Finite Group Databasebeta

← C31×#19021 #21475780

Subgroups — C31×#19021 #21475780

Order 7936.

1012 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3558C211yesmaximal, characteristic
2C31×#3239C211yesmaximal, characteristic, complemented
3C2×C31×#497C211yesmaximal, characteristic, complemented
4C31×#3141C211yesmaximal, characteristic, complemented
5C31×#1661C214yesmaximal, complemented
6C31×#1659C214yesmaximal, complemented
7C31×#1530C211yesmaximal, characteristic, complemented
8C31×#1527C211yesmaximal, characteristic, complemented
9C31×#1524C211yesmaximal, characteristic, complemented
10C22×Q16×C31—21no—
11C22×SD16×C31—21no—
12C31×#562C2212yes—
13C31×H(3,𝔽4)C2212yes—
14C31×#559—21no—
15C31×#548—21no—
16C31×#546—22no—
17C31×#546—21no—
18D42×C31—21no—
19C31×#542—21no—
20C31×#538—22no—
21C31×#538—21no—
22C31×#535—21no—
23C31×#534—21no—
24C31×#532C2211yescharacteristic
25C2×C31×C4⋊Q8C2211yescharacteristic, complemented
26C31×#529C2211yescharacteristic
27C2×C31×#124C2211yescharacteristic
28C31×#501C2212yescomplemented
29C31×#497—22no—
30C31×#497C2214yescomplemented
31C31×#489C2214yescomplemented
32C31×#489C2212yescomplemented
33C31×#487—21no—
34C31×#486—21no—
35C31×#484—21no—
36C31×#481—21no—
37C31×#475—21no—
38C31×#474—21no—
39C31×#432C2211yescharacteristic, complemented
40C31×#417—22no—
41C4×M16×C31C2211yescharacteristic, complemented
42C2×C31×#97C2211yescharacteristic, complemented
43C31×#330C2212yescomplemented
44C31×#329C2212yescomplemented
45C31×#329C2214yescomplemented
46C31×#329C2214yescomplemented
47C2×#41×C31—21no—
48C2×#41×C31D412yes—
49C22×Q8×C31—21no—
50C22×Q8×C31D411yescharacteristic
51C22×D4×C31—22no—
52C22×D4×C31—21no—
53C22×D4×C31D411yescharacteristic
54C2×Q16×C31—24no—
55C2×Q16×C31—44no—
56C2×SD16×C31—24no—
57C2×SD16×C31—44no—
58C2×M16×C31—21no—
59C22×C8×C31—21no—
60C31×C4⋊Q8—21no—
61C31×C4⋊Q8—21no—
62C31×C4⋊Q8—21no—
63C31×C4⋊Q8D411yescharacteristic
64C31×C4⋊Q8C2312yes—
65C31×#127—21no—
66C31×#127D411yescharacteristic
67C31×#126—22no—
68C31×#126—21no—
69C31×#125C2311yescharacteristic
70C31×#124—21no—
71C31×#124—22no—
72C31×#124—22no—
73C31×#124—22no—
74C31×#124—21no—
75C31×#124—21no—
76C31×#124D412yes—
77C31×#124C2311yescharacteristic
78C31×#124C2312yes—
79C31×#124C2312yes—
80C31×#123—22no—
81C31×#123—22no—
82C31×#123—21no—
83C31×#123—22no—
84C31×#123—22no—
85C31×C22⋊Q8—21no—
86C31×C22⋊Q8—22no—
87C31×C22⋊Q8—22no—
88C31×C22⋊Q8D411yescharacteristic
89C31×#121—22no—
90C31×#121—22no—
91C31×#121—22no—
92C31×#121—21no—
93C31×#121—22no—
94C31×#121—22no—
95C31×#121—21no—
96C31×#121—21no—
97C31×#121D411yescharacteristic
98C31×#120—22no—
99C31×#120—22no—
100C31×#120—22no—
101C31×#120—22no—
102C31×#120—22no—
103C4×Q8×C31—21no—
