Finite Group Databasebeta

← C31×#3337 #16843270

Subgroups — C31×#3337 #16843270

Order 3968.

354 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#553C211yesmaximal, characteristic
2C31×#552C211yesmaximal, characteristic, complemented
3C31×#499C211yesmaximal, characteristic, complemented
4C31×#485C211yesmaximal, characteristic, complemented
5C31×#484C211yesmaximal, characteristic, complemented
6C31×#482C211yesmaximal, characteristic, complemented
7C31×#480C211yesmaximal, characteristic, complemented
8C31×#479C211yesmaximal, characteristic, complemented
9C31×#478C211yesmaximal, characteristic, complemented
10C31×#465C211yesmaximal, characteristic, complemented
11C31×#465C211yesmaximal, characteristic, complemented
12C4×SD16×C31C211yesmaximal, characteristic, complemented
13C8×D4×C31C211yesmaximal, characteristic, complemented
14C31×#427C211yesmaximal, characteristic, complemented
15C2×C31×#106C211yesmaximal, characteristic, complemented
16C2×SD16×C31C2211yescharacteristic
17C22×C8×C31C2211yescharacteristic
18C22×C8×C31C2211yescharacteristic
19C31×#126—21no—
20C31×#126—21no—
21C31×#125—21no—
22C31×#125C2211yescharacteristic
23C31×#125C2211yescharacteristic
24C31×#123—21no—
25C31×#123—21no—
26C31×C22⋊Q8C2211yescharacteristic
27C31×C22⋊Q8C2211yescharacteristic
28C31×#121—21no—
29C31×#121C2211yescharacteristic
30C31×#121C2211yescharacteristic
31C4×Q8×C31C2211yescharacteristic
32C4×D4×C31—21no—
33C4×D4×C31C2211yescharacteristic
34C4×D4×C31C2211yescharacteristic
35C31×#117—21no—
36C31×#117C2211yescharacteristic
37C31×#117C2211yescharacteristic
38C2×C4⋊C4×C31C2211yescharacteristic
39C2×C4⋊C4×C31C2211yescharacteristic
40C31×#107C2211yescharacteristic
41C31×#107C2211yescharacteristic
42C31×#107C2211yescharacteristic
43C31×#107C2211yescharacteristic
44C31×#106C2211yescharacteristic
45C31×#106C2211yescharacteristic
46C31×#106C2211yescharacteristic
47C31×#106C2211yescharacteristic
48C31×#106C2211yescharacteristic
49C31×C4⋊C8C2211yescharacteristic
50C31×#103C2211yescharacteristic
51C31×#103C2211yescharacteristic
52C31×#103C2211yescharacteristic
53C31×#102C2211yescharacteristic
54C31×#102C2211yescharacteristic
55C31×#102C2211yescharacteristic
56C31×C22⋊C8C2211yescharacteristic
57C31×C22⋊C8C2211yescharacteristic
58C4×C8×C31C2211yescharacteristic
59C2×Q8×C31C2311yescharacteristic
60C2×D4×C31—21no—
61C2×D4×C31—21no—
62C2×D4×C31D411yescharacteristic
63C2×D4×C31C2311yescharacteristic
64C22×C4×C31—21no—
65C22×C4×C31—21no—
66C22×C4×C31—21no—
67C22×C4×C31C2311yescharacteristic
68C22×C4×C31C2311yescharacteristic
69SD16×C31—22no—
70C2×C8×C31—22no—
71C2×C8×C31—22no—
72C2×C8×C31C2311yescharacteristic
73C2×C8×C31C2311yescharacteristic
74C2×C8×C31C2311yescharacteristic
75C2×C8×C31C2311yescharacteristic
76C4⋊C4×C31—21no—
77C4⋊C4×C31—21no—
78C4⋊C4×C31—21no—
79C4⋊C4×C31—21no—
80C4⋊C4×C31—21no—
81C4⋊C4×C31—21no—
82C4⋊C4×C31—21no—
83C4⋊C4×C31—21no—
84C4⋊C4×C31—21no—
85C4⋊C4×C31D411yescharacteristic
86C4⋊C4×C31C2311yescharacteristic
87C4⋊C4×C31C2311yescharacteristic
88C4⋊C4×C31C2311yescharacteristic
89C4⋊C4×C31C2311yescharacteristic
90C4⋊C4×C31C2311yescharacteristic
91C4⋊C4×C31C2311yescharacteristic
92C22⋊C4×C31—21no—
93C22⋊C4×C31—21no—
94C22⋊C4×C31—21no—
95C22⋊C4×C31—21no—
96C22⋊C4×C31—21no—
97C22⋊C4×C31—21no—
98C22⋊C4×C31—21no—
99C22⋊C4×C31D411yescharacteristic
