Finite Group Databasebeta

← C31×#9108 #21465867

Subgroups — C31×#9108 #21465867

Order 7936.

586 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#2972C211yesmaximal, characteristic
2C31×#2322C211yesmaximal, characteristic
3C31×#1459C211yesmaximal, characteristic
4C2×C31×#324C212yesmaximal, complemented
5C31×#1282C212yesmaximal, complemented
6C2×C31×#123C411yescharacteristic
7C2×C4×D4×C31C2211yescharacteristic
8C22×C4⋊C4×C31C2211yescharacteristic
9C8×D4×C31—24no—
10C31×#431—21no—
11C2×C31×C22⋊C8—21no—
12C2×C31×C22⋊C8C2212yes—
13C2×C31×C22⋊C8C2212yes—
14C31×#389—21no—
15C31×#388—21no—
16C31×#387—21no—
17C31×#386C411yescharacteristic
18C31×#385—21no—
19C31×#385C411yescharacteristic
20C31×#381C411yescharacteristic
21C4×C22⋊C4×C31C2211yescharacteristic
22C31×#324—21no—
23C31×#324—24no—
24C31×#323—21no—
25C22×D4×C31C2×C411yescharacteristic
26C23×C4×C31C2×C411yescharacteristic
27C23×C4×C31C2×C411yescharacteristic
28C23×C4×C31D411yescharacteristic
29C23×C4×C31C2311yescharacteristic
30C22×C8×C31—22no—
31C22×C8×C31—22no—
32C22×C8×C31—22no—
33C31×#123—22no—
34C31×#123C814yescomplemented
35C4×D4×C31—22no—
36C4×D4×C31—22no—
37C2×C4⋊C4×C31—21no—
38C2×C4⋊C4×C31—22no—
39C2×C4⋊C4×C31—21no—
40C2×C4⋊C4×C31—21no—
41C2×C4⋊C4×C31—22no—
42C2×C4⋊C4×C31—21no—
43C2×C4⋊C4×C31D411yescharacteristic
44C2×C22⋊C4×C31—21no—
45C2×C22⋊C4×C31—21no—
46C2×C22⋊C4×C31C2×C411yescharacteristic
47C2×C22⋊C4×C31C2×C411yescharacteristic
48C2×C22⋊C4×C31C2×C411yescharacteristic
49C2×C22⋊C4×C31D411yescharacteristic
50C2×C42×C31—21no—
51C2×C42×C31D411yescharacteristic
52C31×C4⋊C8—22no—
53C31×C22⋊C8—22no—
54C31×C22⋊C8—22no—
55C31×C22⋊C8—22no—
56C31×C22⋊C8—22no—
57C31×C22⋊C8D414yes—
58C4×C8×C31—22no—
59C31×#95—21no—
60C31×#95—21no—
61C31×#95—21no—
62C31×#95—21no—
63C31×#95—21no—
64C24×C31C22⋊C411yescharacteristic
65C24×C31C22×C411yescharacteristic
66C2×D4×C31—22no—
67C2×D4×C31—22no—
68C2×D4×C31C2×C812yes—
69C2×D4×C31M1612yes—
70C22×C4×C31—21no—
71C22×C4×C31—21no—
72C22×C4×C31—22no—
73C22×C4×C31—21no—
74C22×C4×C31—22no—
75C22×C4×C31—21no—
76C22×C4×C31—21no—
77C22×C4×C31—21no—
78C22×C4×C31—21no—
79C22×C4×C31—22no—
80C22×C4×C31—81no—
81C22×C4×C31—81no—
82C22×C4×C31—81no—
83C22×C4×C31C22⋊C411yescharacteristic
84C22×C4×C31C22⋊C411yescharacteristic
85C22×C4×C31C22⋊C411yescharacteristic
86C22×C4×C31C2×C812yescomplemented
87C22×C4×C31C2×D412yes—
88C22×C4×C31C2×D411yescharacteristic
89C22×C4×C31C2×D411yescharacteristic
90C22×C4×C31#4112yes—
91C2×C8×C31—24no—
92C2×C8×C31—42no—
93C2×C8×C31—82no—
94C2×C8×C31—82no—
95C2×C8×C31—82no—
96C4⋊C4×C31—24no—
97C4⋊C4×C31—22no—
98C4⋊C4×C31—41no—
99C4⋊C4×C31—42no—
100C4⋊C4×C31—41no—
101C4⋊C4×C31—81no—
102C22⋊C4×C31—24no—
103C22⋊C4×C31—22no—
104C22⋊C4×C31—21no—
105C22⋊C4×C31—22no—
106C22⋊C4×C31—22no—
107C22⋊C4×C31—81no—
108C22⋊C4×C31C2×C812yescomplemented
109C42×C31—21no—
110C42×C31—21no—
111C42×C31—81no—
112#9108C3111yesmaximal, characteristic, direct product factor, Sylow
113C23×C31—21no—
114C23×C31—21no—
