Finite Group Databasebeta

← C31×#18770 #21475529

Subgroups — C31×#18770 #21475529

Order 7936.

1288 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#550C211yesmaximal, characteristic, complemented
2C31×#3205C211yesmaximal, characteristic, complemented
3C31×#3103C211yesmaximal, characteristic, complemented
4C31×#3036C211yesmaximal, characteristic, complemented
5C31×#1642C214yesmaximal, complemented
6C31×#1621C214yesmaximal, complemented
7C31×#1545C211yesmaximal, characteristic, complemented
8C31×#1487C211yesmaximal, characteristic, complemented
9C31×#1478C211yesmaximal, characteristic, complemented
10C22×#41×C31—21no—
11C2×C31×AGL(1,ℤ/8)—21no—
12C31×#550—22no—
13C31×#550—22no—
14C31×#550—22no—
15C31×#550C2214yes—
16C31×#532C2211yescharacteristic
17C2×C31×#127—21no—
18C2×C31×#127C2211yescharacteristic
19C2×C31×#121—22no—
20C2×C31×#121—21no—
21C2×C4×Q8×C31C2211yescharacteristic
22C2×C4×D4×C31—21no—
23C2×C4×D4×C31C2211yescharacteristic
24C31×#493C2212yescomplemented
25C31×#486C2212yescomplemented
26C31×#481—21no—
27C31×#475C2212yescomplemented
28C31×#463—22no—
29C31×#461C2212yescomplemented
30C31×#460C2212yescomplemented
31C31×#459—22no—
32C31×#459C2212yescomplemented
33C31×#448—21no—
34C31×#447—22no—
35C4×SD16×C31—22no—
36C31×#431C2211yescharacteristic
37C31×#431C2211yescharacteristic
38C31×#423C2211yescharacteristic
39C31×#417—21no—
40C31×#331C2214yescomplemented
41C31×Q8⋊C8C2214yescomplemented
42C31×D4⋊C8C2214yescomplemented
43C2×#41×C31—24no—
44C2×#41×C31—24no—
45C2×#41×C31—21no—
46C2×#41×C31—48no—
47C22×Q8×C31D411yescharacteristic
48C22×D4×C31—22no—
49C22×D4×C31—22no—
50C22×D4×C31—21no—
51C22×D4×C31—21no—
52C22×D4×C31D411yescharacteristic
53C22×D4×C31D412yes—
54C23×C4×C31—22no—
55C23×C4×C31—21no—
56C31×AGL(1,ℤ/8)—44no—
57C2×SD16×C31—22no—
58C2×SD16×C31—22no—
59C2×D8×C31—22no—
60C2×D8×C31—24no—
61C2×M16×C31—21no—
62C31×C4⋊Q8C2311yescharacteristic
63C31×#127—24no—
64C31×#127—24no—
65C31×#127—21no—
66C31×#127—21no—
67C31×#127D412yes—
68C31×#127C2311yescharacteristic
69C31×#127C2312yes—
70C31×#124—21no—
71C31×#121—24no—
72C31×#121—28no—
73C31×#121—28no—
74C31×#121—24no—
75C31×#121—21no—
76C31×#121—21no—
77C4×Q8×C31—21no—
78C4×Q8×C31—21no—
79C4×Q8×C31D412yes—
80C4×Q8×C31C2312yes—
81C4×D4×C31—24no—
82C4×D4×C31—24no—
83C4×D4×C31—21no—
84C4×D4×C31—21no—
85C4×D4×C31—21no—
86C4×D4×C31—21no—
87C4×D4×C31D412yes—
88C4×D4×C31C2312yes—
89C2×C4⋊C4×C31—21no—
90C2×C4⋊C4×C31D411yescharacteristic
91C2×C4⋊C4×C31D411yescharacteristic
92C2×C22⋊C4×C31—22no—
93C2×C22⋊C4×C31—21no—
94C2×C42×C31—21no—
