Finite Group Databasebeta

← C31×#3319 #16843252

Subgroups — C31×#3319 #16843252

Order 3968.

414 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#555C211yesmaximal, characteristic, complemented
2C31×#546C211yesmaximal, characteristic, complemented
3C31×#489C211yesmaximal, characteristic, complemented
4C31×#485C211yesmaximal, characteristic, complemented
5C31×#482C211yesmaximal, characteristic, complemented
6C31×#474C211yesmaximal, characteristic, complemented
7C31×#468C211yesmaximal, characteristic, complemented
8C31×#466C211yesmaximal, characteristic, complemented
9C31×#463C211yesmaximal, characteristic, complemented
10C31×#449C211yesmaximal, characteristic, complemented
11C31×#447C211yesmaximal, characteristic, complemented
12C31×#442C211yesmaximal, characteristic, complemented
13C31×#435C211yesmaximal, characteristic, complemented
14C31×#418C211yesmaximal, characteristic, complemented
15C31×#416C211yesmaximal, characteristic, complemented
16C2×#41×C31C2211yescharacteristic
17C22×D4×C31C2211yescharacteristic
18C2×SD16×C31C2211yescharacteristic, complemented
19C2×D8×C31C2211yescharacteristic
20C2×D8×C31C2211yescharacteristic
21C2×M16×C31C2211yescharacteristic
22C22×C8×C31C2211yescharacteristic, complemented
23C31×C4⋊Q8C2211yescharacteristic, complemented
24C31×#124—21no—
25C31×#124—21no—
26C31×#124C2211yescharacteristic
27C31×#123—21no—
28C31×C22⋊Q8C2211yescharacteristic
29C31×#121—21no—
30C31×#121—21no—
31C31×#121C2211yescharacteristic
32C31×#121C2211yescharacteristic
33C31×#121C2211yescharacteristic, complemented
34C31×#120—21no—
35C4×D4×C31C2211yescharacteristic
36C4×D4×C31C2211yescharacteristic
37C4×D4×C31C2211yescharacteristic, complemented
38C31×#117—21no—
39C31×#117C2211yescharacteristic
40C31×#117C2211yescharacteristic, complemented
41C2×C22⋊C4×C31—21no—
42C31×#107C2211yescharacteristic, complemented
43C31×#107C2211yescharacteristic, complemented
44C31×#106C2211yescharacteristic, complemented
45C31×C4⋊C8C2211yescharacteristic, complemented
46C31×#103C2211yescharacteristic, complemented
47C31×#103C2211yescharacteristic, complemented
48C31×#103C2211yescharacteristic, complemented
49C31×#102C2211yescharacteristic
50C31×#102C2211yescharacteristic
51C31×#102C2211yescharacteristic
52C31×#102C2211yescharacteristic
53C31×#102C2211yescharacteristic
54C31×#102C2211yescharacteristic
55C31×#102C2211yescharacteristic, complemented
56C31×C22⋊C8C2211yescharacteristic
57C31×C22⋊C8C2211yescharacteristic, complemented
58C31×#97C2211yescharacteristic, complemented
59C24×C31—21no—
60#41×C31—22no—
61#41×C31—41no—
62C2×Q8×C31—21no—
63C2×Q8×C31C2311yescharacteristic
64C2×D4×C31—22no—
65C2×D4×C31—21no—
66C2×D4×C31—21no—
67C2×D4×C31—21no—
68C2×D4×C31—21no—
69C2×D4×C31—21no—
70C2×D4×C31—41no—
71C2×D4×C31D411yescharacteristic
72C2×D4×C31C2311yescharacteristic
73C2×D4×C31C2311yescharacteristic
74C2×D4×C31C2311yescharacteristic
75C22×C4×C31—21no—
76C22×C4×C31—21no—
77C22×C4×C31—21no—
78C22×C4×C31C2311yescharacteristic
79C22×C4×C31C2311yescharacteristic
80SD16×C31—41no—