115C23×C31—41no—
116C23×C31—41no—
117C23×C31—41no—
118C23×C31—41no—
119C23×C31C22⋊C812yes—
120C23×C31C2×C22⋊C411yescharacteristic
121C23×C31C4×D412yes—
122C23×C31C22×C811yescharacteristic
123C23×C31C2×M1611yescharacteristic
124C23×C31#13112yes—
125D4×C31—24no—
126D4×C31—24no—
127C2×C4×C31—21no—
128C2×C4×C31—21no—
129C2×C4×C31—21no—
130C2×C4×C31—21no—
131C2×C4×C31—22no—
132C2×C4×C31—22no—
133C2×C4×C31—24no—
134C2×C4×C31—41no—
135C2×C4×C31—42no—
136C2×C4×C31—42no—
137C2×C4×C31—41no—
138C2×C4×C31—42no—
139C2×C4×C31—41no—
140C2×C4×C31—41no—
141C2×C4×C31—41no—
142C2×C4×C31—41no—
143C2×C4×C31—42no—
144C2×C4×C31—41no—
145C2×C4×C31—41no—
146C2×C4×C31—41no—
147C2×C4×C31—41no—
148C2×C4×C31—41no—
149C2×C4×C31—41no—
150C2×C4×C31—41no—
151C2×C4×C31—41no—
152C2×C4×C31—81no—
153C2×C4×C31—82no—
154C2×C4×C31C22⋊C812yes—
155C2×C4×C31#12011yescharacteristic
156C2×C4×C31#12112yes—
157C2×C4×C31#12311yescharacteristic
158C8×C31—44no—
159C8×C31—82no—
160#2972C2×C3111yescharacteristic
161#2322C2×C3111yescharacteristic
162#1459C2×C3111yescharacteristic
163C2×#324C2×C3112yescomplemented
164#1282C2×C3112yescomplemented
165C22×C31—21no—
166C22×C31—22no—
167C22×C31—21no—
168C22×C31—22no—
169C22×C31—22no—
170C22×C31—41no—
171C22×C31—41no—
172C22×C31—42no—
173C22×C31—42no—
174C22×C31—41no—
175C22×C31—41no—
176C22×C31—81no—
177C22×C31#38611yescharacteristic
178C22×C31C2×C22⋊C811yescharacteristic
179C22×C31#40811yescharacteristic
180C22×C31C8×D412yes—
181C22×C31#43512yes—
182C4×C31—22no—
183C4×C31—21no—
184C4×C31—22no—
185C4×C31—21no—
186C4×C31—42no—
187C4×C31—42no—
188C4×C31—41no—
189C4×C31—41no—
190C4×C31—81no—
191C4×C31—81no—
192C2×#123C4×C3111yescharacteristic
193C2×C4×D4C22×C3111yescharacteristic
194C22×C4⋊C4C22×C3111yescharacteristic
195C8×D4—24no—
196#431—21no—
197C2×C22⋊C8—21no—
198C2×C22⋊C8C22×C3112yes—
199C2×C22⋊C8C22×C3112yes—
200#389—21no—
201#388—21no—
202#387—21no—
203#386C4×C3111yescharacteristic
204#385—21no—
205#385C4×C3111yescharacteristic
206#381C4×C3111yescharacteristic
207C4×C22⋊C4C22×C3111yescharacteristic
208#324—21no—
209#324—24no—
210#323—21no—
211C2×C31—21no—
212C2×C31—21no—
213C2×C31—22no—
214C2×C31—22no—
215C2×C31—42no—
216C2×C31#186911yescharacteristic
217C2×C31#191311yescharacteristic
218C2×C31#191511yescharacteristic
219C22×D4C2×C4×C3111yescharacteristic
220C23×C4C2×C4×C3111yescharacteristic
221C23×C4C2×C4×C3111yescharacteristic
222C23×C4D4×C3111yescharacteristic
223C23×C4C23×C3111yescharacteristic
224C22×C8—22no—
225C22×C8—22no—
226C22×C8—22no—
227#123—22no—
228#123C8×C3114yescomplemented
229C4×D4—22no—
230C4×D4—22no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4D4×C3111yescharacteristic
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4C2×C4×C3111yescharacteristic
241C2×C22⋊C4C2×C4×C3111yescharacteristic
242C2×C22⋊C4C2×C4×C3111yescharacteristic
243C2×C22⋊C4D4×C3111yescharacteristic
244C2×C42—21no—
245C2×C42D4×C3111yescharacteristic
246C4⋊C8—22no—
247C22⋊C8—22no—
