Finite Group Databasebeta

← C31×#3327 #16843260

Subgroups — C31×#3327 #16843260

Order 3968.

380 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#552C211yesmaximal, characteristic, complemented
2C31×#548C211yesmaximal, characteristic
3C31×#488C211yesmaximal, characteristic, complemented
4C31×#484C211yesmaximal, characteristic, complemented
5C31×#482C211yesmaximal, characteristic, complemented
6C31×#479C211yesmaximal, characteristic, complemented
7C31×#470C211yesmaximal, characteristic, complemented
8C31×#468C211yesmaximal, characteristic, complemented
9C31×#464C211yesmaximal, characteristic, complemented
10C31×#452C211yesmaximal, characteristic, complemented
11C31×#451C211yesmaximal, characteristic, complemented
12C4×Q16×C31C211yesmaximal, characteristic, complemented
13C31×#436C211yesmaximal, characteristic, complemented
14C31×#419C211yesmaximal, characteristic, complemented
15C31×#417C211yesmaximal, characteristic, complemented
16C2×#41×C31C2211yescharacteristic
17C22×Q8×C31C2211yescharacteristic
18C2×Q16×C31C2211yescharacteristic
19C2×Q16×C31C2211yescharacteristic
20C2×SD16×C31C2211yescharacteristic
21C2×M16×C31C2211yescharacteristic
22C2×M16×C31C2211yescharacteristic
23C31×#126—21no—
24C31×#125—21no—
25C31×#125C2211yescharacteristic
26C31×#124—21no—
27C31×#124C2211yescharacteristic
28C31×#123—21no—
29C31×C22⋊Q8—21no—
30C31×C22⋊Q8C2211yescharacteristic
31C31×C22⋊Q8C2211yescharacteristic
32C31×C22⋊Q8C2211yescharacteristic
33C31×#121—21no—
34C31×#121C2211yescharacteristic
35C4×Q8×C31C2211yescharacteristic
36C4×Q8×C31C2211yescharacteristic
37C4×D4×C31—21no—
38C4×D4×C31C2211yescharacteristic
39C31×#117—21no—
40C31×#117C2211yescharacteristic
41C2×C4⋊C4×C31C2211yescharacteristic
42C31×#107C2211yescharacteristic
43C31×#107C2211yescharacteristic
44C31×#106C2211yescharacteristic
45C31×C4⋊C8C2211yescharacteristic
46C31×#103C2211yescharacteristic
47C31×#103C2211yescharacteristic
48C31×#103C2211yescharacteristic
49C31×#103C2211yescharacteristic
50C31×#103C2211yescharacteristic
51C31×#103C2211yescharacteristic
52C31×#103C2211yescharacteristic
53C31×#102C2211yescharacteristic
54C31×#102C2211yescharacteristic
55C31×#102C2211yescharacteristic
56C31×C22⋊C8C2211yescharacteristic
57C31×C22⋊C8C2211yescharacteristic
58C4×C8×C31C2211yescharacteristic
59#41×C31—22no—
60#41×C31—41no—
61C2×Q8×C31—21no—
62C2×Q8×C31—22no—
63C2×Q8×C31—41no—
64C2×Q8×C31C2311yescharacteristic
65C2×Q8×C31C2311yescharacteristic
66C2×Q8×C31C2311yescharacteristic
67C2×D4×C31—21no—
68C2×D4×C31—21no—
69C2×D4×C31D411yescharacteristic
70C2×D4×C31C2311yescharacteristic
71C22×C4×C31—21no—
72C22×C4×C31—21no—
73C22×C4×C31—21no—
74C22×C4×C31C2311yescharacteristic
75C22×C4×C31C2311yescharacteristic
76Q16×C31—22no—
77Q16×C31—41no—
78SD16×C31—41no—
79M16×C31—41no—
80M16×C31—41no—
81C2×C8×C31C2311yescharacteristic
82C2×C8×C31C2311yescharacteristic
83C2×C8×C31C2311yescharacteristic
84C2×C8×C31C2311yescharacteristic
85C4⋊C4×C31—21no—
86C4⋊C4×C31—21no—
87C4⋊C4×C31—21no—
88C4⋊C4×C31—21no—
89C4⋊C4×C31—21no—
90C4⋊C4×C31—21no—
91C4⋊C4×C31—21no—
92C4⋊C4×C31—21no—
93C4⋊C4×C31D411yescharacteristic
94C4⋊C4×C31C2311yescharacteristic
95C4⋊C4×C31C2311yescharacteristic
96C4⋊C4×C31C2311yescharacteristic
97C4⋊C4×C31C2311yescharacteristic
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×C31C2×D411yescharacteristic