95C2×C42×C31C2311yescharacteristic
96C31×#106—22no—
97C31×C4⋊C8—21no—
98C31×C4⋊C8—21no—
99C31×C4⋊C8C2312yes—
100C31×C4⋊C8C2312yes—
101C31×#103—22no—
102C31×#103—22no—
103C31×#102—24no—
104C31×#102—22no—
105C31×#102—22no—
106C31×C22⋊C8—21no—
107C31×C22⋊C8—21no—
108C4×C8×C31—21no—
109C4×C8×C31C2312yes—
110C24×C31—21no—
111C24×C31—22no—
112#41×C31—416no—
113#41×C31—48no—
114#41×C31—48no—
115#41×C31—42no—
116#41×C31—42no—
117C2×Q8×C31—24no—
118C2×Q8×C31—21no—
119C2×Q8×C31—21no—
120C2×Q8×C31C2×D412yes—
121C2×D4×C31—28no—
122C2×D4×C31—28no—
123C2×D4×C31—21no—
124C2×D4×C31—22no—
125C2×D4×C31—22no—
126C2×D4×C31—22no—
127C2×D4×C31—22no—
128C2×D4×C31—22no—
129C2×D4×C31—22no—
130C2×D4×C31—22no—
131C2×D4×C31—21no—
132C2×D4×C31—21no—
133C2×D4×C31—21no—
134C2×D4×C31—48no—
135C2×D4×C31—44no—
136C2×D4×C31—41no—
137C2×D4×C31—42no—
138C2×D4×C31—42no—
139C2×D4×C31—82no—
140C2×D4×C31—84no—
141C2×D4×C31C2×D412yes—
142C2×D4×C31C2×D414yes—
143C22×C4×C31—21no—
144C22×C4×C31—28no—
145C22×C4×C31—21no—
146C22×C4×C31—21no—
147C22×C4×C31—22no—
148C22×C4×C31—22no—
149C22×C4×C31—48no—
150C22×C4×C31—44no—
151C22×C4×C31—41no—
152C22×C4×C31C2×D411yescharacteristic
153C22×C4×C31C2×D411yescharacteristic
154C22×C4×C31C2×D411yescharacteristic
155SD16×C31—44no—
156SD16×C31—44no—
157D8×C31—44no—
158D8×C31—84no—
159M16×C31—42no—
160C2×C8×C31—22no—
161C2×C8×C31—22no—
162C2×C8×C31—22no—
163C4⋊C4×C31—22no—
164C4⋊C4×C31—21no—
165C4⋊C4×C31—21no—
166C4⋊C4×C31—22no—
167C4⋊C4×C31—21no—
168C4⋊C4×C31—21no—
169C4⋊C4×C31C2×D412yes—
170C4⋊C4×C31C2×D412yes—
171C22⋊C4×C31—28no—
172C22⋊C4×C31—22no—
173C22⋊C4×C31—22no—
174C22⋊C4×C31—41no—
175C22⋊C4×C31—41no—
176C42×C31—22no—
177C42×C31—22no—
178C42×C31C2×D411yescharacteristic
179C42×C31C2×D411yescharacteristic
180C42×C31C2×D411yescharacteristic
181C42×C31C2411yescharacteristic
182#18770C3111yesmaximal, characteristic, direct product factor, Sylow
183C23×C31—22no—
184C23×C31—24no—
185C23×C31—44no—
186C23×C31—42no—
187C23×C31—44no—
188C23×C31—42no—
189C23×C31—42no—
190C23×C31—44no—
191C23×C31—41no—
192C23×C31#12011yescharacteristic
193Q8×C31—24no—
194Q8×C31—22no—
195Q8×C31—41no—
196Q8×C31—41no—
197Q8×C31AGL(1,ℤ/8)14yescomplemented
198D4×C31—22no—
199D4×C31—22no—
200D4×C31—44no—
201D4×C31—416no—
202D4×C31—44no—
203D4×C31—42no—
204D4×C31—42no—
205D4×C31—41no—
206D4×C31—42no—
207D4×C31—44no—
208D4×C31—41no—
209D4×C31—42no—
210D4×C31—42no—
211D4×C31—44no—
212D4×C31—82no—
213D4×C31—88no—
214D4×C31—82no—
215C2×C4×C31—24no—
216C2×C4×C31—22no—
217C2×C4×C31—22no—
218C2×C4×C31—22no—
219C2×C4×C31—22no—
220C2×C4×C31—41no—
