Finite Group Databasebeta

← C31×#8325 #21465084

Subgroups — C31×#8325 #21465084

Order 7936.

748 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#542C211yesmaximal, characteristic
2C31×#1479C212yesmaximal, complemented
3C31×#1478C212yesmaximal, complemented
4C31×#1458C211yesmaximal, characteristic, complemented
5C31×#1458C211yesmaximal, characteristic, complemented
6C31×#542—24no—
7C31×#542—24no—
8C2×C31×C4⋊Q8C411yescharacteristic
9C2×C31×#126—21no—
10C2×C31×#124—21no—
11C2×C31×#124C411yescharacteristic
12C2×C31×#123—21no—
13C2×C31×C22⋊Q8—21no—
14C2×C31×C22⋊Q8C411yescharacteristic
15C2×C31×#121C411yescharacteristic
16C2×C4×Q8×C31C2211yescharacteristic
17C2×C4×D4×C31C2211yescharacteristic
18C2×C31×#117C2211yescharacteristic
19C31×#431C2212yes—
20C31×#431C2212yes—
21C31×Q8⋊C8—24no—
22C31×D4⋊C8—24no—
23C31×#324—21no—
24C31×#324—21no—
25C31×#323—21no—
26C31×#323—21no—
27C22×Q8×C31—21no—
28C22×Q8×C31C2×C411yescharacteristic
29C22×D4×C31—21no—
30C22×D4×C31C2×C411yescharacteristic
31C23×C4×C31—21no—
32C23×C4×C31C2×C411yescharacteristic
33C31×C4⋊Q8—22no—
34C31×C4⋊Q8—22no—
35C31×#126—28no—
36C31×#124—28no—
37C31×#124—22no—
38C31×#124—22no—
39C31×#123—28no—
40C31×C22⋊Q8—28no—
41C31×C22⋊Q8—22no—
42C31×C22⋊Q8—22no—
43C31×#121—22no—
44C31×#121—22no—
45C4×Q8×C31—22no—
46C4×Q8×C31D414yes—
47C4×D4×C31—22no—
48C4×D4×C31D414yes—
49C31×#117—22no—
50C31×#117—22no—
51C2×C4⋊C4×C31—21no—
52C2×C4⋊C4×C31—21no—
53C2×C4⋊C4×C31—21no—
54C2×C4⋊C4×C31C2×C411yescharacteristic
55C2×C4⋊C4×C31C2×C411yescharacteristic
56C2×C4⋊C4×C31D411yescharacteristic
57C2×C4⋊C4×C31D411yescharacteristic
58C2×C22⋊C4×C31—21no—
59C2×C22⋊C4×C31—21no—
60C2×C22⋊C4×C31—21no—
61C2×C22⋊C4×C31—21no—
62C2×C22⋊C4×C31—21no—
63C2×C22⋊C4×C31D411yescharacteristic
64C2×C22⋊C4×C31D411yescharacteristic
65C2×C42×C31—21no—
66C2×C42×C31C2×C411yescharacteristic
67C2×C42×C31C2311yescharacteristic
68C31×C4⋊C8—22no—
69C31×C4⋊C8—22no—
70C31×C22⋊C8—22no—
71C31×C22⋊C8—22no—
72C4×C8×C31—22no—
73C4×C8×C31—22no—
74C24×C31—21no—
75C24×C31C22⋊C411yescharacteristic
76C2×Q8×C31—24no—
77C2×Q8×C31—21no—
78C2×Q8×C31—24no—
79C2×Q8×C31—82no—
80C2×Q8×C31C22⋊C412yes—
81C2×D4×C31—24no—
82C2×D4×C31—21no—
83C2×D4×C31—24no—
84C2×D4×C31—82no—
85C2×D4×C31C22⋊C412yes—
86C22×C4×C31—21no—
87C22×C4×C31—21no—
88C22×C4×C31—21no—
89C22×C4×C31—21no—
90C22×C4×C31—21no—
91C22×C4×C31—24no—
92C22×C4×C31—24no—
93C22×C4×C31—82no—
94C22×C4×C31C22⋊C411yescharacteristic
95C22×C4×C31C22⋊C411yescharacteristic
96C22×C4×C31C22⋊C411yescharacteristic
97C22×C4×C31C22×C411yescharacteristic
98C22×C4×C31C2×D411yescharacteristic
99C22×C4×C31C2×D411yescharacteristic
100C2×C8×C31—42no—
101C2×C8×C31—42no—
102C4⋊C4×C31—24no—
103C4⋊C4×C31—24no—
104C4⋊C4×C31—21no—
105C4⋊C4×C31—21no—
106C4⋊C4×C31—24no—
107C4⋊C4×C31—22no—
108C4⋊C4×C31—22no—
109C4⋊C4×C31C22⋊C412yes—
110C4⋊C4×C31C22⋊C412yes—
111C22⋊C4×C31—24no—
112C22⋊C4×C31—24no—