113C23×C31C2×D411yescharacteristic
114Q8×C31—21no—
115Q8×C31—21no—
116Q8×C31—21no—
117Q8×C31—21no—
118Q8×C31—21no—
119Q8×C31—42no—
120Q8×C31#4112yes—
121D4×C31—21no—
122D4×C31—21no—
123D4×C31—41no—
124D4×C31—42no—
125D4×C31—42no—
126C2×C4×C31—21no—
127C2×C4×C31—21no—
128C2×C4×C31—21no—
129C2×C4×C31—21no—
130C2×C4×C31—21no—
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—42no—
138C2×C4×C31—42no—
139C2×C4×C31—42no—
140C2×C4×C31C2×D411yescharacteristic
141C2×C4×C31C2×D411yescharacteristic
142C2×C4×C31C2×D411yescharacteristic
143C2×C4×C31C2×D411yescharacteristic
144C2×C4×C31C2411yescharacteristic
145C8×C31—21no—
146C8×C31—21no—
147C8×C31—21no—
148C8×C31—21no—
149#3327C3111yesmaximal, characteristic, direct product factor, Sylow
150C22×C31—21no—
151C22×C31—21no—
152C22×C31—21no—
153C22×C31—21no—
154C22×C31—21no—
155C22×C31—21no—
156C22×C31—41no—
157C22×C31—41no—
158C22×C31—41no—
159C22×C31C22×D411yescharacteristic
160C4×C31—21no—
161C4×C31—21no—
162C4×C31—22no—
163C4×C31—21no—
164C4×C31—21no—
165C4×C31—41no—
166C4×C31—41no—
167C4×C31—41no—
168C4×C31—41no—
169C4×C31—41no—
170C4×C31—41no—
171C4×C31C2×#4111yescharacteristic
172C4×C31H(5,𝔽2)11yescharacteristic
173#552C2×C3111yescharacteristic, complemented
174#548C2×C3111yescharacteristic
175#488C2×C3111yescharacteristic, complemented
176#484C2×C3111yescharacteristic, complemented
177#482C2×C3111yescharacteristic, complemented
178#479C2×C3111yescharacteristic, complemented
179#470C2×C3111yescharacteristic, complemented
180#468C2×C3111yescharacteristic, complemented
181#464C2×C3111yescharacteristic, complemented
182#452C2×C3111yescharacteristic, complemented
183#451C2×C3111yescharacteristic, complemented
184C4×Q16C2×C3111yescharacteristic, complemented
185#436C2×C3111yescharacteristic, complemented
186#419C2×C3111yescharacteristic, complemented
187#417C2×C3111yescharacteristic, complemented
188C2×C31—41no—
189C2×C31—41no—
190C2×C31—81no—
191C2×C31#54611yescharacteristic
192C2×C31#57511yescharacteristic
193C2×C31#57811yescharacteristic
194C2×#41C22×C3111yescharacteristic
195C22×Q8C22×C3111yescharacteristic
196C2×Q16C22×C3111yescharacteristic
197C2×Q16C22×C3111yescharacteristic
198C2×SD16C22×C3111yescharacteristic
199C2×M16C22×C3111yescharacteristic
200C2×M16C22×C3111yescharacteristic
201#126—21no—
202#125—21no—
203#125C22×C3111yescharacteristic
204#124—21no—
205#124C22×C3111yescharacteristic
206#123—21no—
207C22⋊Q8—21no—
208C22⋊Q8C22×C3111yescharacteristic
209C22⋊Q8C22×C3111yescharacteristic
210C22⋊Q8C22×C3111yescharacteristic
211#121—21no—
212#121C22×C3111yescharacteristic
213C4×Q8C22×C3111yescharacteristic
214C4×Q8C22×C3111yescharacteristic
215C4×D4—21no—
216C4×D4C22×C3111yescharacteristic
217#117—21no—
218#117C22×C3111yescharacteristic
219C2×C4⋊C4C22×C3111yescharacteristic
220#107C22×C3111yescharacteristic
221#107C22×C3111yescharacteristic
222#106C22×C3111yescharacteristic
223C4⋊C8C22×C3111yescharacteristic
224#103C22×C3111yescharacteristic
225#103C22×C3111yescharacteristic
226#103C22×C3111yescharacteristic
227#103C22×C3111yescharacteristic
228#103C22×C3111yescharacteristic
229#103C22×C3111yescharacteristic
230#103C22×C3111yescharacteristic
231#102C22×C3111yescharacteristic
232#102C22×C3111yescharacteristic
233#102C22×C3111yescharacteristic