104C4×Q8×C31D411yescharacteristic
105C4×D4×C31—22no—
106C4×D4×C31—22no—
107C4×D4×C31—21no—
108C4×D4×C31—21no—
109C4×D4×C31—21no—
110C4×D4×C31—21no—
111C4×D4×C31D411yescharacteristic
112C31×#117—21no—
113C2×C4⋊C4×C31—21no—
114C2×C4⋊C4×C31—21no—
115C2×C22⋊C4×C31—22no—
116C2×C22⋊C4×C31—21no—
117C2×C22⋊C4×C31—21no—
118C2×C42×C31C2311yescharacteristic
119C31×#107—22no—
120C31×#106—22no—
121C31×C4⋊C8—21no—
122C31×#103—24no—
123C31×#103—22no—
124C31×#102—24no—
125C31×#102—22no—
126C31×C22⋊C8—21no—
127C31×#97—21no—
128C31×#97C2312yes—
129C31×#97C2312yes—
130C31×#97C2312yes—
131C4×C8×C31—21no—
132C24×C31—22no—
133C24×C31—21no—
134#41×C31—44no—
135#41×C31—82no—
136C2×Q8×C31—21no—
137C2×Q8×C31—22no—
138C2×Q8×C31—42no—
139C2×Q8×C31—42no—
140C2×Q8×C31—42no—
141C2×Q8×C31—41no—
142C2×Q8×C31C2×D411yescharacteristic
143C2×Q8×C31C2×D412yes—
144C2×Q8×C31C2×D412yes—
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—42no—
156C2×D4×C31—82no—
157C2×D4×C31—84no—
158C2×D4×C31C2×D411yescharacteristic
159C2×D4×C31C2×D412yes—
160C2×D4×C31C2×D412yes—
161C22×C4×C31—22no—
162C22×C4×C31—22no—
163C22×C4×C31—21no—
164C22×C4×C31—21no—
165C22×C4×C31—22no—
166C22×C4×C31—41no—
167C22×C4×C31C2×D411yescharacteristic
168C22×C4×C31C2×D411yescharacteristic
169C22×C4×C31C2×D411yescharacteristic
170Q16×C31—48no—
171SD16×C31—48no—
172M16×C31—42no—
173C2×C8×C31—24no—
174C2×C8×C31—22no—
175C2×C8×C31—22no—
176C2×C8×C31—22no—
177C4⋊C4×C31—22no—
178C4⋊C4×C31—22no—
179C4⋊C4×C31—21no—
180C4⋊C4×C31—21no—
181C4⋊C4×C31—21no—
182C4⋊C4×C31—21no—
183C4⋊C4×C31—21no—
184C4⋊C4×C31—21no—
185C4⋊C4×C31—22no—
186C4⋊C4×C31—22no—
187C4⋊C4×C31—41no—
188C4⋊C4×C31—41no—
189C4⋊C4×C31C2×D411yescharacteristic
190C4⋊C4×C31C2×D411yescharacteristic
191C22⋊C4×C31—22no—
192C22⋊C4×C31—22no—
193C22⋊C4×C31—22no—
194C22⋊C4×C31—22no—
195C22⋊C4×C31—22no—
196C22⋊C4×C31—21no—
197C22⋊C4×C31—22no—
198C22⋊C4×C31—21no—
199C22⋊C4×C31—22no—
200C22⋊C4×C31—21no—
201C22⋊C4×C31—22no—
202C22⋊C4×C31—22no—
203C22⋊C4×C31—21no—
204C22⋊C4×C31—21no—
205C22⋊C4×C31—22no—
206C22⋊C4×C31—22no—
207C42×C31—22no—
208C42×C31—21no—
209C42×C31C2×D411yescharacteristic
210C42×C31C2×D411yescharacteristic
211C42×C31C2×D411yescharacteristic
212C42×C31C2411yescharacteristic
213#19021C3111yesmaximal, characteristic, direct product factor, Sylow
214C23×C31—21no—
215C23×C31—22no—
216C23×C31—22no—
217C23×C31—42no—
218C23×C31—81no—
219C23×C31—81no—
220C23×C31—82no—
221C23×C31—82no—
222C23×C31—82no—
223C23×C31#12011yescharacteristic
224Q8×C31—42no—
225Q8×C31—41no—
226Q8×C31—42no—
227Q8×C31—42no—
228Q8×C31—42no—
229Q8×C31—42no—
230Q8×C31—82no—
231D4×C31—42no—
232D4×C31—41no—
233D4×C31—42no—
234D4×C31—42no—
235D4×C31—44no—
236D4×C31—42no—
237D4×C31—42no—
238D4×C31—81no—
239D4×C31—82no—
