Finite Group Databasebeta

← C31×#18142 #21474901

Subgroups — C31×#18142 #21474901

Order 7936.

1450 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#537C211yesmaximal, characteristic, complemented
2C31×#3006C212yesmaximal, complemented
3C31×#2406C211yesmaximal, characteristic, complemented
4C31×#1604C214yesmaximal, complemented
5C31×#1598C214yesmaximal, complemented
6C31×#1510C212yesmaximal, complemented
7C31×#1463C211yesmaximal, characteristic, complemented
8C22×#41×C31—21no—
9C22×#41×C31C2211yescharacteristic
10C2×C31×AGL(1,ℤ/8)—22no—
11C31×#537—24no—
12C31×#537—22no—
13C31×#537C2214yes—
14C2×C31×#123—21no—
15C2×C31×C22⋊Q8—22no—
16C2×C31×#121—22no—
17C2×C31×#121—21no—
18C2×C31×#121C2212yes—
19C22×C4⋊C4×C31C2211yescharacteristic
20C31×#449—24no—
21C31×#448C2214yescomplemented
22C31×#447C2214yescomplemented
23C31×#417—22no—
24C31×#407C2212yes—
25C31×#395C2212yescomplemented
26C31×#394C2212yescomplemented
27C31×#386—21no—
28C31×#385—22no—
29C31×#381C2211yescharacteristic
30C31×#327C2218yescomplemented
31C31×#324C2214yescomplemented
32C2×#41×C31—24no—
33C2×#41×C31—24no—
34C2×#41×C31—22no—
35C2×#41×C31—22no—
36C2×#41×C31—48no—
37C2×#41×C31—44no—
38C22×Q8×C31—21no—
39C22×Q8×C31C2311yescharacteristic
40C22×D4×C31—21no—
41C22×D4×C31—22no—
42C22×D4×C31—21no—
43C22×D4×C31—21no—
44C22×D4×C31D412yes—
45C22×D4×C31C2311yescharacteristic
46C23×C4×C31—22no—
47C23×C4×C31—21no—
48C23×C4×C31D411yescharacteristic
49C23×C4×C31D411yescharacteristic
50C23×C4×C31C2311yescharacteristic
51C31×AGL(1,ℤ/8)—48no—
52C2×SD16×C31—24no—
53C2×D8×C31—24no—
54C2×M16×C31—22no—
55C31×#123—44no—
56C31×C22⋊Q8—28no—
57C31×C22⋊Q8—28no—
58C31×#121—24no—
59C31×#121—28no—
60C31×#121—28no—
61C31×#121—24no—
62C31×#121—22no—
63C31×#121—22no—
64C31×#121D414yes—
65C31×#121C2314yes—
66C2×C4⋊C4×C31—21no—
67C2×C4⋊C4×C31—22no—
68C2×C4⋊C4×C31—22no—
69C2×C4⋊C4×C31—22no—
70C2×C4⋊C4×C31D412yes—
71C2×C4⋊C4×C31D412yes—
72C2×C4⋊C4×C31C2312yes—
73C2×C22⋊C4×C31—22no—
74C2×C22⋊C4×C31—22no—
75C2×C22⋊C4×C31—21no—
76C2×C22⋊C4×C31—22no—
77C2×C22⋊C4×C31—22no—
78C31×#103—24no—
79C31×#102—24no—
80C31×C22⋊C8—22no—
81C31×C22⋊C8C2314yes—
82C31×#95—22no—
83C31×#95—21no—
84C31×#95C2312yes—
85C24×C31—22no—
86C24×C31—21no—
87C24×C31C2×D411yescharacteristic
88#41×C31—28no—
89#41×C31—416no—
90#41×C31—48no—
91#41×C31—44no—
92#41×C31—44no—
93#41×C31—44no—
94#41×C31—48no—
95C2×Q8×C31—24no—
96C2×Q8×C31—22no—
97C2×Q8×C31—42no—
98C2×Q8×C31—44no—
99C2×D4×C31—24no—
100C2×D4×C31—24no—
101C2×D4×C31—24no—
102C2×D4×C31—24no—
103C2×D4×C31—24no—
104C2×D4×C31—22no—
105C2×D4×C31—22no—
106C2×D4×C31—44no—
107C2×D4×C31—44no—
108C2×D4×C31—42no—
109C2×D4×C31—48no—
