋊C4×C31—48no—
183C4⋊C4×C31—41no—
184C4⋊C4×C31—41no—
185C4⋊C4×C31C2×D412yes—
186C4⋊C4×C31C2×D411yescharacteristic
187C4⋊C4×C31C2×D411yescharacteristic
188C22⋊C4×C31—22no—
189C22⋊C4×C31—22no—
190C22⋊C4×C31—22no—
191C22⋊C4×C31—22no—
192C22⋊C4×C31—22no—
193C22⋊C4×C31—21no—
194C22⋊C4×C31—21no—
195C22⋊C4×C31—22no—
196C22⋊C4×C31—22no—
197C22⋊C4×C31—21no—
198C22⋊C4×C31—21no—
199C22⋊C4×C31—22no—
200C22⋊C4×C31—22no—
201C22⋊C4×C31—22no—
202C22⋊C4×C31—21no—
203C22⋊C4×C31—22no—
204C22⋊C4×C31—48no—
205C42×C31—24no—
206C42×C31—22no—
207C42×C31—21no—
208C42×C31C2×D411yescharacteristic
209C42×C31C2×D411yescharacteristic
210C42×C31C2×D411yescharacteristic
211C42×C31C2×D411yescharacteristic
212#18437C3111yesmaximal, characteristic, direct product factor, Sylow
213C23×C31—22no—
214C23×C31—21no—
215C23×C31—22no—
216C23×C31—24no—
217C23×C31—42no—
218C23×C31—42no—
219C23×C31—82no—
220C23×C31—82no—
221C23×C31—82no—
222C23×C31C22×D411yescharacteristic
223Q8×C31—41no—
224Q8×C31—42no—
225Q8×C31—81no—
226Q8×C31—81no—
227Q8×C31—82no—
228Q8×C31—82no—
229D4×C31—42no—
230D4×C31—41no—
231D4×C31—42no—
232D4×C31—44no—
233D4×C31—42no—
234D4×C31—42no—
235D4×C31—81no—
236D4×C31—81no—
237D4×C31—82no—
238D4×C31—82no—
239D4×C31—82no—
240D4×C31—82no—
241D4×C31—82no—
242C2×C4×C31—22no—
243C2×C4×C31—22no—
244C2×C4×C31—24no—
245C2×C4×C31—21no—
246C2×C4×C31—22no—
247C2×C4×C31—22no—
248C2×C4×C31—21no—
249C2×C4×C31—21no—
250C2×C4×C31—41no—
251C2×C4×C31—42no—
252C2×C4×C31—42no—
253C2×C4×C31—42no—
254C2×C4×C31—42no—
255C2×C4×C31—42no—
256C2×C4×C31—42no—
257C2×C4×C31—42no—
258C2×C4×C31—82no—
259C2×C4×C31—82no—
260C2×C4×C31—82no—
261C2×C4×C31#12011yescharacteristic
262C2×C4×C31#12011yescharacteristic
263C2×C4×C31#12011yescharacteristic
264C2×C4×C31#12011yescharacteristic
265C2×C4×C31C22×D411yescharacteristic
266C2×C4×C31C22×D411yescharacteristic
267C8×C31—42no—
268C8×C31—42no—
269#3558C2×C3111yescharacteristic
270#3239C2×C3111yescharacteristic, complemented
271#3227C2×C3111yescharacteristic, complemented
272#2658C2×C3111yescharacteristic, complemented
273#1618C2×C3114yescomplemented
274#1613C2×C3114yescomplemented
275#1505C2×C3111yescharacteristic, complemented
276#1500C2×C3111yescharacteristic, complemented
277#1469C2×C3111yescharacteristic, complemented
278C22×C31—21no—
279C22×C31—21no—
280C22×C31—42no—
281C22×C31—42no—
282C22×C31—41no—
283C22×C31—41no—
284C22×C31—42no—
285C22×C31—42no—
286C22×C31—42no—
287C22×C31—41no—
288C22×C31—42no—
289C22×C31—84no—
290C22×C31—164no—
291C22×C31C2×#12011yescharacteristic
292C22×C31#57712yes—
293C4×C31—21no—
294C4×C31—21no—
295C4×C31—21no—
296C4×C31—21no—
297C4×C31—21no—
298C4×C31—44no—
299C4×C31—41no—
300C4×C31—42no—
301C4×C31—41no—
302C4×C31—42no—
303C4×C31—41no—
304C4×C31—82no—
305C4×C31#45312yes—
306#562—21no—
307H(3,𝔽4)—21no—
308#559—21no—
309#548—21no—
310#546—21no—
311#546—22no—
312D42—21no—
313#542—21no—
314#538—22no—
315#538C22×C3112yes—
316#535—21no—
317#534C22×C3112yes—
318#532C22×C3111yescharacteristic
319C2×C4⋊Q8—21no—
320C2×C4⋊Q8C22×C3111yescharacteristic, complemented
321#529C22×C3111yescharacteristic
322C2×#124—21no—