240D4×C31—82no—
241D4×C31—82no—
242D4×C31—82no—
243D4×C31—82no—
244C2×C4×C31—22no—
245C2×C4×C31—22no—
246C2×C4×C31—21no—
247C2×C4×C31—22no—
248C2×C4×C31—22no—
249C2×C4×C31—21no—
250C2×C4×C31—21no—
251C2×C4×C31—41no—
252C2×C4×C31—42no—
253C2×C4×C31—42no—
254C2×C4×C31—82no—
255C2×C4×C31—81no—
256C2×C4×C31—82no—
257C2×C4×C31—82no—
258C2×C4×C31—82no—
259C2×C4×C31#12011yescharacteristic
260C2×C4×C31#12011yescharacteristic
261C2×C4×C31#12011yescharacteristic
262C2×C4×C31C22×D411yescharacteristic
263C2×C4×C31C22×D411yescharacteristic
264C2×C4×C31C22×D411yescharacteristic
265C8×C31—24no—
266C8×C31—42no—
267C8×C31—42no—
268#3558C2×C3111yescharacteristic
269#3239C2×C3111yescharacteristic, complemented
270C2×#497C2×C3111yescharacteristic, complemented
271#3141C2×C3111yescharacteristic, complemented
272#1661C2×C3114yescomplemented
273#1659C2×C3114yescomplemented
274#1530C2×C3111yescharacteristic, complemented
275#1527C2×C3111yescharacteristic, complemented
276#1524C2×C3111yescharacteristic, complemented
277C22×C31—21no—
278C22×C31—21no—
279C22×C31—41no—
280C22×C31—42no—
281C22×C31—42no—
282C22×C31—42no—
283C22×C31—42no—
284C22×C31—41no—
285C22×C31—41no—
286C22×C31—42no—
287C22×C31—42no—
288C22×C31—162no—
289C22×C31—164no—
290C22×C31#45512yes—
291C22×C31C2×#12011yescharacteristic
292C4×C31—21no—
293C4×C31—21no—
294C4×C31—21no—
295C4×C31—21no—
296C4×C31—21no—
297C4×C31—41no—
298C4×C31—42no—
299C4×C31—41no—
300C4×C31—42no—
301C4×C31—41no—
302C4×C31—82no—
303C4×C31#45312yes—
304C22×Q16—21no—
305C22×SD16—21no—
306#562C22×C3112yes—
307H(3,𝔽4)C22×C3112yes—
308#559—21no—
309#548—21no—
310#546—22no—
311#546—21no—
312D42—21no—
313#542—21no—
314#538—22no—
315#538—21no—
316#535—21no—
317#534—21no—
318#532C22×C3111yescharacteristic
319C2×C4⋊Q8C22×C3111yescharacteristic, complemented
320#529C22×C3111yescharacteristic
321C2×#124C22×C3111yescharacteristic
322#501C22×C3112yescomplemented
323#497—22no—
324#497C22×C3114yescomplemented
325#489C22×C3114yescomplemented
326#489C22×C3112yescomplemented
327#487—21no—
328#486—21no—
329#484—21no—
330#481—21no—
331#475—21no—
332#474—21no—
333#432C22×C3111yescharacteristic, complemented
334#417—22no—
335C4×M16C22×C3111yescharacteristic, complemented
336C2×#97C22×C3111yescharacteristic, complemented
337#330C22×C3112yescomplemented
338#329C22×C3112yescomplemented
339#329C22×C3114yescomplemented
340#329C22×C3114yescomplemented
341C2×C31—22no—
342C2×C31—82no—
343C2×C31—81no—
344C2×C31—82no—
345C2×C31C2×#45311yescharacteristic
346C2×C31C2×#45511yescharacteristic
347C2×C31#302211yescharacteristic
348C2×#41—21no—
349C2×#41D4×C3112yes—
350C22×Q8—21no—
351C22×Q8D4×C3111yescharacteristic
352C22×D4—22no—
353C22×D4—21no—
354C22×D4D4×C3111yescharacteristic
