Finite Group Databasebeta

← C31×#3311 #16843244

Subgroups — C31×#3311 #16843244

Order 3968.

402 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#556C211yesmaximal, characteristic, complemented
2C31×#546C211yesmaximal, characteristic
3C31×#489C211yesmaximal, characteristic, complemented
4C31×#485C211yesmaximal, characteristic, complemented
5C31×#482C211yesmaximal, characteristic, complemented
6C31×#476C211yesmaximal, characteristic, complemented
7C31×#470C211yesmaximal, characteristic, complemented
8C31×#465C211yesmaximal, characteristic, complemented
9C31×#464C211yesmaximal, characteristic, complemented
10C31×#452C211yesmaximal, characteristic, complemented
11C31×#450C211yesmaximal, characteristic, complemented
12C31×#440C211yesmaximal, characteristic, complemented
13C31×#435C211yesmaximal, characteristic, complemented
14C31×#418C211yesmaximal, characteristic, complemented
15C31×#416C211yesmaximal, characteristic, complemented
16C2×#41×C31C2211yescharacteristic
17C22×D4×C31C2211yescharacteristic
18C2×Q16×C31C2211yescharacteristic, complemented
19C2×SD16×C31C2211yescharacteristic
20C2×SD16×C31C2211yescharacteristic
21C2×M16×C31C2211yescharacteristic
22C22×C8×C31C2211yescharacteristic, complemented
23C31×C4⋊Q8C2211yescharacteristic, complemented
24C31×#126—21no—
25C31×#126—21no—
26C31×#124—21no—
27C31×#124C2211yescharacteristic
28C31×#123—21no—
29C31×C22⋊Q8C2211yescharacteristic
30C31×C22⋊Q8C2211yescharacteristic
31C31×C22⋊Q8C2211yescharacteristic, complemented
32C31×#121—21no—
33C31×#121C2211yescharacteristic
34C31×#120—21no—
35C4×Q8×C31—21no—
36C4×Q8×C31C2211yescharacteristic, complemented
37C4×D4×C31C2211yescharacteristic
38C4×D4×C31C2211yescharacteristic
39C31×#117C2211yescharacteristic
40C31×#117C2211yescharacteristic, complemented
41C2×C22⋊C4×C31—21no—
42C31×#107C2211yescharacteristic, complemented
43C31×#106C2211yescharacteristic, complemented
44C31×#106C2211yescharacteristic, 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×C31C2311yescharacteristic
76C22×C4×C31C2311yescharacteristic
77Q16×C31—41no—
78SD16×C31—22no—
79SD16×C31—41no—
80M16×C31—41no—
81C2×C8×C31—41no—
82C2×C8×C31C2311yescharacteristic
83C2×C8×C31C2311yescharacteristic
84C2×C8×C31C2311yescharacteristic
85C2×C8×C31C2311yescharacteristic
86C4⋊C4×C31—21no—
87C4⋊C4×C31—21no—
88C4⋊C4×C31—21no—
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×C31D411yescharacteristic
106C22⋊C4×C31D411yescharacteristic
107C42×C31—21no—
108C42×C31—21no—
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—42no—
123D4×C31—21no—
124D4×C31—21no—
125D4×C31—21no—
126D4×C31—41no—
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×C31C2×D411yescharacteristic
144C2×C4×C31C2×D411yescharacteristic
145C2×C4×C31C2×D411yescharacteristic
146C2×C4×C31C2×D411yescharacteristic
147C2×C4×C31C2411yescharacteristic
148C8×C31—21no—
149C8×C31—21no—
150C8×C31—21no—
151C8×C31—21no—
152#3311C3111yesmaximal, characteristic, direct product factor, Sylow
153C22×C31—21no—
154C22×C31—21no—
155C22×C31—22no—
156C22×C31—21no—
157C22×C31—21no—
158C22×C31—21no—
159C22×C31—21no—
160C22×C31—41no—
161C22×C31—41no—
162C22×C31—41no—
163C22×C31—41no—
164C22×C31—41no—
165C22×C31—84no—
166C22×C31C22×D411yescharacteristic
167C4×C31—21no—
168C4×C31—21no—
169C4×C31—21no—
170C4×C31—21no—
171C4×C31—41no—
172C4×C31—41no—
