Finite Group Databasebeta

← C31×#18137 #21474896

Subgroups — C31×#18137 #21474896

Order 7936.

1398 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#534C211yesmaximal, characteristic
2C31×#3094C212yesmaximal, complemented
3C31×#2482C211yesmaximal, characteristic, complemented
4C31×#1613C214yesmaximal, complemented
5C31×#1607C214yesmaximal, complemented
6C31×#1510C212yesmaximal, complemented
7C31×#1462C211yesmaximal, characteristic, complemented
8C23×D4×C31—21no—
9C31×#534—24no—
10C31×#534—22no—
11C31×#534C2214yes—
12C2×C31×#123—22no—
13C2×C31×#121—21no—
14C2×C31×#121C2212yes—
15C2×C31×#120—22no—
16C2×C31×#117C2211yescharacteristic, complemented
17C22×C22⋊C4×C31C2211yescharacteristic
18C31×#482—24no—
19C31×#481C2214yescomplemented
20C31×#480C2214yescomplemented
21C31×#428—22no—
22C31×#418—22no—
23C31×#407C2212yescomplemented
24C31×#399C2212yescomplemented
25C31×#397C2212yescomplemented
26C31×#388—22no—
27C31×#385—21no—
28C31×#381C2211yescharacteristic, complemented
29C31×#379—21no—
30C31×#327C2218yescomplemented
31C31×#323C2214yescomplemented
32C25×C31—21no—
33C25×C31D411yescharacteristic
34C22×D4×C31—22no—
35C22×D4×C31—24no—
36C22×D4×C31—22no—
37C22×D4×C31—24no—
38C22×D4×C31—21no—
39C22×D4×C31—48no—
40C22×D4×C31D412yes—
41C23×C4×C31—21no—
42C23×C4×C31D411yescharacteristic
43C23×C4×C31C2311yescharacteristic
44C2×M16×C31—22no—
45C31×#123—28no—
46C31×#123—28no—
47C31×#121—24no—
48C31×#121—24no—
49C31×#121—22no—
50C31×#121—22no—
51C31×#121D414yes—
52C31×#121C2314yes—
53C31×#120—28no—
54C31×#120—28no—
55C31×#117—22no—
56C31×#117—22no—
57C2×C4⋊C4×C31—21no—
58C2×C4⋊C4×C31—22no—
59C2×C4⋊C4×C31D412yes—
60C2×C4⋊C4×C31C2311yescharacteristic, complemented
61C2×C4⋊C4×C31C2311yescharacteristic, complemented
62C2×C22⋊C4×C31—22no—
63C2×C22⋊C4×C31—22no—
64C2×C22⋊C4×C31—22no—
65C2×C22⋊C4×C31—22no—
66C2×C22⋊C4×C31—22no—
67C2×C22⋊C4×C31—21no—
68C2×C22⋊C4×C31—21no—
69C2×C22⋊C4×C31—22no—
70C2×C22⋊C4×C31D412yes—
71C2×C22⋊C4×C31C2312yes—
72C2×C42×C31—21no—
73C31×#107—24no—
74C31×#106—24no—
75C31×#102—24no—
76C31×#102—24no—
77C31×C22⋊C8—22no—
78C31×C22⋊C8C2314yescomplemented
79C31×#95—22no—
80C31×#95—21no—
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×D4×C31—28no—
95C2×D4×C31—24no—
96C2×D4×C31—24no—
97C2×D4×C31—24no—
98C2×D4×C31—22no—
99C2×D4×C31—44no—
100C2×D4×C31—48no—
101C2×D4×C31—416no—
102C2×D4×C31—48no—
103C2×D4×C31—48no—
104C2×D4×C31—48no—
105C2×D4×C31—82no—
106C2×D4×C31C2×D414yes—
107C22×C4×C31—24no—
108C22×C4×C31—21no—
109C22×C4×C31—22no—
110C22×C4×C31—21no—
111C22×C4×C31—22no—
112C22×C4×C31—21no—
113C22×C4×C31—21no—
114C22×C4×C31—42no—
115C22×C4×C31—44no—
116C22×C4×C31—81no—
117C22×C4×C31C2×D412yes—
118C22×C4×C31C2×D412yes—
119C22×C4×C31C2×D411yescharacteristic
120C22×C4×C31C2×D412yes—
121C22×C4×C31C2411yescharacteristic
122M16×C31—44no—
123C2×C8×C31—24no—
124C4⋊C4×C31—24no—
125C4⋊C4×C31—22no—
126C4⋊C4×C31—22no—
127C4⋊C4×C31—22no—
128C4⋊C4×C31—82no—
129C4⋊C4×C31C2×D414yes—
130C22⋊C4×C31—28no—
131C22⋊C4×C31—24no—
132C22⋊C4×C31—24no—
133C22⋊C4×C31—24no—
134C22⋊C4×C31—24no—
135C22⋊C4×C31—42no—
136C22⋊C4×C31—41no—
137C22⋊C4×C31—41no—
138C22⋊C4×C31—44no—
139C22⋊C4×C31—82no—
140C42×C31—42no—
141#18137C3111yesmaximal, characteristic, direct product factor, Sylow
142C23×C31—24no—
143C23×C31—24no—
144C23×C31—22no—
145C23×C31—22no—
146C23×C31—22no—
147C23×C31—22no—
148C23×C31—24no—
149C23×C31—41no—
150C23×C31—42no—
151C23×C31—42no—
152C23×C31—42no—
153C23×C31—44no—
154C23×C31—48no—
