Finite Group Databasebeta

← C31×#3257 #16843190

Subgroups — C31×#3257 #16843190

Order 3968.

358 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×#542C211yesmaximal, characteristic, complemented
2C31×#540C211yesmaximal, characteristic, complemented
3C31×#479C211yesmaximal, characteristic, complemented
4C31×#478C211yesmaximal, characteristic, complemented
5C31×#476C211yesmaximal, characteristic, complemented
6C31×#474C211yesmaximal, characteristic, complemented
7C31×#469C211yesmaximal, characteristic, complemented
8C31×#468C211yesmaximal, characteristic, complemented
9C31×#466C211yesmaximal, characteristic, complemented
10C31×#465C211yesmaximal, characteristic, complemented
11C31×#463C211yesmaximal, characteristic, complemented
12C31×#463C211yesmaximal, characteristic, complemented
13C31×#460C211yesmaximal, characteristic, complemented
14C31×#459C211yesmaximal, characteristic, complemented
15C31×#424C211yesmaximal, characteristic, complemented
16C2×SD16×C31C2211yescharacteristic
17C2×SD16×C31C2211yescharacteristic
18C2×D8×C31C2211yescharacteristic
19C2×D8×C31C2211yescharacteristic
20C2×M16×C31C2211yescharacteristic
21C22×C8×C31C2211yescharacteristic
22C31×C4⋊Q8C2211yescharacteristic
23C31×#127C2211yescharacteristic
24C31×#126—21no—
25C31×#125C2211yescharacteristic
26C31×#124—21no—
27C31×#124C2211yescharacteristic
28C31×#123—21no—
29C31×#123—21no—
30C31×#123—21no—
31C31×C22⋊Q8—21no—
32C31×C22⋊Q8C2211yescharacteristic
33C31×#121—21no—
34C31×#121—21no—
35C31×#121C2211yescharacteristic
36C31×#121C2211yescharacteristic
37C31×#121C2211yescharacteristic
38C4×Q8×C31C2211yescharacteristic
39C4×D4×C31C2211yescharacteristic
40C4×D4×C31C2211yescharacteristic
41C4×D4×C31C2211yescharacteristic
42C31×#117C2211yescharacteristic
43C31×#107C2211yescharacteristic
44C31×#107C2211yescharacteristic
45C31×#106C2211yescharacteristic
46C31×#106C2211yescharacteristic
47C31×C4⋊C8C2211yescharacteristic
48C31×C4⋊C8C2211yescharacteristic
49C31×C4⋊C8C2211yescharacteristic
50C31×C4⋊C8C2211yescharacteristic
51C31×#103C2211yescharacteristic
52C31×#103C2211yescharacteristic
53C31×#102C2211yescharacteristic
54C31×#102C2211yescharacteristic
55C31×#102C2211yescharacteristic
56C31×#102C2211yescharacteristic
57C31×#102C2211yescharacteristic
58C31×#102C2211yescharacteristic
59C2×Q8×C31—21no—
60C2×Q8×C31C2311yescharacteristic
61C2×D4×C31—21no—
62C2×D4×C31—21no—
63C2×D4×C31—21no—
64C2×D4×C31—21no—
65C2×D4×C31—21no—
66C2×D4×C31C2311yescharacteristic
67C2×D4×C31C2311yescharacteristic
68C2×D4×C31C2311yescharacteristic
69C22×C4×C31—21no—
70C22×C4×C31—21no—
71C22×C4×C31—21no—
72C22×C4×C31C2311yescharacteristic
73SD16×C31—41no—
74SD16×C31—41no—
75D8×C31—41no—
76D8×C31—41no—
77M16×C31—41no—
78C2×C8×C31—41no—
79C2×C8×C31C2311yescharacteristic
80C2×C8×C31C2311yescharacteristic
81C2×C8×C31C2311yescharacteristic
82C2×C8×C31C2311yescharacteristic
83C4⋊C4×C31—21no—
84C4⋊C4×C31—21no—
85C4⋊C4×C31—21no—
86C4⋊C4×C31—21no—
87C4⋊C4×C31—21no—
88C4⋊C4×C31D411yescharacteristic
89C4⋊C4×C31D411yescharacteristic
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×C31—21no—
100C22⋊C4×C31—21no—
101C22⋊C4×C31—21no—
102C22⋊C4×C31—21no—
103C22⋊C4×C31D411yescharacteristic
104C22⋊C4×C31D411yescharacteristic
105C42×C31—21no—
106C42×C31C2311yescharacteristic
107C42×C31C2311yescharacteristic
