Finite Group Databasebeta

← C31×#3120 #16843053

Subgroups — C31×#3120 #16843053

Order 3968.

372 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#541C211yesmaximal, characteristic, complemented
2C31×#520C211yesmaximal, characteristic
3C31×#484C211yesmaximal, characteristic, complemented
4C31×#483C211yesmaximal, characteristic, complemented
5C31×#479C211yesmaximal, characteristic, complemented
6C31×#478C211yesmaximal, characteristic, complemented
7C31×#462C211yesmaximal, characteristic, complemented
8C31×#461C211yesmaximal, characteristic, complemented
9C31×#452C211yesmaximal, characteristic
10C31×#452C211yesmaximal, characteristic
11C31×#441C211yesmaximal, characteristic, complemented
12C31×#440C211yesmaximal, characteristic, complemented
13C4×Q16×C31C211yesmaximal, characteristic, complemented
14C4×SD16×C31C211yesmaximal, characteristic, complemented
15C31×#433C211yesmaximal, characteristic, complemented
16C2×#41×C31C2211yescharacteristic
17C2×Q16×C31C2211yescharacteristic
18C2×Q16×C31C2211yescharacteristic
19C2×SD16×C31C2211yescharacteristic
20C2×SD16×C31C2211yescharacteristic
21C31×C4⋊Q8C2211yescharacteristic
22C31×#126—21no—
23C31×#126—21no—
24C31×#125—21no—
25C31×#125—21no—
26C31×#125C2211yescharacteristic
27C31×C22⋊Q8C2211yescharacteristic
28C31×C22⋊Q8C2211yescharacteristic
29C4×Q8×C31C2211yescharacteristic
30C4×Q8×C31C2211yescharacteristic
31C4×Q8×C31C2211yescharacteristic
32C4×Q8×C31C2211yescharacteristic
33C4×D4×C31—21no—
34C4×D4×C31—21no—
35C4×D4×C31C2211yescharacteristic
36C4×D4×C31C2211yescharacteristic
37C31×#117—21no—
38C31×#117C2211yescharacteristic
39C2×C4⋊C4×C31—21no—
40C2×C4⋊C4×C31C2211yescharacteristic
41C31×#107C2211yescharacteristic
42C31×#107C2211yescharacteristic
43C31×#106C2211yescharacteristic
44C31×#106C2211yescharacteristic
45C31×C4⋊C8C2211yescharacteristic
46C31×C4⋊C8C2211yescharacteristic
47C31×#103C2211yescharacteristic
48C31×#103C2211yescharacteristic
49C31×#103C2211yescharacteristic
50C31×#103C2211yescharacteristic
51C31×#103C2211yescharacteristic
52C31×#103C2211yescharacteristic
53C31×#102C2211yescharacteristic
54C31×#102C2211yescharacteristic
55C31×C22⋊C8C2211yescharacteristic
56C31×C22⋊C8C2211yescharacteristic
57C31×#97C2211yescharacteristic
58C4×C8×C31C2211yescharacteristic
59#41×C31—22no—
60#41×C31—22no—
61C2×Q8×C31C2311yescharacteristic
62C2×Q8×C31C2311yescharacteristic
63C2×Q8×C31C2311yescharacteristic
64C2×D4×C31—21no—
65C2×D4×C31C2311yescharacteristic
66C22×C4×C31—21no—
67C22×C4×C31—21no—
68C22×C4×C31—21no—
69C22×C4×C31—21no—
70C22×C4×C31C2311yescharacteristic
71Q16×C31—22no—
72Q16×C31—22no—
73SD16×C31—22no—
74SD16×C31—22no—
75C2×C8×C31C2311yescharacteristic
76C2×C8×C31C2311yescharacteristic
77C2×C8×C31C2311yescharacteristic
78C2×C8×C31C2311yescharacteristic
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×C31—21no—
86C4⋊C4×C31—21no—
87C4⋊C4×C31—21no—
88C4⋊C4×C31—21no—
89C4⋊C4×C31—21no—
90C4⋊C4×C31D411yescharacteristic
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×C31D411yescharacteristic
101C22⋊C4×C31D411yescharacteristic
102C42×C31—21no—
103C42×C31—21no—
104C42×C31—21no—
105C42×C31—21no—
106C42×C31C2311yescharacteristic
