Finite Group Databasebeta

← C31×#3314 #16843247

Subgroups — C31×#3314 #16843247

Order 3968.

410 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#554C211yesmaximal, characteristic, complemented
2C31×#546C211yesmaximal, characteristic
3C31×#489C211yesmaximal, characteristic, complemented
4C31×#484C211yesmaximal, characteristic, complemented
5C31×#480C211yesmaximal, characteristic, complemented
6C31×#474C211yesmaximal, characteristic, complemented
7C31×#468C211yesmaximal, characteristic, complemented
8C31×#467C211yesmaximal, characteristic, complemented
9C31×#464C211yesmaximal, characteristic, complemented
10C31×#452C211yesmaximal, characteristic, complemented
11C31×#450C211yesmaximal, characteristic, complemented
12C31×#440C211yesmaximal, characteristic, complemented
13C31×#435C211yesmaximal, characteristic, complemented
14C31×#417C211yesmaximal, characteristic, complemented
15C2×C31×#102C211yesmaximal, characteristic, complemented
16C2×#41×C31C2211yescharacteristic
17C22×D4×C31C2211yescharacteristic
18C2×Q16×C31C2211yescharacteristic, complemented
19C2×SD16×C31C2211yescharacteristic
20C2×SD16×C31C2211yescharacteristic
21C2×M16×C31C2211yescharacteristic, complemented
22C22×C8×C31C2211yescharacteristic
23C31×C4⋊Q8C2211yescharacteristic, complemented
24C31×#125—21no—
25C31×#124C2211yescharacteristic
26C31×#123—21no—
27C31×C22⋊Q8—21no—
28C31×C22⋊Q8—21no—
29C31×C22⋊Q8C2211yescharacteristic
30C31×C22⋊Q8C2211yescharacteristic
31C31×C22⋊Q8C2211yescharacteristic, complemented
32C31×#121—21no—
33C31×#121C2211yescharacteristic
34C31×#120—21no—
35C4×Q8×C31C2211yescharacteristic, complemented
36C4×D4×C31—21no—
37C4×D4×C31C2211yescharacteristic
38C4×D4×C31C2211yescharacteristic
39C2×C4⋊C4×C31C2211yescharacteristic
40C2×C4⋊C4×C31C2211yescharacteristic, complemented
41C2×C22⋊C4×C31—21no—
42C31×#107C2211yescharacteristic, complemented
43C31×#107C2211yescharacteristic, complemented
44C31×#107C2211yescharacteristic, complemented
45C31×C4⋊C8C2211yescharacteristic, complemented
46C31×#103C2211yescharacteristic, complemented
47C31×#103C2211yescharacteristic, complemented
48C31×#103C2211yescharacteristic, complemented
49C31×#103C2211yescharacteristic, complemented
50C31×#102C2211yescharacteristic
51C31×#102C2211yescharacteristic
52C31×#102C2211yescharacteristic
53C31×#102C2211yescharacteristic
54C31×#102C2211yescharacteristic
55C31×#102C2211yescharacteristic
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×Q8×C31C2311yescharacteristic
65C2×D4×C31—22no—
66C2×D4×C31—21no—
67C2×D4×C31—21no—
68C2×D4×C31—21no—
69C2×D4×C31—41no—
70C2×D4×C31D411yescharacteristic
71C2×D4×C31C2311yescharacteristic
72C2×D4×C31C2311yescharacteristic
73C22×C4×C31—21no—
74C22×C4×C31—21no—
75C22×C4×C31—21no—
76C22×C4×C31—21no—
77C22×C4×C31C2311yescharacteristic
78C22×C4×C31C2311yescharacteristic
79Q16×C31—41no—
80SD16×C31—22no—
81SD16×C31—41no—
82M16×C31—41no—
83C2×C8×C31—41no—
84C2×C8×C31C2311yescharacteristic
85C2×C8×C31C2311yescharacteristic
86C2×C8×C31C2311yescharacteristic
87C2×C8×C31C2311yescharacteristic
88C4⋊C4×C31—21no—
89C4⋊C4×C31—21no—
90C4⋊C4×C31—21no—
91C4⋊C4×C31—21no—
92C4⋊C4×C31—21no—
93C4⋊C4×C31—21no—
94C4⋊C4×C31—21no—
95C4⋊C4×C31D411yescharacteristic
96C4⋊C4×C31C2311yescharacteristic
97C4⋊C4×C31C2311yescharacteristic
98C4⋊C4×C31C2311yescharacteristic
99C4⋊C4×C31C2311yescharacteristic
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×C31—21no—
107C22⋊C4×C31D411yescharacteristic
108C22⋊C4×C31D411yescharacteristic
109C42×C31—21no—
110C42×C31C2311yescharacteristic
111C23×C31—21no—
112C23×C31—21no—
113C23×C31—42no—
114C23×C31—42no—
115C23×C31—81no—
116C23×C31C2×D411yescharacteristic
117C23×C31C2×D411yescharacteristic
118Q8×C31—21no—
119Q8×C31—21no—
120Q8×C31—21no—
121Q8×C31—21no—
122Q8×C31—81no—
123D4×C31—22no—
124D4×C31—21no—
125D4×C31—21no—
