Finite Group Databasebeta

← C31×#18436 #21475195

Subgroups — C31×#18436 #21475195

Order 7936.

972 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×#3238C211yesmaximal, characteristic, complemented
3C31×#3226C211yesmaximal, characteristic, complemented
4C31×#2640C211yesmaximal, characteristic, complemented
5C31×#1612C214yesmaximal, complemented
6C31×#1607C214yesmaximal, 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⋊Q8C2211yescharacteristic, complemented
24C31×#529C2211yescharacteristic
25C2×C31×#124C2211yescharacteristic
26C31×#501—21no—
27C31×#500—21no—
28C31×#489—21no—
29C31×#486—21no—
30C31×#483C2212yescomplemented
31C31×#480C2212yes—
32C31×#480C2214yescomplemented
33C31×#477—21no—
34C31×#476—21no—
35C31×#474—22no—
36C31×#432C2211yescharacteristic, complemented
37C31×#431C2211yescharacteristic, complemented
38C31×#397C2214yescomplemented
39C31×#393—22no—
40C31×#391—21no—
41C31×#390—21no—
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⋊Q8D411yescharacteristic
59C31×#127—21no—
60C31×#127D411yescharacteristic
61C31×#126—22no—
62C31×#126—21no—
63C31×#125D411yescharacteristic
64C31×#124—21no—
65C31×#124—22no—
66C31×#124—22no—
67C31×#124—22no—
68C31×#124—21no—
69C31×#124—21no—
70C31×#124—21no—
71C31×#124—21no—
72C31×#124—21no—
73C31×#124D411yescharacteristic
74C31×#123—22no—
75C31×#123—21no—
76C31×#123—22no—
77C31×#123—22no—
78C31×#123—22no—
79C31×C22⋊Q8—22no—
80C31×C22⋊Q8—21no—
81C31×C22⋊Q8—22no—
82C31×C22⋊Q8C2311yescharacteristic
83C31×#121—22no—
84C31×#121—22no—
85C31×#121—22no—
86C31×#121—21no—
87C31×#121—22no—
88C31×#121—21no—
89C31×#121—22no—
90C31×#121C2311yescharacteristic
91C31×#121C2312yes—
92C31×#120—22no—
93C31×#120—22no—
94C31×#120—22no—
95C31×#120—22no—
96C31×#120—22no—
97C4×Q8×C31—21no—
98C4×Q8×C31D411yescharacteristic
99C4×D4×C31—22no—
100C4×D4×C31—21no—
101C4×D4×C31—21no—
102C4×D4×C31—22no—
103C4×D4×C31—21no—
104C4×D4×C31D412yes—
105C4×D4×C31D411yescharacteristic
106C31×#117—21no—
107C2×C4⋊C4×C31—21no—
108C2×C4⋊C4×C31C2312yes—
109C2×C22⋊C4×C31—22no—
110C2×C22⋊C4×C31—24no—
111C2×C22⋊C4×C31D412yes—
112C2×C22⋊C4×C31C2312yes—
113C2×C42×C31—21no—
114C2×C42×C31C2311yescharacteristic
115C31×#107—24no—
116C31×#107—22no—
117C31×#106—22no—
118C31×C4⋊C8—21no—
119C31×C4⋊C8—21no—
120C31×#103—22no—
121C31×#102—24no—
122C31×#102—22no—
123C31×C22⋊C8C2312yes—
124C31×C22⋊C8C2312yes—
125C31×#97—21no—
126C4×C8×C31—21no—
127C31×#95—24no—
128C31×#95—21no—
129C31×#95C2312yes—
130C24×C31—22no—
131C24×C31C2×D412yes—
132#41×C31—82no—
133#41×C31—82no—
134C2×Q8×C31—21no—
135C2×Q8×C31—21no—
136C2×Q8×C31—21no—
137C2×Q8×C31—42no—
138C2×Q8×C31—41no—
139C2×Q8×C31—41no—
140C2×Q8×C31—82no—
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—21no—
153C2×D4×C31—42no—
154C2×D4×C31—82no—
155C2×D4×C31—84no—
156C2×D4×C31C2×D411yescharacteristic
157C2×D4×C31C2×D412yes—
158C22×C4×C31—22no—
159C22×C4×C31—22no—
160C22×C4×C31—21no—
161C22×C4×C31—22no—
162C22×C4×C31—22no—
163C22×C4×C31—22no—
164C22×C4×C31C2×D411yescharacteristic
