Finite Group Databasebeta

← C31×#18317 #21475076

Subgroups — C31×#18317 #21475076

Order 7936.

1402 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#534C211yesmaximal, characteristic
2C31×#3193C212yesmaximal, complemented
3C31×#2561C211yesmaximal, characteristic, complemented
4C31×#1607C214yesmaximal, complemented
5C31×#1601C214yesmaximal, complemented
6C31×#1507C212yesmaximal, complemented
7C31×#1462C211yesmaximal, characteristic, complemented
8C23×D4×C31—21no—
9C31×#534—24no—
10C31×#534—22no—
11C31×#534C2214yes—
12C2×C31×#123—22no—
13C2×C31×C22⋊Q8C2211yescharacteristic, complemented
14C2×C31×#121—21no—
15C2×C31×#121C2212yes—
16C2×C31×#120—22no—
17C22×C22⋊C4×C31C2211yescharacteristic
18C31×#484—22no—
19C31×#480C2214yescomplemented
20C31×#468—24no—
21C31×#450—22no—
22C31×#448C2214yescomplemented
23C31×#407C2212yescomplemented
24C31×#397C2212yescomplemented
25C31×#393C2212yescomplemented
26C31×#388—22no—
27C31×#381—21no—
28C31×#379—21no—
29C4×C22⋊C4×C31C2211yescharacteristic, complemented
30C31×#327C2218yescomplemented
31C31×#323C2214yescomplemented
32C25×C31—21no—
33C25×C31D411yescharacteristic
34C22×Q8×C31C2311yescharacteristic, complemented
35C22×D4×C31—22no—
36C22×D4×C31—24no—
37C22×D4×C31—22no—
38C22×D4×C31—24no—
39C22×D4×C31—21no—
40C22×D4×C31—48no—
41C22×D4×C31D412yes—
42C23×C4×C31—21no—
43C23×C4×C31D411yescharacteristic
44C23×C4×C31C2311yescharacteristic
45C2×SD16×C31—24no—
46C2×M16×C31—22no—
47C31×#123—28no—
48C31×#123—28no—
49C31×C22⋊Q8—22no—
50C31×C22⋊Q8—22no—
51C31×#121—24no—
52C31×#121—24no—
53C31×#121—22no—
54C31×#121—22no—
55C31×#121D414yes—
56C31×#121C2314yes—
57C31×#120—28no—
58C31×#120—28no—
59C2×C4⋊C4×C31—21no—
60C2×C4⋊C4×C31—21no—
61C2×C4⋊C4×C31D412yes—
62C2×C4⋊C4×C31C2311yescharacteristic, complemented
63C2×C22⋊C4×C31—22no—
64C2×C22⋊C4×C31—22no—
65C2×C22⋊C4×C31—22no—
66C2×C22⋊C4×C31—22no—
67C2×C22⋊C4×C31—22no—
68C2×C22⋊C4×C31—21no—
69C2×C22⋊C4×C31—21no—
70C2×C22⋊C4×C31—22no—
71C2×C22⋊C4×C31D412yes—
72C2×C22⋊C4×C31C2312yes—
73C2×C42×C31—21no—
74C31×#107—24no—
75C31×#103—24no—
76C31×#102—24no—
77C31×C22⋊C8—22no—
78C31×C22⋊C8C2314yescomplemented
79C31×#95—22no—
80C31×#95—22no—
81C31×#95C2312yescomplemented
82C24×C31—21no—
83C24×C31—21no—
84C24×C31—24no—
85C24×C31—24no—
86C24×C31—22no—
87C24×C31—44no—
88C24×C31—42no—
89C24×C31—81no—
90C24×C31—82no—
91C24×C31—82no—
92C24×C31C2×D411yescharacteristic
93C24×C31C2×D412yes—
94C2×Q8×C31—22no—
95C2×Q8×C31—81no—
96C2×D4×C31—28no—
97C2×D4×C31—24no—
98C2×D4×C31—24no—
99C2×D4×C31—24no—
100C2×D4×C31—22no—
101C2×D4×C31—48no—
102C2×D4×C31—416no—
103C2×D4×C31—48no—
104C2×D4×C31—48no—
105C2×D4×C31—48no—
106C2×D4×C31—82no—
107C2×D4×C31—82no—
108C2×D4×C31C2×D414yes—