113C22⋊C4×C31—24no—
114C22⋊C4×C31—24no—
115C22⋊C4×C31—24no—
116C22⋊C4×C31—22no—
117C22⋊C4×C31—22no—
118C42×C31—21no—
119C42×C31—21no—
120C42×C31—24no—
121C42×C31C2×D412yes—
122C42×C31C2×D412yes—
123#8325C3111yesmaximal, characteristic, direct product factor, Sylow
124C23×C31—21no—
125C23×C31—22no—
126C23×C31—22no—
127C23×C31—42no—
128C23×C31—42no—
129C23×C31—41no—
130C23×C31—41no—
131C23×C31—42no—
132C23×C31C2×C22⋊C411yescharacteristic
133Q8×C31—24no—
134Q8×C31—42no—
135Q8×C31—84no—
136D4×C31—24no—
137D4×C31—42no—
138D4×C31—84no—
139C2×C4×C31—21no—
140C2×C4×C31—22no—
141C2×C4×C31—21no—
142C2×C4×C31—22no—
143C2×C4×C31—21no—
144C2×C4×C31—22no—
145C2×C4×C31—22no—
146C2×C4×C31—41no—
147C2×C4×C31—41no—
148C2×C4×C31—42no—
149C2×C4×C31—42no—
150C2×C4×C31—42no—
151C2×C4×C31—42no—
152C2×C4×C31—42no—
153C2×C4×C31—41no—
154C2×C4×C31—41no—
155C2×C4×C31—84no—
156C2×C4×C31C2×C22⋊C411yescharacteristic
157C2×C4×C31C2×C22⋊C411yescharacteristic
158C2×C4×C31#12012yes—
159C2×C4×C31#12012yes—
160C2×C4×C31#13114yes—
161C8×C31—82no—
162C8×C31—82no—
163C2×#542C2×C3111yescharacteristic
164#1479C2×C3112yescomplemented
165#1478C2×C3112yescomplemented
166#1458C2×C3111yescharacteristic, complemented
167#1458C2×C3111yescharacteristic, complemented
168C22×C31—21no—
169C22×C31—21no—
170C22×C31—42no—
171C22×C31—41no—
172C22×C31—41no—
173C22×C31—42no—
174C22×C31—41no—
175C22×C31—42no—
176C22×C31—82no—
177C22×C31—84no—
178C22×C31#37911yescharacteristic
179C22×C31#40812yes—
180C4×C31—22no—
181C4×C31—22no—
182C4×C31—42no—
183C4×C31—42no—
184C4×C31—41no—
185C4×C31—41no—
186C4×C31—41no—
187C4×C31—42no—
188C4×C31#40812yes—
189C4×C31#40812yes—
190#542—24no—
191#542—24no—
192C2×C4⋊Q8C4×C3111yescharacteristic
193C2×#126—21no—
194C2×#124—21no—
195C2×#124C4×C3111yescharacteristic
196C2×#123—21no—
197C2×C22⋊Q8—21no—
198C2×C22⋊Q8C4×C3111yescharacteristic
199C2×#121C4×C3111yescharacteristic
200C2×C4×Q8C22×C3111yescharacteristic
201C2×C4×D4C22×C3111yescharacteristic
202C2×#117C22×C3111yescharacteristic
203#431C22×C3112yes—
204#431C22×C3112yes—
205Q8⋊C8—24no—
206D4⋊C8—24no—
207#324—21no—
208#324—21no—
209#323—21no—
210#323—21no—
211C2×C31—22no—
212C2×C31—81no—
213C2×C31—82no—
214C2×C31#178011yescharacteristic
215C2×C31#179511yescharacteristic
216C2×C31#179611yescharacteristic
217C22×Q8—21no—
218C22×Q8C2×C4×C3111yescharacteristic
219C22×D4—21no—
220C22×D4C2×C4×C3111yescharacteristic
221C23×C4—21no—
222C23×C4C2×C4×C3111yescharacteristic
223C4⋊Q8—22no—
224C4⋊Q8—22no—
225#126—28no—
226#124—28no—
227#124—22no—
228#124—22no—
229#123—28no—
230C22⋊Q8—28no—
231C22⋊Q8—22no—
232C22⋊Q8—22no—
233#121—22no—
234#121—22no—
235C4×Q8—22no—
236C4×Q8D4×C3114yes—
237C4×D4—22no—
238C4×D4D4×C3114yes—
239#117—22no—
240#117—22no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4C2×C4×C3111yescharacteristic
245C2×C4⋊C4C2×C4×C3111yescharacteristic
246C2×C4⋊C4D4×C3111yescharacteristic
