Finite Group Databasebeta

← C31×#18772 #21475531

Subgroups — C31×#18772 #21475531

Order 7936.

1040 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#555C211yesmaximal, characteristic, complemented
2C31×#3206C211yesmaximal, characteristic, complemented
3C31×#3100C211yesmaximal, characteristic, complemented
4C31×#3036C211yesmaximal, characteristic, complemented
5C31×#1646C214yesmaximal, complemented
6C31×#1624C214yesmaximal, complemented
7C31×#1548C211yesmaximal, characteristic, complemented
8C31×#1486C211yesmaximal, characteristic, complemented
9C31×#1478C211yesmaximal, characteristic, complemented
10C2×C31×AGL(1,ℤ/8)—21no—
11C31×#555—22no—
12C31×#555—22no—
13C31×#555—22no—
14C31×#555C2214yes—
15C31×#532C2211yescharacteristic
16C2×C31×C4⋊Q8C2211yescharacteristic
17C2×C31×#124—22no—
18C2×C31×#121—21no—
19C2×C31×#121—21no—
20C2×C4×D4×C31C2211yescharacteristic
21C2×C4×D4×C31C2211yescharacteristic
22C2×C31×#117—21no—
23C2×C31×#117—21no—
24C31×#492C2212yescomplemented
25C31×#487C2212yescomplemented
26C31×#480—21no—
27C31×#474C2212yescomplemented
28C31×#463—22no—
29C31×#461—22no—
30C31×#461C2212yescomplemented
31C31×#459C2212yescomplemented
32C31×#459C2212yescomplemented
33C31×#448—22no—
34C31×#447—21no—
35C4×D8×C31—22no—
36C31×#431C2211yescharacteristic
37C31×#431C2211yescharacteristic
38C31×#423C2211yescharacteristic
39C31×#417—21no—
40C31×#333C2214yescomplemented
41C31×D4⋊C8C2214yescomplemented
42C31×D4⋊C8C2214yescomplemented
43C2×#41×C31—21no—
44C22×Q8×C31D412yes—
45C22×D4×C31—22no—
46C22×D4×C31D411yescharacteristic
47C22×D4×C31D411yescharacteristic
48C23×C4×C31—21no—
49C23×C4×C31—21no—
50C31×AGL(1,ℤ/8)—44no—
51C2×SD16×C31—22no—
52C2×SD16×C31—24no—
53C2×D8×C31—22no—
54C2×D8×C31—22no—
55C2×M16×C31—21no—
56C31×C4⋊Q8—21no—
57C31×C4⋊Q8—21no—
58C31×C4⋊Q8D412yes—
59C31×C4⋊Q8C2311yescharacteristic
60C31×C4⋊Q8C2312yes—
61C31×#127C2311yescharacteristic
62C31×#124—28no—
63C31×#124—28no—
64C31×#124—21no—
65C31×#121—24no—
66C31×#121—24no—
67C31×#121—24no—
68C31×#121—24no—
69C31×#121—21no—
70C31×#121—21no—
71C4×D4×C31—21no—
72C4×D4×C31—21no—
73C4×D4×C31—21no—
74C4×D4×C31—21no—
75C4×D4×C31—21no—
76C4×D4×C31—21no—
77C4×D4×C31D412yes—
78C4×D4×C31D412yes—
79C4×D4×C31C2312yes—
80C4×D4×C31C2312yes—
81C31×#117—24no—
82C31×#117—24no—
83C31×#117—24no—
84C31×#117—24no—
85C2×C4⋊C4×C31—21no—
86C2×C4⋊C4×C31—21no—
87C2×C4⋊C4×C31D411yescharacteristic
88C2×C4⋊C4×C31D411yescharacteristic
89C2×C22⋊C4×C31—22no—
90C2×C22⋊C4×C31—22no—
91C2×C22⋊C4×C31—21no—
92C2×C22⋊C4×C31—21no—
93C2×C42×C31—22no—
94C2×C42×C31C2311yescharacteristic
95C31×#107—22no—
96C31×C4⋊C8—21no—
97C31×C4⋊C8—21no—
98C31×C4⋊C8C2312yes—
99C31×C4⋊C8C2312yes—
100C31×#103—24no—
101C31×#103—22no—
102C31×#102—22no—
103C31×#102—22no—
104C31×#102—22no—
105C31×C22⋊C8—21no—
106C31×C22⋊C8—21no—
107C4×C8×C31—21no—
108C4×C8×C31C2312yes—
109C24×C31—21no—
110C24×C31—21no—
111#41×C31—42no—
112#41×C31—42no—
113C2×Q8×C31—22no—
114C2×Q8×C31—21no—
115C2×Q8×C31—82no—
116C2×Q8×C31C2×D414yes—
117C2×D4×C31—28no—
118C2×D4×C31—21no—
119C2×D4×C31—21no—