126D4×C31—21no—
127D4×C31—42no—
128D4×C31—42no—
129D4×C31—81no—
130D4×C31#4112yes—
131C2×C4×C31—21no—
132C2×C4×C31—21no—
133C2×C4×C31—21no—
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—42no—
142C2×C4×C31—81no—
143C2×C4×C31—81no—
144C2×C4×C31—81no—
145C2×C4×C31C2×D411yescharacteristic
146C2×C4×C31C2×D411yescharacteristic
147C2×C4×C31C2×D411yescharacteristic
148C2×C4×C31C2×D411yescharacteristic
149C2×C4×C31C2411yescharacteristic
150C8×C31—21no—
151C8×C31—21no—
152C8×C31—21no—
153C8×C31—21no—
154#3314C3111yesmaximal, characteristic, direct product factor, Sylow
155C22×C31—21no—
156C22×C31—21no—
157C22×C31—21no—
158C22×C31—22no—
159C22×C31—21no—
160C22×C31—21no—
161C22×C31—41no—
162C22×C31—41no—
163C22×C31—41no—
164C22×C31—41no—
165C22×C31—41no—
166C22×C31—84no—
167C22×C31#13712yes—
168C22×C31C22×D411yescharacteristic
169C4×C31—21no—
170C4×C31—21no—
171C4×C31—21no—
172C4×C31—21no—
173C4×C31—41no—
174C4×C31—41no—
175C4×C31—41no—
176C4×C31—41no—
177C4×C31—41no—
178C4×C31—41no—
179C4×C31C2×#4111yescharacteristic
180C4×C31H(5,𝔽2)11yescharacteristic
181#554C2×C3111yescharacteristic, complemented
182#546C2×C3111yescharacteristic
183#489C2×C3111yescharacteristic, complemented
184#484C2×C3111yescharacteristic, complemented
185#480C2×C3111yescharacteristic, complemented
186#474C2×C3111yescharacteristic, complemented
187#468C2×C3111yescharacteristic, complemented
188#467C2×C3111yescharacteristic, complemented
189#464C2×C3111yescharacteristic, complemented
190#452C2×C3111yescharacteristic, complemented
191#450C2×C3111yescharacteristic, complemented
192#440C2×C3111yescharacteristic, complemented
193#435C2×C3111yescharacteristic, complemented
194#417C2×C3111yescharacteristic, complemented
195C2×#102C2×C3111yescharacteristic, complemented
196C2×C31—22no—
197C2×C31—41no—
198C2×C31—42no—
199C2×C31—81no—
200C2×C31#54611yescharacteristic
201C2×C31C2×#13711yescharacteristic
202C2×C31#57611yescharacteristic
203C2×#41C22×C3111yescharacteristic
204C22×D4C22×C3111yescharacteristic
205C2×Q16C22×C3111yescharacteristic, complemented
206C2×SD16C22×C3111yescharacteristic
207C2×SD16C22×C3111yescharacteristic
208C2×M16C22×C3111yescharacteristic, complemented
209C22×C8C22×C3111yescharacteristic
210C4⋊Q8C22×C3111yescharacteristic, complemented
211#125—21no—
212#124C22×C3111yescharacteristic
213#123—21no—
214C22⋊Q8—21no—
215C22⋊Q8—21no—
216C22⋊Q8C22×C3111yescharacteristic
217C22⋊Q8C22×C3111yescharacteristic
218C22⋊Q8C22×C3111yescharacteristic, complemented
219#121—21no—
220#121C22×C3111yescharacteristic
221#120—21no—
222C4×Q8C22×C3111yescharacteristic, complemented
223C4×D4—21no—
224C4×D4C22×C3111yescharacteristic
225C4×D4C22×C3111yescharacteristic
226C2×C4⋊C4C22×C3111yescharacteristic
227C2×C4⋊C4C22×C3111yescharacteristic, complemented
228C2×C22⋊C4—21no—
229#107C22×C3111yescharacteristic, complemented
230#107C22×C3111yescharacteristic, complemented
231#107C22×C3111yescharacteristic, complemented
232C4⋊C8C22×C3111yescharacteristic, complemented
233#103C22×C3111yescharacteristic, complemented
234#103C22×C3111yescharacteristic, complemented
235#103C22×C3111yescharacteristic, complemented
236#103C22×C3111yescharacteristic, complemented
237#102C22×C3111yescharacteristic
238#102C22×C3111yescharacteristic
239#102C22×C3111yescharacteristic
240#102C22×C3111yescharacteristic
241#102C22×C3111yescharacteristic
242#102C22×C3111yescharacteristic
243C22⋊C8C22×C3111yescharacteristic
244C22⋊C8C22×C3111yescharacteristic, complemented
245#97C22×C3111yescharacteristic, complemented
246C31#331411yescharacteristic, direct product factor, Sylow
247C24—21no—
248#41—22no—
249#41—41no—
250C2×Q8—21no—
251C2×Q8C23×C3111yescharacteristic
252C2×Q8C23×C3111yescharacteristic
253C2×D4—22no—
254C2×D4—21no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—41no—