110C2×D4×C31—42no—
111C2×D4×C31—41no—
112C2×D4×C31—41no—
113C2×D4×C31—82no—
114C2×D4×C31C2×D414yes—
115C22×C4×C31—21no—
116C22×C4×C31—21no—
117C22×C4×C31—24no—
118C22×C4×C31—28no—
119C22×C4×C31—21no—
120C22×C4×C31—22no—
121C22×C4×C31—21no—
122C22×C4×C31—21no—
123C22×C4×C31—22no—
124C22×C4×C31—42no—
125C22×C4×C31—48no—
126C22×C4×C31—42no—
127C22×C4×C31—42no—
128C22×C4×C31—42no—
129C22×C4×C31—42no—
130C22×C4×C31C2×D412yes—
131C22×C4×C31C2×D412yes—
132C22×C4×C31C2×D412yes—
133C22×C4×C31C2×D411yescharacteristic
134C22×C4×C31C2×D412yes—
135C22×C4×C31C2411yescharacteristic
136SD16×C31—48no—
137D8×C31—48no—
138M16×C31—44no—
139C2×C8×C31—24no—
140C4⋊C4×C31—24no—
141C4⋊C4×C31—24no—
142C4⋊C4×C31—22no—
143C4⋊C4×C31—24no—
144C4⋊C4×C31—44no—
145C4⋊C4×C31C2×D414yes—
146C22⋊C4×C31—28no—
147C22⋊C4×C31—24no—
148C22⋊C4×C31—24no—
149C22⋊C4×C31—44no—
150C22⋊C4×C31—42no—
151C22⋊C4×C31—44no—
152#18142C3111yesmaximal, characteristic, direct product factor, Sylow
153C23×C31—22no—
154C23×C31—22no—
155C23×C31—22no—
156C23×C31—24no—
157C23×C31—41no—
158C23×C31—44no—
159C23×C31—41no—
160C23×C31—42no—
161C23×C31—44no—
162C23×C31—41no—
163C23×C31—42no—
164C23×C31—44no—
165C23×C31#12011yescharacteristic
166C23×C31#12011yescharacteristic
167C23×C31C22×D411yescharacteristic
168Q8×C31—24no—
169Q8×C31—48no—
170Q8×C31—42no—
171D4×C31—24no—
172D4×C31—44no—
173D4×C31—48no—
174D4×C31—42no—
175D4×C31—48no—
176D4×C31—44no—
177D4×C31—48no—
178D4×C31—44no—
179D4×C31—82no—
180D4×C31—84no—
181C2×C4×C31—24no—
182C2×C4×C31—24no—
183C2×C4×C31—22no—
184C2×C4×C31—22no—
185C2×C4×C31—24no—
186C2×C4×C31—22no—
187C2×C4×C31—22no—
188C2×C4×C31—24no—
189C2×C4×C31—41no—
190C2×C4×C31—42no—
191C2×C4×C31—41no—
192C2×C4×C31—416no—
193C2×C4×C31—48no—
194C2×C4×C31—44no—
195C2×C4×C31—41no—
196C2×C4×C31—41no—
197C2×C4×C31—42no—
198C2×C4×C31—42no—
199C2×C4×C31—42no—
200C2×C4×C31—42no—
201C2×C4×C31—42no—
202C2×C4×C31—81no—
203C2×C4×C31#12012yes—
204C2×C4×C31AGL(1,ℤ/8)14yescomplemented
205C2×C4×C31C22×D412yes—
206C8×C31—44no—
207C2×#537C2×C3111yescharacteristic, complemented
208#3006C2×C3112yescomplemented
209#2406C2×C3111yescharacteristic, complemented
210#1604C2×C3114yescomplemented
211#1598C2×C3114yescomplemented
212#1510C2×C3112yescomplemented
213#1463C2×C3111yescharacteristic, complemented
214C22×C31—21no—
215C22×C31—21no—
216C22×C31—21no—
217C22×C31—21no—
218C22×C31—21no—
219C22×C31—21no—
220C22×C31—21no—
221C22×C31—44no—
222C22×C31—44no—
223C22×C31—44no—
224C22×C31—44no—
225C22×C31—42no—
226C22×C31—44no—
227C22×C31—82no—
228C22×C31—88no—
229C22×C31—81no—
230C22×C31—81no—
231C22×C31B(4,𝔽2)12yes—
232C22×C31C2×#12011yescharacteristic
