Finite Group Databasebeta

← C31×#8320 #21465079

Subgroups — C31×#8320 #21465079

Order 7936.

794 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3460C211yesmaximal, characteristic
2C31×#1954C211yesmaximal, characteristic, complemented
3C31×#1477C212yesmaximal, complemented
4C31×#1476C212yesmaximal, complemented
5C31×#1460C211yesmaximal, characteristic, complemented
6C31×#544C411yescharacteristic
7C31×#542—22no—
8C31×#540—22no—
9C31×#538—22no—
10C31×#537C411yescharacteristic
11C31×#535C412yes—
12C31×#532—22no—
13C31×#529—22no—
14C31×#525—22no—
15C4×#41×C31C2212yes—
16C2×C4×D4×C31C2211yescharacteristic
17C31×#431C2212yes—
18C2×C31×C4⋊C8C2211yescharacteristic
19C2×C31×C22⋊C8—22no—
20C2×C4×C8×C31C2211yescharacteristic
21C31×Q8⋊C8—22no—
22C31×Q8⋊C8—22no—
23C31×D4⋊C8—22no—
24C31×D4⋊C8—22no—
25C31×#323—22no—
26C2×#41×C31—22no—
27C2×#41×C31C2×C412yes—
28C22×D4×C31C2×C411yescharacteristic
29C23×C4×C31D412yes—
30C22×C8×C31—21no—
31C22×C8×C31—21no—
32C31×C4⋊Q8—22no—
33C31×#127—22no—
34C31×#126—24no—
35C31×#125—22no—
36C31×#124—24no—
37C31×#124—22no—
38C31×#123—24no—
39C31×#123—24no—
40C31×C22⋊Q8—22no—
41C31×C22⋊Q8—22no—
42C31×C22⋊Q8—22no—
43C31×#121—22no—
44C31×#121—24no—
45C31×#121—22no—
46C31×#121—22no—
47C31×#121—22no—
48C31×#121—22no—
49C31×#120—24no—
50C4×Q8×C31D412yes—
51C4×Q8×C31D412yes—
52C4×D4×C31—24no—
53C4×D4×C31—22no—
54C4×D4×C31—22no—
55C4×D4×C31—22no—
56C4×D4×C31—22no—
57C4×D4×C31D412yes—
58C4×D4×C31D412yes—
59C31×#117—22no—
60C31×#117—22no—
61C31×#117C2×C412yes—
62C2×C4⋊C4×C31C2×C411yescharacteristic
63C2×C22⋊C4×C31D412yes—
64C2×C42×C31—22no—
65C2×C42×C31C2311yescharacteristic
66C31×C4⋊C8—22no—
67C31×C4⋊C8—22no—
68C31×C22⋊C8—22no—
69C31×C22⋊C8—44no—
70C4×C8×C31—22no—
71C4×C8×C31—22no—
72C24×C31C22⋊C412yes—
73#41×C31—24no—
74#41×C31—24no—
75#41×C31—82no—
76#41×C31—82no—
77C2×Q8×C31—22no—
78C2×Q8×C31C22⋊C412yes—
79C2×D4×C31—22no—
80C2×D4×C31—24no—
81C2×D4×C31—24no—
82C2×D4×C31—22no—
83C2×D4×C31—22no—
84C2×D4×C31—22no—
85C2×D4×C31—22no—
86C2×D4×C31C22⋊C412yes—
87C22×C4×C31—22no—
88C22×C4×C31—22no—
89C22×C4×C31—22no—
90C22×C4×C31—22no—
91C22×C4×C31—22no—
92C22×C4×C31—42no—
93C22×C4×C31—42no—
94C22×C4×C31C22⋊C412yes—
95C22×C4×C31C22×C411yescharacteristic
96C22×C4×C31C2×D411yescharacteristic
97C22×C4×C31C2×D411yescharacteristic
98C2×C8×C31—22no—
99C2×C8×C31—41no—
100C2×C8×C31—41no—
101C2×C8×C31—42no—
102C2×C8×C31—42no—
103C2×C8×C31—41no—
104C4⋊C4×C31—22no—
105C4⋊C4×C31—22no—
106C4⋊C4×C31—24no—
107C4⋊C4×C31—22no—
108C4⋊C4×C31—22no—
109C4⋊C4×C31—22no—
110C4⋊C4×C31C22⋊C412yes—
111C4⋊C4×C31C22⋊C412yes—
112C22⋊C4×C31—24no—
113C22⋊C4×C31—24no—
114C22⋊C4×C31—24no—
115C22⋊C4×C31—22no—
116C22⋊C4×C31—24no—
117C22⋊C4×C31—22no—
118C22⋊C4×C31—22no—
119C42×C31—22no—
120C42×C31—22no—
121C42×C31—22no—
