Finite Group Databasebeta

← C31×#3224 #16843157

Subgroups — C31×#3224 #16843157

Order 3968.

408 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#542C211yesmaximal, characteristic, complemented
2C31×#535C211yesmaximal, characteristic, complemented
3C31×#496C211yesmaximal, characteristic, complemented
4C31×#495C211yesmaximal, characteristic, complemented
5C31×#489C211yesmaximal, characteristic, complemented
6C31×#486C211yesmaximal, characteristic, complemented
7C31×#464C211yesmaximal, characteristic, complemented
8C31×#463C211yesmaximal, characteristic, complemented
9C31×#460C211yesmaximal, characteristic, complemented
10C31×#459C211yesmaximal, characteristic, complemented
11C31×#452C211yesmaximal, characteristic, complemented
12C31×#450C211yesmaximal, characteristic, complemented
13C31×#449C211yesmaximal, characteristic, complemented
14C31×#447C211yesmaximal, characteristic, complemented
15C31×#433C211yesmaximal, characteristic, complemented
16C2×#41×C31C2211yescharacteristic
17C22×D4×C31C2211yescharacteristic
18C2×Q16×C31C2211yescharacteristic, complemented
19C2×SD16×C31C2211yescharacteristic
20C2×SD16×C31C2211yescharacteristic
21C2×SD16×C31C2211yescharacteristic, complemented
22C2×SD16×C31C2211yescharacteristic, complemented
23C2×D8×C31C2211yescharacteristic
24C2×D8×C31C2211yescharacteristic
25C2×D8×C31C2211yescharacteristic, complemented
26C31×C4⋊Q8C2211yescharacteristic, complemented
27C31×#127C2211yescharacteristic
28C31×#127C2211yescharacteristic
29C31×#126—21no—
30C31×#124—21no—
31C31×#124C2211yescharacteristic
32C31×#124C2211yescharacteristic
33C31×#124C2211yescharacteristic, complemented
34C31×#123—21no—
35C31×C22⋊Q8—21no—
36C31×C22⋊Q8C2211yescharacteristic
37C31×#121—21no—
38C31×#121—21no—
39C31×#121C2211yescharacteristic
40C31×#120—21no—
41C31×#120—21no—
42C4×Q8×C31C2211yescharacteristic, complemented
43C4×D4×C31C2211yescharacteristic, complemented
44C31×#117C2211yescharacteristic
45C31×C4⋊C8C2211yescharacteristic, complemented
46C31×C4⋊C8C2211yescharacteristic, complemented
47C31×#103C2211yescharacteristic, complemented
48C31×#103C2211yescharacteristic, complemented
49C31×#102C2211yescharacteristic
50C31×#102C2211yescharacteristic
51C31×#102C2211yescharacteristic
52C31×#102C2211yescharacteristic
53C31×#102C2211yescharacteristic, complemented
54C31×#102C2211yescharacteristic, complemented
55C31×C22⋊C8C2211yescharacteristic
56C31×C22⋊C8C2211yescharacteristic
57C31×#97C2211yescharacteristic, complemented
58C4×C8×C31C2211yescharacteristic, complemented
59C24×C31—21no—
60#41×C31—41no—
61#41×C31—41no—
62C2×Q8×C31—21no—
63C2×Q8×C31C2311yescharacteristic
64C2×Q8×C31C2311yescharacteristic
65C2×D4×C31—21no—
66C2×D4×C31—21no—
67C2×D4×C31—21no—
68C2×D4×C31—21no—
69C2×D4×C31—21no—
70C2×D4×C31—21no—
71C2×D4×C31—21no—
72C2×D4×C31—41no—
73C2×D4×C31—41no—
74C2×D4×C31C2311yescharacteristic
75C2×D4×C31C2311yescharacteristic
76C2×D4×C31C2311yescharacteristic
77C2×D4×C31C2311yescharacteristic
78C22×C4×C31—21no—
79C22×C4×C31—21no—