323C2×#124C22×C3111yescharacteristic
324#501—21no—
325#498—21no—
326#489—21no—
327#486—21no—
328#484C22×C3112yescomplemented
329#481C22×C3112yes—
330#481C22×C3114yescomplemented
331#476—22no—
332#475—21no—
333#474—21no—
334#432C22×C3111yescharacteristic, complemented
335#431C22×C3111yescharacteristic, complemented
336#399—22no—
337#399C22×C3114yescomplemented
338#376C22×C3111yescharacteristic, complemented
339#328C22×C3112yescomplemented
340#327C22×C3112yes—
341#327C22×C3114yescomplemented
342#323C22×C3114yescomplemented
343C2×C31—22no—
344C2×C31—82no—
345C2×C31—82no—
346C2×C31—81no—
347C2×C31#300911yescharacteristic
348C2×C31C2×#45311yescharacteristic
349C2×C31#302811yescharacteristic
350C2×#41—21no—
351C2×#41—21no—
352C22×Q8—21no—
353C22×Q8D4×C3111yescharacteristic
354C22×D4—21no—
355C22×D4—22no—
356C22×D4D4×C3111yescharacteristic
357C4⋊Q8—21no—
358C4⋊Q8—21no—
359C4⋊Q8—21no—
360C4⋊Q8—21no—
361C4⋊Q8—44no—
362C4⋊Q8D4×C3111yescharacteristic
363#127—21no—
364#127D4×C3111yescharacteristic
365#126—22no—
366#126—21no—
367#125D4×C3111yescharacteristic
368#124—21no—
369#124—22no—
370#124—22no—
371#124—22no—
372#124—21no—
373#124—21no—
374#124—21no—
375#124—21no—
376#124—21no—
377#124—44no—
378#124D4×C3111yescharacteristic
379#123—22no—
380#123—21no—
381#123—22no—
382#123—22no—
383#123—22no—
384C22⋊Q8—22no—
385C22⋊Q8—21no—
386C22⋊Q8—22no—
387C22⋊Q8C23×C3111yescharacteristic
388#121—22no—
389#121—22no—
390#121—22no—
391#121—21no—
392#121—22no—
393#121—21no—
394#121—22no—
395#121C23×C3111yescharacteristic
396#121C23×C3112yes—
397#120—22no—
398#120—22no—
399#120—22no—
400#120—22no—
401#120—22no—
402C4×Q8—21no—
403C4×Q8D4×C3111yescharacteristic
404C4×D4—22no—
405C4×D4—21no—
406C4×D4—21no—
407C4×D4—22no—
408C4×D4—21no—
409C4×D4D4×C3111yescharacteristic
410C4×D4D4×C3112yes—
411#117—21no—
412C2×C4⋊C4—24no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4C23×C3112yes—
415C2×C22⋊C4—22no—
416C2×C22⋊C4—24no—
417C2×C22⋊C4D4×C3112yes—
418C2×C22⋊C4C23×C3112yes—
419C2×C42—21no—
420C2×C42C23×C3111yescharacteristic
421#107—22no—
422#106—22no—
423#106—24no—
424C4⋊C8—21no—
425C4⋊C8—21no—
426#103—22no—
427#102—22no—
428#102—24no—
429C22⋊C8C23×C3112yes—
430C22⋊C8C23×C3112yes—
431#97—21no—
432C4×C8—21no—
433#95—21no—
434#95C23×C3112yes—
435C31#1843711yescharacteristic, direct product factor, Sylow
436C24—22no—
437C24C2×D4×C3112yes—
438#41—82no—
439#41—82no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—42no—
444C2×Q8—42no—
445C2×Q8—42no—
446C2×Q8—41no—
447C2×Q8—41no—
448C2×Q8C2×D4×C3111yescharacteristic
449C2×D4—22no—
450C2×D4—21no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—22no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—42no—
461C2×D4—82no—
462C2×D4—84no—
463C2×D4C2×D4×C3111yescharacteristic
464C2×D4C2×D4×C3112yes—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—21no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4C2×D4×C3111yescharacteristic
472C22×C4C2×D4×C3111yescharacteristic
473C22×C4C2×D4×C3112yes—
474C22×C4C24×C3111yescharacteristic
475C2×C8—22no—
476C2×C8—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—48no—
487C4⋊C4—41no—
