Finite Group Databasebeta

← C31×#19020 #21475779

Subgroups — C31×#19020 #21475779

Order 7936.

960 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3558C211yesmaximal, characteristic
2C31×#3218C211yesmaximal, characteristic, complemented
3C31×#3141C211yesmaximal, characteristic, complemented
4C2×C31×#489C211yesmaximal, characteristic, complemented
5C31×#1662C214yesmaximal, complemented
6C31×#1659C214yesmaximal, complemented
7C31×#1530C211yesmaximal, characteristic, complemented
8C31×#1527C211yesmaximal, characteristic, complemented
9C31×#1524C211yesmaximal, characteristic, complemented
10C31×#562C2212yes—
11C31×H(3,𝔽4)C2212yes—
12C31×#559—21no—
13C31×#548—21no—
14C31×#546—22no—
15C31×#546—21no—
16D42×C31—21no—
17C31×#542—21no—
18C31×#538—22no—
19C31×#538—21no—
20C31×#535—21no—
21C31×#534—21no—
22C31×#532C2211yescharacteristic
23C2×C31×C4⋊Q8C2211yescharacteristic, complemented
24C31×#529C2211yescharacteristic
25C2×C31×#124C2211yescharacteristic
26C31×#497C2212yescomplemented
27C31×#491C2212yescomplemented
28C31×#489—22no—
29C31×#489C2214yescomplemented
30C31×#489C2214yescomplemented
31C31×#487—21no—
32C31×#486—21no—
33C31×#462—21no—
34C31×#461—21no—
35C31×#451—21no—
36C31×#448—21no—
37C31×#432C2211yescharacteristic, complemented
38C31×#417—22no—
39C2×C31×#103—21no—
40C2×C31×#102—21no—
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—22no—
55C2×SD16×C31—22no—
56C2×M16×C31—21no—
57C22×C8×C31—21no—
58C31×C4⋊Q8—21no—
59C31×C4⋊Q8—21no—
60C31×C4⋊Q8—21no—
61C31×C4⋊Q8D411yescharacteristic
62C31×C4⋊Q8C2312yes—
63C31×#127—21no—
64C31×#127D411yescharacteristic
65C31×#126—22no—
66C31×#126—21no—
67C31×#125C2311yescharacteristic
68C31×#124—21no—
69C31×#124—22no—
70C31×#124—22no—
71C31×#124—22no—
72C31×#124—21no—
73C31×#124—21no—
74C31×#124D412yes—
75C31×#124C2312yes—
76C31×#124C2312yes—
77C31×#124C2311yescharacteristic
78C31×#123—22no—
79C31×#123—22no—
80C31×#123—21no—
81C31×#123—22no—
82C31×#123—22no—
83C31×C22⋊Q8—21no—
84C31×C22⋊Q8—22no—
85C31×C22⋊Q8—22no—
86C31×C22⋊Q8D411yescharacteristic
87C31×#121—22no—
88C31×#121—22no—
89C31×#121—22no—
90C31×#121—21no—
91C31×#121—22no—
92C31×#121—22no—
93C31×#121—21no—
94C31×#121—21no—
95C31×#121D411yescharacteristic
96C31×#120—22no—
97C31×#120—22no—
98C31×#120—22no—
99C31×#120—22no—
100C31×#120—22no—
101C4×Q8×C31—21no—
102C4×Q8×C31D411yescharacteristic
103C4×D4×C31—22no—
104C4×D4×C31—22no—
105C4×D4×C31—21no—
106C4×D4×C31—21no—
107C4×D4×C31—21no—
108C4×D4×C31—21no—
109C4×D4×C31D411yescharacteristic
110C31×#117—21no—
111C2×C4⋊C4×C31—21no—
112C2×C4⋊C4×C31—21no—
113C2×C22⋊C4×C31—22no—
114C2×C22⋊C4×C31—21no—
115C2×C22⋊C4×C31—21no—
