Finite Group Databasebeta

← C31×#19002 #21475761

Subgroups — C31×#19002 #21475761

Order 7936.

884 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#563C211yesmaximal, characteristic, complemented
2C31×#3234C211yesmaximal, characteristic, complemented
3C31×#3213C211yesmaximal, characteristic
4C31×#3142C211yesmaximal, characteristic, complemented
5C31×#1664C214yesmaximal, complemented
6C31×#1657C214yesmaximal, complemented
7C31×#1534C211yesmaximal, characteristic, complemented
8C31×#1528C211yesmaximal, characteristic, complemented
9C31×#1527C211yesmaximal, characteristic, complemented
10C31×#563—22no—
11C31×#563—22no—
12C31×#563—22no—
13C31×#563C2214yes—
14C31×#532C2211yescharacteristic
15C2×C31×C4⋊Q8—21no—
16C2×C31×#126—21no—
17C2×C31×#126—21no—
18C2×C31×#125C2211yescharacteristic
19C2×C31×#125C2211yescharacteristic
20C2×C31×#124C2211yescharacteristic
21C2×C31×#123—21no—
22C2×C31×C22⋊Q8—21no—
23C2×C31×C22⋊Q8—21no—
24C31×#501C2212yescomplemented
25C31×#496C2212yes—
26C31×#491C2212yescomplemented
27C31×#490C2212yescomplemented
28C31×#490C2212yescomplemented
29C31×#489C2212yes—
30C31×#488—22no—
31C31×#481—21no—
32C31×#480—21no—
33C31×#478—22no—
34C31×#463—22no—
35C31×#448—21no—
36C31×#447—21no—
37C31×#432C2211yescharacteristic
38C31×#432C2211yescharacteristic
39C31×#417—21no—
40C31×#417—21no—
41C4×M16×C31C2211yescharacteristic
42C31×#330C2214yescomplemented
43C31×#330C2214yescomplemented
44C31×#329C2214yes—
45C2×#41×C31—21no—
46C22×Q8×C31—21no—
47C22×Q8×C31D411yescharacteristic
48C22×D4×C31D411yescharacteristic
49C23×C4×C31—21no—
50C2×SD16×C31—22no—
51C2×D8×C31—22no—
52C2×M16×C31—21no—
53C31×C4⋊Q8—24no—
54C31×C4⋊Q8—24no—
55C31×C4⋊Q8C2311yescharacteristic
56C31×#127C2311yescharacteristic
57C31×#126—24no—
58C31×#126—24no—
59C31×#126—24no—
60C31×#126—24no—
61C31×#125—21no—
62C31×#125—21no—
63C31×#125—21no—
64C31×#125—21no—
65C31×#125D412yes—
66C31×#125D412yes—
67C31×#125C2312yes—
68C31×#125C2312yes—
69C31×#124—21no—
70C31×#124—21no—
71C31×#124—21no—
72C31×#124D412yes—
73C31×#124C2312yes—
74C31×#123—24no—
75C31×#123—24no—
76C31×C22⋊Q8—24no—
77C31×C22⋊Q8—24no—
78C31×C22⋊Q8—24no—
79C31×C22⋊Q8—24no—
80C31×#121—21no—
81C31×#121—21no—
82C4×D4×C31—21no—
83C4×D4×C31—21no—
84C2×C4⋊C4×C31—21no—
85C2×C4⋊C4×C31—21no—
86C2×C4⋊C4×C31—21no—
87C2×C4⋊C4×C31—21no—
88C2×C4⋊C4×C31—21no—
89C2×C4⋊C4×C31—21no—
90C2×C4⋊C4×C31D411yescharacteristic
91C2×C4⋊C4×C31D411yescharacteristic
92C2×C4⋊C4×C31D411yescharacteristic
93C2×C4⋊C4×C31D411yescharacteristic
94C2×C22⋊C4×C31—21no—
95C2×C22⋊C4×C31—21no—
96C2×C22⋊C4×C31—21no—
97C2×C22⋊C4×C31—21no—
98C2×C22⋊C4×C31—21no—
99C2×C42×C31—21no—
100C2×C42×C31C2311yescharacteristic
101C31×#107—22no—
102C31×#106—22no—
103C31×C4⋊C8—21no—
104C31×C4⋊C8—21no—
105C31×#103—22no—
106C31×#103—22no—
107C31×#103—22no—
108C31×#102—22no—
109C31×#102—22no—
110C31×#102—22no—
111C31×#102—22no—
112C31×#102—22no—