100C22⋊C4×C31D411yescharacteristic
101C42×C31—21no—
102C42×C31—21no—
103C42×C31—21no—
104C42×C31C2311yescharacteristic
105C23×C31—21no—
106C23×C31C2×D411yescharacteristic
107C23×C31C2×D411yescharacteristic
108Q8×C31—21no—
109Q8×C31—21no—
110D4×C31—22no—
111D4×C31—21no—
112D4×C31—21no—
113D4×C31—42no—
114D4×C31—81no—
115C2×C4×C31—21no—
116C2×C4×C31—21no—
117C2×C4×C31—21no—
118C2×C4×C31—21no—
119C2×C4×C31—21no—
120C2×C4×C31—21no—
121C2×C4×C31—21no—
122C2×C4×C31—21no—
123C2×C4×C31—21no—
124C2×C4×C31—21no—
125C2×C4×C31—21no—
126C2×C4×C31—42no—
127C2×C4×C31—81no—
128C2×C4×C31—81no—
129C2×C4×C31C2×D411yescharacteristic
130C2×C4×C31C2×D411yescharacteristic
131C2×C4×C31C2×D411yescharacteristic
132C2×C4×C31C2×D411yescharacteristic
133C2×C4×C31C2411yescharacteristic
134C8×C31—21no—
135C8×C31—21no—
136C8×C31—21no—
137C8×C31#4112yes—
138#3337C3111yesmaximal, characteristic, direct product factor, Sylow
139C22×C31—21no—
140C22×C31—21no—
141C22×C31—21no—
142C22×C31—21no—
143C22×C31—21no—
144C22×C31—41no—
145C22×C31—41no—
146C22×C31—41no—
147C22×C31#13512yes—
148C22×C31C22×D411yescharacteristic
149C4×C31—21no—
150C4×C31—21no—
151C4×C31—21no—
152C4×C31—21no—
153C4×C31—41no—
154C4×C31—41no—
155C4×C31—41no—
156C4×C31—41no—
157C4×C31—41no—
158C4×C31—41no—
159C4×C31—41no—
160C4×C31C2×#4111yescharacteristic
161C4×C312−1+411yescharacteristic
162#553C2×C3111yescharacteristic
163#552C2×C3111yescharacteristic, complemented
164#499C2×C3111yescharacteristic, complemented
165#485C2×C3111yescharacteristic, complemented
166#484C2×C3111yescharacteristic, complemented
167#482C2×C3111yescharacteristic, complemented
168#480C2×C3111yescharacteristic, complemented
169#479C2×C3111yescharacteristic, complemented
170#478C2×C3111yescharacteristic, complemented
171#465C2×C3111yescharacteristic, complemented
172#465C2×C3111yescharacteristic, complemented
173C4×SD16C2×C3111yescharacteristic, complemented
174C8×D4C2×C3111yescharacteristic, complemented
175#427C2×C3111yescharacteristic, complemented
176C2×#106C2×C3111yescharacteristic, complemented
177C2×C31—22no—
178C2×C31—41no—
179C2×C31—81no—
180C2×C31#54711yescharacteristic
181C2×C31C2×#13511yescharacteristic
182C2×C31#57711yescharacteristic
183C2×SD16C22×C3111yescharacteristic
184C22×C8C22×C3111yescharacteristic
185C22×C8C22×C3111yescharacteristic
186#126—21no—
187#126—21no—
188#125—21no—
189#125C22×C3111yescharacteristic
190#125C22×C3111yescharacteristic
191#123—21no—
192#123—21no—
193C22⋊Q8C22×C3111yescharacteristic
194C22⋊Q8C22×C3111yescharacteristic
195#121—21no—
196#121C22×C3111yescharacteristic
197#121C22×C3111yescharacteristic
198C4×Q8C22×C3111yescharacteristic
199C4×D4—21no—
200C4×D4C22×C3111yescharacteristic
201C4×D4C22×C3111yescharacteristic
202#117—21no—
203#117C22×C3111yescharacteristic
204#117C22×C3111yescharacteristic
205C2×C4⋊C4C22×C3111yescharacteristic
206C2×C4⋊C4C22×C3111yescharacteristic
207#107C22×C3111yescharacteristic
208#107C22×C3111yescharacteristic
209#107C22×C3111yescharacteristic
210#107C22×C3111yescharacteristic
211#106C22×C3111yescharacteristic
212#106C22×C3111yescharacteristic
213#106C22×C3111yescharacteristic
214#106C22×C3111yescharacteristic
215#106C22×C3111yescharacteristic
216C4⋊C8C22×C3111yescharacteristic
217#103C22×C3111yescharacteristic
218#103C22×C3111yescharacteristic
219#103C22×C3111yescharacteristic