116C2×C42×C31C2311yescharacteristic
117C31×C4⋊C8—21no—
118C31×#103—24no—
119C31×#103—24no—
120C31×#103—22no—
121C31×#103—22no—
122C31×#102—24no—
123C31×#102—24no—
124C31×C22⋊C8—21no—
125C31×#97—21no—
126C31×#97C2312yes—
127C31×#97C2312yes—
128C31×#97C2312yes—
129C4×C8×C31—21no—
130C24×C31—22no—
131C24×C31—21no—
132#41×C31—44no—
133#41×C31—82no—
134C2×Q8×C31—21no—
135C2×Q8×C31—22no—
136C2×Q8×C31—42no—
137C2×Q8×C31—41no—
138C2×Q8×C31—82no—
139C2×Q8×C31C2×D412yes—
140C2×Q8×C31C2×D412yes—
141C2×Q8×C31C2×D411yescharacteristic
142C2×D4×C31—22no—
143C2×D4×C31—21no—
144C2×D4×C31—22no—
145C2×D4×C31—22no—
146C2×D4×C31—22no—
147C2×D4×C31—22no—
148C2×D4×C31—22no—
149C2×D4×C31—21no—
150C2×D4×C31—21no—
151C2×D4×C31—21no—
152C2×D4×C31—42no—
153C2×D4×C31—82no—
154C2×D4×C31—84no—
155C2×D4×C31C2×D412yes—
156C2×D4×C31C2×D412yes—
157C2×D4×C31C2×D411yescharacteristic
158C22×C4×C31—22no—
159C22×C4×C31—22no—
160C22×C4×C31—21no—
161C22×C4×C31—21no—
162C22×C4×C31—22no—
163C22×C4×C31—41no—
164C22×C4×C31C2×D411yescharacteristic
165C22×C4×C31C2×D411yescharacteristic
166C22×C4×C31C2×D411yescharacteristic
167Q16×C31—82no—
168SD16×C31—82no—
169M16×C31—42no—
170C2×C8×C31—22no—
171C2×C8×C31—22no—
172C2×C8×C31—22no—
173C2×C8×C31—42no—
174C4⋊C4×C31—22no—
175C4⋊C4×C31—22no—
176C4⋊C4×C31—21no—
177C4⋊C4×C31—21no—
178C4⋊C4×C31—21no—
179C4⋊C4×C31—21no—
180C4⋊C4×C31—21no—
181C4⋊C4×C31—21no—
182C4⋊C4×C31—22no—
183C4⋊C4×C31—22no—
184C4⋊C4×C31—41no—
185C4⋊C4×C31—41no—
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—22no—
195C22⋊C4×C31—21no—
196C22⋊C4×C31—22no—
197C22⋊C4×C31—21no—
198C22⋊C4×C31—22no—
199C22⋊C4×C31—22no—
200C22⋊C4×C31—21no—
201C22⋊C4×C31—21no—
202C22⋊C4×C31—22no—
203C22⋊C4×C31—22no—
204C42×C31—22no—
205C42×C31—21no—
206C42×C31C2×D411yescharacteristic
207C42×C31C2×D411yescharacteristic
208C42×C31C2×D411yescharacteristic
209C42×C31C2411yescharacteristic
210#19020C3111yesmaximal, characteristic, direct product factor, Sylow
211C23×C31—21no—
212C23×C31—22no—
213C23×C31—22no—
214C23×C31—42no—
215C23×C31—81no—
216C23×C31—81no—
217C23×C31—82no—
218C23×C31—82no—
219C23×C31—82no—
220C23×C31#12011yescharacteristic
221Q8×C31—41no—
222Q8×C31—42no—
223Q8×C31—42no—
224Q8×C31—42no—
225Q8×C31—42no—
226Q8×C31—42no—
227Q8×C31—82no—
228D4×C31—41no—
229D4×C31—42no—
230D4×C31—42no—
231D4×C31—42no—
232D4×C31—44no—
233D4×C31—42no—
234D4×C31—42no—
235D4×C31—81no—
236D4×C31—82no—
237D4×C31—82no—
238D4×C31—82no—
239D4×C31—82no—
240D4×C31—82no—
241C2×C4×C31—22no—
242C2×C4×C31—22no—
243C2×C4×C31—21no—
244C2×C4×C31—22no—
245C2×C4×C31—22no—
246C2×C4×C31—21no—
247C2×C4×C31—21no—
248C2×C4×C31—41no—