81D8×C31—22no—
82D8×C31—41no—
83M16×C31—41no—
84C2×C8×C31—41no—
85C2×C8×C31C2311yescharacteristic
86C2×C8×C31C2311yescharacteristic
87C2×C8×C31C2311yescharacteristic
88C2×C8×C31C2311yescharacteristic
89C4⋊C4×C31—21no—
90C4⋊C4×C31—21no—
91C4⋊C4×C31D411yescharacteristic
92C4⋊C4×C31C2311yescharacteristic
93C4⋊C4×C31C2311yescharacteristic
94C4⋊C4×C31C2311yescharacteristic
95C4⋊C4×C31C2311yescharacteristic
96C22⋊C4×C31—21no—
97C22⋊C4×C31—21no—
98C22⋊C4×C31—21no—
99C22⋊C4×C31—21no—
100C22⋊C4×C31—21no—
101C22⋊C4×C31—21no—
102C22⋊C4×C31—21no—
103C22⋊C4×C31—21no—
104C22⋊C4×C31—21no—
105C22⋊C4×C31—21no—
106C22⋊C4×C31D411yescharacteristic
107C22⋊C4×C31D411yescharacteristic
108C42×C31—21no—
109C42×C31—21no—
110C42×C31C2311yescharacteristic
111C23×C31—21no—
112C23×C31—21no—
113C23×C31—21no—
114C23×C31—42no—
115C23×C31—42no—
116C23×C31—81no—
117C23×C31C2×D411yescharacteristic
118C23×C31C2×D411yescharacteristic
119Q8×C31—21no—
120Q8×C31—21no—
121Q8×C31—81no—
122D4×C31—21no—
123D4×C31—21no—
124D4×C31—21no—
125D4×C31—21no—
126D4×C31—21no—
127D4×C31—41no—
128D4×C31—42no—
129D4×C31—42no—
130D4×C31—81no—
131D4×C31—81no—
132D4×C31—81no—
133D4×C31#4112yes—
134C2×C4×C31—21no—
135C2×C4×C31—21no—
136C2×C4×C31—21no—
137C2×C4×C31—21no—
138C2×C4×C31—21no—
139C2×C4×C31—21no—
140C2×C4×C31—21no—
141C2×C4×C31—21no—
142C2×C4×C31—21no—
143C2×C4×C31—42no—
144C2×C4×C31—42no—
145C2×C4×C31—81no—
146C2×C4×C31C2×D411yescharacteristic
147C2×C4×C31C2×D411yescharacteristic
148C2×C4×C31C2×D411yescharacteristic
149C2×C4×C31C2×D411yescharacteristic
150C2×C4×C31C2411yescharacteristic
151C8×C31—21no—
152C8×C31—21no—
153C8×C31—21no—
154C8×C31—21no—
155#3319C3111yesmaximal, characteristic, direct product factor, Sylow
156C22×C31—21no—
157C22×C31—21no—
158C22×C31—22no—
159C22×C31—21no—
160C22×C31—21no—
161C22×C31—21no—
162C22×C31—21no—
163C22×C31—41no—
164C22×C31—41no—
165C22×C31—41no—
166C22×C31—41no—
167C22×C31—41no—
168C22×C31—41no—
169C22×C31—41no—
170C22×C31—41no—
171C22×C31—84no—
172C22×C31C22×D411yescharacteristic
173C4×C31—21no—
174C4×C31—21no—
175C4×C31—21no—
176C4×C31—21no—
177C4×C31—41no—
178C4×C31—41no—
179C4×C31—41no—
180C4×C31—41no—
181C4×C31—41no—
182C4×C31C2×#4111yescharacteristic
183C4×C31H(5,𝔽2)11yescharacteristic
184#555C2×C3111yescharacteristic, complemented
185#546C2×C3111yescharacteristic, complemented
186#489C2×C3111yescharacteristic, complemented
187#485C2×C3111yescharacteristic, complemented
188#482C2×C3111yescharacteristic, complemented
189#474C2×C3111yescharacteristic, complemented
190#468C2×C3111yescharacteristic, complemented
191#466C2×C3111yescharacteristic, complemented
192#463C2×C3111yescharacteristic, complemented
193#449C2×C3111yescharacteristic, complemented
194#447C2×C3111yescharacteristic, complemented
195#442C2×C3111yescharacteristic, complemented
196#435C2×C3111yescharacteristic, complemented
197#418C2×C3111yescharacteristic, complemented
198#416C2×C3111yescharacteristic, complemented
199C2×C31—41no—
200C2×C31—41no—
201C2×C31—42no—
202C2×C31—81no—
203C2×C31—81no—