122C42×C31C2×D412yes—
123C42×C31C2×D412yes—
124#8320C3111yesmaximal, characteristic, direct product factor, Sylow
125C23×C31—22no—
126C23×C31—22no—
127C23×C31—22no—
128C23×C31—22no—
129C23×C31—42no—
130C23×C31—42no—
131C23×C31C2×C22⋊C411yescharacteristic
132Q8×C31—24no—
133Q8×C31—82no—
134D4×C31—24no—
135D4×C31—42no—
136D4×C31—42no—
137D4×C31—44no—
138D4×C31—84no—
139D4×C31—82no—
140D4×C31—84no—
141C2×C4×C31—21no—
142C2×C4×C31—21no—
143C2×C4×C31—22no—
144C2×C4×C31—22no—
145C2×C4×C31—22no—
146C2×C4×C31—22no—
147C2×C4×C31—41no—
148C2×C4×C31—44no—
149C2×C4×C31—42no—
150C2×C4×C31—42no—
151C2×C4×C31—42no—
152C2×C4×C31—42no—
153C2×C4×C31—42no—
154C2×C4×C31—82no—
155C2×C4×C31—82no—
156C2×C4×C31C2×C22⋊C412yes—
157C2×C4×C31#12012yes—
158C2×C4×C31#12011yescharacteristic
159C2×C4×C31#12011yescharacteristic
160C2×C4×C31#13112yes—
161C2×C4×C31#13112yes—
162C8×C31—42no—
163C8×C31—82no—
164C8×C31—81no—
165#3460C2×C3111yescharacteristic
166#1954C2×C3111yescharacteristic, complemented
167#1477C2×C3112yescomplemented
168#1476C2×C3112yescomplemented
169#1460C2×C3111yescharacteristic, complemented
170C22×C31—21no—
171C22×C31—21no—
172C22×C31—44no—
173C22×C31—42no—
174C22×C31—42no—
175C22×C31—42no—
176C22×C31—42no—
177C22×C31—42no—
178C22×C31—42no—
179C22×C31—82no—
180C22×C31#37911yescharacteristic
181C22×C31#40811yescharacteristic
182C22×C31#40811yescharacteristic
183C4×C31—21no—
184C4×C31—22no—
185C4×C31—42no—
186C4×C31—42no—
187C4×C31—42no—
188C4×C31—42no—
189C4×C31#40814yes—
190C4×C31#45311yescharacteristic
191C4×C31#45511yescharacteristic
192#544C4×C3111yescharacteristic
193#542—22no—
194#540—22no—
195#538—22no—
196#537C4×C3111yescharacteristic
197#535C4×C3112yes—
198#532—22no—
199#529—22no—
200#525—22no—
201C4×#41C22×C3112yes—
202C2×C4×D4C22×C3111yescharacteristic
203#431C22×C3112yes—
204C2×C4⋊C8C22×C3111yescharacteristic
205C2×C22⋊C8—22no—
206C2×C4×C8C22×C3111yescharacteristic
207Q8⋊C8—22no—
208Q8⋊C8—22no—
209D4⋊C8—22no—
210D4⋊C8—22no—
211#323—22no—
212C2×C31—21no—
213C2×C31—21no—
214C2×C31—82no—
215C2×C31—82no—
216C2×C31#178011yescharacteristic
217C2×C31#179211yescharacteristic
218C2×C31#179411yescharacteristic
219C2×#41—22no—
220C2×#41C2×C4×C3112yes—
221C22×D4C2×C4×C3111yescharacteristic
222C23×C4D4×C3112yes—
223C22×C8—21no—
224C22×C8—21no—
225C4⋊Q8—22no—
226#127—22no—
227#126—24no—
228#125—22no—
229#124—24no—
230#124—22no—
231#123—24no—
232#123—24no—
233C22⋊Q8—22no—
234C22⋊Q8—22no—
235C22⋊Q8—22no—
236#121—22no—
237#121—24no—
238#121—22no—
239#121—22no—
240#121—22no—
241#121—22no—
242#120—24no—
243C4×Q8D4×C3112yes—
244C4×Q8D4×C3112yes—
245C4×D4—24no—
246C4×D4—22no—
247C4×D4—22no—
248C4×D4—22no—
249C4×D4—22no—
250C4×D4D4×C3112yes—
251C4×D4D4×C3112yes—
252#117—22no—
253#117—22no—
254#117C2×C4×C3112yes—
