Finite Group Databasebeta

← C31×#18577 #21475336

Subgroups — C31×#18577 #21475336

Order 7936.

916 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3511C211yesmaximal, characteristic
2C31×#3254C211yesmaximal, characteristic, complemented
3C31×#3139C211yesmaximal, characteristic, complemented
4C31×#3107C211yesmaximal, characteristic, complemented
5C31×#1646C212yesmaximal, complemented
6C31×#1646C212yesmaximal, complemented
7C31×#1644C212yesmaximal, complemented
8C31×#1643C212yesmaximal, complemented
9C31×#1548C211yesmaximal, characteristic, complemented
10C31×#1489C211yesmaximal, characteristic, complemented
11C31×#1484C211yesmaximal, characteristic, complemented
12C31×#556—21no—
13C31×#555C2212yes—
14C31×#554C2212yes—
15C31×#553—22no—
16C31×#548—22no—
17C31×#547—21no—
18C31×#542—22no—
19C31×#537—21no—
20C31×#533—21no—
21C31×#533C2211yescharacteristic
22C31×#532—21no—
23C31×#532C2211yescharacteristic
24C2×C31×C4⋊Q8C2211yescharacteristic
25C31×#520—21no—
26C4×#41×C31C2211yescharacteristic
27C31×#491—22no—
28C31×#487C2212yes—
29C31×#487C2212yes—
30C31×#485—21no—
31C31×#477—21no—
32C31×#475C2211yescharacteristic
33C31×#474C2211yescharacteristic
34C31×#474C2212yes—
35C31×#468—22no—
36C31×#462C2211yescharacteristic
37C31×#461C2211yescharacteristic
38C31×#461C2212yes—
39C31×#460—21no—
40C31×#452—21no—
41C4×Q16×C31—21no—
42C4×SD16×C31—21no—
43C31×#431C2211yescharacteristic
44C31×#423C2211yescharacteristic
45C31×#419—22no—
46C4×M16×C31C2211yescharacteristic
47C31×#333C2212yes—
48C31×#333C2212yes—
49C31×Q8⋊C8C2212yes—
50C31×D4⋊C8C2212yes—
51C31×D4⋊C8C2212yes—
52C31×D4⋊C8C2212yes—
53C2×#41×C31—22no—
54C2×#41×C31—21no—
55C2×#41×C31D411yescharacteristic
56C22×Q8×C31D412yes—
57C2×Q16×C31—21no—
58C2×SD16×C31—21no—
59C2×SD16×C31—22no—
60C2×M16×C31—21no—
61C2×M16×C31—21no—
62C31×C4⋊Q8—21no—
63C31×C4⋊Q8—21no—
64C31×C4⋊Q8D412yes—
65C31×C4⋊Q8D411yescharacteristic
66C31×C4⋊Q8C2311yescharacteristic
67C31×C4⋊Q8C2311yescharacteristic
68C31×C4⋊Q8C2312yes—
69C31×#127D411yescharacteristic
70C31×#126—24no—
71C31×#125—22no—
72C31×#125—21no—
73C31×#124—22no—
74C31×#124—24no—
75C31×#124—21no—
76C31×#123—24no—
77C31×C22⋊Q8—22no—
78C31×C22⋊Q8—22no—
79C31×C22⋊Q8—22no—
80C31×C22⋊Q8—24no—
81C31×C22⋊Q8—21no—
82C31×C22⋊Q8—21no—
83C31×#121—22no—
84C31×#121—24no—
85C31×#121—22no—
86C31×#121—22no—
87C31×#121—21no—
88C31×#121D412yes—
89C4×Q8×C31—22no—
90C4×Q8×C31—22no—
91C4×Q8×C31—21no—
92C4×Q8×C31—21no—
93C4×Q8×C31D411yescharacteristic
94C4×Q8×C31C2311yescharacteristic
95C4×D4×C31—22no—
96C4×D4×C31—24no—
97C4×D4×C31—22no—
98C4×D4×C31—21no—
99C4×D4×C31—21no—