173C4×C31—41no—
174C4×C31—41no—
175C4×C31—41no—
176C4×C31—41no—
177C4×C31C2×#4111yescharacteristic
178C4×C31H(5,𝔽2)11yescharacteristic
179#556C2×C3111yescharacteristic, complemented
180#546C2×C3111yescharacteristic
181#489C2×C3111yescharacteristic, complemented
182#485C2×C3111yescharacteristic, complemented
183#482C2×C3111yescharacteristic, complemented
184#476C2×C3111yescharacteristic, complemented
185#470C2×C3111yescharacteristic, complemented
186#465C2×C3111yescharacteristic, complemented
187#464C2×C3111yescharacteristic, complemented
188#452C2×C3111yescharacteristic, complemented
189#450C2×C3111yescharacteristic, complemented
190#440C2×C3111yescharacteristic, complemented
191#435C2×C3111yescharacteristic, complemented
192#418C2×C3111yescharacteristic, complemented
193#416C2×C3111yescharacteristic, complemented
194C2×C31—41no—
195C2×C31—41no—
196C2×C31—42no—
197C2×C31—81no—
198C2×C31#54611yescharacteristic
199C2×C31#57511yescharacteristic
200C2×C31#57711yescharacteristic
201C2×#41C22×C3111yescharacteristic
202C22×D4C22×C3111yescharacteristic
203C2×Q16C22×C3111yescharacteristic, complemented
204C2×SD16C22×C3111yescharacteristic
205C2×SD16C22×C3111yescharacteristic
206C2×M16C22×C3111yescharacteristic
207C22×C8C22×C3111yescharacteristic, complemented
208C4⋊Q8C22×C3111yescharacteristic, complemented
209#126—21no—
210#126—21no—
211#124—21no—
212#124C22×C3111yescharacteristic
213#123—21no—
214C22⋊Q8C22×C3111yescharacteristic
215C22⋊Q8C22×C3111yescharacteristic
216C22⋊Q8C22×C3111yescharacteristic, complemented
217#121—21no—
218#121C22×C3111yescharacteristic
219#120—21no—
220C4×Q8—21no—
221C4×Q8C22×C3111yescharacteristic, complemented
222C4×D4C22×C3111yescharacteristic
223C4×D4C22×C3111yescharacteristic
224#117C22×C3111yescharacteristic
225#117C22×C3111yescharacteristic, complemented
226C2×C22⋊C4—21no—
227#107C22×C3111yescharacteristic, complemented
228#106C22×C3111yescharacteristic, complemented
229#106C22×C3111yescharacteristic, complemented
230C4⋊C8C22×C3111yescharacteristic, complemented
231#103C22×C3111yescharacteristic, complemented
232#103C22×C3111yescharacteristic, complemented
233#103C22×C3111yescharacteristic, complemented
234#103C22×C3111yescharacteristic, complemented
235#102C22×C3111yescharacteristic
236#102C22×C3111yescharacteristic
237#102C22×C3111yescharacteristic
238#102C22×C3111yescharacteristic
239#102C22×C3111yescharacteristic
240#102C22×C3111yescharacteristic
241C22⋊C8C22×C3111yescharacteristic
242C22⋊C8C22×C3111yescharacteristic, complemented
243#97C22×C3111yescharacteristic, complemented
244C31#331111yescharacteristic, direct product factor, Sylow
245C24—21no—
246#41—22no—
247#41—41no—
248C2×Q8—21no—
249C2×Q8C23×C3111yescharacteristic
250C2×Q8C23×C3111yescharacteristic
251C2×D4—22no—
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—21no—
255C2×D4—41no—
256C2×D4D4×C3111yescharacteristic
257C2×D4C23×C3111yescharacteristic
258C2×D4C23×C3111yescharacteristic
259C22×C4—21no—
260C22×C4—21no—
261C22×C4C23×C3111yescharacteristic
262C22×C4C23×C3111yescharacteristic
263Q16—41no—
264SD16—22no—
265SD16—41no—
266M16—41no—
267C2×C8—41no—
268C2×C8C23×C3111yescharacteristic
269C2×C8C23×C3111yescharacteristic
270C2×C8C23×C3111yescharacteristic
271C2×C8C23×C3111yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4D4×C3111yescharacteristic