221C2×C4×C31—416no—
222C2×C4×C31—44no—
223C2×C4×C31—42no—
224C2×C4×C31—42no—
225C2×C4×C31—44no—
226C2×C4×C31—41no—
227C2×C4×C31—82no—
228C2×C4×C31#12011yescharacteristic
229C2×C4×C31#12011yescharacteristic
230C2×C4×C31#12011yescharacteristic
231C2×C4×C31#13512yes—
232C2×C4×C31AGL(1,ℤ/8)12yes—
233C2×C4×C31C22×D411yescharacteristic
234C2×C4×C31C22×D411yescharacteristic
235C2×C4×C31C22×D411yescharacteristic
236C8×C31—42no—
237C8×C31—42no—
238C8×C31—42no—
239C2×#550C2×C3111yescharacteristic, complemented
240#3205C2×C3111yescharacteristic, complemented
241#3103C2×C3111yescharacteristic, complemented
242#3036C2×C3111yescharacteristic, complemented
243#1642C2×C3114yescomplemented
244#1621C2×C3114yescomplemented
245#1545C2×C3111yescharacteristic, complemented
246#1487C2×C3111yescharacteristic, complemented
247#1478C2×C3111yescharacteristic, complemented
248C22×C31—21no—
249C22×C31—21no—
250C22×C31—42no—
251C22×C31—44no—
252C22×C31—44no—
253C22×C31—42no—
254C22×C31—44no—
255C22×C31—42no—
256C22×C31—84no—
257C22×C31—88no—
258C22×C31—81no—
259C22×C31—81no—
260C22×C31—81no—
261C22×C31#44912yes—
262C22×C31C2×#12011yescharacteristic
263C4×C31—24no—
264C4×C31—21no—
265C4×C31—21no—
266C4×C31—21no—
267C4×C31—42no—
268C4×C31—42no—
269C4×C31—81no—
270C4×C31#44912yes—
271C4×C31C2×#13511yescharacteristic
272C4×C31C2×AGL(1,ℤ/8)11yescharacteristic
273C4×C31C2×AGL(1,ℤ/8)12yes—
274C22×#41—21no—
275C2×AGL(1,ℤ/8)—21no—
276#550—22no—
277#550—22no—
278#550—22no—
279#550C22×C3114yes—
280#532C22×C3111yescharacteristic
281C2×#127—21no—
282C2×#127C22×C3111yescharacteristic
283C2×#121—22no—
284C2×#121—21no—
285C2×C4×Q8C22×C3111yescharacteristic
286C2×C4×D4—21no—
287C2×C4×D4C22×C3111yescharacteristic
288#493C22×C3112yescomplemented
289#486C22×C3112yescomplemented
290#481—21no—
291#475C22×C3112yescomplemented
292#463—22no—
293#461C22×C3112yescomplemented
294#460C22×C3112yescomplemented
295#459—22no—
296#459C22×C3112yescomplemented
297#448—21no—
298#447—22no—
299C4×SD16—22no—
300#431C22×C3111yescharacteristic
301#431C22×C3111yescharacteristic
302#423C22×C3111yescharacteristic
303#417—21no—
304#331C22×C3114yescomplemented
305Q8⋊C8C22×C3114yescomplemented
306D4⋊C8C22×C3114yescomplemented
307C2×C31—22no—
308C2×C31—82no—
309C2×C31—84no—
310C2×C31—161no—
311C2×C31C2×#44911yescharacteristic
312C2×C31#300311yescharacteristic
313C2×C31#301911yescharacteristic
314C2×#41—24no—
315C2×#41—24no—
316C2×#41—21no—
317C2×#41—48no—
318C22×Q8D4×C3111yescharacteristic
319C22×D4—22no—
320C22×D4—22no—
321C22×D4—21no—
322C22×D4—21no—
