Finite Group Databasebeta

← C31×#61892 #21518651

Subgroups — C31×#61892 #21518651

Order 7936.

1770 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3502C211yesmaximal, characteristic, complemented
2C2×C31×#559C211yesmaximal, characteristic
3C31×#3400C218yesmaximal, complemented
4C31×#3375C218yesmaximal, complemented
5C31×#3215C214yesmaximal, complemented
6C31×#3208C212yesmaximal
7C31×#3110C214yesmaximal, complemented
8C31×#3104C212yesmaximal, complemented
9C31×#2983C211yesmaximal, characteristic, complemented
10C31×#559—24no—
11C31×#559C2218yes—
12Q82×C31C2214yescomplemented
13C31×#557—24no—
14C31×#557C2214yescomplemented
15C31×#544—21no—
16C31×#544—22no—
17C31×#533C2212yes—
18C31×#533C2214yes—
19C2×C31×#127C2211yescharacteristic
20C2×C31×#124—21no—
21C2×C31×#124C2212yes—
22C2×C31×#123—22no—
23C31×#519—21no—
24C2×C4×Q8×C31C2211yescharacteristic
25C2×C4×Q8×C31C2211yescharacteristic
26C2×C4×D4×C31—21no—
27C2×C4×D4×C31C2212yes—
28C31×#497C2218yescomplemented
29C31×#492C2214yes—
30C31×#489C2218yescomplemented
31C31×#486C2214yes—
32C31×#484C2214yes—
33C31×#477C2214yescomplemented
34C31×#474C2218yescomplemented
35C31×#464C22116yescomplemented
36C31×#462C2214yescomplemented
37C31×#461C2218yescomplemented
38C31×#460C2218yes—
39C31×#451C2214yes—
40C31×#450C2218yes—
41C31×#446C2218yescomplemented
42C31×#440C22116yescomplemented
43C4×Q16×C31C2218yescomplemented
44C31×#432C2214yes—
45C31×#431C2212yes—
46C22×Q8×C31D411yescharacteristic
47C22×Q8×C31C2311yescharacteristic
48C22×D4×C31—21no—
49C22×D4×C31—21no—
50C22×D4×C31C2312yes—
51C23×C4×C31—22no—
52C2×Q16×C31C2318yes—
53C2×SD16×C31C23116yes—
54C31×C4⋊Q8—24no—
55C31×C4⋊Q8—28no—
56C31×C4⋊Q8C2314yes—
57C31×C4⋊Q8C2312yes—
58C31×C4⋊Q8C2314yes—
59C31×C4⋊Q8C2312yes—
60C31×#127—22no—
61C31×#127C2314yes—
62C31×#125—24no—
63C31×#125—28no—
64C31×#125—24no—
65C31×#125—22no—
66C31×#124—28no—
67C31×#124—24no—
68C31×#124C2318yes—
69C31×#123—216no—
70C31×C22⋊Q8—24no—
71C31×C22⋊Q8—22no—
72C31×C22⋊Q8C2314yes—
73C4×Q8×C31—24no—
74C4×Q8×C31—24no—
75C4×Q8×C31—22no—
76C4×Q8×C31—24no—
77C4×Q8×C31—22no—
78C4×Q8×C31D414yes—
79C4×Q8×C31C2314yes—
80C4×Q8×C31C2314yes—
81C4×Q8×C31C2318yes—
82C4×Q8×C31C2314yes—
83C4×D4×C31—28no—
84C4×D4×C31—24no—
85C4×D4×C31C2318yes—
86C31×#117—24no—
87C31×#117—22no—
88C2×C4⋊C4×C31—21no—
89C2×C4⋊C4×C31D411yescharacteristic
90C2×C4⋊C4×C31D412yes—
91C2×C4⋊C4×C31C2311yescharacteristic
92C2×C22⋊C4×C31—22no—
93C2×C22⋊C4×C31—22no—
94C2×C22⋊C4×C31—22no—
95C2×C42×C31—21no—
96C2×C42×C31C2311yescharacteristic
97C2×C42×C31C2312yes—
98C31×#107C2318yes—
99C31×C4⋊C8C2314yes—
100C31×C4⋊C8C2318yes—
101C31×#103C2318yes—
102C31×#103C2318yes—
103C31×#102C23116yes—
104C31×C22⋊C8C2314yes—
105C31×#97C2318yes—
106C4×C8×C31C2314yes—
107C24×C31—22no—
108C2×Q8×C31—24no—