113C31×C22⋊C8—21no—
114C31×C22⋊C8—21no—
115C31×#97C2312yes—
116C31×#97C2312yes—
117C31×#97C2312yes—
118C4×C8×C31—21no—
119C24×C31—21no—
120#41×C31—44no—
121C2×Q8×C31—24no—
122C2×Q8×C31—21no—
123C2×Q8×C31—21no—
124C2×Q8×C31—81no—
125C2×Q8×C31—82no—
126C2×Q8×C31C2×D412yes—
127C2×D4×C31—21no—
128C2×D4×C31—21no—
129C2×D4×C31—21no—
130C2×D4×C31—21no—
131C2×D4×C31—41no—
132C2×D4×C31—81no—
133C2×D4×C31C2×D412yes—
134C22×C4×C31—21no—
135C22×C4×C31—21no—
136C22×C4×C31—21no—
137C22×C4×C31—21no—
138C22×C4×C31—21no—
139C22×C4×C31—24no—
140C22×C4×C31—41no—
141C22×C4×C31—82no—
142C22×C4×C31C2×D411yescharacteristic
143C22×C4×C31C2×D411yescharacteristic
144C22×C4×C31C2×D411yescharacteristic
145SD16×C31—82no—
146D8×C31—82no—
147M16×C31—42no—
148C2×C8×C31—22no—
149C2×C8×C31—22no—
150C2×C8×C31—22no—
151C4⋊C4×C31—24no—
152C4⋊C4×C31—24no—
153C4⋊C4×C31—22no—
154C4⋊C4×C31—22no—
155C4⋊C4×C31—22no—
156C4⋊C4×C31—22no—
157C4⋊C4×C31—21no—
158C4⋊C4×C31—21no—
159C4⋊C4×C31—21no—
160C4⋊C4×C31—21no—
161C4⋊C4×C31—22no—
162C4⋊C4×C31—22no—
163C4⋊C4×C31—22no—
164C4⋊C4×C31—22no—
165C4⋊C4×C31—21no—
166C4⋊C4×C31—21no—
167C4⋊C4×C31C2×D412yes—
168C4⋊C4×C31C2×D412yes—
169C4⋊C4×C31C2×D412yes—
170C4⋊C4×C31C2×D412yes—
171C22⋊C4×C31—24no—
172C22⋊C4×C31—24no—
173C22⋊C4×C31—24no—
174C22⋊C4×C31—22no—
175C22⋊C4×C31—22no—
176C22⋊C4×C31—22no—
177C22⋊C4×C31—22no—
178C22⋊C4×C31—41no—
179C22⋊C4×C31—41no—
180C42×C31—24no—
181C42×C31C2×D411yescharacteristic
182C42×C31C2×D411yescharacteristic
183C42×C31C2×D411yescharacteristic
184C42×C31C2411yescharacteristic
185#19002C3111yesmaximal, characteristic, direct product factor, Sylow
186C23×C31—22no—
187C23×C31—42no—
188C23×C31—42no—
189C23×C31—42no—
190C23×C31—41no—
191C23×C31#12011yescharacteristic
192Q8×C31—42no—
193Q8×C31—42no—
194Q8×C31—81no—
195Q8×C31—84no—
196D4×C31—42no—
197D4×C31—42no—
198D4×C31—42no—
199D4×C31—42no—
200D4×C31—81no—
201D4×C31—82no—
202C2×C4×C31—22no—
203C2×C4×C31—22no—
204C2×C4×C31—22no—
205C2×C4×C31—22no—
206C2×C4×C31—22no—
207C2×C4×C31—22no—
208C2×C4×C31—41no—
209C2×C4×C31—41no—
210C2×C4×C31—42no—
211C2×C4×C31—42no—
212C2×C4×C31—42no—
213C2×C4×C31—42no—
214C2×C4×C31—42no—
215C2×C4×C31—41no—
216C2×C4×C31—82no—
217C2×C4×C31—84no—
218C2×C4×C31#12011yescharacteristic
219C2×C4×C31#12011yescharacteristic
220C2×C4×C31#12011yescharacteristic
221C2×C4×C31C22×D411yescharacteristic
222C2×C4×C31C22×D411yescharacteristic
223C2×C4×C31C22×D411yescharacteristic
224C8×C31—42no—
225C8×C31—42no—
226C8×C31—42no—
227C2×#563C2×C3111yescharacteristic, complemented
228#3234C2×C3111yescharacteristic, complemented
229#3213C2×C3111yescharacteristic
230#3142C2×C3111yescharacteristic, complemented
231#1664C2×C3114yescomplemented
232#1657C2×C3114yescomplemented
233#1534C2×C3111yescharacteristic, complemented
234#1528C2×C3111yescharacteristic, complemented
235#1527C2×C3111yescharacteristic, complemented
236C22×C31—21no—
237C22×C31—21no—