165C22×C4×C31C2×D411yescharacteristic
166C22×C4×C31C2×D412yes—
167C22×C4×C31C2411yescharacteristic
168C2×C8×C31—22no—
169C2×C8×C31—22no—
170C4⋊C4×C31—22no—
171C4⋊C4×C31—22no—
172C4⋊C4×C31—21no—
173C4⋊C4×C31—21no—
174C4⋊C4×C31—21no—
175C4⋊C4×C31—21no—
176C4⋊C4×C31—21no—
177C4⋊C4×C31—22no—
178C4⋊C4×C31—22no—
179C4⋊C4×C31—41no—
180C4⋊C4×C31—41no—
181C4⋊C4×C31C2×D411yescharacteristic
182C4⋊C4×C31C2×D412yes—
183C4⋊C4×C31C2×D411yescharacteristic
184C22⋊C4×C31—22no—
185C22⋊C4×C31—22no—
186C22⋊C4×C31—22no—
187C22⋊C4×C31—22no—
188C22⋊C4×C31—22no—
189C22⋊C4×C31—21no—
190C22⋊C4×C31—21no—
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—22no—
198C22⋊C4×C31—21no—
199C22⋊C4×C31—22no—
200C22⋊C4×C31—84no—
201C42×C31—22no—
202C42×C31—21no—
203C42×C31—42no—
204C42×C31C2×D411yescharacteristic
205C42×C31C2×D411yescharacteristic
206C42×C31C2×D411yescharacteristic
207C42×C31C2×D411yescharacteristic
208#18436C3111yesmaximal, characteristic, direct product factor, Sylow
209C23×C31—22no—
210C23×C31—21no—
211C23×C31—22no—
212C23×C31—24no—
213C23×C31—42no—
214C23×C31—42no—
215C23×C31—82no—
216C23×C31—82no—
217C23×C31—82no—
218C23×C31C22×D411yescharacteristic
219Q8×C31—41no—
220Q8×C31—42no—
221Q8×C31—81no—
222Q8×C31—81no—
223Q8×C31—82no—
224Q8×C31—82no—
225D4×C31—42no—
226D4×C31—41no—
227D4×C31—42no—
228D4×C31—44no—
229D4×C31—42no—
230D4×C31—42no—
231D4×C31—81no—
232D4×C31—81no—
233D4×C31—82no—
234D4×C31—82no—
235D4×C31—82no—
236D4×C31—82no—
237D4×C31—82no—
238C2×C4×C31—22no—
239C2×C4×C31—22no—
240C2×C4×C31—21no—
241C2×C4×C31—22no—
242C2×C4×C31—22no—
243C2×C4×C31—21no—
244C2×C4×C31—21no—
245C2×C4×C31—41no—
246C2×C4×C31—42no—
247C2×C4×C31—42no—
248C2×C4×C31—42no—
249C2×C4×C31—42no—
250C2×C4×C31—42no—
251C2×C4×C31—42no—
252C2×C4×C31—44no—
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×C31#12011yescharacteristic
260C2×C4×C31C22×D411yescharacteristic
261C2×C4×C31C22×D411yescharacteristic
262C8×C31—42no—
263C8×C31—42no—
264#3558C2×C3111yescharacteristic
265#3238C2×C3111yescharacteristic, complemented
266#3226C2×C3111yescharacteristic, complemented
267#2640C2×C3111yescharacteristic, complemented
268#1612C2×C3114yescomplemented
269#1607C2×C3114yescomplemented
270#1505C2×C3111yescharacteristic, complemented
271#1500C2×C3111yescharacteristic, complemented
272#1469C2×C3111yescharacteristic, complemented
273C22×C31—21no—
274C22×C31—21no—
275C22×C31—42no—
276C22×C31—42no—
277C22×C31—41no—
278C22×C31—41no—
279C22×C31—42no—
280C22×C31—42no—
281C22×C31—42no—
282C22×C31—41no—
283C22×C31—42no—
284C22×C31—84no—
285C22×C31—164no—
286C22×C31C2×#12011yescharacteristic
287C22×C31#57612yes—
288C4×C31—21no—
289C4×C31—21no—
290C4×C31—21no—
291C4×C31—21no—
292C4×C31—21no—
293C4×C31—41no—
294C4×C31—42no—
295C4×C31—41no—
296C4×C31—42no—
297C4×C31—41no—
298C4×C31—82no—
299C4×C31—82no—
300C4×C31#45512yes—
301#562—21no—
302H(3,𝔽4)—21no—
303#559—21no—
304#548—21no—
305#546—21no—
306#546—22no—
307D42—21no—
308#542—21no—
309#538—22no—
310#538C22×C3112yes—
311#535—21no—
312#534C22×C3112yes—