109C2×Q8×C31—21no—
110C2×Q8×C31—24no—
111C2×Q8×C31—22no—
112C2×Q8×C31—42no—
113C2×Q8×C31C2×D414yes—
114C2×Q8×C31C2412yes—
115C2×Q8×C31C2414yes—
116C2×D4×C31—22no—
117C2×D4×C31—24no—
118C2×D4×C31—24no—
119C2×D4×C31—44no—
120C2×D4×C31—44no—
121C2×D4×C31—82no—
122C2×D4×C31C2414yes—
123C22×C4×C31—21no—
124C22×C4×C31—21no—
125C22×C4×C31—28no—
126C22×C4×C31—84no—
127C22×C4×C31C2×D412yes—
128C22×C4×C31C2×D412yes—
129C22×C4×C31C2×D411yescharacteristic
130C22×C4×C31C2×D411yescharacteristic
131C22×C4×C31C2411yescharacteristic
132Q16×C31—48no—
133SD16×C31—416no—
134C2×C8×C31C2418yes—
135C4⋊C4×C31—24no—
136C4⋊C4×C31—21no—
137C4⋊C4×C31—24no—
138C4⋊C4×C31—22no—
139C4⋊C4×C31—24no—
140C4⋊C4×C31—22no—
141C4⋊C4×C31—24no—
142C4⋊C4×C31—24no—
143C4⋊C4×C31—22no—
144C4⋊C4×C31—28no—
145C4⋊C4×C31—24no—
146C4⋊C4×C31C2×D414yes—
147C4⋊C4×C31C2×D418yes—
148C4⋊C4×C31C2414yes—
149C4⋊C4×C31C2412yes—
150C22⋊C4×C31—24no—
151C22⋊C4×C31—28no—
152C22⋊C4×C31—28no—
153C22⋊C4×C31—28no—
154C42×C31—24no—
155C42×C31—28no—
156C42×C31—24no—
157C42×C31C2×D414yes—
158C42×C31C2×D412yes—
159C42×C31C2412yes—
160C42×C31C2414yes—
161#61892C3111yesmaximal, characteristic, direct product factor, Sylow
162C23×C31—24no—
163C23×C31—44no—
164C23×C31—44no—
165C23×C31—44no—
166C23×C31C22×D411yescharacteristic
167Q8×C31—28no—
168Q8×C31—28no—
169Q8×C31—24no—
170Q8×C31—42no—
171Q8×C31—84no—
172Q8×C31—82no—
173D4×C31—28no—
174D4×C31—44no—
175D4×C31—48no—
176D4×C31—84no—
177C2×C4×C31—24no—
178C2×C4×C31—22no—
179C2×C4×C31—24no—
180C2×C4×C31—24no—
181C2×C4×C31—22no—
182C2×C4×C31—22no—
183C2×C4×C31—42no—
184C2×C4×C31—41no—
185C2×C4×C31—42no—
186C2×C4×C31—42no—
187C2×C4×C31—88no—
188C2×C4×C31#13714yes—
189C2×C4×C31C22×D414yes—
190C2×C4×C31C22×D411yescharacteristic
191C2×C4×C31C22×D414yes—
192C2×C4×C31C22×D412yes—
193C2×C4×C31C22×D412yes—
194C2×C4×C31C2511yescharacteristic
195C8×C31—28no—
196#3502C2×C3111yescharacteristic, complemented
197C2×#559C2×C3111yescharacteristic
198#3400C2×C3118yescomplemented
199#3375C2×C3118yescomplemented
200#3215C2×C3114yescomplemented
201#3208C2×C3112yes—
202#3110C2×C3114yescomplemented
203#3104C2×C3112yescomplemented
204#2983C2×C3111yescharacteristic, complemented
205C22×C31—21no—
206C22×C31—21no—
207C22×C31—44no—
208C22×C31—44no—
209C22×C31—44no—
210C22×C31—88no—
211C22×C31C2×#13712yes—
212C22×C31C23×D411yescharacteristic
213C4×C31—24no—
214C4×C31—22no—
215C4×C31—21no—
216C4×C31—24no—
217C4×C31—42no—
218C4×C31—44no—
219C4×C31—42no—
220C4×C31—44no—
221C4×C31C2×#13714yes—
222C4×C31#58512yes—
223#559—24no—
224#559C22×C3118yes—
225Q82C22×C3114yescomplemented
226#557—24no—
227#557C22×C3114yescomplemented
228#544—21no—
229#544—22no—
230#533C22×C3112yes—
231#533C22×C3114yes—
232C2×#127C22×C3111yescharacteristic
233C2×#124—21no—