323C22×D4D4×C3111yescharacteristic
324C22×D4D4×C3112yes—
325C23×C4—22no—
326C23×C4—21no—
327AGL(1,ℤ/8)—44no—
328C2×SD16—22no—
329C2×SD16—22no—
330C2×D8—22no—
331C2×D8—24no—
332C2×M16—21no—
333C4⋊Q8C23×C3111yescharacteristic
334#127—24no—
335#127—24no—
336#127—21no—
337#127—21no—
338#127D4×C3112yes—
339#127C23×C3111yescharacteristic
340#127C23×C3112yes—
341#124—21no—
342#121—24no—
343#121—28no—
344#121—28no—
345#121—24no—
346#121—21no—
347#121—21no—
348C4×Q8—21no—
349C4×Q8—21no—
350C4×Q8D4×C3112yes—
351C4×Q8C23×C3112yes—
352C4×D4—24no—
353C4×D4—24no—
354C4×D4—21no—
355C4×D4—21no—
356C4×D4—21no—
357C4×D4—21no—
358C4×D4D4×C3112yes—
359C4×D4C23×C3112yes—
360C2×C4⋊C4—21no—
361C2×C4⋊C4D4×C3111yescharacteristic
362C2×C4⋊C4D4×C3111yescharacteristic
363C2×C22⋊C4—22no—
364C2×C22⋊C4—21no—
365C2×C42—21no—
366C2×C42C23×C3111yescharacteristic
367#106—22no—
368C4⋊C8—21no—
369C4⋊C8—21no—
370C4⋊C8C23×C3112yes—
371C4⋊C8C23×C3112yes—
372#103—22no—
373#103—22no—
374#102—24no—
375#102—22no—
376#102—22no—
377C22⋊C8—21no—
378C22⋊C8—21no—
379C4×C8—21no—
380C4×C8C23×C3112yes—
381C31#1877011yescharacteristic, direct product factor, Sylow
382C24—21no—
383C24—22no—
384#41—416no—
385#41—48no—
386#41—48no—
387#41—42no—
388#41—42no—
389C2×Q8—24no—
390C2×Q8—21no—
391C2×Q8—21no—
392C2×Q8C2×D4×C3112yes—
393C2×D4—28no—
394C2×D4—28no—
395C2×D4—21no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—48no—
407C2×D4—44no—
408C2×D4—41no—
409C2×D4—42no—
410C2×D4—42no—
411C2×D4—82no—
412C2×D4—84no—
413C2×D4C2×D4×C3112yes—
414C2×D4C2×D4×C3114yes—
415C22×C4—21no—
416C22×C4—28no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—48no—
422C22×C4—44no—
423C22×C4—41no—
424C22×C4C2×D4×C3111yescharacteristic
425C22×C4C2×D4×C3111yescharacteristic
426C22×C4C2×D4×C3111yescharacteristic
427SD16—44no—
428SD16—44no—
429D8—44no—
430D8—84no—
431M16—42no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C4⋊C4—22no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—22no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4C2×D4×C3112yes—
442C4⋊C4C2×D4×C3112yes—
443C22⋊C4—28no—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C22⋊C4—41no—
447C22⋊C4—41no—
448C42—22no—
449C42—22no—
450C42C2×D4×C3111yescharacteristic
451C42C2×D4×C3111yescharacteristic
452C42C2×D4×C3111yescharacteristic
453C42C24×C3111yescharacteristic
454C23—22no—
455C23—24no—
456C23—44no—
457C23—42no—
458C23—44no—
459C23—42no—
460C23—42no—
461C23—44no—
462C23—41no—
463C23C31×#12011yescharacteristic
464Q8—24no—
465Q8—22no—
466Q8—41no—
467Q8—41no—
468Q8C31×AGL(1,ℤ/8)14yescomplemented
469D4—22no—