155C23×C31—44no—
156C23×C31—42no—
157C23×C31—44no—
158C23×C31—81no—
159C23×C31—81no—
160C23×C31—88no—
161C23×C31—84no—
162C23×C31—84no—
163C23×C31—816no—
164C23×C31#12011yescharacteristic
165C23×C31#12011yescharacteristic
166C23×C31AGL(1,ℤ/8)14yes—
167C23×C31C22×D411yescharacteristic
168D4×C31—44no—
169D4×C31—416no—
170D4×C31—48no—
171D4×C31—48no—
172D4×C31—82no—
173D4×C31—84no—
174C2×C4×C31—24no—
175C2×C4×C31—22no—
176C2×C4×C31—24no—
177C2×C4×C31—42no—
178C2×C4×C31—41no—
179C2×C4×C31—41no—
180C2×C4×C31—44no—
181C2×C4×C31—48no—
182C2×C4×C31—42no—
183C2×C4×C31—42no—
184C2×C4×C31—42no—
185C2×C4×C31—42no—
186C2×C4×C31—42no—
187C2×C4×C31—41no—
188C2×C4×C31—41no—
189C2×C4×C31—42no—
190C2×C4×C31—42no—
191C2×C4×C31—81no—
192C2×C4×C31#12012yes—
193C2×C4×C31C22×D412yes—
194C8×C31—44no—
195C2×#534C2×C3111yescharacteristic
196#3094C2×C3112yescomplemented
197#2482C2×C3111yescharacteristic, complemented
198#1613C2×C3114yescomplemented
199#1607C2×C3114yescomplemented
200#1510C2×C3112yescomplemented
201#1462C2×C3111yescharacteristic, complemented
202C22×C31—21no—
203C22×C31—21no—
204C22×C31—21no—
205C22×C31—21no—
206C22×C31—24no—
207C22×C31—21no—
208C22×C31—21no—
209C22×C31—21no—
210C22×C31—44no—
211C22×C31—44no—
212C22×C31—48no—
213C22×C31—42no—
214C22×C31—42no—
215C22×C31—44no—
216C22×C31—44no—
217C22×C31—44no—
218C22×C31—82no—
219C22×C31—88no—
220C22×C31—84no—
221C22×C31—816no—
222C22×C31#45812yes—
223C22×C31C2×#12011yescharacteristic
224C22×C31C2×AGL(1,ℤ/8)12yes—
225C4×C31—22no—
226C4×C31—22no—
227C4×C31—44no—
228C4×C31—42no—
229C4×C31—82no—
230C4×C31—81no—
231C4×C31—81no—
232C23×D4—21no—
233#534—24no—
234#534—22no—
235#534C22×C3114yes—
236C2×#123—22no—
237C2×#121—21no—
238C2×#121C22×C3112yes—
239C2×#120—22no—
240C2×#117C22×C3111yescharacteristic, complemented
241C22×C22⋊C4C22×C3111yescharacteristic
242#482—24no—
243#481C22×C3114yescomplemented
244#480C22×C3114yescomplemented
245#428—22no—
246#418—22no—
247#407C22×C3112yescomplemented
248#399C22×C3112yescomplemented
249#397C22×C3112yescomplemented
250#388—22no—
251#385—21no—
252#381C22×C3111yescharacteristic, complemented
253#379—21no—
254#327C22×C3118yescomplemented
255#323C22×C3114yescomplemented
256C2×C31—22no—
257C2×C31—22no—
258C2×C31—41no—
259C2×C31—44no—
260C2×C31—84no—
261C2×C31#300311yescharacteristic
262C2×C31#301911yescharacteristic
263C2×C31C2×#45811yescharacteristic
264C25—21no—
265C25D4×C3111yescharacteristic
266C22×D4—22no—
267C22×D4—24no—
268C22×D4—22no—
269C22×D4—24no—
270C22×D4—21no—
271C22×D4—48no—
272C22×D4D4×C3112yes—
273C23×C4—21no—
274C23×C4D4×C3111yescharacteristic
275C23×C4C23×C3111yescharacteristic
276C2×M16—22no—
277#123—28no—
278#123—28no—
279#121—24no—
280#121—24no—
281#121—22no—
282#121—22no—
283#121D4×C3114yes—
284#121C23×C3114yes—
285#120—28no—
286#120—28no—
287#117—22no—
288#117—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4D4×C3112yes—
292C2×C4⋊C4C23×C3111yescharacteristic, complemented
293C2×C4⋊C4C23×C3111yescharacteristic, complemented
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4—21no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4D4×C3112yes—
303C2×C22⋊C4C23×C3112yes—
304C2×C42—21no—
305#107—24no—
306#106—24no—
307#102—24no—
308#102—24no—
309C22⋊C8—22no—
310C22⋊C8C23×C3114yescomplemented
311#95—21no—
312#95—22no—
313#95C23×C3112yescomplemented
314C31#1813711yescharacteristic, direct product factor, Sylow
315C24—21no—
316C24—21no—
317C24—24no—
318C24—24no—
319C24—22no—
320C24—44no—
321C24—42no—