234C2×#124C22×C3112yes—
235C2×#123—22no—
236#519—21no—
237C2×C4×Q8C22×C3111yescharacteristic
238C2×C4×Q8C22×C3111yescharacteristic
239C2×C4×D4—21no—
240C2×C4×D4C22×C3112yes—
241#497C22×C3118yescomplemented
242#492C22×C3114yes—
243#489C22×C3118yescomplemented
244#486C22×C3114yes—
245#484C22×C3114yes—
246#477C22×C3114yescomplemented
247#474C22×C3118yescomplemented
248#464C22×C31116yescomplemented
249#462C22×C3114yescomplemented
250#461C22×C3118yescomplemented
251#460C22×C3118yes—
252#451C22×C3114yes—
253#450C22×C3118yes—
254#446C22×C3118yescomplemented
255#440C22×C31116yescomplemented
256C4×Q16C22×C3118yescomplemented
257#432C22×C3114yes—
258#431C22×C3112yes—
259C2×C31—22no—
260C2×C31—84no—
261C2×C31#348711yescharacteristic
262C2×C31C22×#13711yescharacteristic
263C2×C31#358511yescharacteristic
264C22×Q8D4×C3111yescharacteristic
265C22×Q8C23×C3111yescharacteristic
266C22×D4—21no—
267C22×D4—21no—
268C22×D4C23×C3112yes—
269C23×C4—22no—
270C2×Q16C23×C3118yes—
271C2×SD16C23×C31116yes—
272C4⋊Q8—24no—
273C4⋊Q8—28no—
274C4⋊Q8C23×C3114yes—
275C4⋊Q8C23×C3112yes—
276C4⋊Q8C23×C3114yes—
277C4⋊Q8C23×C3112yes—
278#127—22no—
279#127C23×C3114yes—
280#125—24no—
281#125—28no—
282#125—24no—
283#125—22no—
284#124—28no—
285#124—24no—
286#124C23×C3118yes—
287#123—216no—
288C22⋊Q8—24no—
289C22⋊Q8—22no—
290C22⋊Q8C23×C3114yes—
291C4×Q8—24no—
292C4×Q8—24no—
293C4×Q8—22no—
294C4×Q8—24no—
295C4×Q8—22no—
296C4×Q8D4×C3114yes—
297C4×Q8C23×C3114yes—
298C4×Q8C23×C3114yes—
299C4×Q8C23×C3118yes—
300C4×Q8C23×C3114yes—
301C4×D4—28no—
302C4×D4—24no—
303C4×D4C23×C3118yes—
304#117—24no—
305#117—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4D4×C3111yescharacteristic
308C2×C4⋊C4D4×C3112yes—
309C2×C4⋊C4C23×C3111yescharacteristic
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C42—21no—
314C2×C42C23×C3112yes—
315C2×C42C23×C3111yescharacteristic
316#107C23×C3118yes—
317C4⋊C8C23×C3114yes—
318C4⋊C8C23×C3118yes—
319#103C23×C3118yes—
320#103C23×C3118yes—
321#102C23×C31116yes—
322C22⋊C8C23×C3114yes—
323#97C23×C3118yes—
324C4×C8C23×C3114yes—
325C31#6189211yescharacteristic, direct product factor, Sylow
326C24—22no—
327C2×Q8—24no—
328C2×Q8—21no—
329C2×Q8—24no—
330C2×Q8—22no—
331C2×Q8—42no—
332C2×Q8C2×D4×C3114yes—
333C2×Q8C24×C3112yes—
334C2×Q8C24×C3114yes—
335C2×D4—22no—
336C2×D4—24no—
337C2×D4—24no—
338C2×D4—44no—
339C2×D4—44no—
340C2×D4—82no—
341C2×D4C24×C3114yes—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—28no—
345C22×C4—84no—
346C22×C4C2×D4×C3112yes—
347C22×C4C2×D4×C3112yes—
348C22×C4C2×D4×C3111yescharacteristic
349C22×C4C2×D4×C3111yescharacteristic
350C22×C4C24×C3111yescharacteristic
351Q16—48no—
352SD16—416no—
353C2×C8C24×C3118yes—
354C4⋊C4—24no—
355C4⋊C4—21no—
356C4⋊C4—24no—
357C4⋊C4—22no—
358C4⋊C4—24no—
359C4⋊C4—22no—