120C2×D4×C31—22no—
121C2×D4×C31—22no—
122C2×D4×C31—21no—
123C2×D4×C31—21no—
124C2×D4×C31—21no—
125C2×D4×C31—42no—
126C2×D4×C31—41no—
127C2×D4×C31—84no—
128C2×D4×C31C2×D412yes—
129C2×D4×C31C2×D412yes—
130C22×C4×C31—21no—
131C22×C4×C31—21no—
132C22×C4×C31—22no—
133C22×C4×C31—22no—
134C22×C4×C31—22no—
135C22×C4×C31—22no—
136C22×C4×C31—22no—
137C22×C4×C31—44no—
138C22×C4×C31—42no—
139C22×C4×C31—42no—
140C22×C4×C31—41no—
141C22×C4×C31C2×D411yescharacteristic
142C22×C4×C31C2×D411yescharacteristic
143C22×C4×C31C2×D411yescharacteristic
144SD16×C31—44no—
145SD16×C31—84no—
146D8×C31—44no—
147D8×C31—44no—
148M16×C31—42no—
149C2×C8×C31—22no—
150C2×C8×C31—22no—
151C2×C8×C31—22no—
152C4⋊C4×C31—22no—
153C4⋊C4×C31—22no—
154C4⋊C4×C31—21no—
155C4⋊C4×C31—21no—
156C4⋊C4×C31—22no—
157C4⋊C4×C31—22no—
158C4⋊C4×C31—21no—
159C4⋊C4×C31—21no—
160C4⋊C4×C31C2×D412yes—
161C4⋊C4×C31C2×D412yes—
162C22⋊C4×C31—28no—
163C22⋊C4×C31—28no—
164C22⋊C4×C31—22no—
165C22⋊C4×C31—22no—
166C22⋊C4×C31—22no—
167C22⋊C4×C31—22no—
168C22⋊C4×C31—41no—
169C22⋊C4×C31—41no—
170C42×C31—28no—
171C42×C31C2×D411yescharacteristic
172C42×C31C2×D411yescharacteristic
173C42×C31C2×D411yescharacteristic
174C42×C31C2411yescharacteristic
175#18772C3111yesmaximal, characteristic, direct product factor, Sylow
176C23×C31—22no—
177C23×C31—22no—
178C23×C31—42no—
179C23×C31—42no—
180C23×C31—42no—
181C23×C31—42no—
182C23×C31—42no—
183C23×C31—42no—
184C23×C31—41no—
185C23×C31#12011yescharacteristic
186Q8×C31—44no—
187Q8×C31—41no—
188Q8×C31—41no—
189Q8×C31—82no—
190D4×C31—22no—
191D4×C31—22no—
192D4×C31—22no—
193D4×C31—22no—
194D4×C31—42no—
195D4×C31—42no—
196D4×C31—41no—
197D4×C31—41no—
198D4×C31—42no—
199D4×C31—42no—
200D4×C31—88no—
201D4×C31—82no—
202C2×C4×C31—22no—
203C2×C4×C31—22no—
204C2×C4×C31—22no—
205C2×C4×C31—24no—
206C2×C4×C31—41no—
207C2×C4×C31—44no—
208C2×C4×C31—42no—
209C2×C4×C31—44no—
210C2×C4×C31—42no—
211C2×C4×C31—42no—
212C2×C4×C31—42no—
213C2×C4×C31—44no—
214C2×C4×C31—44no—
215C2×C4×C31—41no—
216C2×C4×C31—82no—
217C2×C4×C31#12011yescharacteristic
218C2×C4×C31#12011yescharacteristic
219C2×C4×C31#12011yescharacteristic
220C2×C4×C31#13512yes—
221C2×C4×C31AGL(1,ℤ/8)12yes—
222C2×C4×C31C22×D411yescharacteristic
223C2×C4×C31C22×D411yescharacteristic
224C2×C4×C31C22×D411yescharacteristic
225C8×C31—42no—
226C8×C31—42no—
227C8×C31—42no—
228C2×#555C2×C3111yescharacteristic, complemented
229#3206C2×C3111yescharacteristic, complemented
230#3100C2×C3111yescharacteristic, complemented
231#3036C2×C3111yescharacteristic, complemented
232#1646C2×C3114yescomplemented
233#1624C2×C3114yescomplemented
234#1548C2×C3111yescharacteristic, complemented
235#1486C2×C3111yescharacteristic, complemented
236#1478C2×C3111yescharacteristic, complemented
237C22×C31—21no—
238C22×C31—21no—
239C22×C31—42no—
240C22×C31—42no—
241C22×C31—42no—
242C22×C31—42no—
243C22×C31—42no—
244C22×C31—42no—
245C22×C31—84no—
246C22×C31—84no—
247C22×C31—81no—
248C22×C31—81no—
249C22×C31—81no—
250C22×C31#44912yes—
251C22×C31C2×#12011yescharacteristic
252C4×C31—21no—
253C4×C31—21no—