100C4×D4×C31—21no—
101C4×D4×C31D411yescharacteristic
102C4×D4×C31C2311yescharacteristic
103C4×D4×C31C2312yes—
104C31×#117—22no—
105C31×#117—22no—
106C31×#117—21no—
107C31×#117—21no—
108C31×#117D411yescharacteristic
109C2×C4⋊C4×C31—21no—
110C2×C4⋊C4×C31—21no—
111C2×C4⋊C4×C31—22no—
112C2×C42×C31—21no—
113C2×C42×C31C2311yescharacteristic
114C31×#107—21no—
115C31×#107—22no—
116C31×#106—21no—
117C31×C4⋊C8—21no—
118C31×C4⋊C8C2312yes—
119C31×C4⋊C8C2311yescharacteristic
120C31×C4⋊C8C2311yescharacteristic
121C31×#103—24no—
122C31×#103—24no—
123C31×#103—21no—
124C31×#103—21no—
125C31×#103—21no—
126C31×#103—22no—
127C31×#102—21no—
128C31×#102—22no—
129C31×C22⋊C8—21no—
130C31×#97—21no—
131C4×C8×C31—21no—
132C4×C8×C31C2312yes—
133#41×C31—41no—
134#41×C31—41no—
135#41×C31—42no—
136#41×C31—42no—
137#41×C31—84no—
138C2×Q8×C31—21no—
139C2×Q8×C31—22no—
140C2×Q8×C31—21no—
141C2×Q8×C31—21no—
142C2×Q8×C31—44no—
143C2×Q8×C31C2×D411yescharacteristic
144C2×Q8×C31C2×D414yes—
145C2×D4×C31—21no—
146C2×D4×C31—22no—
147C2×D4×C31—24no—
148C2×D4×C31—21no—
149C2×D4×C31—21no—
150C2×D4×C31C2×D412yes—
151C2×D4×C31C2×D411yescharacteristic
152C22×C4×C31—24no—
153C22×C4×C31—22no—
154C22×C4×C31—21no—
155C22×C4×C31—21no—
156C22×C4×C31—21no—
157C22×C4×C31—22no—
158C22×C4×C31C2×D411yescharacteristic
159C22×C4×C31C2×D411yescharacteristic
160C22×C4×C31C2×D411yescharacteristic
161Q16×C31—42no—
162SD16×C31—42no—
163SD16×C31—82no—
164M16×C31—42no—
165M16×C31—81no—
166C2×C8×C31—21no—
167C2×C8×C31—21no—
168C2×C8×C31—22no—
169C2×C8×C31—22no—
170C4⋊C4×C31—24no—
171C4⋊C4×C31—24no—
172C4⋊C4×C31—22no—
173C4⋊C4×C31—22no—
174C4⋊C4×C31—21no—
175C4⋊C4×C31—21no—
176C4⋊C4×C31—22no—
177C4⋊C4×C31—22no—
178C4⋊C4×C31—21no—
179C4⋊C4×C31—21no—
180C4⋊C4×C31—21no—
181C4⋊C4×C31—21no—
182C4⋊C4×C31—21no—
183C4⋊C4×C31—41no—
184C4⋊C4×C31—41no—
185C4⋊C4×C31C2×D411yescharacteristic
186C4⋊C4×C31C2×D411yescharacteristic
187C4⋊C4×C31C2×D412yes—
188C22⋊C4×C31—24no—
189C22⋊C4×C31—24no—
190C22⋊C4×C31—24no—
191C22⋊C4×C31—21no—
192C22⋊C4×C31—21no—
193C22⋊C4×C31—22no—
194C22⋊C4×C31—21no—
195C22⋊C4×C31—22no—
196C22⋊C4×C31—41no—
197C42×C31—24no—
198C42×C31—21no—
199C42×C31—21no—
200C42×C31—21no—
201C42×C31—41no—
202C42×C31C2×D411yescharacteristic
203C42×C31C2×D411yescharacteristic
204C42×C31C2×D411yescharacteristic
205C42×C31C2411yescharacteristic
206#18577C3111yesmaximal, characteristic, direct product factor, Sylow
207C23×C31—21no—
208C23×C31—22no—
209C23×C31#12011yescharacteristic
210Q8×C31—21no—
211Q8×C31—21no—
212Q8×C31—44no—
213Q8×C31—42no—
214Q8×C31—44no—
215Q8×C31—41no—
216Q8×C31—41no—
217Q8×C31—41no—