204C2×C31#54611yescharacteristic
205C2×C31#57511yescharacteristic
206C2×C31#57611yescharacteristic
207C2×#41C22×C3111yescharacteristic
208C22×D4C22×C3111yescharacteristic
209C2×SD16C22×C3111yescharacteristic, complemented
210C2×D8C22×C3111yescharacteristic
211C2×D8C22×C3111yescharacteristic
212C2×M16C22×C3111yescharacteristic
213C22×C8C22×C3111yescharacteristic, complemented
214C4⋊Q8C22×C3111yescharacteristic, complemented
215#124—21no—
216#124—21no—
217#124C22×C3111yescharacteristic
218#123—21no—
219C22⋊Q8C22×C3111yescharacteristic
220#121—21no—
221#121—21no—
222#121C22×C3111yescharacteristic
223#121C22×C3111yescharacteristic
224#121C22×C3111yescharacteristic, complemented
225#120—21no—
226C4×D4C22×C3111yescharacteristic
227C4×D4C22×C3111yescharacteristic
228C4×D4C22×C3111yescharacteristic, complemented
229#117—21no—
230#117C22×C3111yescharacteristic
231#117C22×C3111yescharacteristic, complemented
232C2×C22⋊C4—21no—
233#107C22×C3111yescharacteristic, complemented
234#107C22×C3111yescharacteristic, complemented
235#106C22×C3111yescharacteristic, complemented
236C4⋊C8C22×C3111yescharacteristic, complemented
237#103C22×C3111yescharacteristic, complemented
238#103C22×C3111yescharacteristic, complemented
239#103C22×C3111yescharacteristic, complemented
240#102C22×C3111yescharacteristic
241#102C22×C3111yescharacteristic
242#102C22×C3111yescharacteristic
243#102C22×C3111yescharacteristic
244#102C22×C3111yescharacteristic
245#102C22×C3111yescharacteristic
246#102C22×C3111yescharacteristic, complemented
247C22⋊C8C22×C3111yescharacteristic
248C22⋊C8C22×C3111yescharacteristic, complemented
249#97C22×C3111yescharacteristic, complemented
250C31#331911yescharacteristic, direct product factor, Sylow
251C24—21no—
252#41—22no—
253#41—41no—
254C2×Q8—21no—
255C2×Q8C23×C3111yescharacteristic
256C2×D4—22no—
257C2×D4—21no—
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—21no—
261C2×D4—21no—
262C2×D4—41no—
263C2×D4D4×C3111yescharacteristic
264C2×D4C23×C3111yescharacteristic
265C2×D4C23×C3111yescharacteristic
266C2×D4C23×C3111yescharacteristic
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4C23×C3111yescharacteristic
271C22×C4C23×C3111yescharacteristic
272SD16—41no—
273D8—22no—
274D8—41no—
275M16—41no—
276C2×C8—41no—
277C2×C8C23×C3111yescharacteristic
278C2×C8C23×C3111yescharacteristic
279C2×C8C23×C3111yescharacteristic
280C2×C8C23×C3111yescharacteristic
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4D4×C3111yescharacteristic
284C4⋊C4C23×C3111yescharacteristic
285C4⋊C4C23×C3111yescharacteristic
286C4⋊C4C23×C3111yescharacteristic
287C4⋊C4C23×C3111yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—21no—
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—21no—
295C22⋊C4—21no—
296C22⋊C4—21no—
297C22⋊C4—21no—
298C22⋊C4D4×C3111yescharacteristic
299C22⋊C4D4×C3111yescharacteristic
300C42—21no—
301C42—21no—
302C42C23×C3111yescharacteristic
303C23—21no—
304C23—21no—
305C23—21no—
306C23—42no—
307C23—42no—
308C23—81no—
309C23C2×D4×C3111yescharacteristic