238C22×C31—42no—
239C22×C31—42no—
240C22×C31—42no—
241C22×C31—84no—
242C22×C31—81no—
243C22×C31—81no—
244C22×C31—81no—
245C22×C31#45412yes—
246C22×C31C2×#12011yescharacteristic
247C4×C31—21no—
248C4×C31—21no—
249C4×C31—21no—
250C4×C31—21no—
251C4×C31—21no—
252C4×C31—21no—
253C4×C31—42no—
254C4×C31—42no—
255C4×C31—42no—
256C4×C31—42no—
257C4×C31—42no—
258C4×C31—81no—
259#563—22no—
260#563—22no—
261#563—22no—
262#563C22×C3114yes—
263#532C22×C3111yescharacteristic
264C2×C4⋊Q8—21no—
265C2×#126—21no—
266C2×#126—21no—
267C2×#125C22×C3111yescharacteristic
268C2×#125C22×C3111yescharacteristic
269C2×#124C22×C3111yescharacteristic
270C2×#123—21no—
271C2×C22⋊Q8—21no—
272C2×C22⋊Q8—21no—
273#501C22×C3112yescomplemented
274#496C22×C3112yes—
275#491C22×C3112yescomplemented
276#490C22×C3112yescomplemented
277#490C22×C3112yescomplemented
278#489C22×C3112yes—
279#488—22no—
280#481—21no—
281#480—21no—
282#478—22no—
283#463—22no—
284#448—21no—
285#447—21no—
286#432C22×C3111yescharacteristic
287#432C22×C3111yescharacteristic
288#417—21no—
289#417—21no—
290C4×M16C22×C3111yescharacteristic
291#330C22×C3114yescomplemented
292#330C22×C3114yescomplemented
293#329C22×C3114yes—
294C2×C31—22no—
295C2×C31—82no—
296C2×C31—161no—
297C2×C31C2×#45411yescharacteristic
298C2×C31#301911yescharacteristic
299C2×C31#302011yescharacteristic
300C2×#41—21no—
301C22×Q8—21no—
302C22×Q8D4×C3111yescharacteristic
303C22×D4D4×C3111yescharacteristic
304C23×C4—21no—
305C2×SD16—22no—
306C2×D8—22no—
307C2×M16—21no—
308C4⋊Q8—24no—
309C4⋊Q8—24no—
310C4⋊Q8C23×C3111yescharacteristic
311#127C23×C3111yescharacteristic
312#126—24no—
313#126—24no—
314#126—24no—
315#126—24no—
316#125—21no—
317#125—21no—
318#125—21no—
319#125—21no—
320#125D4×C3112yes—
321#125D4×C3112yes—
322#125C23×C3112yes—
323#125C23×C3112yes—
324#124—21no—
325#124—21no—
326#124—21no—
327#124D4×C3112yes—
328#124C23×C3112yes—
329#123—24no—
330#123—24no—
331C22⋊Q8—24no—
332C22⋊Q8—24no—
333C22⋊Q8—24no—
334C22⋊Q8—24no—
335#121—21no—
336#121—21no—
337C4×D4—21no—
338C4×D4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4D4×C3111yescharacteristic
346C2×C4⋊C4D4×C3111yescharacteristic
347C2×C4⋊C4D4×C3111yescharacteristic
348C2×C4⋊C4D4×C3111yescharacteristic
349C2×C22⋊C4—21no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C42—21no—
355C2×C42C23×C3111yescharacteristic
356#107—22no—
357#106—22no—
358C4⋊C8—21no—
359C4⋊C8—21no—
360#103—22no—
361#103—22no—
362#103—22no—
363#102—22no—
364#102—22no—
365#102—22no—
366#102—22no—
367#102—22no—
368C22⋊C8—21no—
369C22⋊C8—21no—
370#97C23×C3112yes—
371#97C23×C3112yes—
372#97C23×C3112yes—
373C4×C8—21no—
374C31#1900211yescharacteristic, direct product factor, Sylow
375C24—21no—
376#41—44no—
377C2×Q8—24no—
378C2×Q8—21no—
379C2×Q8—21no—
380C2×Q8—81no—
381C2×Q8—82no—
382C2×Q8C2×D4×C3112yes—
383C2×D4—21no—
384C2×D4—21no—
385C2×D4—21no—
386C2×D4—21no—
387C2×D4—41no—
388C2×D4—81no—