249C2×C4×C31—42no—
250C2×C4×C31—42no—
251C2×C4×C31—82no—
252C2×C4×C31—81no—
253C2×C4×C31—82no—
254C2×C4×C31—82no—
255C2×C4×C31—82no—
256C2×C4×C31#12011yescharacteristic
257C2×C4×C31#12011yescharacteristic
258C2×C4×C31#12011yescharacteristic
259C2×C4×C31C22×D411yescharacteristic
260C2×C4×C31C22×D411yescharacteristic
261C2×C4×C31C22×D411yescharacteristic
262C8×C31—42no—
263C8×C31—42no—
264C8×C31—42no—
265#3558C2×C3111yescharacteristic
266#3218C2×C3111yescharacteristic, complemented
267#3141C2×C3111yescharacteristic, complemented
268C2×#489C2×C3111yescharacteristic, complemented
269#1662C2×C3114yescomplemented
270#1659C2×C3114yescomplemented
271#1530C2×C3111yescharacteristic, complemented
272#1527C2×C3111yescharacteristic, complemented
273#1524C2×C3111yescharacteristic, complemented
274C22×C31—21no—
275C22×C31—21no—
276C22×C31—41no—
277C22×C31—42no—
278C22×C31—42no—
279C22×C31—42no—
280C22×C31—42no—
281C22×C31—41no—
282C22×C31—41no—
283C22×C31—42no—
284C22×C31—42no—
285C22×C31—162no—
286C22×C31—164no—
287C22×C31#45512yes—
288C22×C31C2×#12011yescharacteristic
289C4×C31—21no—
290C4×C31—21no—
291C4×C31—21no—
292C4×C31—21no—
293C4×C31—21no—
294C4×C31—41no—
295C4×C31—41no—
296C4×C31—42no—
297C4×C31—42no—
298C4×C31—41no—
299C4×C31—82no—
300C4×C31#45512yes—
301#562C22×C3112yes—
302H(3,𝔽4)C22×C3112yes—
303#559—21no—
304#548—21no—
305#546—22no—
306#546—21no—
307D42—21no—
308#542—21no—
309#538—22no—
310#538—21no—
311#535—21no—
312#534—21no—
313#532C22×C3111yescharacteristic
314C2×C4⋊Q8C22×C3111yescharacteristic, complemented
315#529C22×C3111yescharacteristic
316C2×#124C22×C3111yescharacteristic
317#497C22×C3112yescomplemented
318#491C22×C3112yescomplemented
319#489—22no—
320#489C22×C3114yescomplemented
321#489C22×C3114yescomplemented
322#487—21no—
323#486—21no—
324#462—21no—
325#461—21no—
326#451—21no—
327#448—21no—
328#432C22×C3111yescharacteristic, complemented
329#417—22no—
330C2×#103—21no—
331C2×#102—21no—
332C4×M16C22×C3111yescharacteristic, complemented
333C2×#97C22×C3111yescharacteristic, complemented
334#330C22×C3112yescomplemented
335#329C22×C3112yescomplemented
336#329C22×C3114yescomplemented
337#329C22×C3114yescomplemented
338C2×C31—22no—
339C2×C31—82no—
340C2×C31—81no—
341C2×C31—82no—
342C2×C31C2×#45511yescharacteristic
343C2×C31C2×#45511yescharacteristic
344C2×C31#302111yescharacteristic
345C2×#41—21no—
346C2×#41D4×C3112yes—
347C22×Q8—21no—
348C22×Q8D4×C3111yescharacteristic
349C22×D4—22no—
350C22×D4—21no—
351C22×D4D4×C3111yescharacteristic
352C2×Q16—22no—
353C2×SD16—22no—
354C2×M16—21no—
355C22×C8—21no—
356C4⋊Q8—21no—
357C4⋊Q8—21no—
358C4⋊Q8—21no—