218Q8×C31—41no—
219D4×C31—21no—
220D4×C31—21no—
221D4×C31—22no—
222D4×C31—22no—
223D4×C31—42no—
224D4×C31—42no—
225D4×C31—44no—
226D4×C31—42no—
227D4×C31—84no—
228C2×C4×C31—21no—
229C2×C4×C31—21no—
230C2×C4×C31—21no—
231C2×C4×C31—21no—
232C2×C4×C31—22no—
233C2×C4×C31—21no—
234C2×C4×C31—21no—
235C2×C4×C31—22no—
236C2×C4×C31—21no—
237C2×C4×C31—24no—
238C2×C4×C31—42no—
239C2×C4×C31—42no—
240C2×C4×C31—42no—
241C2×C4×C31—41no—
242C2×C4×C31—41no—
243C2×C4×C31—42no—
244C2×C4×C31—41no—
245C2×C4×C31—41no—
246C2×C4×C31—82no—
247C2×C4×C31—84no—
248C2×C4×C31#12011yescharacteristic
249C2×C4×C31#12011yescharacteristic
250C2×C4×C31#12011yescharacteristic
251C2×C4×C31C22×D411yescharacteristic
252C2×C4×C31C22×D411yescharacteristic
253C2×C4×C31C22×D411yescharacteristic
254C8×C31—42no—
255C8×C31—41no—
256C8×C31—41no—
257C8×C31—42no—
258#3511C2×C3111yescharacteristic
259#3254C2×C3111yescharacteristic, complemented
260#3139C2×C3111yescharacteristic, complemented
261#3107C2×C3111yescharacteristic, complemented
262#1646C2×C3112yescomplemented
263#1646C2×C3112yescomplemented
264#1644C2×C3112yescomplemented
265#1643C2×C3112yescomplemented
266#1548C2×C3111yescharacteristic, complemented
267#1489C2×C3111yescharacteristic, complemented
268#1484C2×C3111yescharacteristic, complemented
269C22×C31—21no—
270C22×C31—21no—
271C22×C31—21no—
272C22×C31—41no—
273C22×C31—42no—
274C22×C31—41no—
275C22×C31—42no—
276C22×C31—41no—
277C22×C31—42no—
278C22×C31C2×#12011yescharacteristic
279C4×C31—21no—
280C4×C31—21no—
281C4×C31—21no—
282C4×C31—41no—
283C4×C31—42no—
284C4×C31—41no—
285C4×C31—41no—
286C4×C31—44no—
287C4×C31—81no—
288C4×C31—81no—
289C4×C31#45512yes—
290C4×C31#57511yescharacteristic
291C4×C31#57511yescharacteristic
292C4×C31#57711yescharacteristic
293C4×C31#57811yescharacteristic
294#556—21no—
295#555C22×C3112yes—
296#554C22×C3112yes—
297#553—22no—
298#548—22no—
299#547—21no—
300#542—22no—
301#537—21no—
302#533—21no—
303#533C22×C3111yescharacteristic
304#532—21no—
305#532C22×C3111yescharacteristic
306C2×C4⋊Q8C22×C3111yescharacteristic
307#520—21no—
308C4×#41C22×C3111yescharacteristic
309#491—22no—
310#487C22×C3112yes—
311#487C22×C3112yes—
312#485—21no—
313#477—21no—
314#475C22×C3111yescharacteristic
315#474C22×C3111yescharacteristic
316#474C22×C3112yes—
317#468—22no—
318#462C22×C3111yescharacteristic
319#461C22×C3111yescharacteristic
320#461C22×C3112yes—
321#460—21no—
322#452—21no—
323C4×Q16—21no—
324C4×SD16—21no—
325#431C22×C3111yescharacteristic
326#423C22×C3111yescharacteristic
327#419—22no—
328C4×M16C22×C3111yescharacteristic
329#333C22×C3112yes—
330#333C22×C3112yes—