470D4—22no—
471D4—44no—
472D4—416no—
473D4—44no—
474D4—42no—
475D4—42no—
476D4—44no—
477D4—42no—
478D4—41no—
479D4—42no—
480D4—44no—
481D4—41no—
482D4—42no—
483D4—82no—
484D4—88no—
485D4—82no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—41no—
492C2×C4—416no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—41no—
498C2×C4—82no—
499C2×C4C31×#12011yescharacteristic
500C2×C4C31×#12011yescharacteristic
501C2×C4C31×#12011yescharacteristic
502C2×C4C31×#13512yes—
503C2×C4C31×AGL(1,ℤ/8)12yes—
504C2×C4C22×D4×C3111yescharacteristic
505C2×C4C22×D4×C3111yescharacteristic
506C2×C4C22×D4×C3111yescharacteristic
507C8—42no—
508C8—42no—
509C8—42no—
510C22—21no—
511C22—21no—
512C22—42no—
513C22—44no—
514C22—44no—
515C22—42no—
516C22—44no—
517C22—42no—
518C22—84no—
519C22—88no—
520C22—81no—
521C22—81no—
522C22—81no—
523C22C31×#44912yes—
524C22C2×C31×#12011yescharacteristic
525C4—24no—
526C4—21no—
527C4—21no—
528C4—21no—
529C4—42no—
530C4—42no—
531C4—81no—
532C4C31×#44912yes—
533C4C2×C31×#13511yescharacteristic
534C4C2×C31×AGL(1,ℤ/8)11yescharacteristic
535C4C2×C31×AGL(1,ℤ/8)12yes—
536C2—22no—
537C2—82no—
538C2—84no—
539C2—161no—
540C2C2×C31×#44911yescharacteristic
541C2C31×#300311yescharacteristic
542C2C31×#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×#550C2semidirect
2. C31×#3205C2semidirect
3. C31×#3103C2semidirect
4. C31×#3036C2semidirect
5. C31×#1642C2semidirect
6. C31×#1621C2semidirect
7. C31×#1545C2semidirect
8. C31×#1487C2semidirect
9. C31×#1478C2semidirect
24. C31×#493C22semidirect
25. C31×#486C22semidirect
27. C31×#475C22semidirect
29. C31×#461C22semidirect
30. C31×#460C22semidirect
32. C31×#459C22semidirect
40. C31×#331C22semidirect
41. C31×Q8⋊C8C22semidirect
42. C31×D4⋊C8C22semidirect
182. #18770C31direct (N × H)
197. Q8×C31AGL(1,ℤ/8)semidirect
239. C2×#550C2×C31semidirect
240. #3205C2×C31semidirect
241. #3103C2×C31semidirect
242. #3036C2×C31semidirect
243. #1642C2×C31semidirect
244. #1621C2×C31semidirect
245. #1545C2×C31semidirect
246. #1487C2×C31semidirect
247. #1478C2×C31semidirect
288. #493C22×C31semidirect
289. #486C22×C31semidirect
291. #475C22×C31semidirect
293. #461C22×C31semidirect
294. #460C22×C31semidirect
296. #459C22×C31semidirect
304. #331C22×C31semidirect
305. Q8⋊C8C22×C31semidirect
306. D4⋊C8C22×C31semidirect
381. C31#18770direct (N × H)
468. Q8C31×AGL(1,ℤ/8)semidirect

Maximal subgroups

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

subgroupindex
C2×C31×#5502
C31×#32052
C31×#31032
C31×#30362
C31×#16422
C31×#16212
C31×#15452
C31×#14872
C31×#14782
#1877031