355C2×Q16—24no—
356C2×Q16—44no—
357C2×SD16—24no—
358C2×SD16—44no—
359C2×M16—21no—
360C22×C8—21no—
361C4⋊Q8—21no—
362C4⋊Q8—21no—
363C4⋊Q8—21no—
364C4⋊Q8D4×C3111yescharacteristic
365C4⋊Q8C23×C3112yes—
366#127—21no—
367#127D4×C3111yescharacteristic
368#126—22no—
369#126—21no—
370#125C23×C3111yescharacteristic
371#124—21no—
372#124—22no—
373#124—22no—
374#124—22no—
375#124—21no—
376#124—21no—
377#124D4×C3112yes—
378#124C23×C3112yes—
379#124C23×C3111yescharacteristic
380#124C23×C3112yes—
381#123—22no—
382#123—22no—
383#123—21no—
384#123—22no—
385#123—22no—
386C22⋊Q8—21no—
387C22⋊Q8—22no—
388C22⋊Q8—22no—
389C22⋊Q8D4×C3111yescharacteristic
390#121—22no—
391#121—22no—
392#121—22no—
393#121—21no—
394#121—22no—
395#121—22no—
396#121—21no—
397#121—21no—
398#121D4×C3111yescharacteristic
399#120—22no—
400#120—22no—
401#120—22no—
402#120—22no—
403#120—22no—
404C4×Q8—21no—
405C4×Q8D4×C3111yescharacteristic
406C4×D4—22no—
407C4×D4—22no—
408C4×D4—21no—
409C4×D4—21no—
410C4×D4—21no—
411C4×D4—21no—
412C4×D4D4×C3111yescharacteristic
413#117—21no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—21no—
416C2×C22⋊C4—22no—
417C2×C22⋊C4—21no—
418C2×C22⋊C4—21no—
419C2×C42C23×C3111yescharacteristic
420#107—22no—
421#106—22no—
422C4⋊C8—21no—
423#103—24no—
424#103—22no—
425#102—24no—
426#102—22no—
427C22⋊C8—21no—
428#97—21no—
429#97C23×C3112yes—
430#97C23×C3112yes—
431#97C23×C3112yes—
432C4×C8—21no—
433C31#1902111yescharacteristic, direct product factor, Sylow
434C24—22no—
435C24—21no—
436#41—44no—
437#41—82no—
438C2×Q8—21no—
439C2×Q8—22no—
440C2×Q8—42no—
441C2×Q8—42no—
442C2×Q8—42no—
443C2×Q8—41no—
444C2×Q8C2×D4×C3111yescharacteristic
445C2×Q8C2×D4×C3112yes—
446C2×Q8C2×D4×C3112yes—
447C2×D4—22no—
448C2×D4—21no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—42no—
458C2×D4—82no—
459C2×D4—84no—
460C2×D4C2×D4×C3111yescharacteristic
461C2×D4C2×D4×C3112yes—
462C2×D4C2×D4×C3112yes—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—22no—
468C22×C4—41no—
469C22×C4C2×D4×C3111yescharacteristic
470C22×C4C2×D4×C3111yescharacteristic
471C22×C4C2×D4×C3111yescharacteristic
472Q16—48no—
473SD16—48no—
474M16—42no—
475C2×C8—24no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—41no—
490C4⋊C4—41no—
491C4⋊C4C2×D4×C3111yescharacteristic
492C4⋊C4C2×D4×C3111yescharacteristic
493C22⋊C4—22no—
494C22⋊C4—22no—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—22no—
498C22⋊C4—21no—
499C22⋊C4—22no—
500C22⋊C4—21no—
501C22⋊C4—22no—
502C22⋊C4—21no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—21no—
506C22⋊C4—21no—
507C22⋊C4—22no—
508C22⋊C4—22no—
509C42—22no—
510C42—21no—
511C42C2×D4×C3111yescharacteristic
512C42C2×D4×C3111yescharacteristic
513C42C2×D4×C3111yescharacteristic