254C4×C31—21no—
255C4×C31—42no—
256C4×C31—42no—
257C4×C31—44no—
258C4×C31—81no—
259C4×C31#44912yes—
260C4×C31C2×#13511yescharacteristic
261C4×C31C2×AGL(1,ℤ/8)11yescharacteristic
262C4×C31#57512yes—
263C2×AGL(1,ℤ/8)—21no—
264#555—22no—
265#555—22no—
266#555—22no—
267#555C22×C3114yes—
268#532C22×C3111yescharacteristic
269C2×C4⋊Q8C22×C3111yescharacteristic
270C2×#124—22no—
271C2×#121—21no—
272C2×#121—21no—
273C2×C4×D4C22×C3111yescharacteristic
274C2×C4×D4C22×C3111yescharacteristic
275C2×#117—21no—
276C2×#117—21no—
277#492C22×C3112yescomplemented
278#487C22×C3112yescomplemented
279#480—21no—
280#474C22×C3112yescomplemented
281#463—22no—
282#461—22no—
283#461C22×C3112yescomplemented
284#459C22×C3112yescomplemented
285#459C22×C3112yescomplemented
286#448—22no—
287#447—21no—
288C4×D8—22no—
289#431C22×C3111yescharacteristic
290#431C22×C3111yescharacteristic
291#423C22×C3111yescharacteristic
292#417—21no—
293#333C22×C3114yescomplemented
294D4⋊C8C22×C3114yescomplemented
295D4⋊C8C22×C3114yescomplemented
296C2×C31—22no—
297C2×C31—82no—
298C2×C31—82no—
299C2×C31—161no—
300C2×C31C2×#44911yescharacteristic
301C2×C31#300511yescharacteristic
302C2×C31#301911yescharacteristic
303C2×#41—21no—
304C22×Q8D4×C3112yes—
305C22×D4—22no—
306C22×D4D4×C3111yescharacteristic
307C22×D4D4×C3111yescharacteristic
308C23×C4—21no—
309C23×C4—21no—
310AGL(1,ℤ/8)—44no—
311C2×SD16—22no—
312C2×SD16—24no—
313C2×D8—22no—
314C2×D8—22no—
315C2×M16—21no—
316C4⋊Q8—21no—
317C4⋊Q8—21no—
318C4⋊Q8D4×C3112yes—
319C4⋊Q8C23×C3111yescharacteristic
320C4⋊Q8C23×C3112yes—
321#127C23×C3111yescharacteristic
322#124—28no—
323#124—28no—
324#124—21no—
325#121—24no—
326#121—24no—
327#121—24no—
328#121—24no—
329#121—21no—
330#121—21no—
331C4×D4—21no—
332C4×D4—21no—
333C4×D4—21no—
334C4×D4—21no—
335C4×D4—21no—
336C4×D4—21no—
337C4×D4D4×C3112yes—
338C4×D4D4×C3112yes—
339C4×D4C23×C3112yes—
340C4×D4C23×C3112yes—
341#117—24no—
342#117—24no—
343#117—24no—
344#117—24no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4D4×C3111yescharacteristic
348C2×C4⋊C4D4×C3111yescharacteristic
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C42—22no—
354C2×C42C23×C3111yescharacteristic
355#107—22no—
356C4⋊C8—21no—
357C4⋊C8—21no—
358C4⋊C8C23×C3112yes—
359C4⋊C8C23×C3112yes—
360#103—24no—
361#103—22no—
362#102—22no—
363#102—22no—
364#102—22no—
365C22⋊C8—21no—
366C22⋊C8—21no—
367C4×C8—21no—
368C4×C8C23×C3112yes—
369C31#1877211yescharacteristic, direct product factor, Sylow
370C24—21no—
371C24—21no—
372#41—42no—
373#41—42no—
374C2×Q8—22no—
375C2×Q8—21no—
376C2×Q8—82no—
377C2×Q8C2×D4×C3114yes—
378C2×D4—28no—
379C2×D4—21no—
380C2×D4—21no—
381C2×D4—22no—
382C2×D4—22no—
383C2×D4—21no—
384C2×D4—21no—
385C2×D4—21no—
386C2×D4—42no—
387C2×D4—41no—
388C2×D4—84no—
389C2×D4C2×D4×C3112yes—
390C2×D4C2×D4×C3112yes—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—44no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—41no—
402C22×C4C2×D4×C3111yescharacteristic
403C22×C4C2×D4×C3111yescharacteristic
404C22×C4C2×D4×C3111yescharacteristic
405SD16—44no—
406SD16—84no—