247C2×C4⋊C4D4×C3111yescharacteristic
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4D4×C3111yescharacteristic
254C2×C22⋊C4D4×C3111yescharacteristic
255C2×C42—21no—
256C2×C42C2×C4×C3111yescharacteristic
257C2×C42C23×C3111yescharacteristic
258C4⋊C8—22no—
259C4⋊C8—22no—
260C22⋊C8—22no—
261C22⋊C8—22no—
262C4×C8—22no—
263C4×C8—22no—
264C31#832511yescharacteristic, direct product factor, Sylow
265C24—21no—
266C24C22⋊C4×C3111yescharacteristic
267C2×Q8—24no—
268C2×Q8—21no—
269C2×Q8—24no—
270C2×Q8—82no—
271C2×Q8C22⋊C4×C3112yes—
272C2×D4—24no—
273C2×D4—21no—
274C2×D4—24no—
275C2×D4—82no—
276C2×D4C22⋊C4×C3112yes—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—82no—
285C22×C4C22⋊C4×C3111yescharacteristic
286C22×C4C22⋊C4×C3111yescharacteristic
287C22×C4C22⋊C4×C3111yescharacteristic
288C22×C4C22×C4×C3111yescharacteristic
289C22×C4C2×D4×C3111yescharacteristic
290C22×C4C2×D4×C3111yescharacteristic
291C2×C8—42no—
292C2×C8—42no—
293C4⋊C4—24no—
294C4⋊C4—24no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—24no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4C22⋊C4×C3112yes—
301C4⋊C4C22⋊C4×C3112yes—
302C22⋊C4—24no—
303C22⋊C4—24no—
304C22⋊C4—24no—
305C22⋊C4—24no—
306C22⋊C4—24no—
307C22⋊C4—22no—
308C22⋊C4—22no—
309C42—21no—
310C42—21no—
311C42—24no—
312C42C2×D4×C3112yes—
313C42C2×D4×C3112yes—
314C23—21no—
315C23—22no—
316C23—22no—
317C23—42no—
318C23—42no—
319C23—41no—
320C23—41no—
321C23—42no—
322C23C2×C22⋊C4×C3111yescharacteristic
323Q8—24no—
324Q8—42no—
325Q8—84no—
326D4—24no—
327D4—42no—
328D4—84no—
329C2×C4—22no—
330C2×C4—21no—
331C2×C4—22no—
332C2×C4—21no—
333C2×C4—22no—
334C2×C4—21no—
335C2×C4—22no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—84no—
346C2×C4C2×C22⋊C4×C3111yescharacteristic
347C2×C4C2×C22⋊C4×C3111yescharacteristic
348C2×C4C31×#12012yes—
349C2×C4C31×#12012yes—
350C2×C4C31×#13114yes—
351C8—82no—
352C8—82no—
353C22—21no—
354C22—21no—
355C22—42no—
356C22—41no—
357C22—42no—
358C22—41no—
359C22—41no—
360C22—42no—
361C22—82no—
362C22—84no—
363C22C31×#37911yescharacteristic
364C22C31×#40812yes—
365C4—22no—
366C4—22no—
367C4—42no—
368C4—42no—
369C4—41no—
370C4—41no—
371C4—41no—
372C4—42no—
373C4C31×#40812yes—
374C4C31×#40812yes—
375C2—22no—
376C2—82no—
377C2—81no—
378C2C31×#178011yescharacteristic
379C2C31×#179511yescharacteristic
380C2C31×#179611yescharacteristic
subgroupG/H
center, socleC22×C31#379
commutatorC22×C4C22×C4×C31
Frattini, cosocleC2×C42C23×C31
prime radicalC2×C31×#121C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#1479C2semidirect
3. C31×#1478C2semidirect
4. C31×#1458C2semidirect
5. C31×#1458C2semidirect
123. #8325C31direct (N × H)
164. #1479C2×C31semidirect
165. #1478C2×C31semidirect
166. #1458C2×C31semidirect
167. #1458C2×C31semidirect
264. C31#8325direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5422
C31×#14792
C31×#14782
C31×#14582
#832531