Finite Group Databasebeta

← C31×#9112 #21465871

Subgroups — C31×#9112 #21465871

Order 7936.

632 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#431C211yesmaximal, characteristic
2C31×#2855C211yesmaximal, characteristic
3C2×C31×#328C212yesmaximal
4C31×#1434C211yesmaximal, characteristic
5C31×#1294C212yesmaximal
6C2×C31×C22⋊Q8C2211yescharacteristic
7C22×C42×C31C2211yescharacteristic
8C31×#431—24no—
9C2×C31×C4⋊C8—21no—
10C2×C31×C22⋊C8C2212yes—
11C2×C4×C8×C31—21no—
12C31×#391C411yescharacteristic, complemented
13C31×#391C411yescharacteristic, complemented
14C31×#389C411yescharacteristic, complemented
15C31×#389C411yescharacteristic, complemented
16C31×#387—21no—
17C31×#387—21no—
18C31×#385—21no—
19C31×#381—21no—
20C31×#380C2211yescharacteristic
21C2×C31×#95—21no—
22C2×C31×#95—21no—
23C2×C31×#95C2212yes—
24C31×#328—24no—
25C22×Q8×C31C2×C411yescharacteristic
26C23×C4×C31—22no—
27C23×C4×C31D411yescharacteristic
28C23×C4×C31D411yescharacteristic
29C23×C4×C31C2311yescharacteristic
30C22×C8×C31—22no—
31C31×C22⋊Q8—22no—
32C31×C22⋊Q8—22no—
33C2×C4⋊C4×C31—21no—
34C2×C4⋊C4×C31—21no—
35C2×C4⋊C4×C31—21no—
36C2×C4⋊C4×C31C2×C411yescharacteristic
37C2×C4⋊C4×C31C2×C411yescharacteristic
38C2×C4⋊C4×C31C2×C411yescharacteristic
39C2×C22⋊C4×C31—21no—
40C2×C22⋊C4×C31—21no—
41C2×C42×C31—22no—
42C2×C42×C31—22no—
43C2×C42×C31C2×C411yescharacteristic
44C2×C42×C31C2×C411yescharacteristic
45C2×C42×C31D411yescharacteristic
46C2×C42×C31D411yescharacteristic
47C31×C4⋊C8—24no—
48C31×C22⋊C8—22no—
49C31×C22⋊C8D414yes—
50C4×C8×C31—24no—
51C31×#95—24no—
52C31×#95—21no—
53C31×#95—21no—
54C31×#95—21no—
55C31×#95—21no—
56C31×#95—21no—
57C31×#95—21no—
58C31×#95—21no—
59C31×#95—22no—
60C31×#95—22no—
61C31×#95—42no—
62C24×C31C2×D411yescharacteristic
63C2×Q8×C31—22no—
64C2×Q8×C31—41no—
65C2×Q8×C31—41no—
66C22×C4×C31—22no—
67C22×C4×C31—22no—
68C22×C4×C31—22no—
69C22×C4×C31—22no—
70C22×C4×C31—24no—
71C22×C4×C31—21no—
72C22×C4×C31—21no—
73C22×C4×C31—21no—
74C22×C4×C31—21no—
75C22×C4×C31—41no—
76C22×C4×C31—41no—
77C22×C4×C31—41no—
78C22×C4×C31—42no—
79C22×C4×C31—42no—
80C22×C4×C31—42no—
81C22×C4×C31—42no—
82C22×C4×C31—42no—
83C22×C4×C31—42no—
84C22×C4×C31C22⋊C411yescharacteristic
85C22×C4×C31C22⋊C411yescharacteristic
86C22×C4×C31C22⋊C411yescharacteristic
87C22×C4×C31C22⋊C411yescharacteristic
88C22×C4×C31SD1612yes—
89C22×C4×C31Q1612yes—
90C22×C4×C31C22×C411yescharacteristic
91C22×C4×C31C2×D411yescharacteristic
92C22×C4×C31C2×D412yes—
93C22×C4×C31#4112yes—
94C2×C8×C31—24no—
95C2×C8×C31—42no—
96C2×C8×C31—42no—
97C4⋊C4×C31—22no—
98C4⋊C4×C31—42no—
99C4⋊C4×C31—41no—
100C4⋊C4×C31—41no—
101C4⋊C4×C31—42no—
102C4⋊C4×C31—41no—
103C4⋊C4×C31—41no—
104C22⋊C4×C31—42no—
105C22⋊C4×C31—42no—
106C42×C31—22no—
107C42×C31—22no—
108C42×C31—21no—
109C42×C31—21no—
110C42×C31—21no—
111C42×C31—21no—
112#9112C3111yesmaximal, characteristic, direct product factor, Sylow
113C23×C31—21no—
114C23×C31—21no—
115C23×C31—21no—