233C22×C31C2×AGL(1,ℤ/8)12yes—
234C4×C31—24no—
235C4×C31—22no—
236C4×C31—22no—
237C4×C31—44no—
238C4×C31—42no—
239C4×C31—42no—
240C4×C31—82no—
241C22×#41—21no—
242C22×#41C22×C3111yescharacteristic
243C2×AGL(1,ℤ/8)—22no—
244#537—24no—
245#537—22no—
246#537C22×C3114yes—
247C2×#123—21no—
248C2×C22⋊Q8—22no—
249C2×#121—22no—
250C2×#121—21no—
251C2×#121C22×C3112yes—
252C22×C4⋊C4C22×C3111yescharacteristic
253#449—24no—
254#448C22×C3114yescomplemented
255#447C22×C3114yescomplemented
256#417—22no—
257#407C22×C3112yes—
258#395C22×C3112yescomplemented
259#394C22×C3112yescomplemented
260#386—21no—
261#385—22no—
262#381C22×C3111yescharacteristic
263#327C22×C3118yescomplemented
264#324C22×C3114yescomplemented
265C2×C31—22no—
266C2×C31—22no—
267C2×C31—41no—
268C2×C31—82no—
269C2×C31—84no—
270C2×C31#300311yescharacteristic
271C2×C31#302011yescharacteristic
272C2×C31C2×B(4,𝔽2)11yescharacteristic
273C2×#41—24no—
274C2×#41—24no—
275C2×#41—22no—
276C2×#41—22no—
277C2×#41—48no—
278C2×#41—44no—
279C22×Q8—21no—
280C22×Q8C23×C3111yescharacteristic
281C22×D4—21no—
282C22×D4—22no—
283C22×D4—21no—
284C22×D4—21no—
285C22×D4D4×C3112yes—
286C22×D4C23×C3111yescharacteristic
287C23×C4—22no—
288C23×C4—21no—
289C23×C4D4×C3111yescharacteristic
290C23×C4D4×C3111yescharacteristic
291C23×C4C23×C3111yescharacteristic
292AGL(1,ℤ/8)—48no—
293C2×SD16—24no—
294C2×D8—24no—
295C2×M16—22no—
296#123—44no—
297C22⋊Q8—28no—
298C22⋊Q8—28no—
299#121—24no—
300#121—28no—
301#121—28no—
302#121—24no—
303#121—22no—
304#121—22no—
305#121D4×C3114yes—
306#121C23×C3114yes—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4D4×C3112yes—
312C2×C4⋊C4D4×C3112yes—
313C2×C4⋊C4C23×C3112yes—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319#103—24no—
320#102—24no—
321C22⋊C8—22no—
322C22⋊C8C23×C3114yes—
323#95—21no—
324#95—22no—
325#95C23×C3112yes—
326C31#1814211yescharacteristic, direct product factor, Sylow
327C24—22no—
328C24—21no—
329C24C2×D4×C3111yescharacteristic
330#41—28no—
331#41—416no—
332#41—44no—
333#41—48no—
334#41—44no—
335#41—44no—
336#41—48no—
337C2×Q8—24no—
338C2×Q8—22no—
339C2×Q8—42no—
340C2×Q8—44no—
341C2×D4—24no—
342C2×D4—24no—
343C2×D4—24no—
344C2×D4—24no—
345C2×D4—24no—
346C2×D4—22no—
347C2×D4—22no—
348C2×D4—44no—
349C2×D4—44no—
350C2×D4—42no—
351C2×D4—42no—
352C2×D4—48no—
353C2×D4—41no—
354C2×D4—41no—
355C2×D4—82no—
356C2×D4C2×D4×C3114yes—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—24no—
360C22×C4—28no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—42no—
367C22×C4—48no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4C2×D4×C3112yes—
373C22×C4C2×D4×C3112yes—
374C22×C4C2×D4×C3112yes—