109C22×C4×C31—24no—
110C22×C4×C31—21no—
111C22×C4×C31—22no—
112C22×C4×C31—21no—
113C22×C4×C31—22no—
114C22×C4×C31—21no—
115C22×C4×C31—21no—
116C22×C4×C31—44no—
117C22×C4×C31—42no—
118C22×C4×C31—81no—
119C22×C4×C31C2×D412yes—
120C22×C4×C31C2×D411yescharacteristic
121C22×C4×C31C2×D412yes—
122C22×C4×C31C2×D412yes—
123C22×C4×C31C2411yescharacteristic
124SD16×C31—84no—
125M16×C31—82no—
126C2×C8×C31—24no—
127C4⋊C4×C31—24no—
128C4⋊C4×C31—22no—
129C4⋊C4×C31—22no—
130C4⋊C4×C31—42no—
131C4⋊C4×C31C2×D414yes—
132C22⋊C4×C31—28no—
133C22⋊C4×C31—24no—
134C22⋊C4×C31—24no—
135C22⋊C4×C31—24no—
136C22⋊C4×C31—24no—
137C22⋊C4×C31—42no—
138C22⋊C4×C31—41no—
139C22⋊C4×C31—41no—
140C22⋊C4×C31—44no—
141C22⋊C4×C31—82no—
142C42×C31—81no—
143#18317C3111yesmaximal, characteristic, direct product factor, Sylow
144C23×C31—24no—
145C23×C31—24no—
146C23×C31—22no—
147C23×C31—22no—
148C23×C31—22no—
149C23×C31—22no—
150C23×C31—24no—
151C23×C31—41no—
152C23×C31—42no—
153C23×C31—42no—
154C23×C31—42no—
155C23×C31—44no—
156C23×C31—48no—
157C23×C31—44no—
158C23×C31—42no—
159C23×C31—44no—
160C23×C31—81no—
161C23×C31—81no—
162C23×C31—88no—
163C23×C31—84no—
164C23×C31—84no—
165C23×C31—816no—
166C23×C31#12011yescharacteristic
167C23×C31#12011yescharacteristic
168C23×C31AGL(1,ℤ/8)14yes—
169C23×C31C22×D411yescharacteristic
170Q8×C31—44no—
171D4×C31—44no—
172D4×C31—416no—
173D4×C31—48no—
174D4×C31—48no—
175D4×C31—82no—
176D4×C31—84no—
177C2×C4×C31—22no—
178C2×C4×C31—24no—
179C2×C4×C31—41no—
180C2×C4×C31—42no—
181C2×C4×C31—42no—
182C2×C4×C31—41no—
183C2×C4×C31—44no—
184C2×C4×C31—48no—
185C2×C4×C31—42no—
186C2×C4×C31—42no—
187C2×C4×C31—42no—
188C2×C4×C31—42no—
189C2×C4×C31—42no—
190C2×C4×C31—41no—
191C2×C4×C31—41no—
192C2×C4×C31—42no—
193C2×C4×C31—42no—
194C2×C4×C31—81no—
195C2×C4×C31#12012yes—
196C2×C4×C31C22×D412yes—
197C8×C31—44no—
198C2×#534C2×C3111yescharacteristic
199#3193C2×C3112yescomplemented
200#2561C2×C3111yescharacteristic, complemented
201#1607C2×C3114yescomplemented
202#1601C2×C3114yescomplemented
203#1507C2×C3112yescomplemented
204#1462C2×C3111yescharacteristic, complemented
205C22×C31—21no—
206C22×C31—21no—
207C22×C31—21no—
208C22×C31—21no—
209C22×C31—24no—
210C22×C31—21no—
211C22×C31—21no—
212C22×C31—21no—
213C22×C31—44no—
214C22×C31—44no—
215C22×C31—48no—
216C22×C31—42no—
217C22×C31—42no—
218C22×C31—44no—
219C22×C31—44no—
220C22×C31—44no—
221C22×C31—82no—
222C22×C31—88no—
223C22×C31—84no—
224C22×C31—816no—
225C22×C31#45512yes—
226C22×C31C2×#12011yescharacteristic
227C22×C31C2×AGL(1,ℤ/8)12yes—
228C4×C31—22no—
229C4×C31—22no—
230C4×C31—44no—
231C4×C31—42no—
232C4×C31—82no—
233C4×C31—81no—
234C4×C31—81no—
235C23×D4—21no—
236#534—24no—