107C42×C31C2311yescharacteristic
108C23×C31—21no—
109C23×C31C2×D411yescharacteristic
110Q8×C31—21no—
111Q8×C31—21no—
112Q8×C31—21no—
113Q8×C31—21no—
114Q8×C31—21no—
115Q8×C31#4112yes—
116D4×C31—24no—
117D4×C31—21no—
118D4×C31#4112yes—
119C2×C4×C31—24no—
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—21no—
127C2×C4×C31—21no—
128C2×C4×C31—21no—
129C2×C4×C31—81no—
130C2×C4×C31—81no—
131C2×C4×C31—81no—
132C2×C4×C31C2×D411yescharacteristic
133C2×C4×C31C2×D411yescharacteristic
134C2×C4×C31C2×D411yescharacteristic
135C2×C4×C31C2×D411yescharacteristic
136C2×C4×C31C2×D411yescharacteristic
137C2×C4×C31C2411yescharacteristic
138C8×C31—21no—
139C8×C31—21no—
140C8×C31—21no—
141C8×C31—21no—
142#3120C3111yesmaximal, characteristic, direct product factor, Sylow
143C22×C31—21no—
144C22×C31—22no—
145C22×C31—21no—
146C22×C31—21no—
147C22×C31—41no—
148C22×C31—41no—
149C22×C31C22×D411yescharacteristic
150C4×C31—21no—
151C4×C31—22no—
152C4×C31—21no—
153C4×C31—21no—
154C4×C31—21no—
155C4×C31—21no—
156C4×C31—41no—
157C4×C31—41no—
158C4×C31—41no—
159C4×C31—41no—
160C4×C31—41no—
161C4×C31—41no—
162C4×C31C2×#4111yescharacteristic
163C4×C31C2×#4111yescharacteristic
164#541C2×C3111yescharacteristic, complemented
165#520C2×C3111yescharacteristic
166#484C2×C3111yescharacteristic, complemented
167#483C2×C3111yescharacteristic, complemented
168#479C2×C3111yescharacteristic, complemented
169#478C2×C3111yescharacteristic, complemented
170#462C2×C3111yescharacteristic, complemented
171#461C2×C3111yescharacteristic, complemented
172#452C2×C3111yescharacteristic
173#452C2×C3111yescharacteristic
174#441C2×C3111yescharacteristic, complemented
175#440C2×C3111yescharacteristic, complemented
176C4×Q16C2×C3111yescharacteristic, complemented
177C4×SD16C2×C3111yescharacteristic, complemented
178#433C2×C3111yescharacteristic, complemented
179C2×C31—41no—
180C2×C31—42no—
181C2×C31#52511yescharacteristic
182C2×C31#57711yescharacteristic
183C2×C31#57811yescharacteristic
184C2×#41C22×C3111yescharacteristic
185C2×Q16C22×C3111yescharacteristic
186C2×Q16C22×C3111yescharacteristic
187C2×SD16C22×C3111yescharacteristic
188C2×SD16C22×C3111yescharacteristic
189C4⋊Q8C22×C3111yescharacteristic
190#126—21no—
191#126—21no—
192#125—21no—
193#125—21no—
194#125C22×C3111yescharacteristic
195C22⋊Q8C22×C3111yescharacteristic
196C22⋊Q8C22×C3111yescharacteristic
197C4×Q8C22×C3111yescharacteristic
198C4×Q8C22×C3111yescharacteristic
199C4×Q8C22×C3111yescharacteristic
200C4×Q8C22×C3111yescharacteristic
201C4×D4—21no—
202C4×D4—21no—
203C4×D4C22×C3111yescharacteristic
204C4×D4C22×C3111yescharacteristic
205#117—21no—
206#117C22×C3111yescharacteristic
207C2×C4⋊C4—21no—
208C2×C4⋊C4C22×C3111yescharacteristic
209#107C22×C3111yescharacteristic
210#107C22×C3111yescharacteristic
211#106C22×C3111yescharacteristic
212#106C22×C3111yescharacteristic
213C4⋊C8C22×C3111yescharacteristic
214C4⋊C8C22×C3111yescharacteristic
215#103C22×C3111yescharacteristic
216#103C22×C3111yescharacteristic
217#103C22×C3111yescharacteristic
218#103C22×C3111yescharacteristic
219#103C22×C3111yescharacteristic
220#103C22×C3111yescharacteristic
221#102C22×C3111yescharacteristic
222#102C22×C3111yescharacteristic