278C4⋊C4C23×C3111yescharacteristic
279C4⋊C4C23×C3111yescharacteristic
280C4⋊C4C23×C3111yescharacteristic
281C4⋊C4C23×C3111yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—21no—
288C22⋊C4—21no—
289C22⋊C4—21no—
290C22⋊C4—21no—
291C22⋊C4D4×C3111yescharacteristic
292C22⋊C4D4×C3111yescharacteristic
293C42—21no—
294C42—21no—
295C42—21no—
296C42C23×C3111yescharacteristic
297C23—21no—
298C23—21no—
299C23—42no—
300C23—42no—
301C23—81no—
302C23C2×D4×C3111yescharacteristic
303C23C2×D4×C3111yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—21no—
308Q8—42no—
309D4—21no—
310D4—21no—
311D4—21no—
312D4—41no—
313D4—42no—
314D4—42no—
315D4—81no—
316D4#41×C3112yes—
317C2×C4—21no—
318C2×C4—21no—
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—42no—
328C2×C4—81no—
329C2×C4C2×D4×C3111yescharacteristic
330C2×C4C2×D4×C3111yescharacteristic
331C2×C4C2×D4×C3111yescharacteristic
332C2×C4C2×D4×C3111yescharacteristic
333C2×C4C24×C3111yescharacteristic
334C8—21no—
335C8—21no—
336C8—21no—
337C8—21no—
338C22—21no—
339C22—21no—
340C22—22no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—84no—
351C22C22×D4×C3111yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—41no—
361C4—41no—
362C4C2×#41×C3111yescharacteristic
363C4C31×H(5,𝔽2)11yescharacteristic
364C2—41no—
365C2—41no—
366C2—42no—
367C2—81no—
368C2C31×#54611yescharacteristic
369C2C31×#57511yescharacteristic
370C2C31×#57711yescharacteristic
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×#556C2semidirect
3. C31×#489C2semidirect
4. C31×#485C2semidirect
5. C31×#482C2semidirect
6. C31×#476C2semidirect
7. C31×#470C2semidirect
8. C31×#465C2semidirect
9. C31×#464C2semidirect
10. C31×#452C2semidirect
11. C31×#450C2semidirect
12. C31×#440C2semidirect
13. C31×#435C2semidirect
14. C31×#418C2semidirect
15. C31×#416C2semidirect
18. C2×Q16×C31C22semidirect
22. C22×C8×C31C22semidirect
23. C31×C4⋊Q8C22semidirect
31. C31×C22⋊Q8C22semidirect
36. C4×Q8×C31C22semidirect
40. C31×#117C22semidirect
42. C31×#107C22semidirect
43. C31×#106C22semidirect
44. C31×#106C22semidirect
45. C31×C4⋊C8C22semidirect
46. C31×#103C22semidirect
47. C31×#103C22semidirect
48. C31×#103C22semidirect
49. C31×#103C22semidirect
57. C31×C22⋊C8C22semidirect
58. C31×#97C22semidirect
152. #3311C31direct (N × H)
179. #556C2×C31semidirect
181. #489C2×C31semidirect
182. #485C2×C31semidirect
183. #482C2×C31semidirect
184. #476C2×C31semidirect
185. #470C2×C31semidirect
186. #465C2×C31semidirect
187. #464C2×C31semidirect
188. #452C2×C31semidirect
189. #450C2×C31semidirect
190. #440C2×C31semidirect
191. #435C2×C31semidirect
192. #418C2×C31semidirect
193. #416C2×C31semidirect
203. C2×Q16C22×C31semidirect
207. C22×C8C22×C31semidirect
208. C4⋊Q8C22×C31semidirect
216. C22⋊Q8C22×C31semidirect
221. C4×Q8C22×C31semidirect
225. #117C22×C31semidirect
227. #107C22×C31semidirect
228. #106C22×C31semidirect
229. #106C22×C31semidirect
230. C4⋊C8C22×C31semidirect
231. #103C22×C31semidirect
232. #103C22×C31semidirect
233. #103C22×C31semidirect
234. #103C22×C31semidirect
242. C22⋊C8C22×C31semidirect
243. #97C22×C31semidirect
244. C31#3311direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#5562
C31×#5462
C31×#4892
C31×#4852
C31×#4822
C31×#4762
C31×#4702
C31×#4652
C31×#4642
C31×#4522
C31×#4502
C31×#4402
C31×#4352
C31×#4182
C31×#4162
#331131