116C23×C31—21no—
117C23×C31C2×C22⋊C411yescharacteristic
118C23×C31#12112yes—
119C23×C31C2×SD1611yescharacteristic
120C23×C31C2×Q1611yescharacteristic
121C23×C31#13512yes—
122Q8×C31—82no—
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—21no—
130C2×C4×C31—21no—
131C2×C4×C31—21no—
132C2×C4×C31—21no—
133C2×C4×C31—24no—
134C2×C4×C31—24no—
135C2×C4×C31—21no—
136C2×C4×C31—21no—
137C2×C4×C31—21no—
138C2×C4×C31—42no—
139C2×C4×C31—42no—
140C2×C4×C31—44no—
141C2×C4×C31—44no—
142C2×C4×C31—44no—
143C2×C4×C31—44no—
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—41no—
153C2×C4×C31—41no—
154C2×C4×C31—41no—
155C2×C4×C31—41no—
156C2×C4×C31—82no—
157C2×C4×C31—82no—
158C2×C4×C31—82no—
159C2×C4×C31—82no—
160C2×C4×C31#10312yes—
161C2×C4×C31#10312yes—
162C2×C4×C31#10412yes—
163C2×C4×C31C4×D412yes—
164C2×C4×C31#12011yescharacteristic
165C2×C4×C31#12311yescharacteristic
166C8×C31—44no—
167C2×#431C2×C3111yescharacteristic
168#2855C2×C3111yescharacteristic
169C2×#328C2×C3112yes—
170#1434C2×C3111yescharacteristic
171#1294C2×C3112yes—
172C22×C31—22no—
173C22×C31—22no—
174C22×C31—22no—
175C22×C31—21no—
176C22×C31—21no—
177C22×C31—21no—
178C22×C31—21no—
179C22×C31—22no—
180C22×C31#38611yescharacteristic
181C22×C31#41011yescharacteristic
182C22×C31#41011yescharacteristic
183C22×C31C2×#10311yescharacteristic
184C22×C31#41611yescharacteristic
185C22×C31C2×#10411yescharacteristic
186C22×C31#42111yescharacteristic
187C22×C31#46512yes—
188C22×C31#46712yes—
189C4×C31—22no—
190C4×C31—22no—
191C4×C31—22no—
192C4×C31—21no—
193C4×C31—22no—
194C4×C31—21no—
195C4×C31—41no—
196C4×C31—84no—
197C4×C31—81no—
198C4×C31—81no—
199C4×C31—81no—
200C4×C31—81no—
201C2×C22⋊Q8C22×C3111yescharacteristic
202C22×C42C22×C3111yescharacteristic
203#431—24no—
204C2×C4⋊C8—21no—
205C2×C22⋊C8C22×C3112yes—
206C2×C4×C8—21no—
207#391C4×C3111yescharacteristic, complemented
208#391C4×C3111yescharacteristic, complemented
209#389C4×C3111yescharacteristic, complemented
210#389C4×C3111yescharacteristic, complemented
211#387—21no—
212#387—21no—
213#385—21no—
214#381—21no—
215#380C22×C3111yescharacteristic
216C2×#95—21no—
217C2×#95—21no—
218C2×#95C22×C3112yes—
219#328—24no—
220C2×C31—22no—
221C2×C31—22no—
222C2×C31#150011yescharacteristic
223C2×C31#150311yescharacteristic
224C2×C31#151811yescharacteristic
225C2×C31#152111yescharacteristic
226C2×C31#187111yescharacteristic
227C2×C31#189211yescharacteristic
228C2×C31#189911yescharacteristic
229C22×Q8C2×C4×C3111yescharacteristic
230C23×C4—22no—
231C23×C4D4×C3111yescharacteristic
232C23×C4D4×C3111yescharacteristic
233C23×C4C23×C3111yescharacteristic
234C22×C8—22no—
235C22⋊Q8—22no—
236C22⋊Q8—22no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4C2×C4×C3111yescharacteristic
241C2×C4⋊C4C2×C4×C3111yescharacteristic
242C2×C4⋊C4C2×C4×C3111yescharacteristic
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42C2×C4×C3111yescharacteristic
248C2×C42C2×C4×C3111yescharacteristic
