Finite Group Databasebeta

← C31×#3344 #16843277

Subgroups — C31×#3344 #16843277

Order 3968.

344 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#554C211yesmaximal, characteristic
2C31×#552C211yesmaximal, characteristic
3C31×#501C211yesmaximal, characteristic, complemented
4C31×#485C211yesmaximal, characteristic, complemented
5C31×#484C211yesmaximal, characteristic, complemented
6C31×#483C211yesmaximal, characteristic, complemented
7C31×#483C211yesmaximal, characteristic, complemented
8C31×#479C211yesmaximal, characteristic, complemented
9C31×#475C211yesmaximal, characteristic, complemented
10C31×#470C211yesmaximal, characteristic, complemented
11C31×#467C211yesmaximal, characteristic, complemented
12C31×#441C211yesmaximal, characteristic, complemented
13C31×#435C211yesmaximal, characteristic
14C31×#428C211yesmaximal, characteristic, complemented
15C2×C31×#106C211yesmaximal, characteristic, complemented
16C2×Q16×C31C2211yescharacteristic
17C2×M16×C31C2211yescharacteristic
18C22×C8×C31C2211yescharacteristic
19C31×C4⋊Q8C2211yescharacteristic
20C31×#126—21no—
21C31×#125—21no—
22C31×#125—21no—
23C31×#125C2211yescharacteristic
24C31×#123—21no—
25C31×C22⋊Q8—21no—
26C31×C22⋊Q8—21no—
27C31×C22⋊Q8C2211yescharacteristic
28C31×C22⋊Q8C2211yescharacteristic
29C31×C22⋊Q8C2211yescharacteristic
30C31×C22⋊Q8C2211yescharacteristic
31C4×Q8×C31C2211yescharacteristic
32C4×Q8×C31C2211yescharacteristic
33C4×D4×C31—21no—
34C4×D4×C31C2211yescharacteristic
35C31×#117—21no—
36C31×#117C2211yescharacteristic
37C2×C4⋊C4×C31C2211yescharacteristic
38C2×C4⋊C4×C31C2211yescharacteristic
39C2×C4⋊C4×C31C2211yescharacteristic
40C31×#107C2211yescharacteristic
41C31×#107C2211yescharacteristic
42C31×#107C2211yescharacteristic
43C31×#106C2211yescharacteristic
44C31×#106C2211yescharacteristic
45C31×#106C2211yescharacteristic
46C31×#106C2211yescharacteristic
47C31×#106C2211yescharacteristic
48C31×#106C2211yescharacteristic
49C31×C4⋊C8C2211yescharacteristic
50C31×#103C2211yescharacteristic
51C31×#103C2211yescharacteristic
52C31×#103C2211yescharacteristic
53C31×#103C2211yescharacteristic
54C31×#103C2211yescharacteristic
55C31×#103C2211yescharacteristic
56C31×C22⋊C8C2211yescharacteristic
57C31×C22⋊C8C2211yescharacteristic
58C31×#97C2211yescharacteristic
59C2×Q8×C31—21no—
60C2×Q8×C31C2311yescharacteristic
61C2×Q8×C31C2311yescharacteristic
62C2×D4×C31D411yescharacteristic
63C22×C4×C31—21no—
64C22×C4×C31—21no—
65C22×C4×C31—21no—
66C22×C4×C31C2311yescharacteristic
67C22×C4×C31C2311yescharacteristic
68Q16×C31—22no—
69M16×C31—22no—
70C2×C8×C31—22no—
71C2×C8×C31C2311yescharacteristic
72C2×C8×C31C2311yescharacteristic
73C2×C8×C31C2311yescharacteristic
74C2×C8×C31C2311yescharacteristic
75C4⋊C4×C31—21no—
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×C31—21no—
86C4⋊C4×C31—21no—
87C4⋊C4×C31D411yescharacteristic
88C4⋊C4×C31C2311yescharacteristic
89C4⋊C4×C31C2311yescharacteristic
90C4⋊C4×C31C2311yescharacteristic
91C4⋊C4×C31C2311yescharacteristic
92C4⋊C4×C31C2311yescharacteristic
93C4⋊C4×C31C2311yescharacteristic