234C22⋊C8C22×C3111yescharacteristic
235C22⋊C8C22×C3111yescharacteristic
236C4×C8C22×C3111yescharacteristic
237C31#332711yescharacteristic, direct product factor, Sylow
238#41—22no—
239#41—41no—
240C2×Q8—21no—
241C2×Q8—22no—
242C2×Q8—41no—
243C2×Q8C23×C3111yescharacteristic
244C2×Q8C23×C3111yescharacteristic
245C2×Q8C23×C3111yescharacteristic
246C2×D4—21no—
247C2×D4—21no—
248C2×D4D4×C3111yescharacteristic
249C2×D4C23×C3111yescharacteristic
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4C23×C3111yescharacteristic
254C22×C4C23×C3111yescharacteristic
255Q16—22no—
256Q16—41no—
257SD16—41no—
258M16—41no—
259M16—41no—
260C2×C8C23×C3111yescharacteristic
261C2×C8C23×C3111yescharacteristic
262C2×C8C23×C3111yescharacteristic
263C2×C8C23×C3111yescharacteristic
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4D4×C3111yescharacteristic
273C4⋊C4C23×C3111yescharacteristic
274C4⋊C4C23×C3111yescharacteristic
275C4⋊C4C23×C3111yescharacteristic
276C4⋊C4C23×C3111yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—21no—
280C22⋊C4—21no—
281C22⋊C4—21no—
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4D4×C3111yescharacteristic
285C22⋊C4D4×C3111yescharacteristic
286C42—21no—
287C42—21no—
288C42—21no—
289C42C23×C3111yescharacteristic
290C23—21no—
291C23C2×D4×C3111yescharacteristic
292C23C2×D4×C3111yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—21no—
296Q8—21no—
297Q8—21no—
298Q8—42no—
299Q8#41×C3112yes—
300D4—21no—
301D4—21no—
302D4—41no—
303D4—42no—
304D4—42no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4C2×D4×C3111yescharacteristic
320C2×C4C2×D4×C3111yescharacteristic
321C2×C4C2×D4×C3111yescharacteristic
322C2×C4C2×D4×C3111yescharacteristic
323C2×C4C24×C3111yescharacteristic
324C8—21no—
325C8—21no—
326C8—21no—
327C8—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22C22×D4×C3111yescharacteristic
338C4—21no—
339C4—21no—
340C4—22no—
341C4—21no—
342C4—21no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—41no—
349C4C2×#41×C3111yescharacteristic
350C4C31×H(5,𝔽2)11yescharacteristic
351C2—41no—
352C2—41no—
353C2—81no—
354C2C31×#54611yescharacteristic
355C2C31×#57511yescharacteristic
356C2C31×#57811yescharacteristic
subgroupG/H
center, socleC22×C31C22×D4
commutator, Frattini, cosocleC2×C4C24×C31
prime radicalC31×#548C2
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×#552C2semidirect
3. C31×#488C2semidirect
4. C31×#484C2semidirect
5. C31×#482C2semidirect
6. C31×#479C2semidirect
7. C31×#470C2semidirect
8. C31×#468C2semidirect
9. C31×#464C2semidirect
10. C31×#452C2semidirect
11. C31×#451C2semidirect
12. C4×Q16×C31C2semidirect
13. C31×#436C2semidirect
14. C31×#419C2semidirect
15. C31×#417C2semidirect
149. #3327C31direct (N × H)
173. #552C2×C31semidirect
175. #488C2×C31semidirect
176. #484C2×C31semidirect
177. #482C2×C31semidirect
178. #479C2×C31semidirect
179. #470C2×C31semidirect
180. #468C2×C31semidirect
181. #464C2×C31semidirect
182. #452C2×C31semidirect
183. #451C2×C31semidirect
184. C4×Q16C2×C31semidirect
185. #436C2×C31semidirect
186. #419C2×C31semidirect
187. #417C2×C31semidirect
237. C31#3327direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#5522
C31×#5482
C31×#4882
C31×#4842
C31×#4822
C31×#4792
C31×#4702
C31×#4682
C31×#4642
C31×#4522
C31×#4512
C4×Q16×C312
C31×#4362
C31×#4192
C31×#4172
#332731