108C23×C31—21no—
109C23×C31—21no—
110C23×C31—21no—
111C23×C31C2×D411yescharacteristic
112Q8×C31—21no—
113Q8×C31—21no—
114Q8×C31—81no—
115D4×C31—21no—
116D4×C31—21no—
117D4×C31—21no—
118D4×C31—21no—
119D4×C31—21no—
120D4×C31—21no—
121D4×C31—81no—
122D4×C31—81no—
123D4×C31—81no—
124D4×C31—81no—
125D4×C31—81no—
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—81no—
134C2×C4×C31—81no—
135C2×C4×C31—81no—
136C2×C4×C31C2×D411yescharacteristic
137C2×C4×C31C2×D411yescharacteristic
138C2×C4×C31C2×D411yescharacteristic
139C2×C4×C31C2×D411yescharacteristic
140C2×C4×C31C2×D411yescharacteristic
141C2×C4×C31C2411yescharacteristic
142C8×C31—21no—
143C8×C31—21no—
144C8×C31—21no—
145C8×C31—21no—
146#3257C3111yesmaximal, characteristic, direct product factor, Sylow
147C22×C31—21no—
148C22×C31—21no—
149C22×C31—21no—
150C22×C31—41no—
151C22×C31—41no—
152C22×C31—41no—
153C22×C31—41no—
154C22×C31—41no—
155C22×C31—41no—
156C22×C31—41no—
157C22×C31—41no—
158C22×C31—41no—
159C22×C31C22×D411yescharacteristic
160C4×C31—21no—
161C4×C31—21no—
162C4×C31—21no—
163C4×C31—21no—
164C4×C31—21no—
165C4×C31—41no—
166C4×C31—41no—
167C4×C31—41no—
168C4×C31—41no—
169C4×C31—41no—
170C4×C31H(5,𝔽2)11yescharacteristic
171C4×C312−1+411yescharacteristic
172#542C2×C3111yescharacteristic, complemented
173#540C2×C3111yescharacteristic, complemented
174#479C2×C3111yescharacteristic, complemented
175#478C2×C3111yescharacteristic, complemented
176#476C2×C3111yescharacteristic, complemented
177#474C2×C3111yescharacteristic, complemented
178#469C2×C3111yescharacteristic, complemented
179#468C2×C3111yescharacteristic, complemented
180#466C2×C3111yescharacteristic, complemented
181#465C2×C3111yescharacteristic, complemented
182#463C2×C3111yescharacteristic, complemented
183#463C2×C3111yescharacteristic, complemented
184#460C2×C3111yescharacteristic, complemented
185#459C2×C3111yescharacteristic, complemented
186#424C2×C3111yescharacteristic, complemented
187C2×C31—41no—
188C2×C31—81no—
189C2×C31—81no—
190C2×C31—81no—
191C2×C31#53711yescharacteristic
192C2×C31#57611yescharacteristic
193C2×C31#57711yescharacteristic
194C2×SD16C22×C3111yescharacteristic
195C2×SD16C22×C3111yescharacteristic
196C2×D8C22×C3111yescharacteristic
197C2×D8C22×C3111yescharacteristic
198C2×M16C22×C3111yescharacteristic
199C22×C8C22×C3111yescharacteristic
200C4⋊Q8C22×C3111yescharacteristic
201#127C22×C3111yescharacteristic
202#126—21no—
203#125C22×C3111yescharacteristic
204#124—21no—
205#124C22×C3111yescharacteristic
206#123—21no—
207#123—21no—
208#123—21no—
209C22⋊Q8—21no—
210C22⋊Q8C22×C3111yescharacteristic
211#121—21no—
212#121—21no—
213#121C22×C3111yescharacteristic
214#121C22×C3111yescharacteristic
215#121C22×C3111yescharacteristic
216C4×Q8C22×C3111yescharacteristic
217C4×D4C22×C3111yescharacteristic
218C4×D4C22×C3111yescharacteristic
219C4×D4C22×C3111yescharacteristic
220#117C22×C3111yescharacteristic
221#107C22×C3111yescharacteristic
222#107C22×C3111yescharacteristic
223#106C22×C3111yescharacteristic
224#106C22×C3111yescharacteristic
225C4⋊C8C22×C3111yescharacteristic
226C4⋊C8C22×C3111yescharacteristic
227C4⋊C8C22×C3111yescharacteristic
228C4⋊C8C22×C3111yescharacteristic
229#103C22×C3111yescharacteristic
230#103C22×C3111yescharacteristic
231#102C22×C3111yescharacteristic