80C22×C4×C31C2311yescharacteristic
81Q16×C31—41no—
82SD16×C31—41no—
83SD16×C31—41no—
84SD16×C31—41no—
85SD16×C31—41no—
86D8×C31—41no—
87D8×C31—41no—
88D8×C31—41no—
89C2×C8×C31C2311yescharacteristic
90C2×C8×C31C2311yescharacteristic
91C2×C8×C31C2311yescharacteristic
92C2×C8×C31C2311yescharacteristic
93C4⋊C4×C31—21no—
94C4⋊C4×C31—21no—
95C4⋊C4×C31—21no—
96C4⋊C4×C31D411yescharacteristic
97C4⋊C4×C31D411yescharacteristic
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×C31—21no—
108C22⋊C4×C31—21no—
109C22⋊C4×C31D411yescharacteristic
110C22⋊C4×C31D411yescharacteristic
111C42×C31—21no—
112C42×C31C2311yescharacteristic
113C42×C31C2311yescharacteristic
114C23×C31—21no—
115C23×C31—21no—
116C23×C31—21no—
117C23×C31—21no—
118C23×C31—42no—
119C23×C31—81no—
120C23×C31—81no—
121C23×C31C2×D411yescharacteristic
122Q8×C31—21no—
123Q8×C31—21no—
124Q8×C31—21no—
125Q8×C31—21no—
126Q8×C31—81no—
127D4×C31—21no—
128D4×C31—21no—
129D4×C31—21no—
130D4×C31—21no—
131D4×C31—21no—
132D4×C31—21no—
133D4×C31—21no—
134D4×C31—21no—
135D4×C31—42no—
136D4×C31—42no—
137D4×C31—81no—
138D4×C31—81no—
139D4×C31—81no—
140D4×C31—81no—
141D4×C31—81no—
142C2×C4×C31—21no—
143C2×C4×C31—21no—
144C2×C4×C31—21no—
145C2×C4×C31—21no—
146C2×C4×C31—21no—
147C2×C4×C31—21no—
148C2×C4×C31—42no—
149C2×C4×C31—81no—
150C2×C4×C31C2×D411yescharacteristic
151C2×C4×C31C2×D411yescharacteristic
152C2×C4×C31C2×D411yescharacteristic
153C2×C4×C31C2×D411yescharacteristic
154C2×C4×C31C2×D411yescharacteristic
155C2×C4×C31C2411yescharacteristic
156C8×C31—21no—
157C8×C31—21no—
158C8×C31—21no—
159C8×C31—21no—
160#3224C3111yesmaximal, characteristic, direct product factor, Sylow
161C22×C31—21no—
162C22×C31—21no—
163C22×C31—21no—
164C22×C31—41no—
165C22×C31—41no—
166C22×C31—41no—
167C22×C31—41no—
168C22×C31—41no—
169C22×C31—41no—
170C22×C31—41no—
171C22×C31—41no—
172C22×C31—41no—
173C22×C31—41no—
174C22×C31—41no—
175C22×C31—41no—
176C22×C31—84no—
177C22×C31C22×D411yescharacteristic
178C4×C31—21no—
179C4×C31—21no—
180C4×C31—21no—
181C4×C31—21no—
182C4×C31—21no—
183C4×C31—41no—
184C4×C31—41no—
185C4×C31—41no—
186C4×C31—41no—
187C4×C31H(5,𝔽2)11yescharacteristic
188C4×C31H(5,𝔽2)11yescharacteristic
189#542C2×C3111yescharacteristic, complemented
190#535C2×C3111yescharacteristic, complemented
191#496C2×C3111yescharacteristic, complemented
192#495C2×C3111yescharacteristic, complemented
193#489C2×C3111yescharacteristic, complemented
194#486C2×C3111yescharacteristic, complemented
195#464C2×C3111yescharacteristic, complemented
196#463C2×C3111yescharacteristic, complemented
197#460C2×C3111yescharacteristic, complemented
198#459C2×C3111yescharacteristic, complemented
199#452C2×C3111yescharacteristic, complemented
200#450C2×C3111yescharacteristic, complemented
201#449C2×C3111yescharacteristic, complemented
202#447C2×C3111yescharacteristic, complemented
203#433C2×C3111yescharacteristic, complemented
204C2×C31—41no—
205C2×C31—81no—
206C2×C31—81no—
207C2×C31—81no—
208C2×C31—81no—
209C2×C31#53511yescharacteristic