310C23C2×D4×C3111yescharacteristic
311Q8—21no—
312Q8—21no—
313Q8—81no—
314D4—21no—
315D4—21no—
316D4—21no—
317D4—21no—
318D4—21no—
319D4—41no—
320D4—42no—
321D4—42no—
322D4—81no—
323D4—81no—
324D4—81no—
325D4#41×C3112yes—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—81no—
338C2×C4C2×D4×C3111yescharacteristic
339C2×C4C2×D4×C3111yescharacteristic
340C2×C4C2×D4×C3111yescharacteristic
341C2×C4C2×D4×C3111yescharacteristic
342C2×C4C24×C3111yescharacteristic
343C8—21no—
344C8—21no—
345C8—21no—
346C8—21no—
347C22—21no—
348C22—21no—
349C22—22no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—84no—
363C22C22×D4×C3111yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—41no—
369C4—41no—
370C4—41no—
371C4—41no—
372C4—41no—
373C4C2×#41×C3111yescharacteristic
374C4C31×H(5,𝔽2)11yescharacteristic
375C2—41no—
376C2—41no—
377C2—42no—
378C2—81no—
379C2—81no—
380C2C31×#54611yescharacteristic
381C2C31×#57511yescharacteristic
382C2C31×#57611yescharacteristic
subgroupG/H
center, socleC22×C31C22×D4
commutator, Frattini, cosocleC2×C4C24×C31
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C4×D4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#555C2semidirect
2. C31×#546C2semidirect
3. C31×#489C2semidirect
4. C31×#485C2semidirect
5. C31×#482C2semidirect
6. C31×#474C2semidirect
7. C31×#468C2semidirect
8. C31×#466C2semidirect
9. C31×#463C2semidirect
10. C31×#449C2semidirect
11. C31×#447C2semidirect
12. C31×#442C2semidirect
13. C31×#435C2semidirect
14. C31×#418C2semidirect
15. C31×#416C2semidirect
18. C2×SD16×C31C22semidirect
22. C22×C8×C31C22semidirect
23. C31×C4⋊Q8C22semidirect
33. C31×#121C22semidirect
37. C4×D4×C31C22semidirect
40. C31×#117C22semidirect
42. C31×#107C22semidirect
43. C31×#107C22semidirect
44. C31×#106C22semidirect
45. C31×C4⋊C8C22semidirect
46. C31×#103C22semidirect
47. C31×#103C22semidirect
48. C31×#103C22semidirect
55. C31×#102C22semidirect
57. C31×C22⋊C8C22semidirect
58. C31×#97C22semidirect
155. #3319C31direct (N × H)
184. #555C2×C31semidirect
185. #546C2×C31semidirect
186. #489C2×C31semidirect
187. #485C2×C31semidirect
188. #482C2×C31semidirect
189. #474C2×C31semidirect
190. #468C2×C31semidirect
191. #466C2×C31semidirect
192. #463C2×C31semidirect
193. #449C2×C31semidirect
194. #447C2×C31semidirect
195. #442C2×C31semidirect
196. #435C2×C31semidirect
197. #418C2×C31semidirect
198. #416C2×C31semidirect
209. C2×SD16C22×C31semidirect
213. C22×C8C22×C31semidirect
214. C4⋊Q8C22×C31semidirect
224. #121C22×C31semidirect
228. C4×D4C22×C31semidirect
231. #117C22×C31semidirect
233. #107C22×C31semidirect
234. #107C22×C31semidirect
235. #106C22×C31semidirect
236. C4⋊C8C22×C31semidirect
237. #103C22×C31semidirect
238. #103C22×C31semidirect
239. #103C22×C31semidirect
246. #102C22×C31semidirect
248. C22⋊C8C22×C31semidirect
249. #97C22×C31semidirect
250. C31#3319direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#5552
C31×#5462
C31×#4892
C31×#4852
C31×#4822
C31×#4742
C31×#4682
C31×#4662
C31×#4632
C31×#4492
C31×#4472
C31×#4422
C31×#4352
C31×#4182
C31×#4162
#331931