514C42C24×C3111yescharacteristic
515C23—21no—
516C23—22no—
517C23—22no—
518C23—42no—
519C23—81no—
520C23—81no—
521C23—82no—
522C23—82no—
523C23—82no—
524C23C31×#12011yescharacteristic
525Q8—42no—
526Q8—41no—
527Q8—42no—
528Q8—42no—
529Q8—42no—
530Q8—42no—
531Q8—82no—
532D4—42no—
533D4—41no—
534D4—42no—
535D4—42no—
536D4—44no—
537D4—42no—
538D4—42no—
539D4—81no—
540D4—82no—
541D4—82no—
542D4—82no—
543D4—82no—
544D4—82no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—82no—
556C2×C4—81no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4C31×#12011yescharacteristic
561C2×C4C31×#12011yescharacteristic
562C2×C4C31×#12011yescharacteristic
563C2×C4C22×D4×C3111yescharacteristic
564C2×C4C22×D4×C3111yescharacteristic
565C2×C4C22×D4×C3111yescharacteristic
566C8—24no—
567C8—42no—
568C8—42no—
569C22—21no—
570C22—21no—
571C22—41no—
572C22—42no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—41no—
577C22—41no—
578C22—42no—
579C22—42no—
580C22—162no—
581C22—164no—
582C22C31×#45512yes—
583C22C2×C31×#12011yescharacteristic
584C4—21no—
585C4—21no—
586C4—21no—
587C4—21no—
588C4—21no—
589C4—41no—
590C4—42no—
591C4—41no—
592C4—42no—
593C4—41no—
594C4—82no—
595C4C31×#45312yes—
596C2—22no—
597C2—81no—
598C2—82no—
599C2—82no—
600C2C2×C31×#45311yescharacteristic
601C2C2×C31×#45511yescharacteristic
602C2C31×#302211yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC42C24×C31
prime radicalC31×#3558C2
LCSG ▷ C42 ▷ 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. C2×C31×#497C2semidirect
4. C31×#3141C2semidirect
5. C31×#1661C2semidirect
6. C31×#1659C2semidirect
7. C31×#1530C2semidirect
8. C31×#1527C2semidirect
9. C31×#1524C2semidirect
25. C2×C31×C4⋊Q8C22semidirect
28. C31×#501C22semidirect
30. C31×#497C22semidirect
31. C31×#489C22semidirect
32. C31×#489C22semidirect
39. C31×#432C22semidirect
41. C4×M16×C31C22semidirect
42. C2×C31×#97C22semidirect
43. C31×#330C22semidirect
44. C31×#329C22semidirect
45. C31×#329C22semidirect
46. C31×#329C22semidirect
213. #19021C31direct (N × H)
269. #3239C2×C31semidirect
270. C2×#497C2×C31semidirect
271. #3141C2×C31semidirect
272. #1661C2×C31semidirect
273. #1659C2×C31semidirect
274. #1530C2×C31semidirect
275. #1527C2×C31semidirect
276. #1524C2×C31semidirect
319. C2×C4⋊Q8C22×C31semidirect
322. #501C22×C31semidirect
324. #497C22×C31semidirect
325. #489C22×C31semidirect
326. #489C22×C31semidirect
333. #432C22×C31semidirect
335. C4×M16C22×C31semidirect
336. C2×#97C22×C31semidirect
337. #330C22×C31semidirect
338. #329C22×C31semidirect
339. #329C22×C31semidirect
340. #329C22×C31semidirect
433. C31#19021direct (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
C2×C31×#4972
C31×#31412
C31×#16612
C31×#16592
C31×#15302
C31×#15272
C31×#15242
#1902131