210C2×C31#57611yescharacteristic
211C2×C31#57711yescharacteristic
212C2×#41C22×C3111yescharacteristic
213C22×D4C22×C3111yescharacteristic
214C2×Q16C22×C3111yescharacteristic, complemented
215C2×SD16C22×C3111yescharacteristic
216C2×SD16C22×C3111yescharacteristic
217C2×SD16C22×C3111yescharacteristic, complemented
218C2×SD16C22×C3111yescharacteristic, complemented
219C2×D8C22×C3111yescharacteristic
220C2×D8C22×C3111yescharacteristic
221C2×D8C22×C3111yescharacteristic, complemented
222C4⋊Q8C22×C3111yescharacteristic, complemented
223#127C22×C3111yescharacteristic
224#127C22×C3111yescharacteristic
225#126—21no—
226#124—21no—
227#124C22×C3111yescharacteristic
228#124C22×C3111yescharacteristic
229#124C22×C3111yescharacteristic, complemented
230#123—21no—
231C22⋊Q8—21no—
232C22⋊Q8C22×C3111yescharacteristic
233#121—21no—
234#121—21no—
235#121C22×C3111yescharacteristic
236#120—21no—
237#120—21no—
238C4×Q8C22×C3111yescharacteristic, complemented
239C4×D4C22×C3111yescharacteristic, complemented
240#117C22×C3111yescharacteristic
241C4⋊C8C22×C3111yescharacteristic, complemented
242C4⋊C8C22×C3111yescharacteristic, complemented
243#103C22×C3111yescharacteristic, complemented
244#103C22×C3111yescharacteristic, complemented
245#102C22×C3111yescharacteristic
246#102C22×C3111yescharacteristic
247#102C22×C3111yescharacteristic
248#102C22×C3111yescharacteristic
249#102C22×C3111yescharacteristic, complemented
250#102C22×C3111yescharacteristic, complemented
251C22⋊C8C22×C3111yescharacteristic
252C22⋊C8C22×C3111yescharacteristic
253#97C22×C3111yescharacteristic, complemented
254C4×C8C22×C3111yescharacteristic, complemented
255C31#322411yescharacteristic, direct product factor, Sylow
256C24—21no—
257#41—41no—
258#41—41no—
259C2×Q8—21no—
260C2×Q8C23×C3111yescharacteristic
261C2×Q8C23×C3111yescharacteristic
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—21no—
267C2×D4—21no—
268C2×D4—21no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4C23×C3111yescharacteristic
272C2×D4C23×C3111yescharacteristic
273C2×D4C23×C3111yescharacteristic
274C2×D4C23×C3111yescharacteristic
275C22×C4—21no—
276C22×C4—21no—
277C22×C4C23×C3111yescharacteristic
278Q16—41no—
279SD16—41no—
280SD16—41no—
281SD16—41no—
282SD16—41no—
283D8—41no—
284D8—41no—
285D8—41no—
286C2×C8C23×C3111yescharacteristic
287C2×C8C23×C3111yescharacteristic
288C2×C8C23×C3111yescharacteristic
289C2×C8C23×C3111yescharacteristic
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4D4×C3111yescharacteristic
294C4⋊C4D4×C3111yescharacteristic
295C4⋊C4C23×C3111yescharacteristic
296C4⋊C4C23×C3111yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—21no—
301C22⋊C4—21no—
302C22⋊C4—21no—
303C22⋊C4—21no—
304C22⋊C4—21no—
305C22⋊C4—21no—
306C22⋊C4D4×C3111yescharacteristic
307C22⋊C4D4×C3111yescharacteristic
308C42—21no—
309C42C23×C3111yescharacteristic
310C42C23×C3111yescharacteristic
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—42no—
316C23—81no—
317C23—81no—
318C23C2×D4×C3111yescharacteristic
319Q8—21no—
320Q8—21no—