232#102C22×C3111yescharacteristic
233#102C22×C3111yescharacteristic
234#102C22×C3111yescharacteristic
235#102C22×C3111yescharacteristic
236#102C22×C3111yescharacteristic
237C31#325711yescharacteristic, direct product factor, Sylow
238C2×Q8—21no—
239C2×Q8C23×C3111yescharacteristic
240C2×D4—21no—
241C2×D4—21no—
242C2×D4—21no—
243C2×D4—21no—
244C2×D4—21no—
245C2×D4C23×C3111yescharacteristic
246C2×D4C23×C3111yescharacteristic
247C2×D4C23×C3111yescharacteristic
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4C23×C3111yescharacteristic
252SD16—41no—
253SD16—41no—
254D8—41no—
255D8—41no—
256M16—41no—
257C2×C8—41no—
258C2×C8C23×C3111yescharacteristic
259C2×C8C23×C3111yescharacteristic
260C2×C8C23×C3111yescharacteristic
261C2×C8C23×C3111yescharacteristic
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4D4×C3111yescharacteristic
268C4⋊C4D4×C3111yescharacteristic
269C4⋊C4C23×C3111yescharacteristic
270C4⋊C4C23×C3111yescharacteristic
271C4⋊C4C23×C3111yescharacteristic
272C4⋊C4C23×C3111yescharacteristic
273C22⋊C4—21no—
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—21no—
280C22⋊C4—21no—
281C22⋊C4—21no—
282C22⋊C4D4×C3111yescharacteristic
283C22⋊C4D4×C3111yescharacteristic
284C42—21no—
285C42C23×C3111yescharacteristic
286C42C23×C3111yescharacteristic
287C23—21no—
288C23—21no—
289C23—21no—
290C23C2×D4×C3111yescharacteristic
291Q8—21no—
292Q8—21no—
293Q8—81no—
294D4—21no—
295D4—21no—
296D4—21no—
297D4—21no—
298D4—21no—
299D4—21no—
300D4—81no—
301D4—81no—
302D4—81no—
303D4—81no—
304D4—81no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4C2×D4×C3111yescharacteristic
316C2×C4C2×D4×C3111yescharacteristic
317C2×C4C2×D4×C3111yescharacteristic
318C2×C4C2×D4×C3111yescharacteristic
319C2×C4C2×D4×C3111yescharacteristic
320C2×C4C24×C3111yescharacteristic
321C8—21no—
322C8—21no—
323C8—21no—
324C8—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22C22×D4×C3111yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4C31×H(5,𝔽2)11yescharacteristic
349C4C31×2−1+411yescharacteristic
350C2—41no—
351C2—81no—
352C2—81no—
353C2—81no—
354C2C31×#53711yescharacteristic
355C2C31×#57611yescharacteristic
356C2C31×#57711yescharacteristic
subgroupG/H
center, socleC22×C31C22×D4
commutator, Frattini, cosocleC2×C4C24×C31
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×#542C2semidirect
2. C31×#540C2semidirect
3. C31×#479C2semidirect
4. C31×#478C2semidirect
5. C31×#476C2semidirect
6. C31×#474C2semidirect
7. C31×#469C2semidirect
8. C31×#468C2semidirect
9. C31×#466C2semidirect
10. C31×#465C2semidirect
11. C31×#463C2semidirect
12. C31×#463C2semidirect
13. C31×#460C2semidirect
14. C31×#459C2semidirect
15. C31×#424C2semidirect
146. #3257C31direct (N × H)
172. #542C2×C31semidirect
173. #540C2×C31semidirect
174. #479C2×C31semidirect
175. #478C2×C31semidirect
176. #476C2×C31semidirect
177. #474C2×C31semidirect
178. #469C2×C31semidirect
179. #468C2×C31semidirect
180. #466C2×C31semidirect
181. #465C2×C31semidirect
182. #463C2×C31semidirect
183. #463C2×C31semidirect
184. #460C2×C31semidirect
185. #459C2×C31semidirect
186. #424C2×C31semidirect
237. C31#3257direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#5422
C31×#5402
C31×#4792
C31×#4782
C31×#4762
C31×#4742
C31×#4692
C31×#4682
C31×#4662
C31×#4652
C31×#4632
C31×#4602
C31×#4592
C31×#4242
#325731