Finite Group Databasebeta

← C31×#8461 #21465220

Subgroups — C31×#8461 #21465220

Order 7936.

682 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
1C31×#3507C211yesmaximal, characteristic
2C31×#1985C211yesmaximal, characteristic, complemented
3C31×#1552C211yesmaximal, characteristic, complemented
4C31×#1551C211yesmaximal, characteristic
5C31×#1531C211yesmaximal, characteristic, complemented
6C31×#1530C211yesmaximal, characteristic
7C31×#1467C211yesmaximal, characteristic, complemented
8C31×#559—21no—
9C31×#557—21no—
10C31×#556—21no—
11C31×#556—22no—
12C31×#554—21no—
13C31×#552—22no—
14C31×#539—22no—
15C31×#536C412yes—
16C31×#533C2211yescharacteristic
17C31×#532C2211yescharacteristic
18C31×#529—22no—
19C31×#520C412yes—
20C2×C4×Q8×C31C2211yescharacteristic
21C31×#432C2211yescharacteristic
22C31×#432C2211yescharacteristic
23C2×C31×C4⋊C8C2211yescharacteristic
24C2×C31×#97C2211yescharacteristic
25C31×#336—22no—
26C31×#333—21no—
27C31×#332—21no—
28C31×#332—21no—
29C31×#331—21no—
30C31×#330—22no—
31C31×#329—22no—
32C31×#324—22no—
33C2×#41×C31C2×C412yes—
34C22×Q8×C31C2×C411yescharacteristic
35C22×C8×C31—21no—
36C22×C8×C31—21no—
37C31×C4⋊Q8—22no—
38C31×C4⋊Q8D411yescharacteristic
39C31×C4⋊Q8D411yescharacteristic
40C31×C4⋊Q8D411yescharacteristic
41C31×#127D411yescharacteristic
42C31×#126—24no—
43C31×#126—24no—
44C31×#125—22no—
45C31×#125—24no—
46C31×#125D412yes—
47C31×#124—22no—
48C31×#124—22no—
49C31×#124D412yes—
50C31×#123—24no—
51C31×#123—24no—
52C31×C22⋊Q8—24no—
53C31×C22⋊Q8—21no—
54C31×C22⋊Q8—21no—
55C31×C22⋊Q8—22no—
56C31×C22⋊Q8—22no—
57C31×#121—21no—
58C31×#121—21no—
59C4×Q8×C31—22no—
60C4×Q8×C31—22no—
61C4×Q8×C31—21no—
62C4×Q8×C31—21no—
63C4×Q8×C31—21no—
64C4×Q8×C31—21no—
65C4×Q8×C31—21no—
66C4×Q8×C31—21no—
67C4×D4×C31—22no—
68C4×D4×C31—22no—
69C4×D4×C31—24no—
70C4×D4×C31—21no—
71C4×D4×C31—21no—
72C31×#117—24no—
73C31×#117—22no—
74C31×#117C2×C412yes—
75C2×C4⋊C4×C31—22no—
76C2×C4⋊C4×C31C2×C411yescharacteristic
77C2×C4⋊C4×C31D412yes—
78C2×C42×C31D412yes—
79C2×C42×C31C2311yescharacteristic
80C31×C4⋊C8—21no—
81C31×C4⋊C8—21no—
82C31×C4⋊C8—21no—
83C31×C4⋊C8—21no—
84C31×C22⋊C8—21no—
85C31×C22⋊C8—21no—
86C31×#97—21no—
87C31×#97—21no—
88C31×#97—21no—
89C31×#97—21no—
90#41×C31—44no—
91C2×Q8×C31—21no—
92C2×Q8×C31—21no—
93C2×Q8×C31—21no—
94C2×Q8×C31—21no—
95C2×Q8×C31—42no—
96C2×Q8×C31C22⋊C412yes—
97C2×D4×C31—22no—
98C2×D4×C31—21no—
99C2×D4×C31—21no—
100C2×D4×C31C22⋊C412yes—
101C22×C4×C31—24no—
102C22×C4×C31—22no—
103C22×C4×C31C22⋊C412yes—
104C22×C4×C31C22⋊C412yes—
105C22×C4×C31C22×C411yescharacteristic
106C22×C4×C31C2×D411yescharacteristic
107C22×C4×C31C2×D411yescharacteristic
108C2×C8×C31—22no—
109C2×C8×C31—41no—
110C2×C8×C31—41no—
111C2×C8×C31—42no—
112C2×C8×C31—41no—
113C2×C8×C31—41no—
114C2×C8×C31—41no—
115C4⋊C4×C31—24no—
116C4⋊C4×C31—24no—
117C4⋊C4×C31—24no—
118C4⋊C4×C31—21no—
119C4⋊C4×C31—21no—
120C4⋊C4×C31—22no—
121C4⋊C4×C31—22no—
122C4⋊C4×C31—22no—
123C4⋊C4×C31—21no—
124C4⋊C4×C31—21no—
125C4⋊C4×C31—21no—
126C4⋊C4×C31—22no—
127C4⋊C4×C31—21no—
128C4⋊C4×C31C22⋊C412yes—
129C4⋊C4×C31C22⋊C412yes—
130C22⋊C4×C31—24no—
131C22⋊C4×C31—24no—
132C22⋊C4×C31—24no—
133C22⋊C4×C31—22no—
134C22⋊C4×C31—22no—