249C2×C42D4×C3111yescharacteristic
250C2×C42D4×C3111yescharacteristic
251C4⋊C8—24no—
252C22⋊C8—22no—
253C22⋊C8D4×C3114yes—
254C4×C8—24no—
255#95—24no—
256#95—21no—
257#95—21no—
258#95—21no—
259#95—21no—
260#95—21no—
261#95—21no—
262#95—21no—
263#95—22no—
264#95—22no—
265#95—42no—
266C31#911211yescharacteristic, direct product factor, Sylow
267C24C2×D4×C3111yescharacteristic
268C2×Q8—22no—
269C2×Q8—41no—
270C2×Q8—41no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—24no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4C22⋊C4×C3111yescharacteristic
290C22×C4C22⋊C4×C3111yescharacteristic
291C22×C4C22⋊C4×C3111yescharacteristic
292C22×C4C22⋊C4×C3111yescharacteristic
293C22×C4SD16×C3112yes—
294C22×C4Q16×C3112yes—
295C22×C4C22×C4×C3111yescharacteristic
296C22×C4C2×D4×C3111yescharacteristic
297C22×C4C2×D4×C3112yes—
298C22×C4#41×C3112yes—
299C2×C8—24no—
300C2×C8—42no—
301C2×C8—42no—
302C4⋊C4—22no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C22⋊C4—42no—
310C22⋊C4—42no—
311C42—22no—
312C42—22no—
313C42—21no—
314C42—21no—
315C42—21no—
316C42—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23C2×C22⋊C4×C3111yescharacteristic
322C23C31×#12112yes—
323C23C2×SD16×C3111yescharacteristic
324C23C2×Q16×C3111yescharacteristic
325C23C31×#13512yes—
326Q8—82no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4C31×#10312yes—
365C2×C4C31×#10312yes—
366C2×C4C31×#10412yes—
367C2×C4C4×D4×C3112yes—
368C2×C4C31×#12011yescharacteristic
369C2×C4C31×#12311yescharacteristic
370C8—44no—
371C22—22no—
372C22—22no—
373C22—22no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—22no—
379C22C31×#38611yescharacteristic
380C22C31×#41011yescharacteristic
381C22C31×#41011yescharacteristic
382C22C2×C31×#10311yescharacteristic
383C22C31×#41611yescharacteristic
384C22C2×C31×#10411yescharacteristic
385C22C31×#42111yescharacteristic
386C22C31×#46512yes—
387C22C31×#46712yes—
388C4—22no—
389C4—22no—
390C4—22no—
391C4—21no—
392C4—22no—
393C4—21no—
394C4—41no—
395C4—84no—
396C4—81no—
397C4—81no—
398C4—81no—
399C4—81no—
400C2—22no—
401C2—22no—
402C2C31×#150011yescharacteristic
403C2C31×#150311yescharacteristic
404C2C31×#151811yescharacteristic
405C2C31×#152111yescharacteristic
406C2C31×#187111yescharacteristic
407C2C31×#189211yescharacteristic
408C2C31×#189911yescharacteristic
subgroupG/H
center, socleC23×C31C2×C22⋊C4
commutatorC22×C4C22×C4×C31
Frattini, cosocleC23×C4C23×C31
prime radicalC24×C31C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C31 ◁ C22×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
12. C31×#391C4semidirect
13. C31×#391C4semidirect
14. C31×#389C4semidirect
15. C31×#389C4semidirect
112. #9112C31direct (N × H)
207. #391C4×C31semidirect
208. #391C4×C31semidirect
209. #389C4×C31semidirect
210. #389C4×C31semidirect
266. C31#9112direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#4312
C31×#28552
C2×C31×#3282
C31×#14342
C31×#12942
#911231