331Q8⋊C8C22×C3112yes—
332D4⋊C8C22×C3112yes—
333D4⋊C8C22×C3112yes—
334D4⋊C8C22×C3112yes—
335C2×C31—41no—
336C2×C31—81no—
337C2×C31—82no—
338C2×C31#300511yescharacteristic
339C2×C31#301311yescharacteristic
340C2×C31C2×#45511yescharacteristic
341C2×#41—22no—
342C2×#41—21no—
343C2×#41D4×C3111yescharacteristic
344C22×Q8D4×C3112yes—
345C2×Q16—21no—
346C2×SD16—21no—
347C2×SD16—22no—
348C2×M16—21no—
349C2×M16—21no—
350C4⋊Q8—21no—
351C4⋊Q8—21no—
352C4⋊Q8D4×C3112yes—
353C4⋊Q8D4×C3111yescharacteristic
354C4⋊Q8C23×C3111yescharacteristic
355C4⋊Q8C23×C3111yescharacteristic
356C4⋊Q8C23×C3112yes—
357#127D4×C3111yescharacteristic
358#126—24no—
359#125—22no—
360#125—21no—
361#124—22no—
362#124—24no—
363#124—21no—
364#123—24no—
365C22⋊Q8—22no—
366C22⋊Q8—22no—
367C22⋊Q8—22no—
368C22⋊Q8—24no—
369C22⋊Q8—21no—
370C22⋊Q8—21no—
371#121—22no—
372#121—24no—
373#121—22no—
374#121—22no—
375#121—21no—
376#121D4×C3112yes—
377C4×Q8—22no—
378C4×Q8—22no—
379C4×Q8—21no—
380C4×Q8—21no—
381C4×Q8D4×C3111yescharacteristic
382C4×Q8C23×C3111yescharacteristic
383C4×D4—22no—
384C4×D4—24no—
385C4×D4—22no—
386C4×D4—21no—
387C4×D4—21no—
388C4×D4—21no—
389C4×D4D4×C3111yescharacteristic
390C4×D4C23×C3111yescharacteristic
391C4×D4C23×C3112yes—
392#117—22no—
393#117—22no—
394#117—21no—
395#117—21no—
396#117D4×C3111yescharacteristic
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—22no—
400C2×C42—21no—
401C2×C42C23×C3111yescharacteristic
402#107—21no—
403#107—22no—
404#106—21no—
405C4⋊C8—21no—
406C4⋊C8C23×C3111yescharacteristic
407C4⋊C8C23×C3111yescharacteristic
408C4⋊C8C23×C3112yes—
409#103—24no—
410#103—24no—
411#103—21no—
412#103—21no—
413#103—21no—
414#103—22no—
415#102—21no—
416#102—22no—
417C22⋊C8—21no—
418#97—21no—
419C4×C8—21no—
420C4×C8C23×C3112yes—
421C31#1857711yescharacteristic, direct product factor, Sylow
422#41—41no—
423#41—41no—
424#41—42no—
425#41—42no—
426#41—84no—
427C2×Q8—21no—
428C2×Q8—22no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—44no—
432C2×Q8C2×D4×C3111yescharacteristic
433C2×Q8C2×D4×C3114yes—
434C2×D4—21no—
435C2×D4—22no—
436C2×D4—24no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4C2×D4×C3112yes—
440C2×D4C2×D4×C3111yescharacteristic
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4C2×D4×C3111yescharacteristic
448C22×C4C2×D4×C3111yescharacteristic
449C22×C4C2×D4×C3111yescharacteristic
450Q16—42no—
451SD16—42no—
452SD16—82no—
453M16—42no—
454M16—81no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—22no—
458C2×C8—22no—
459C4⋊C4—24no—
460C4⋊C4—24no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—41no—