258C2×D4D4×C3111yescharacteristic
259C2×D4C23×C3111yescharacteristic
260C2×D4C23×C3111yescharacteristic
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4C23×C3111yescharacteristic
266C22×C4C23×C3111yescharacteristic
267Q16—41no—
268SD16—22no—
269SD16—41no—
270M16—41no—
271C2×C8—41no—
272C2×C8C23×C3111yescharacteristic
273C2×C8C23×C3111yescharacteristic
274C2×C8C23×C3111yescharacteristic
275C2×C8C23×C3111yescharacteristic
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—21no—
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⋊C4D4×C3111yescharacteristic
296C22⋊C4D4×C3111yescharacteristic
297C42—21no—
298C42C23×C3111yescharacteristic
299C23—21no—
300C23—21no—
301C23—42no—
302C23—42no—
303C23—81no—
304C23C2×D4×C3111yescharacteristic
305C23C2×D4×C3111yescharacteristic
306Q8—21no—
307Q8—21no—
308Q8—21no—
309Q8—21no—
310Q8—81no—
311D4—22no—
312D4—21no—
313D4—21no—
314D4—21no—
315D4—42no—
316D4—42no—
317D4—81no—
318D4#41×C3112yes—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—42no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4C2×D4×C3111yescharacteristic
334C2×C4C2×D4×C3111yescharacteristic
335C2×C4C2×D4×C3111yescharacteristic
336C2×C4C2×D4×C3111yescharacteristic
337C2×C4C24×C3111yescharacteristic
338C8—21no—
339C8—21no—
340C8—21no—
341C8—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—22no—
346C22—21no—
347C22—21no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—84no—
354C22C31×#13712yes—
355C22C22×D4×C3111yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—41no—
361C4—41no—
362C4—41no—
363C4—41no—
364C4—41no—
365C4—41no—
366C4C2×#41×C3111yescharacteristic
367C4C31×H(5,𝔽2)11yescharacteristic
368C2—22no—
369C2—41no—
370C2—42no—
371C2—81no—
372C2C31×#54611yescharacteristic
373C2C2×C31×#13711yescharacteristic
374C2C31×#57611yescharacteristic
subgroupG/H
center, socleC22×C31C22×D4
commutator, Frattini, cosocleC2×C4C24×C31
prime radicalC31×#546C2
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×#554C2semidirect
3. C31×#489C2semidirect
4. C31×#484C2semidirect
5. C31×#480C2semidirect
6. C31×#474C2semidirect
7. C31×#468C2semidirect
8. C31×#467C2semidirect
9. C31×#464C2semidirect
10. C31×#452C2semidirect
11. C31×#450C2semidirect
12. C31×#440C2semidirect
13. C31×#435C2semidirect
14. C31×#417C2semidirect
15. C2×C31×#102C2semidirect
18. C2×Q16×C31C22semidirect
21. C2×M16×C31C22semidirect
23. C31×C4⋊Q8C22semidirect
31. C31×C22⋊Q8C22semidirect
35. C4×Q8×C31C22semidirect
40. C2×C4⋊C4×C31C22semidirect
42. C31×#107C22semidirect
43. C31×#107C22semidirect
44. C31×#107C22semidirect
45. C31×C4⋊C8C22semidirect
46. C31×#103C22semidirect
47. C31×#103C22semidirect
48. C31×#103C22semidirect
49. C31×#103C22semidirect
57. C31×C22⋊C8C22semidirect
58. C31×#97C22semidirect
154. #3314C31direct (N × H)
181. #554C2×C31semidirect
183. #489C2×C31semidirect
184. #484C2×C31semidirect
185. #480C2×C31semidirect
186. #474C2×C31semidirect
187. #468C2×C31semidirect
188. #467C2×C31semidirect
189. #464C2×C31semidirect
190. #452C2×C31semidirect
191. #450C2×C31semidirect
192. #440C2×C31semidirect
193. #435C2×C31semidirect
194. #417C2×C31semidirect
195. C2×#102C2×C31semidirect
205. C2×Q16C22×C31semidirect
208. C2×M16C22×C31semidirect
210. C4⋊Q8C22×C31semidirect
218. C22⋊Q8C22×C31semidirect
222. C4×Q8C22×C31semidirect
227. C2×C4⋊C4C22×C31semidirect
229. #107C22×C31semidirect
230. #107C22×C31semidirect
231. #107C22×C31semidirect
232. C4⋊C8C22×C31semidirect
233. #103C22×C31semidirect
234. #103C22×C31semidirect
235. #103C22×C31semidirect
236. #103C22×C31semidirect
244. C22⋊C8C22×C31semidirect
245. #97C22×C31semidirect
246. C31#3314direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#5542
C31×#5462
C31×#4892
C31×#4842
C31×#4802
C31×#4742
C31×#4682
C31×#4672
C31×#4642
C31×#4522
C31×#4502
C31×#4402
C31×#4352
C31×#4172
C2×C31×#1022
#331431