375C22×C4C2×D4×C3111yescharacteristic
376C22×C4C2×D4×C3112yes—
377C22×C4C24×C3111yescharacteristic
378SD16—48no—
379D8—48no—
380M16—44no—
381C2×C8—24no—
382C4⋊C4—24no—
383C4⋊C4—24no—
384C4⋊C4—22no—
385C4⋊C4—24no—
386C4⋊C4—44no—
387C4⋊C4C2×D4×C3114yes—
388C22⋊C4—28no—
389C22⋊C4—24no—
390C22⋊C4—24no—
391C22⋊C4—44no—
392C22⋊C4—42no—
393C22⋊C4—44no—
394C23—22no—
395C23—22no—
396C23—22no—
397C23—24no—
398C23—41no—
399C23—44no—
400C23—41no—
401C23—42no—
402C23—44no—
403C23—44no—
404C23—41no—
405C23—42no—
406C23C31×#12011yescharacteristic
407C23C31×#12011yescharacteristic
408C23C22×D4×C3111yescharacteristic
409Q8—24no—
410Q8—48no—
411Q8—42no—
412D4—24no—
413D4—44no—
414D4—48no—
415D4—42no—
416D4—48no—
417D4—44no—
418D4—44no—
419D4—48no—
420D4—82no—
421D4—84no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—24no—
430C2×C4—41no—
431C2×C4—42no—
432C2×C4—41no—
433C2×C4—416no—
434C2×C4—48no—
435C2×C4—44no—
436C2×C4—41no—
437C2×C4—42no—
438C2×C4—41no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—81no—
444C2×C4C31×#12012yes—
445C2×C4C31×AGL(1,ℤ/8)14yescomplemented
446C2×C4C22×D4×C3112yes—
447C8—44no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—44no—
456C22—44no—
457C22—44no—
458C22—44no—
459C22—42no—
460C22—44no—
461C22—82no—
462C22—88no—
463C22—81no—
464C22—81no—
465C22C31×B(4,𝔽2)12yes—
466C22C2×C31×#12011yescharacteristic
467C22C2×C31×AGL(1,ℤ/8)12yes—
468C4—24no—
469C4—22no—
470C4—22no—
471C4—44no—
472C4—42no—
473C4—42no—
474C4—82no—
475C2—22no—
476C2—22no—
477C2—41no—
478C2—82no—
479C2—84no—
480C2C31×#300311yescharacteristic
481C2C31×#302011yescharacteristic
482C2C2×C31×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
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
1. C2×C31×#537C2semidirect
2. C31×#3006C2semidirect
3. C31×#2406C2semidirect
4. C31×#1604C2semidirect
5. C31×#1598C2semidirect
6. C31×#1510C2semidirect
7. C31×#1463C2semidirect
21. C31×#448C22semidirect
22. C31×#447C22semidirect
25. C31×#395C22semidirect
26. C31×#394C22semidirect
30. C31×#327C22semidirect
31. C31×#324C22semidirect
152. #18142C31direct (N × H)
204. C2×C4×C31AGL(1,ℤ/8)semidirect
207. C2×#537C2×C31semidirect
208. #3006C2×C31semidirect
209. #2406C2×C31semidirect
210. #1604C2×C31semidirect
211. #1598C2×C31semidirect
212. #1510C2×C31semidirect
213. #1463C2×C31semidirect
254. #448C22×C31semidirect
255. #447C22×C31semidirect
258. #395C22×C31semidirect
259. #394C22×C31semidirect
263. #327C22×C31semidirect
264. #324C22×C31semidirect
326. C31#18142direct (N × H)
445. C2×C4C31×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×#5372
C31×#30062
C31×#24062
C31×#16042
C31×#15982
C31×#15102
C31×#14632
#1814231