473C4⋊C4—41no—
474C4⋊C4C2×D4×C3111yescharacteristic
475C4⋊C4C2×D4×C3111yescharacteristic
476C4⋊C4C2×D4×C3112yes—
477C22⋊C4—24no—
478C22⋊C4—24no—
479C22⋊C4—24no—
480C22⋊C4—21no—
481C22⋊C4—21no—
482C22⋊C4—22no—
483C22⋊C4—21no—
484C22⋊C4—22no—
485C22⋊C4—41no—
486C42—24no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—41no—
491C42C2×D4×C3111yescharacteristic
492C42C2×D4×C3111yescharacteristic
493C42C2×D4×C3111yescharacteristic
494C42C24×C3111yescharacteristic
495C23—21no—
496C23—22no—
497C23C31×#12011yescharacteristic
498Q8—21no—
499Q8—21no—
500Q8—44no—
501Q8—42no—
502Q8—44no—
503Q8—41no—
504Q8—41no—
505Q8—41no—
506Q8—41no—
507D4—21no—
508D4—21no—
509D4—22no—
510D4—22no—
511D4—42no—
512D4—42no—
513D4—44no—
514D4—42no—
515D4—84no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—21no—
525C2×C4—24no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—82no—
535C2×C4—84no—
536C2×C4C31×#12011yescharacteristic
537C2×C4C31×#12011yescharacteristic
538C2×C4C31×#12011yescharacteristic
539C2×C4C22×D4×C3111yescharacteristic
540C2×C4C22×D4×C3111yescharacteristic
541C2×C4C22×D4×C3111yescharacteristic
542C8—42no—
543C8—41no—
544C8—41no—
545C8—42no—
546C22—21no—
547C22—21no—
548C22—21no—
549C22—41no—
550C22—42no—
551C22—41no—
552C22—42no—
553C22—41no—
554C22—42no—
555C22C2×C31×#12011yescharacteristic
556C4—21no—
557C4—21no—
558C4—21no—
559C4—41no—
560C4—42no—
561C4—41no—
562C4—41no—
563C4—44no—
564C4—81no—
565C4—81no—
566C4C31×#45512yes—
567C4C31×#57511yescharacteristic
568C4C31×#57511yescharacteristic
569C4C31×#57711yescharacteristic
570C4C31×#57811yescharacteristic
571C2—41no—
572C2—81no—
573C2—82no—
574C2C31×#300511yescharacteristic
575C2C31×#301311yescharacteristic
576C2C2×C31×#45511yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC42C24×C31
prime radicalC31×#3511C2
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×#3254C2semidirect
3. C31×#3139C2semidirect
4. C31×#3107C2semidirect
5. C31×#1646C2semidirect
6. C31×#1646C2semidirect
7. C31×#1644C2semidirect
8. C31×#1643C2semidirect
9. C31×#1548C2semidirect
10. C31×#1489C2semidirect
11. C31×#1484C2semidirect
206. #18577C31direct (N × H)
259. #3254C2×C31semidirect
260. #3139C2×C31semidirect
261. #3107C2×C31semidirect
262. #1646C2×C31semidirect
263. #1646C2×C31semidirect
264. #1644C2×C31semidirect
265. #1643C2×C31semidirect
266. #1548C2×C31semidirect
267. #1489C2×C31semidirect
268. #1484C2×C31semidirect
421. C31#18577direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35112
C31×#32542
C31×#31392
C31×#31072
C31×#16462
C31×#16442
C31×#16432
C31×#15482
C31×#14892
C31×#14842
#1857731