407D8—44no—
408D8—44no—
409M16—42no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4C2×D4×C3112yes—
422C4⋊C4C2×D4×C3112yes—
423C22⋊C4—28no—
424C22⋊C4—28no—
425C22⋊C4—22no—
426C22⋊C4—22no—
427C22⋊C4—22no—
428C22⋊C4—22no—
429C22⋊C4—41no—
430C22⋊C4—41no—
431C42—28no—
432C42C2×D4×C3111yescharacteristic
433C42C2×D4×C3111yescharacteristic
434C42C2×D4×C3111yescharacteristic
435C42C24×C3111yescharacteristic
436C23—22no—
437C23—22no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—41no—
445C23C31×#12011yescharacteristic
446Q8—44no—
447Q8—41no—
448Q8—41no—
449Q8—82no—
450D4—22no—
451D4—22no—
452D4—22no—
453D4—22no—
454D4—42no—
455D4—42no—
456D4—42no—
457D4—41no—
458D4—41no—
459D4—42no—
460D4—88no—
461D4—82no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—24no—
466C2×C4—41no—
467C2×C4—44no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—41no—
476C2×C4—82no—
477C2×C4C31×#12011yescharacteristic
478C2×C4C31×#12011yescharacteristic
479C2×C4C31×#12011yescharacteristic
480C2×C4C31×#13512yes—
481C2×C4C31×AGL(1,ℤ/8)12yes—
482C2×C4C22×D4×C3111yescharacteristic
483C2×C4C22×D4×C3111yescharacteristic
484C2×C4C22×D4×C3111yescharacteristic
485C8—42no—
486C8—42no—
487C8—42no—
488C22—21no—
489C22—21no—
490C22—42no—
491C22—42no—
492C22—42no—
493C22—42no—
494C22—42no—
495C22—42no—
496C22—84no—
497C22—84no—
498C22—81no—
499C22—81no—
500C22—81no—
501C22C31×#44912yes—
502C22C2×C31×#12011yescharacteristic
503C4—21no—
504C4—21no—
505C4—21no—
506C4—42no—
507C4—42no—
508C4—44no—
509C4—81no—
510C4C31×#44912yes—
511C4C2×C31×#13511yescharacteristic
512C4C2×C31×AGL(1,ℤ/8)11yescharacteristic
513C4C31×#57512yes—
514C2—22no—
515C2—82no—
516C2—82no—
517C2—161no—
518C2C2×C31×#44911yescharacteristic
519C2C31×#300511yescharacteristic
520C2C31×#301911yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC42C24×C31
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×#555C2semidirect
2. C31×#3206C2semidirect
3. C31×#3100C2semidirect
4. C31×#3036C2semidirect
5. C31×#1646C2semidirect
6. C31×#1624C2semidirect
7. C31×#1548C2semidirect
8. C31×#1486C2semidirect
9. C31×#1478C2semidirect
24. C31×#492C22semidirect
25. C31×#487C22semidirect
27. C31×#474C22semidirect
30. C31×#461C22semidirect
31. C31×#459C22semidirect
32. C31×#459C22semidirect
40. C31×#333C22semidirect
41. C31×D4⋊C8C22semidirect
42. C31×D4⋊C8C22semidirect
175. #18772C31direct (N × H)
228. C2×#555C2×C31semidirect
229. #3206C2×C31semidirect
230. #3100C2×C31semidirect
231. #3036C2×C31semidirect
232. #1646C2×C31semidirect
233. #1624C2×C31semidirect
234. #1548C2×C31semidirect
235. #1486C2×C31semidirect
236. #1478C2×C31semidirect
277. #492C22×C31semidirect
278. #487C22×C31semidirect
280. #474C22×C31semidirect
283. #461C22×C31semidirect
284. #459C22×C31semidirect
285. #459C22×C31semidirect
293. #333C22×C31semidirect
294. D4⋊C8C22×C31semidirect
295. D4⋊C8C22×C31semidirect
369. C31#18772direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5552
C31×#32062
C31×#31002
C31×#30362
C31×#16462
C31×#16242
C31×#15482
C31×#14862
C31×#14782
#1877231