223C22⋊C8C22×C3111yescharacteristic
224C22⋊C8C22×C3111yescharacteristic
225#97C22×C3111yescharacteristic
226C4×C8C22×C3111yescharacteristic
227C31#312011yescharacteristic, direct product factor, Sylow
228#41—22no—
229#41—22no—
230C2×Q8C23×C3111yescharacteristic
231C2×Q8C23×C3111yescharacteristic
232C2×Q8C23×C3111yescharacteristic
233C2×D4—21no—
234C2×D4C23×C3111yescharacteristic
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4C23×C3111yescharacteristic
240Q16—22no—
241Q16—22no—
242SD16—22no—
243SD16—22no—
244C2×C8C23×C3111yescharacteristic
245C2×C8C23×C3111yescharacteristic
246C2×C8C23×C3111yescharacteristic
247C2×C8C23×C3111yescharacteristic
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4D4×C3111yescharacteristic
260C4⋊C4D4×C3111yescharacteristic
261C4⋊C4C23×C3111yescharacteristic
262C4⋊C4C23×C3111yescharacteristic
263C4⋊C4C23×C3111yescharacteristic
264C4⋊C4C23×C3111yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—21no—
267C22⋊C4—21no—
268C22⋊C4—21no—
269C22⋊C4D4×C3111yescharacteristic
270C22⋊C4D4×C3111yescharacteristic
271C42—21no—
272C42—21no—
273C42—21no—
274C42—21no—
275C42C23×C3111yescharacteristic
276C42C23×C3111yescharacteristic
277C23—21no—
278C23C2×D4×C3111yescharacteristic
279Q8—21no—
280Q8—21no—
281Q8—21no—
282Q8—21no—
283Q8—21no—
284Q8#41×C3112yes—
285D4—24no—
286D4—21no—
287D4#41×C3112yes—
288C2×C4—24no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4C2×D4×C3111yescharacteristic
302C2×C4C2×D4×C3111yescharacteristic
303C2×C4C2×D4×C3111yescharacteristic
304C2×C4C2×D4×C3111yescharacteristic
305C2×C4C2×D4×C3111yescharacteristic
306C2×C4C24×C3111yescharacteristic
307C8—21no—
308C8—21no—
309C8—21no—
310C8—21no—
311C22—21no—
312C22—22no—
313C22—21no—
314C22—21no—
315C22—41no—
316C22—41no—
317C22C22×D4×C3111yescharacteristic
318C4—21no—
319C4—22no—
320C4—21no—
321C4—21no—
322C4—21no—
323C4—21no—
324C4—41no—
325C4—41no—
326C4—41no—
327C4—41no—
328C4—41no—
329C4—41no—
330C4C2×#41×C3111yescharacteristic
331C4C2×#41×C3111yescharacteristic
332C2—41no—
333C2—42no—
334C2C31×#52511yescharacteristic
335C2C31×#57711yescharacteristic
336C2C31×#57811yescharacteristic
subgroupG/H
center, socleC22×C31C22×D4
commutator, Frattini, cosocleC2×C4C24×C31
prime radicalC2×#41×C31C22
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×#541C2semidirect
3. C31×#484C2semidirect
4. C31×#483C2semidirect
5. C31×#479C2semidirect
6. C31×#478C2semidirect
7. C31×#462C2semidirect
8. C31×#461C2semidirect
11. C31×#441C2semidirect
12. C31×#440C2semidirect
13. C4×Q16×C31C2semidirect
14. C4×SD16×C31C2semidirect
15. C31×#433C2semidirect
142. #3120C31direct (N × H)
164. #541C2×C31semidirect
166. #484C2×C31semidirect
167. #483C2×C31semidirect
168. #479C2×C31semidirect
169. #478C2×C31semidirect
170. #462C2×C31semidirect
171. #461C2×C31semidirect
174. #441C2×C31semidirect
175. #440C2×C31semidirect
176. C4×Q16C2×C31semidirect
177. C4×SD16C2×C31semidirect
178. #433C2×C31semidirect
227. C31#3120direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#5412
C31×#5202
C31×#4842
C31×#4832
C31×#4792
C31×#4782
C31×#4622
C31×#4612
C31×#4522
C31×#4412
C31×#4402
C4×Q16×C312
C4×SD16×C312
C31×#4332
#312031