255C2×C4⋊C4C2×C4×C3111yescharacteristic
256C2×C22⋊C4D4×C3112yes—
257C2×C42—22no—
258C2×C42C23×C3111yescharacteristic
259C4⋊C8—22no—
260C4⋊C8—22no—
261C22⋊C8—22no—
262C22⋊C8—44no—
263C4×C8—22no—
264C4×C8—22no—
265C31#832011yescharacteristic, direct product factor, Sylow
266C24C22⋊C4×C3112yes—
267#41—24no—
268#41—24no—
269#41—82no—
270#41—82no—
271C2×Q8—22no—
272C2×Q8C22⋊C4×C3112yes—
273C2×D4—22no—
274C2×D4—24no—
275C2×D4—24no—
276C2×D4—22no—
277C2×D4—22no—
278C2×D4—22no—
279C2×D4—22no—
280C2×D4C22⋊C4×C3112yes—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4C22⋊C4×C3112yes—
289C22×C4C22×C4×C3111yescharacteristic
290C22×C4C2×D4×C3111yescharacteristic
291C22×C4C2×D4×C3111yescharacteristic
292C2×C8—22no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—41no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—24no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4C22⋊C4×C3112yes—
305C4⋊C4C22⋊C4×C3112yes—
306C22⋊C4—24no—
307C22⋊C4—24no—
308C22⋊C4—24no—
309C22⋊C4—22no—
310C22⋊C4—24no—
311C22⋊C4—22no—
312C22⋊C4—22no—
313C42—22no—
314C42—22no—
315C42—22no—
316C42C2×D4×C3112yes—
317C42C2×D4×C3112yes—
318C23—22no—
319C23—22no—
320C23—22no—
321C23—22no—
322C23—42no—
323C23—42no—
324C23C2×C22⋊C4×C3111yescharacteristic
325Q8—24no—
326Q8—82no—
327D4—24no—
328D4—42no—
329D4—42no—
330D4—44no—
331D4—84no—
332D4—82no—
333D4—84no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—41no—
341C2×C4—44no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4C2×C22⋊C4×C3112yes—
350C2×C4C31×#12012yes—
351C2×C4C31×#12011yescharacteristic
352C2×C4C31×#12011yescharacteristic
353C2×C4C31×#13112yes—
354C2×C4C31×#13112yes—
355C8—42no—
356C8—82no—
357C8—81no—
358C22—21no—
359C22—21no—
360C22—44no—
361C22—42no—
362C22—42no—
363C22—42no—
364C22—42no—
365C22—42no—
366C22—42no—
367C22—82no—
368C22C31×#37911yescharacteristic
369C22C31×#40811yescharacteristic
370C22C31×#40811yescharacteristic
371C4—21no—
372C4—22no—
373C4—42no—
374C4—42no—
375C4—42no—
376C4—42no—
377C4C31×#40814yes—
378C4C31×#45311yescharacteristic
379C4C31×#45511yescharacteristic
380C2—21no—
381C2—21no—
382C2—82no—
383C2—82no—
384C2C31×#178011yescharacteristic
385C2C31×#179211yescharacteristic
386C2C31×#179411yescharacteristic
subgroupG/H
centerC2×C4×C31#120
commutatorC22×C4C22×C4×C31
Frattini, cosocleC2×C42C23×C31
socleC22×C31#379
prime radicalC31×#3460C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#1954C2semidirect
3. C31×#1477C2semidirect
4. C31×#1476C2semidirect
5. C31×#1460C2semidirect
124. #8320C31direct (N × H)
166. #1954C2×C31semidirect
167. #1477C2×C31semidirect
168. #1476C2×C31semidirect
169. #1460C2×C31semidirect
265. C31#8320direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#34602
C31×#19542
C31×#14772
C31×#14762
C31×#14602
#832031