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×C31C2×D411yescharacteristic
106C23×C31C2×D411yescharacteristic
107Q8×C31—21no—
108Q8×C31—21no—
109Q8×C31—21no—
110Q8×C31—21no—
111Q8×C31—81no—
112D4×C31—22no—
113C2×C4×C31—21no—
114C2×C4×C31—21no—
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—42no—
126C2×C4×C31—81no—
127C2×C4×C31—81no—
128C2×C4×C31C2×D411yescharacteristic
129C2×C4×C31C2×D411yescharacteristic
130C2×C4×C31C2×D411yescharacteristic
131C2×C4×C31C2×D411yescharacteristic
132C2×C4×C31C2411yescharacteristic
133C8×C31—21no—
134C8×C31—21no—
135C8×C31—21no—
136C8×C31#4112yes—
137#3344C3111yesmaximal, characteristic, direct product factor, Sylow
138C22×C31—21no—
139C22×C31—21no—
140C22×C31—21no—
141C22×C31—21no—
142C22×C31—21no—
143C22×C31#13712yes—
144C22×C31C22×D411yescharacteristic
145C4×C31—21no—
146C4×C31—21no—
147C4×C31—21no—
148C4×C31—21no—
149C4×C31—41no—
150C4×C31—41no—
151C4×C31—41no—
152C4×C31—41no—
153C4×C31—41no—
154C4×C31—41no—
155C4×C31—41no—
156C4×C31—41no—
157C4×C31C2×#4111yescharacteristic
158C4×C312−1+411yescharacteristic
159#554C2×C3111yescharacteristic
160#552C2×C3111yescharacteristic
161#501C2×C3111yescharacteristic, complemented
162#485C2×C3111yescharacteristic, complemented
163#484C2×C3111yescharacteristic, complemented
164#483C2×C3111yescharacteristic, complemented
165#483C2×C3111yescharacteristic, complemented
166#479C2×C3111yescharacteristic, complemented
167#475C2×C3111yescharacteristic, complemented
168#470C2×C3111yescharacteristic, complemented
169#467C2×C3111yescharacteristic, complemented
170#441C2×C3111yescharacteristic, complemented
171#435C2×C3111yescharacteristic
172#428C2×C3111yescharacteristic, complemented
173C2×#106C2×C3111yescharacteristic, complemented
174C2×C31—22no—
175C2×C31—41no—
176C2×C31#54711yescharacteristic
177C2×C31C2×#13711yescharacteristic
178C2×C31#57711yescharacteristic
179C2×Q16C22×C3111yescharacteristic
180C2×M16C22×C3111yescharacteristic
181C22×C8C22×C3111yescharacteristic
182C4⋊Q8C22×C3111yescharacteristic
183#126—21no—
184#125—21no—
185#125—21no—
186#125C22×C3111yescharacteristic
187#123—21no—
188C22⋊Q8—21no—
189C22⋊Q8—21no—
190C22⋊Q8C22×C3111yescharacteristic
191C22⋊Q8C22×C3111yescharacteristic
192C22⋊Q8C22×C3111yescharacteristic
193C22⋊Q8C22×C3111yescharacteristic
194C4×Q8C22×C3111yescharacteristic
195C4×Q8C22×C3111yescharacteristic
196C4×D4—21no—
197C4×D4C22×C3111yescharacteristic
198#117—21no—
199#117C22×C3111yescharacteristic
200C2×C4⋊C4C22×C3111yescharacteristic
201C2×C4⋊C4C22×C3111yescharacteristic
202C2×C4⋊C4C22×C3111yescharacteristic
203#107C22×C3111yescharacteristic
204#107C22×C3111yescharacteristic
205#107C22×C3111yescharacteristic
206#106C22×C3111yescharacteristic
207#106C22×C3111yescharacteristic
208#106C22×C3111yescharacteristic
209#106C22×C3111yescharacteristic
210#106C22×C3111yescharacteristic
211#106C22×C3111yescharacteristic
212C4⋊C8C22×C3111yescharacteristic
213#103C22×C3111yescharacteristic
214#103C22×C3111yescharacteristic
215#103C22×C3111yescharacteristic
216#103C22×C3111yescharacteristic
217#103C22×C3111yescharacteristic