237#534—22no—
238#534C22×C3114yes—
239C2×#123—22no—
240C2×C22⋊Q8C22×C3111yescharacteristic, complemented
241C2×#121—21no—
242C2×#121C22×C3112yes—
243C2×#120—22no—
244C22×C22⋊C4C22×C3111yescharacteristic
245#484—22no—
246#480C22×C3114yescomplemented
247#468—24no—
248#450—22no—
249#448C22×C3114yescomplemented
250#407C22×C3112yescomplemented
251#397C22×C3112yescomplemented
252#393C22×C3112yescomplemented
253#388—22no—
254#381—21no—
255#379—21no—
256C4×C22⋊C4C22×C3111yescharacteristic, complemented
257#327C22×C3118yescomplemented
258#323C22×C3114yescomplemented
259C2×C31—22no—
260C2×C31—22no—
261C2×C31—41no—
262C2×C31—44no—
263C2×C31—84no—
264C2×C31#300311yescharacteristic
265C2×C31C2×#45511yescharacteristic
266C2×C31#302611yescharacteristic
267C25—21no—
268C25D4×C3111yescharacteristic
269C22×Q8C23×C3111yescharacteristic, complemented
270C22×D4—22no—
271C22×D4—24no—
272C22×D4—22no—
273C22×D4—24no—
274C22×D4—21no—
275C22×D4—48no—
276C22×D4D4×C3112yes—
277C23×C4—21no—
278C23×C4D4×C3111yescharacteristic
279C23×C4C23×C3111yescharacteristic
280C2×SD16—24no—
281C2×M16—22no—
282#123—28no—
283#123—28no—
284C22⋊Q8—22no—
285C22⋊Q8—22no—
286#121—24no—
287#121—24no—
288#121—22no—
289#121—22no—
290#121D4×C3114yes—
291#121C23×C3114yes—
292#120—28no—
293#120—28no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4D4×C3112yes—
297C2×C4⋊C4C23×C3111yescharacteristic, complemented
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4D4×C3112yes—
307C2×C22⋊C4C23×C3112yes—
308C2×C42—21no—
309#107—24no—
310#103—24no—
311#102—24no—
312C22⋊C8—22no—
313C22⋊C8C23×C3114yescomplemented
314#95—22no—
315#95—22no—
316#95C23×C3112yescomplemented
317C31#1831711yescharacteristic, direct product factor, Sylow
318C24—21no—
319C24—21no—
320C24—24no—
321C24—24no—
322C24—22no—
323C24—44no—
324C24—42no—
325C24—81no—
326C24—82no—
327C24—82no—
328C24C2×D4×C3111yescharacteristic
329C24C2×D4×C3112yes—
330C2×Q8—22no—
331C2×Q8—81no—
332C2×D4—28no—
333C2×D4—24no—
334C2×D4—24no—
335C2×D4—24no—
336C2×D4—22no—
337C2×D4—48no—
338C2×D4—416no—
339C2×D4—48no—
340C2×D4—48no—
341C2×D4—48no—
342C2×D4—82no—
343C2×D4—82no—
344C2×D4C2×D4×C3114yes—
345C22×C4—24no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—44no—
353C22×C4—42no—
354C22×C4—81no—
355C22×C4C2×D4×C3112yes—
356C22×C4C2×D4×C3111yescharacteristic
357C22×C4C2×D4×C3112yes—
358C22×C4C2×D4×C3112yes—
359C22×C4C24×C3111yescharacteristic
360SD16—84no—
361M16—82no—
362C2×C8—24no—
363C4⋊C4—24no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—42no—
367C4⋊C4C2×D4×C3114yes—
368C22⋊C4—28no—
369C22⋊C4—24no—
370C22⋊C4—24no—
371C22⋊C4—24no—
372C22⋊C4—24no—
373C22⋊C4—42no—
374C22⋊C4—41no—
375C22⋊C4—41no—
376C22⋊C4—44no—