359C4⋊Q8D4×C3111yescharacteristic
360C4⋊Q8C23×C3112yes—
361#127—21no—
362#127D4×C3111yescharacteristic
363#126—22no—
364#126—21no—
365#125C23×C3111yescharacteristic
366#124—21no—
367#124—22no—
368#124—22no—
369#124—22no—
370#124—21no—
371#124—21no—
372#124D4×C3112yes—
373#124C23×C3112yes—
374#124C23×C3111yescharacteristic
375#124C23×C3112yes—
376#123—22no—
377#123—22no—
378#123—21no—
379#123—22no—
380#123—22no—
381C22⋊Q8—21no—
382C22⋊Q8—22no—
383C22⋊Q8—22no—
384C22⋊Q8D4×C3111yescharacteristic
385#121—22no—
386#121—22no—
387#121—22no—
388#121—21no—
389#121—22no—
390#121—22no—
391#121—21no—
392#121—21no—
393#121D4×C3111yescharacteristic
394#120—22no—
395#120—22no—
396#120—22no—
397#120—22no—
398#120—22no—
399C4×Q8—21no—
400C4×Q8D4×C3111yescharacteristic
401C4×D4—22no—
402C4×D4—22no—
403C4×D4—21no—
404C4×D4—21no—
405C4×D4—21no—
406C4×D4—21no—
407C4×D4D4×C3111yescharacteristic
408#117—21no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—21no—
411C2×C22⋊C4—22no—
412C2×C22⋊C4—21no—
413C2×C22⋊C4—21no—
414C2×C42C23×C3111yescharacteristic
415C4⋊C8—21no—
416#103—24no—
417#103—24no—
418#103—22no—
419#103—22no—
420#102—24no—
421#102—24no—
422C22⋊C8—21no—
423#97—21no—
424#97C23×C3112yes—
425#97C23×C3112yes—
426#97C23×C3112yes—
427C4×C8—21no—
428C31#1902011yescharacteristic, direct product factor, Sylow
429C24—22no—
430C24—21no—
431#41—44no—
432#41—82no—
433C2×Q8—21no—
434C2×Q8—22no—
435C2×Q8—42no—
436C2×Q8—41no—
437C2×Q8—82no—
438C2×Q8C2×D4×C3112yes—
439C2×Q8C2×D4×C3112yes—
440C2×Q8C2×D4×C3111yescharacteristic
441C2×D4—22no—
442C2×D4—21no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—42no—
452C2×D4—82no—
453C2×D4—84no—
454C2×D4C2×D4×C3112yes—
455C2×D4C2×D4×C3112yes—
456C2×D4C2×D4×C3111yescharacteristic
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—22no—
462C22×C4—41no—
463C22×C4C2×D4×C3111yescharacteristic
464C22×C4C2×D4×C3111yescharacteristic
465C22×C4C2×D4×C3111yescharacteristic
466Q16—82no—
467SD16—82no—
468M16—42no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—42no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—41no—
484C4⋊C4—41no—
485C4⋊C4C2×D4×C3111yescharacteristic
486C4⋊C4C2×D4×C3111yescharacteristic
487C22⋊C4—22no—
488C22⋊C4—22no—
489C22⋊C4—22no—
490C22⋊C4—22no—
491C22⋊C4—22no—
492C22⋊C4—21no—
493C22⋊C4—22no—
494C22⋊C4—21no—
495C22⋊C4—22no—
496C22⋊C4—21no—
497C22⋊C4—22no—
498C22⋊C4—22no—
499C22⋊C4—21no—
500C22⋊C4—21no—
501C22⋊C4—22no—
502C22⋊C4—22no—
503C42—22no—
504C42—21no—
505C42C2×D4×C3111yescharacteristic
506C42C2×D4×C3111yescharacteristic
507C42C2×D4×C3111yescharacteristic
508C42C24×C3111yescharacteristic
509C23—21no—