218#103C22×C3111yescharacteristic
219C22⋊C8C22×C3111yescharacteristic
220C22⋊C8C22×C3111yescharacteristic
221#97C22×C3111yescharacteristic
222C31#334411yescharacteristic, direct product factor, Sylow
223C2×Q8—21no—
224C2×Q8C23×C3111yescharacteristic
225C2×Q8C23×C3111yescharacteristic
226C2×D4D4×C3111yescharacteristic
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4C23×C3111yescharacteristic
231C22×C4C23×C3111yescharacteristic
232Q16—22no—
233M16—22no—
234C2×C8—22no—
235C2×C8C23×C3111yescharacteristic
236C2×C8C23×C3111yescharacteristic
237C2×C8C23×C3111yescharacteristic
238C2×C8C23×C3111yescharacteristic
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4D4×C3111yescharacteristic
252C4⋊C4C23×C3111yescharacteristic
253C4⋊C4C23×C3111yescharacteristic
254C4⋊C4C23×C3111yescharacteristic
255C4⋊C4C23×C3111yescharacteristic
256C4⋊C4C23×C3111yescharacteristic
257C4⋊C4C23×C3111yescharacteristic
258C22⋊C4—21no—
259C22⋊C4—21no—
260C22⋊C4—21no—
261C22⋊C4—21no—
262C22⋊C4—21no—
263C22⋊C4D4×C3111yescharacteristic
264C22⋊C4D4×C3111yescharacteristic
265C42—21no—
266C42—21no—
267C42—21no—
268C42C23×C3111yescharacteristic
269C23C2×D4×C3111yescharacteristic
270C23C2×D4×C3111yescharacteristic
271Q8—21no—
272Q8—21no—
273Q8—21no—
274Q8—21no—
275Q8—81no—
276D4—22no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—42no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4C2×D4×C3111yescharacteristic
293C2×C4C2×D4×C3111yescharacteristic
294C2×C4C2×D4×C3111yescharacteristic
295C2×C4C2×D4×C3111yescharacteristic
296C2×C4C24×C3111yescharacteristic
297C8—21no—
298C8—21no—
299C8—21no—
300C8#41×C3112yes—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22C31×#13712yes—
307C22C22×D4×C3111yescharacteristic
308C4—21no—
309C4—21no—
310C4—21no—
311C4—21no—
312C4—41no—
313C4—41no—
314C4—41no—
315C4—41no—
316C4—41no—
317C4—41no—
318C4—41no—
319C4—41no—
320C4C2×#41×C3111yescharacteristic
321C4C31×2−1+411yescharacteristic
322C2—22no—
323C2—41no—
324C2C31×#54711yescharacteristic
325C2C2×C31×#13711yescharacteristic
326C2C31×#57711yescharacteristic
subgroupG/H
center, socleC22×C31C22×D4
commutator, Frattini, cosocleC2×C4C24×C31
prime radicalC2×D4×C31D4
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
3. C31×#501C2semidirect
4. C31×#485C2semidirect
5. C31×#484C2semidirect
6. C31×#483C2semidirect
7. C31×#483C2semidirect
8. C31×#479C2semidirect
9. C31×#475C2semidirect
10. C31×#470C2semidirect
11. C31×#467C2semidirect
12. C31×#441C2semidirect
14. C31×#428C2semidirect
15. C2×C31×#106C2semidirect
137. #3344C31direct (N × H)
161. #501C2×C31semidirect
162. #485C2×C31semidirect
163. #484C2×C31semidirect
164. #483C2×C31semidirect
165. #483C2×C31semidirect
166. #479C2×C31semidirect
167. #475C2×C31semidirect
168. #470C2×C31semidirect
169. #467C2×C31semidirect
170. #441C2×C31semidirect
172. #428C2×C31semidirect
173. C2×#106C2×C31semidirect
222. C31#3344direct (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×#5522
C31×#5012
C31×#4852
C31×#4842
C31×#4832
C31×#4792
C31×#4752
C31×#4702
C31×#4672
C31×#4412
C31×#4352
C31×#4282
C2×C31×#1062
#334431