321Q8—21no—
322Q8—21no—
323Q8—81no—
324D4—21no—
325D4—21no—
326D4—21no—
327D4—21no—
328D4—21no—
329D4—21no—
330D4—21no—
331D4—21no—
332D4—42no—
333D4—42no—
334D4—81no—
335D4—81no—
336D4—81no—
337D4—81no—
338D4—81no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—42no—
346C2×C4—81no—
347C2×C4C2×D4×C3111yescharacteristic
348C2×C4C2×D4×C3111yescharacteristic
349C2×C4C2×D4×C3111yescharacteristic
350C2×C4C2×D4×C3111yescharacteristic
351C2×C4C2×D4×C3111yescharacteristic
352C2×C4C24×C3111yescharacteristic
353C8—21no—
354C8—21no—
355C8—21no—
356C8—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—84no—
373C22C22×D4×C3111yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4—41no—
383C4C31×H(5,𝔽2)11yescharacteristic
384C4C31×H(5,𝔽2)11yescharacteristic
385C2—41no—
386C2—81no—
387C2—81no—
388C2—81no—
389C2—81no—
390C2C31×#53511yescharacteristic
391C2C31×#57611yescharacteristic
392C2C31×#57711yescharacteristic
subgroupG/H
center, socleC22×C31C22×D4
commutator, Frattini, cosocleC2×C4C24×C31
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C31×#117 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#542C2semidirect
2. C31×#535C2semidirect
3. C31×#496C2semidirect
4. C31×#495C2semidirect
5. C31×#489C2semidirect
6. C31×#486C2semidirect
7. C31×#464C2semidirect
8. C31×#463C2semidirect
9. C31×#460C2semidirect
10. C31×#459C2semidirect
11. C31×#452C2semidirect
12. C31×#450C2semidirect
13. C31×#449C2semidirect
14. C31×#447C2semidirect
15. C31×#433C2semidirect
18. C2×Q16×C31C22semidirect
21. C2×SD16×C31C22semidirect
22. C2×SD16×C31C22semidirect
25. C2×D8×C31C22semidirect
26. C31×C4⋊Q8C22semidirect
33. C31×#124C22semidirect
42. C4×Q8×C31C22semidirect
43. C4×D4×C31C22semidirect
45. C31×C4⋊C8C22semidirect
46. C31×C4⋊C8C22semidirect
47. C31×#103C22semidirect
48. C31×#103C22semidirect
53. C31×#102C22semidirect
54. C31×#102C22semidirect
57. C31×#97C22semidirect
58. C4×C8×C31C22semidirect
160. #3224C31direct (N × H)
189. #542C2×C31semidirect
190. #535C2×C31semidirect
191. #496C2×C31semidirect
192. #495C2×C31semidirect
193. #489C2×C31semidirect
194. #486C2×C31semidirect
195. #464C2×C31semidirect
196. #463C2×C31semidirect
197. #460C2×C31semidirect
198. #459C2×C31semidirect
199. #452C2×C31semidirect
200. #450C2×C31semidirect
201. #449C2×C31semidirect
202. #447C2×C31semidirect
203. #433C2×C31semidirect
214. C2×Q16C22×C31semidirect
217. C2×SD16C22×C31semidirect
218. C2×SD16C22×C31semidirect
221. C2×D8C22×C31semidirect
222. C4⋊Q8C22×C31semidirect
229. #124C22×C31semidirect
238. C4×Q8C22×C31semidirect
239. C4×D4C22×C31semidirect
241. C4⋊C8C22×C31semidirect
242. C4⋊C8C22×C31semidirect
243. #103C22×C31semidirect
244. #103C22×C31semidirect
249. #102C22×C31semidirect
250. #102C22×C31semidirect
253. #97C22×C31semidirect
254. C4×C8C22×C31semidirect
255. C31#3224direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#5422
C31×#5352
C31×#4962
C31×#4952
C31×#4892
C31×#4862
C31×#4642
C31×#4632
C31×#4602
C31×#4592
C31×#4522
C31×#4502
C31×#4492
C31×#4472
C31×#4332
#322431