322C24—81no—
323C24—82no—
324C24—82no—
325C24C2×D4×C3111yescharacteristic
326C24C2×D4×C3112yes—
327C2×D4—28no—
328C2×D4—24no—
329C2×D4—24no—
330C2×D4—24no—
331C2×D4—22no—
332C2×D4—44no—
333C2×D4—48no—
334C2×D4—416no—
335C2×D4—48no—
336C2×D4—48no—
337C2×D4—48no—
338C2×D4—82no—
339C2×D4C2×D4×C3114yes—
340C22×C4—24no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—42no—
348C22×C4—44no—
349C22×C4—81no—
350C22×C4C2×D4×C3112yes—
351C22×C4C2×D4×C3112yes—
352C22×C4C2×D4×C3111yescharacteristic
353C22×C4C2×D4×C3112yes—
354C22×C4C24×C3111yescharacteristic
355M16—44no—
356C2×C8—24no—
357C4⋊C4—24no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—82no—
362C4⋊C4C2×D4×C3114yes—
363C22⋊C4—28no—
364C22⋊C4—24no—
365C22⋊C4—24no—
366C22⋊C4—24no—
367C22⋊C4—24no—
368C22⋊C4—42no—
369C22⋊C4—41no—
370C22⋊C4—41no—
371C22⋊C4—44no—
372C22⋊C4—82no—
373C42—42no—
374C23—24no—
375C23—24no—
376C23—22no—
377C23—22no—
378C23—22no—
379C23—22no—
380C23—24no—
381C23—41no—
382C23—42no—
383C23—42no—
384C23—42no—
385C23—44no—
386C23—48no—
387C23—44no—
388C23—42no—
389C23—44no—
390C23—81no—
391C23—81no—
392C23—88no—
393C23—84no—
394C23—84no—
395C23—816no—
396C23C31×#12011yescharacteristic
397C23C31×#12011yescharacteristic
398C23C31×AGL(1,ℤ/8)14yes—
399C23C22×D4×C3111yescharacteristic
400D4—44no—
401D4—416no—
402D4—48no—
403D4—48no—
404D4—82no—
405D4—84no—
406C2×C4—24no—
407C2×C4—22no—
408C2×C4—24no—
409C2×C4—42no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—44no—
413C2×C4—48no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—81no—
424C2×C4C31×#12012yes—
425C2×C4C22×D4×C3112yes—
426C8—44no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—24no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—44no—
436C22—44no—
437C22—48no—
438C22—42no—
439C22—42no—
440C22—44no—
441C22—44no—
442C22—44no—
443C22—82no—
444C22—88no—
445C22—84no—
446C22—816no—
447C22C31×#45812yes—
448C22C2×C31×#12011yescharacteristic
449C22C2×C31×AGL(1,ℤ/8)12yes—
450C4—22no—
451C4—22no—
452C4—44no—
453C4—42no—
454C4—82no—
455C4—81no—
456C4—81no—
457C2—22no—
458C2—22no—
459C2—41no—
460C2—44no—
461C2—84no—
462C2C31×#300311yescharacteristic
463C2C31×#301911yescharacteristic
464C2C2×C31×#45811yescharacteristic
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×#3094C2semidirect
3. C31×#2482C2semidirect
4. C31×#1613C2semidirect
5. C31×#1607C2semidirect
6. C31×#1510C2semidirect
7. C31×#1462C2semidirect
16. C2×C31×#117C22semidirect
19. C31×#481C22semidirect
20. C31×#480C22semidirect
23. C31×#407C22semidirect
24. C31×#399C22semidirect
25. C31×#397C22semidirect
28. C31×#381C22semidirect
30. C31×#327C22semidirect
31. C31×#323C22semidirect
60. C2×C4⋊C4×C31C23semidirect
61. C2×C4⋊C4×C31C23semidirect
78. C31×C22⋊C8C23semidirect
81. C31×#95C23semidirect
141. #18137C31direct (N × H)
196. #3094C2×C31semidirect
197. #2482C2×C31semidirect
198. #1613C2×C31semidirect
199. #1607C2×C31semidirect
200. #1510C2×C31semidirect
201. #1462C2×C31semidirect
240. C2×#117C22×C31semidirect
243. #481C22×C31semidirect
244. #480C22×C31semidirect
247. #407C22×C31semidirect
248. #399C22×C31semidirect
249. #397C22×C31semidirect
252. #381C22×C31semidirect
254. #327C22×C31semidirect
255. #323C22×C31semidirect
292. C2×C4⋊C4C23×C31semidirect
293. C2×C4⋊C4C23×C31semidirect
310. C22⋊C8C23×C31semidirect
313. #95C23×C31semidirect
314. C31#18137direct (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×#30942
C31×#24822
C31×#16132
C31×#16072
C31×#15102
C31×#14622
#1813731