510C23—22no—
511C23—22no—
512C23—42no—
513C23—81no—
514C23—81no—
515C23—82no—
516C23—82no—
517C23—82no—
518C23C31×#12011yescharacteristic
519Q8—41no—
520Q8—42no—
521Q8—42no—
522Q8—42no—
523Q8—42no—
524Q8—42no—
525Q8—82no—
526D4—41no—
527D4—42no—
528D4—42no—
529D4—42no—
530D4—44no—
531D4—42no—
532D4—42no—
533D4—81no—
534D4—82no—
535D4—82no—
536D4—82no—
537D4—82no—
538D4—82no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—82no—
550C2×C4—81no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4C31×#12011yescharacteristic
555C2×C4C31×#12011yescharacteristic
556C2×C4C31×#12011yescharacteristic
557C2×C4C22×D4×C3111yescharacteristic
558C2×C4C22×D4×C3111yescharacteristic
559C2×C4C22×D4×C3111yescharacteristic
560C8—42no—
561C8—42no—
562C8—42no—
563C22—21no—
564C22—21no—
565C22—41no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22—41no—
571C22—41no—
572C22—42no—
573C22—42no—
574C22—162no—
575C22—164no—
576C22C31×#45512yes—
577C22C2×C31×#12011yescharacteristic
578C4—21no—
579C4—21no—
580C4—21no—
581C4—21no—
582C4—21no—
583C4—41no—
584C4—41no—
585C4—42no—
586C4—42no—
587C4—41no—
588C4—82no—
589C4C31×#45512yes—
590C2—22no—
591C2—81no—
592C2—82no—
593C2—82no—
594C2C2×C31×#45511yescharacteristic
595C2C2×C31×#45511yescharacteristic
596C2C31×#302111yescharacteristic
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×#3218C2semidirect
3. C31×#3141C2semidirect
4. C2×C31×#489C2semidirect
5. C31×#1662C2semidirect
6. C31×#1659C2semidirect
7. C31×#1530C2semidirect
8. C31×#1527C2semidirect
9. C31×#1524C2semidirect
23. C2×C31×C4⋊Q8C22semidirect
26. C31×#497C22semidirect
27. C31×#491C22semidirect
29. C31×#489C22semidirect
30. C31×#489C22semidirect
37. C31×#432C22semidirect
41. C4×M16×C31C22semidirect
42. C2×C31×#97C22semidirect
43. C31×#330C22semidirect
44. C31×#329C22semidirect
45. C31×#329C22semidirect
46. C31×#329C22semidirect
210. #19020C31direct (N × H)
266. #3218C2×C31semidirect
267. #3141C2×C31semidirect
268. C2×#489C2×C31semidirect
269. #1662C2×C31semidirect
270. #1659C2×C31semidirect
271. #1530C2×C31semidirect
272. #1527C2×C31semidirect
273. #1524C2×C31semidirect
314. C2×C4⋊Q8C22×C31semidirect
317. #497C22×C31semidirect
318. #491C22×C31semidirect
320. #489C22×C31semidirect
321. #489C22×C31semidirect
328. #432C22×C31semidirect
332. C4×M16C22×C31semidirect
333. C2×#97C22×C31semidirect
334. #330C22×C31semidirect
335. #329C22×C31semidirect
336. #329C22×C31semidirect
337. #329C22×C31semidirect
428. C31#19020direct (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×#32182
C31×#31412
C2×C31×#4892
C31×#16622
C31×#16592
C31×#15302
C31×#15272
C31×#15242
#1902031