377C22⋊C4—82no—
378C42—81no—
379C23—24no—
380C23—24no—
381C23—22no—
382C23—22no—
383C23—22no—
384C23—22no—
385C23—24no—
386C23—41no—
387C23—42no—
388C23—42no—
389C23—42no—
390C23—44no—
391C23—48no—
392C23—44no—
393C23—42no—
394C23—44no—
395C23—81no—
396C23—81no—
397C23—88no—
398C23—84no—
399C23—84no—
400C23—816no—
401C23C31×#12011yescharacteristic
402C23C31×#12011yescharacteristic
403C23C31×AGL(1,ℤ/8)14yes—
404C23C22×D4×C3111yescharacteristic
405Q8—44no—
406D4—44no—
407D4—416no—
408D4—48no—
409D4—48no—
410D4—82no—
411D4—84no—
412C2×C4—22no—
413C2×C4—24no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—44no—
419C2×C4—48no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—81no—
430C2×C4C31×#12012yes—
431C2×C4C22×D4×C3112yes—
432C8—44no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—24no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—44no—
442C22—44no—
443C22—48no—
444C22—42no—
445C22—42no—
446C22—44no—
447C22—44no—
448C22—44no—
449C22—82no—
450C22—88no—
451C22—84no—
452C22—816no—
453C22C31×#45512yes—
454C22C2×C31×#12011yescharacteristic
455C22C2×C31×AGL(1,ℤ/8)12yes—
456C4—22no—
457C4—22no—
458C4—44no—
459C4—42no—
460C4—82no—
461C4—81no—
462C4—81no—
463C2—22no—
464C2—22no—
465C2—41no—
466C2—44no—
467C2—84no—
468C2C31×#300311yescharacteristic
469C2C2×C31×#45511yescharacteristic
470C2C31×#302611yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC2×C31×#534C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C31×#3193C2semidirect
3. C31×#2561C2semidirect
4. C31×#1607C2semidirect
5. C31×#1601C2semidirect
6. C31×#1507C2semidirect
7. C31×#1462C2semidirect
13. C2×C31×C22⋊Q8C22semidirect
19. C31×#480C22semidirect
22. C31×#448C22semidirect
23. C31×#407C22semidirect
24. C31×#397C22semidirect
25. C31×#393C22semidirect
29. C4×C22⋊C4×C31C22semidirect
30. C31×#327C22semidirect
31. C31×#323C22semidirect
34. C22×Q8×C31C23semidirect
62. C2×C4⋊C4×C31C23semidirect
78. C31×C22⋊C8C23semidirect
81. C31×#95C23semidirect
143. #18317C31direct (N × H)
199. #3193C2×C31semidirect
200. #2561C2×C31semidirect
201. #1607C2×C31semidirect
202. #1601C2×C31semidirect
203. #1507C2×C31semidirect
204. #1462C2×C31semidirect
240. C2×C22⋊Q8C22×C31semidirect
246. #480C22×C31semidirect
249. #448C22×C31semidirect
250. #407C22×C31semidirect
251. #397C22×C31semidirect
252. #393C22×C31semidirect
256. C4×C22⋊C4C22×C31semidirect
257. #327C22×C31semidirect
258. #323C22×C31semidirect
269. C22×Q8C23×C31semidirect
297. C2×C4⋊C4C23×C31semidirect
313. C22⋊C8C23×C31semidirect
316. #95C23×C31semidirect
317. C31#18317direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5342
C31×#31932
C31×#25612
C31×#16072
C31×#16012
C31×#15072
C31×#14622
#1831731