313#532C22×C3111yescharacteristic
314C2×C4⋊Q8C22×C3111yescharacteristic, complemented
315#529C22×C3111yescharacteristic
316C2×#124C22×C3111yescharacteristic
317#501—21no—
318#500—21no—
319#489—21no—
320#486—21no—
321#483C22×C3112yescomplemented
322#480C22×C3112yes—
323#480C22×C3114yescomplemented
324#477—21no—
325#476—21no—
326#474—22no—
327#432C22×C3111yescharacteristic, complemented
328#431C22×C3111yescharacteristic, complemented
329#397C22×C3114yescomplemented
330#393—22no—
331#391—21no—
332#390—21no—
333#376C22×C3111yescharacteristic, complemented
334#328C22×C3112yescomplemented
335#327C22×C3112yes—
336#327C22×C3114yescomplemented
337#323C22×C3114yescomplemented
338C2×C31—22no—
339C2×C31—82no—
340C2×C31—82no—
341C2×C31—81no—
342C2×C31#300811yescharacteristic
343C2×C31C2×#45511yescharacteristic
344C2×C31#302811yescharacteristic
345C2×#41—21no—
346C2×#41—21no—
347C22×Q8—21no—
348C22×Q8D4×C3111yescharacteristic
349C22×D4—21no—
350C22×D4—22no—
351C22×D4D4×C3111yescharacteristic
352C4⋊Q8—21no—
353C4⋊Q8—21no—
354C4⋊Q8—21no—
355C4⋊Q8—21no—
356C4⋊Q8D4×C3111yescharacteristic
357#127—21no—
358#127D4×C3111yescharacteristic
359#126—22no—
360#126—21no—
361#125D4×C3111yescharacteristic
362#124—21no—
363#124—22no—
364#124—22no—
365#124—22no—
366#124—21no—
367#124—21no—
368#124—21no—
369#124—21no—
370#124—21no—
371#124D4×C3111yescharacteristic
372#123—22no—
373#123—21no—
374#123—22no—
375#123—22no—
376#123—22no—
377C22⋊Q8—22no—
378C22⋊Q8—21no—
379C22⋊Q8—22no—
380C22⋊Q8C23×C3111yescharacteristic
381#121—22no—
382#121—22no—
383#121—22no—
384#121—21no—
385#121—22no—
386#121—21no—
387#121—22no—
388#121C23×C3111yescharacteristic
389#121C23×C3112yes—
390#120—22no—
391#120—22no—
392#120—22no—
393#120—22no—
394#120—22no—
395C4×Q8—21no—
396C4×Q8D4×C3111yescharacteristic
397C4×D4—22no—
398C4×D4—21no—
399C4×D4—21no—
400C4×D4—22no—
401C4×D4—21no—
402C4×D4D4×C3111yescharacteristic
403C4×D4D4×C3112yes—
404#117—21no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4C23×C3112yes—
407C2×C22⋊C4—22no—
408C2×C22⋊C4—24no—
409C2×C22⋊C4D4×C3112yes—
410C2×C22⋊C4C23×C3112yes—
411C2×C42—21no—
412C2×C42C23×C3111yescharacteristic
413#107—24no—
414#107—22no—
415#106—22no—
416C4⋊C8—21no—
417C4⋊C8—21no—
418#103—22no—
419#102—24no—
420#102—22no—
421C22⋊C8C23×C3112yes—
422C22⋊C8C23×C3112yes—
423#97—21no—
424C4×C8—21no—
425#95—21no—
426#95—24no—
427#95C23×C3112yes—
428C31#1843611yescharacteristic, direct product factor, Sylow
429C24—22no—
430C24C2×D4×C3112yes—
431#41—82no—
432#41—82no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—42no—
437C2×Q8—41no—
438C2×Q8—41no—
439C2×Q8—82no—
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—21no—
452C2×D4—42no—
453C2×D4—82no—
454C2×D4—84no—
455C2×D4C2×D4×C3111yescharacteristic
456C2×D4C2×D4×C3112yes—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4C2×D4×C3111yescharacteristic
464C22×C4C2×D4×C3111yescharacteristic
465C22×C4C2×D4×C3112yes—
466C22×C4C24×C3111yescharacteristic
467C2×C8—22no—
468C2×C8—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—41no—