360C4⋊C4—24no—
361C4⋊C4—24no—
362C4⋊C4—22no—
363C4⋊C4—28no—
364C4⋊C4—24no—
365C4⋊C4C2×D4×C3114yes—
366C4⋊C4C2×D4×C3118yes—
367C4⋊C4C24×C3114yes—
368C4⋊C4C24×C3112yes—
369C22⋊C4—24no—
370C22⋊C4—28no—
371C22⋊C4—28no—
372C22⋊C4—28no—
373C42—24no—
374C42—28no—
375C42—24no—
376C42C2×D4×C3114yes—
377C42C2×D4×C3112yes—
378C42C24×C3112yes—
379C42C24×C3114yes—
380C23—24no—
381C23—44no—
382C23—44no—
383C23—44no—
384C23C22×D4×C3111yescharacteristic
385Q8—28no—
386Q8—28no—
387Q8—24no—
388Q8—42no—
389Q8—84no—
390Q8—82no—
391D4—28no—
392D4—44no—
393D4—48no—
394D4—84no—
395C2×C4—24no—
396C2×C4—22no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—42no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—88no—
406C2×C4C31×#13714yes—
407C2×C4C22×D4×C3114yes—
408C2×C4C22×D4×C3111yescharacteristic
409C2×C4C22×D4×C3114yes—
410C2×C4C22×D4×C3112yes—
411C2×C4C22×D4×C3112yes—
412C2×C4C25×C3111yescharacteristic
413C8—28no—
414C22—21no—
415C22—21no—
416C22—44no—
417C22—44no—
418C22—44no—
419C22—88no—
420C22C2×C31×#13712yes—
421C22C23×D4×C3111yescharacteristic
422C4—21no—
423C4—24no—
424C4—24no—
425C4—22no—
426C4—42no—
427C4—44no—
428C4—42no—
429C4—44no—
430C4C2×C31×#13714yes—
431C4C31×#58512yes—
432C2—22no—
433C2—84no—
434C2C31×#348711yescharacteristic
435C2C22×C31×#13711yescharacteristic
436C2C31×#358511yescharacteristic
subgroupG/H
center, socleC22×C31C23×D4
commutator, Frattini, cosocleC2×C4C25×C31
prime radicalC2×C31×#127C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C4×Q8×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#3502C2semidirect
3. C31×#3400C2semidirect
4. C31×#3375C2semidirect
5. C31×#3215C2semidirect
7. C31×#3110C2semidirect
8. C31×#3104C2semidirect
9. C31×#2983C2semidirect
12. Q82×C31C22semidirect
14. C31×#557C22semidirect
28. C31×#497C22semidirect
30. C31×#489C22semidirect
33. C31×#477C22semidirect
34. C31×#474C22semidirect
35. C31×#464C22semidirect
36. C31×#462C22semidirect
37. C31×#461C22semidirect
41. C31×#446C22semidirect
42. C31×#440C22semidirect
43. C4×Q16×C31C22semidirect
161. #61892C31direct (N × H)
196. #3502C2×C31semidirect
198. #3400C2×C31semidirect
199. #3375C2×C31semidirect
200. #3215C2×C31semidirect
202. #3110C2×C31semidirect
203. #3104C2×C31semidirect
204. #2983C2×C31semidirect
225. Q82C22×C31semidirect
227. #557C22×C31semidirect
241. #497C22×C31semidirect
243. #489C22×C31semidirect
246. #477C22×C31semidirect
247. #474C22×C31semidirect
248. #464C22×C31semidirect
249. #462C22×C31semidirect
250. #461C22×C31semidirect
254. #446C22×C31semidirect
255. #440C22×C31semidirect
256. C4×Q16C22×C31semidirect
325. C31#61892direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35022
C2×C31×#5592
C31×#34002
C31×#33752
C31×#32152
C31×#32082
C31×#31102
C31×#31042
C31×#29832
#6189231