220#102C22×C3111yescharacteristic
221#102C22×C3111yescharacteristic
222#102C22×C3111yescharacteristic
223C22⋊C8C22×C3111yescharacteristic
224C22⋊C8C22×C3111yescharacteristic
225C4×C8C22×C3111yescharacteristic
226C31#333711yescharacteristic, direct product factor, Sylow
227C2×Q8C23×C3111yescharacteristic
228C2×D4—21no—
229C2×D4—21no—
230C2×D4D4×C3111yescharacteristic
231C2×D4C23×C3111yescharacteristic
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4C23×C3111yescharacteristic
236C22×C4C23×C3111yescharacteristic
237SD16—22no—
238C2×C8—22no—
239C2×C8—22no—
240C2×C8C23×C3111yescharacteristic
241C2×C8C23×C3111yescharacteristic
242C2×C8C23×C3111yescharacteristic
243C2×C8C23×C3111yescharacteristic
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4D4×C3111yescharacteristic
254C4⋊C4C23×C3111yescharacteristic
255C4⋊C4C23×C3111yescharacteristic
256C4⋊C4C23×C3111yescharacteristic
257C4⋊C4C23×C3111yescharacteristic
258C4⋊C4C23×C3111yescharacteristic
259C4⋊C4C23×C3111yescharacteristic
260C22⋊C4—21no—
261C22⋊C4—21no—
262C22⋊C4—21no—
263C22⋊C4—21no—
264C22⋊C4—21no—
265C22⋊C4—21no—
266C22⋊C4—21no—
267C22⋊C4D4×C3111yescharacteristic
268C22⋊C4D4×C3111yescharacteristic
269C42—21no—
270C42—21no—
271C42—21no—
272C42C23×C3111yescharacteristic
273C23—21no—
274C23C2×D4×C3111yescharacteristic
275C23C2×D4×C3111yescharacteristic
276Q8—21no—
277Q8—21no—
278D4—22no—
279D4—21no—
280D4—21no—
281D4—42no—
282D4—81no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—42no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4C2×D4×C3111yescharacteristic
298C2×C4C2×D4×C3111yescharacteristic
299C2×C4C2×D4×C3111yescharacteristic
300C2×C4C2×D4×C3111yescharacteristic
301C2×C4C24×C3111yescharacteristic
302C8—21no—
303C8—21no—
304C8—21no—
305C8#41×C3112yes—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22C31×#13512yes—
315C22C22×D4×C3111yescharacteristic
316C4—21no—
317C4—21no—
318C4—21no—
319C4—21no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—41no—
325C4—41no—
326C4—41no—
327C4C2×#41×C3111yescharacteristic
328C4C31×2−1+411yescharacteristic
329C2—22no—
330C2—41no—
331C2—81no—
332C2C31×#54711yescharacteristic
333C2C2×C31×#13511yescharacteristic
334C2C31×#57711yescharacteristic
subgroupG/H
center, socleC22×C31C22×D4
commutator, Frattini, cosocleC2×C4C24×C31
prime radicalC31×#553C2
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
2. C31×#552C2semidirect
3. C31×#499C2semidirect
4. C31×#485C2semidirect
5. C31×#484C2semidirect
6. C31×#482C2semidirect
7. C31×#480C2semidirect
8. C31×#479C2semidirect
9. C31×#478C2semidirect
10. C31×#465C2semidirect
11. C31×#465C2semidirect
12. C4×SD16×C31C2semidirect
13. C8×D4×C31C2semidirect
14. C31×#427C2semidirect
15. C2×C31×#106C2semidirect
138. #3337C31direct (N × H)
163. #552C2×C31semidirect
164. #499C2×C31semidirect
165. #485C2×C31semidirect
166. #484C2×C31semidirect
167. #482C2×C31semidirect
168. #480C2×C31semidirect
169. #479C2×C31semidirect
170. #478C2×C31semidirect
171. #465C2×C31semidirect
172. #465C2×C31semidirect
173. C4×SD16C2×C31semidirect
174. C8×D4C2×C31semidirect
175. #427C2×C31semidirect
176. C2×#106C2×C31semidirect
226. C31#3337direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#5532
C31×#5522
C31×#4992
C31×#4852
C31×#4842
C31×#4822
C31×#4802
C31×#4792
C31×#4782
C31×#4652
C4×SD16×C312
C8×D4×C312
C31×#4272
C2×C31×#1062
#333731