389C2×D4C2×D4×C3112yes—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—24no—
396C22×C4—41no—
397C22×C4—82no—
398C22×C4C2×D4×C3111yescharacteristic
399C22×C4C2×D4×C3111yescharacteristic
400C22×C4C2×D4×C3111yescharacteristic
401SD16—82no—
402D8—82no—
403M16—42no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—22no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4C2×D4×C3112yes—
424C4⋊C4C2×D4×C3112yes—
425C4⋊C4C2×D4×C3112yes—
426C4⋊C4C2×D4×C3112yes—
427C22⋊C4—24no—
428C22⋊C4—24no—
429C22⋊C4—24no—
430C22⋊C4—22no—
431C22⋊C4—22no—
432C22⋊C4—22no—
433C22⋊C4—22no—
434C22⋊C4—41no—
435C22⋊C4—41no—
436C42—24no—
437C42C2×D4×C3111yescharacteristic
438C42C2×D4×C3111yescharacteristic
439C42C2×D4×C3111yescharacteristic
440C42C24×C3111yescharacteristic
441C23—22no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—41no—
446C23C31×#12011yescharacteristic
447Q8—42no—
448Q8—42no—
449Q8—81no—
450Q8—84no—
451D4—42no—
452D4—42no—
453D4—42no—
454D4—42no—
455D4—81no—
456D4—82no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—41no—
471C2×C4—82no—
472C2×C4—84no—
473C2×C4C31×#12011yescharacteristic
474C2×C4C31×#12011yescharacteristic
475C2×C4C31×#12011yescharacteristic
476C2×C4C22×D4×C3111yescharacteristic
477C2×C4C22×D4×C3111yescharacteristic
478C2×C4C22×D4×C3111yescharacteristic
479C8—42no—
480C8—42no—
481C8—42no—
482C22—21no—
483C22—21no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22—84no—
488C22—81no—
489C22—81no—
490C22—81no—
491C22C31×#45412yes—
492C22C2×C31×#12011yescharacteristic
493C4—21no—
494C4—21no—
495C4—21no—
496C4—21no—
497C4—21no—
498C4—21no—
499C4—42no—
500C4—42no—
501C4—42no—
502C4—42no—
503C4—42no—
504C4—81no—
505C2—22no—
506C2—82no—
507C2—161no—
508C2C2×C31×#45411yescharacteristic
509C2C31×#301911yescharacteristic
510C2C31×#302011yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC42C24×C31
prime radicalC31×#3213C2
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
1. C2×C31×#563C2semidirect
2. C31×#3234C2semidirect
4. C31×#3142C2semidirect
5. C31×#1664C2semidirect
6. C31×#1657C2semidirect
7. C31×#1534C2semidirect
8. C31×#1528C2semidirect
9. C31×#1527C2semidirect
24. C31×#501C22semidirect
26. C31×#491C22semidirect
27. C31×#490C22semidirect
28. C31×#490C22semidirect
42. C31×#330C22semidirect
43. C31×#330C22semidirect
185. #19002C31direct (N × H)
227. C2×#563C2×C31semidirect
228. #3234C2×C31semidirect
230. #3142C2×C31semidirect
231. #1664C2×C31semidirect
232. #1657C2×C31semidirect
233. #1534C2×C31semidirect
234. #1528C2×C31semidirect
235. #1527C2×C31semidirect
273. #501C22×C31semidirect
275. #491C22×C31semidirect
276. #490C22×C31semidirect
277. #490C22×C31semidirect
291. #330C22×C31semidirect
292. #330C22×C31semidirect
374. C31#19002direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5632
C31×#32342
C31×#32132
C31×#31422
C31×#16642
C31×#16572
C31×#15342
C31×#15282
C31×#15272
#1900231