488C4⋊C4—41no—
489C4⋊C4C2×D4×C3112yes—
490C4⋊C4C2×D4×C3111yescharacteristic
491C4⋊C4C2×D4×C3111yescharacteristic
492C22⋊C4—22no—
493C22⋊C4—22no—
494C22⋊C4—22no—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—21no—
498C22⋊C4—21no—
499C22⋊C4—22no—
500C22⋊C4—22no—
501C22⋊C4—21no—
502C22⋊C4—21no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—21no—
507C22⋊C4—22no—
508C22⋊C4—48no—
509C42—24no—
510C42—22no—
511C42—21no—
512C42C2×D4×C3111yescharacteristic
513C42C2×D4×C3111yescharacteristic
514C42C2×D4×C3111yescharacteristic
515C42C2×D4×C3111yescharacteristic
516C23—22no—
517C23—21no—
518C23—22no—
519C23—24no—
520C23—42no—
521C23—42no—
522C23—82no—
523C23—82no—
524C23—82no—
525C23C22×D4×C3111yescharacteristic
526Q8—41no—
527Q8—42no—
528Q8—81no—
529Q8—81no—
530Q8—82no—
531Q8—82no—
532D4—42no—
533D4—41no—
534D4—42no—
535D4—44no—
536D4—42no—
537D4—42no—
538D4—81no—
539D4—81no—
540D4—82no—
541D4—82no—
542D4—82no—
543D4—82no—
544D4—82no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—82no—
562C2×C4—82no—
563C2×C4—82no—
564C2×C4C31×#12011yescharacteristic
565C2×C4C31×#12011yescharacteristic
566C2×C4C31×#12011yescharacteristic
567C2×C4C31×#12011yescharacteristic
568C2×C4C22×D4×C3111yescharacteristic
569C2×C4C22×D4×C3111yescharacteristic
570C8—42no—
571C8—42no—
572C22—21no—
573C22—21no—
574C22—42no—
575C22—42no—
576C22—41no—
577C22—41no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—41no—
582C22—42no—
583C22—84no—
584C22—164no—
585C22C2×C31×#12011yescharacteristic
586C22C31×#57712yes—
587C4—21no—
588C4—21no—
589C4—21no—
590C4—21no—
591C4—21no—
592C4—44no—
593C4—41no—
594C4—42no—
595C4—41no—
596C4—42no—
597C4—41no—
598C4—82no—
599C4C31×#45312yes—
600C2—22no—
601C2—82no—
602C2—81no—
603C2—82no—
604C2C31×#300911yescharacteristic
605C2C2×C31×#45311yescharacteristic
606C2C31×#302811yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#3558C2
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×#3239C2semidirect
3. C31×#3227C2semidirect
4. C31×#2658C2semidirect
5. C31×#1618C2semidirect
6. C31×#1613C2semidirect
7. C31×#1505C2semidirect
8. C31×#1500C2semidirect
9. C31×#1469C2semidirect
24. C2×C31×C4⋊Q8C22semidirect
32. C31×#484C22semidirect
34. C31×#481C22semidirect
38. C31×#432C22semidirect
39. C31×#431C22semidirect
41. C31×#399C22semidirect
42. C31×#376C22semidirect
43. C31×#328C22semidirect
45. C31×#327C22semidirect
46. C31×#323C22semidirect
212. #18437C31direct (N × H)
270. #3239C2×C31semidirect
271. #3227C2×C31semidirect
272. #2658C2×C31semidirect
273. #1618C2×C31semidirect
274. #1613C2×C31semidirect
275. #1505C2×C31semidirect
276. #1500C2×C31semidirect
277. #1469C2×C31semidirect
320. C2×C4⋊Q8C22×C31semidirect
328. #484C22×C31semidirect
330. #481C22×C31semidirect
334. #432C22×C31semidirect
335. #431C22×C31semidirect
337. #399C22×C31semidirect
338. #376C22×C31semidirect
339. #328C22×C31semidirect
341. #327C22×C31semidirect
342. #323C22×C31semidirect
435. C31#18437direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35582
C31×#32392
C31×#32272
C31×#26582
C31×#16182
C31×#16132
C31×#15052
C31×#15002
C31×#14692
#1843731