248C22⋊C8—22no—
249C22⋊C8—22no—
250C22⋊C8—22no—
251C22⋊C8D4×C3114yes—
252C4×C8—22no—
253#95—21no—
254#95—21no—
255#95—21no—
256#95—21no—
257#95—21no—
258C31#910811yescharacteristic, direct product factor, Sylow
259C24C22⋊C4×C3111yescharacteristic
260C24C22×C4×C3111yescharacteristic
261C2×D4—22no—
262C2×D4—22no—
263C2×D4C2×C8×C3112yes—
264C2×D4M16×C3112yes—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—22no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—81no—
276C22×C4—81no—
277C22×C4—81no—
278C22×C4C22⋊C4×C3111yescharacteristic
279C22×C4C22⋊C4×C3111yescharacteristic
280C22×C4C22⋊C4×C3111yescharacteristic
281C22×C4C2×C8×C3112yescomplemented
282C22×C4C2×D4×C3112yes—
283C22×C4C2×D4×C3111yescharacteristic
284C22×C4C2×D4×C3111yescharacteristic
285C22×C4#41×C3112yes—
286C2×C8—24no—
287C2×C8—42no—
288C2×C8—82no—
289C2×C8—82no—
290C2×C8—82no—
291C4⋊C4—24no—
292C4⋊C4—22no—
293C4⋊C4—41no—
294C4⋊C4—42no—
295C4⋊C4—41no—
296C4⋊C4—81no—
297C22⋊C4—24no—
298C22⋊C4—22no—
299C22⋊C4—21no—
300C22⋊C4—22no—
301C22⋊C4—22no—
302C22⋊C4—81no—
303C22⋊C4C2×C8×C3112yescomplemented
304C42—21no—
305C42—21no—
306C42—81no—
307C23—21no—
308C23—21no—
309C23—41no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23C31×C22⋊C812yes—
314C23C2×C22⋊C4×C3111yescharacteristic
315C23C4×D4×C3112yes—
316C23C22×C8×C3111yescharacteristic
317C23C2×M16×C3111yescharacteristic
318C23C31×#13112yes—
319D4—24no—
320D4—24no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—24no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—41no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—81no—
347C2×C4—82no—
348C2×C4C31×C22⋊C812yes—
349C2×C4C31×#12011yescharacteristic
350C2×C4C31×#12112yes—
351C2×C4C31×#12311yescharacteristic
352C8—44no—
353C8—82no—
354C22—21no—
355C22—22no—
356C22—21no—
357C22—22no—
358C22—22no—
359C22—41no—
360C22—41no—
361C22—42no—
362C22—42no—
363C22—41no—
364C22—41no—
365C22—81no—
366C22C31×#38611yescharacteristic
367C22C2×C31×C22⋊C811yescharacteristic
368C22C31×#40811yescharacteristic
369C22C8×D4×C3112yes—
370C22C31×#43512yes—
371C4—22no—
372C4—21no—
373C4—22no—
374C4—21no—
375C4—42no—
376C4—42no—
377C4—41no—
378C4—41no—
379C4—81no—
380C4—81no—
381C2—21no—
382C2—21no—
383C2—22no—
384C2—22no—
385C2—42no—
386C2C31×#186911yescharacteristic
387C2C31×#191311yescharacteristic
388C2C31×#191511yescharacteristic
subgroupG/H
center, socleC22×C31#386
commutatorC23C22×C8×C31
Frattini, cosocleC23×C4C23×C31
prime radicalC22×D4×C31C2×C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. C2×C31×#324C2semidirect
5. C31×#1282C2semidirect
34. C31×#123C8semidirect
86. C22×C4×C31C2×C8semidirect
108. C22⋊C4×C31C2×C8semidirect
112. #9108C31direct (N × H)
163. C2×#324C2×C31semidirect
164. #1282C2×C31semidirect
228. #123C8×C31semidirect
258. C31#9108direct (N × H)
281. C22×C4C2×C8×C31semidirect
303. C22⋊C4C2×C8×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#29722
C31×#23222
C31×#14592
C2×C31×#3242
C31×#12822
#910831