479C4⋊C4—41no—
480C4⋊C4C2×D4×C3111yescharacteristic
481C4⋊C4C2×D4×C3112yes—
482C4⋊C4C2×D4×C3111yescharacteristic
483C22⋊C4—22no—
484C22⋊C4—22no—
485C22⋊C4—22no—
486C22⋊C4—22no—
487C22⋊C4—22no—
488C22⋊C4—21no—
489C22⋊C4—21no—
490C22⋊C4—22no—
491C22⋊C4—22no—
492C22⋊C4—21no—
493C22⋊C4—21no—
494C22⋊C4—22no—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—21no—
498C22⋊C4—22no—
499C22⋊C4—84no—
500C42—22no—
501C42—21no—
502C42—42no—
503C42C2×D4×C3111yescharacteristic
504C42C2×D4×C3111yescharacteristic
505C42C2×D4×C3111yescharacteristic
506C42C2×D4×C3111yescharacteristic
507C23—22no—
508C23—21no—
509C23—22no—
510C23—24no—
511C23—42no—
512C23—42no—
513C23—82no—
514C23—82no—
515C23—82no—
516C23C22×D4×C3111yescharacteristic
517Q8—41no—
518Q8—42no—
519Q8—81no—
520Q8—81no—
521Q8—82no—
522Q8—82no—
523D4—42no—
524D4—41no—
525D4—42no—
526D4—44no—
527D4—42no—
528D4—42no—
529D4—81no—
530D4—81no—
531D4—82no—
532D4—82no—
533D4—82no—
534D4—82no—
535D4—82no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4C31×#12011yescharacteristic
555C2×C4C31×#12011yescharacteristic
556C2×C4C31×#12011yescharacteristic
557C2×C4C31×#12011yescharacteristic
558C2×C4C22×D4×C3111yescharacteristic
559C2×C4C22×D4×C3111yescharacteristic
560C8—42no—
561C8—42no—
562C22—21no—
563C22—21no—
564C22—42no—
565C22—42no—
566C22—41no—
567C22—41no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—41no—
572C22—42no—
573C22—84no—
574C22—164no—
575C22C2×C31×#12011yescharacteristic
576C22C31×#57612yes—
577C4—21no—
578C4—21no—
579C4—21no—
580C4—21no—
581C4—21no—
582C4—41no—
583C4—42no—
584C4—41no—
585C4—42no—
586C4—41no—
587C4—82no—
588C4—82no—
589C4C31×#45512yes—
590C2—22no—
591C2—82no—
592C2—81no—
593C2—82no—
594C2C31×#300811yescharacteristic
595C2C2×C31×#45511yescharacteristic
596C2C31×#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×#3238C2semidirect
3. C31×#3226C2semidirect
4. C31×#2640C2semidirect
5. C31×#1612C2semidirect
6. C31×#1607C2semidirect
7. C31×#1505C2semidirect
8. C31×#1500C2semidirect
9. C31×#1469C2semidirect
23. C2×C31×C4⋊Q8C22semidirect
30. C31×#483C22semidirect
32. C31×#480C22semidirect
36. C31×#432C22semidirect
37. C31×#431C22semidirect
38. C31×#397C22semidirect
42. C31×#376C22semidirect
43. C31×#328C22semidirect
45. C31×#327C22semidirect
46. C31×#323C22semidirect
208. #18436C31direct (N × H)
265. #3238C2×C31semidirect
266. #3226C2×C31semidirect
267. #2640C2×C31semidirect
268. #1612C2×C31semidirect
269. #1607C2×C31semidirect
270. #1505C2×C31semidirect
271. #1500C2×C31semidirect
272. #1469C2×C31semidirect
314. C2×C4⋊Q8C22×C31semidirect
321. #483C22×C31semidirect
323. #480C22×C31semidirect
327. #432C22×C31semidirect
328. #431C22×C31semidirect
329. #397C22×C31semidirect
333. #376C22×C31semidirect
334. #328C22×C31semidirect
336. #327C22×C31semidirect
337. #323C22×C31semidirect
428. C31#18436direct (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×#32382
C31×#32262
C31×#26402
C31×#16122
C31×#16072
C31×#15052
C31×#15002
C31×#14692
#1843631