135C22⋊C4×C31—22no—
136C42×C31—24no—
137C42×C31—22no—
138C42×C31—22no—
139C42×C31—22no—
140C42×C31C2×D411yescharacteristic
141C42×C31C2×D411yescharacteristic
142C42×C31C2×D411yescharacteristic
143C42×C31C2×D411yescharacteristic
144#8461C3111yesmaximal, characteristic, direct product factor, Sylow
145C23×C31—22no—
146C23×C31C2×C22⋊C411yescharacteristic
147Q8×C31—42no—
148Q8×C31—42no—
149Q8×C31—42no—
150Q8×C31—41no—
151Q8×C31—41no—
152Q8×C31—81no—
153D4×C31—42no—
154D4×C31—44no—
155D4×C31—41no—
156D4×C31—41no—
157D4×C31—81no—
158C2×C4×C31—22no—
159C2×C4×C31—21no—
160C2×C4×C31—21no—
161C2×C4×C31—22no—
162C2×C4×C31—22no—
163C2×C4×C31—22no—
164C2×C4×C31—22no—
165C2×C4×C31—22no—
166C2×C4×C31—24no—
167C2×C4×C31—41no—
168C2×C4×C31—44no—
169C2×C4×C31—42no—
170C2×C4×C31—84no—
171C2×C4×C31C2×C22⋊C411yescharacteristic
172C2×C4×C31C2×C22⋊C411yescharacteristic
173C2×C4×C31#12011yescharacteristic
174C2×C4×C31#12011yescharacteristic
175C2×C4×C31#12011yescharacteristic
176C2×C4×C31#12011yescharacteristic
177C8×C31—42no—
178C8×C31—81no—
179C8×C31—81no—
180C8×C31—81no—
181#3507C2×C3111yescharacteristic
182#1985C2×C3111yescharacteristic, complemented
183#1552C2×C3111yescharacteristic, complemented
184#1551C2×C3111yescharacteristic
185#1531C2×C3111yescharacteristic, complemented
186#1530C2×C3111yescharacteristic
187#1467C2×C3111yescharacteristic, complemented
188C22×C31—21no—
189C22×C31—21no—
190C22×C31—42no—
191C22×C31—42no—
192C22×C31—42no—
193C22×C31#37911yescharacteristic
194C22×C31#41312yes—
195C4×C31—21no—
196C4×C31—21no—
197C4×C31—21no—
198C4×C31—21no—
199C4×C31—21no—
200C4×C31—42no—
201C4×C31—42no—
202C4×C31—42no—
203C4×C31—42no—
204C4×C31—42no—
205C4×C31#45411yescharacteristic
206C4×C31#45611yescharacteristic
207#559—21no—
208#557—21no—
209#556—21no—
210#556—22no—
211#554—21no—
212#552—22no—
213#539—22no—
214#536C4×C3112yes—
215#533C22×C3111yescharacteristic
216#532C22×C3111yescharacteristic
217#529—22no—
218#520C4×C3112yes—
219C2×C4×Q8C22×C3111yescharacteristic
220#432C22×C3111yescharacteristic
221#432C22×C3111yescharacteristic
222C2×C4⋊C8C22×C3111yescharacteristic
223C2×#97C22×C3111yescharacteristic
224#336—22no—
225#333—21no—
226#332—21no—
227#332—21no—
228#331—21no—
229#330—22no—
230#329—22no—
231#324—22no—
232C2×C31—22no—
233C2×C31—82no—
234C2×C31#178511yescharacteristic
235C2×C31#179311yescharacteristic
236C2×C31#179411yescharacteristic
237C2×#41C2×C4×C3112yes—
238C22×Q8C2×C4×C3111yescharacteristic
239C22×C8—21no—
240C22×C8—21no—
241C4⋊Q8—22no—
242C4⋊Q8D4×C3111yescharacteristic
243C4⋊Q8D4×C3111yescharacteristic
244C4⋊Q8D4×C3111yescharacteristic
245#127D4×C3111yescharacteristic
246#126—24no—
247#126—24no—
248#125—22no—
249#125—24no—
250#125D4×C3112yes—
251#124—22no—
252#124—22no—
253#124D4×C3112yes—
254#123—24no—
255#123—24no—
256C22⋊Q8—24no—
257C22⋊Q8—21no—
258C22⋊Q8—21no—
259C22⋊Q8—22no—
260C22⋊Q8—22no—
261#121—21no—
262#121—21no—
263C4×Q8—22no—
264C4×Q8—22no—
265C4×Q8—21no—
266C4×Q8—21no—
267C4×Q8—21no—
268C4×Q8—21no—
269C4×Q8—21no—
270C4×Q8—21no—
271C4×D4—22no—
272C4×D4—22no—
273C4×D4—24no—
274C4×D4—21no—
275C4×D4—21no—
276#117—24no—
277#117—22no—
278#117C2×C4×C3112yes—
279C2×C4⋊C4—22no—
280C2×C4⋊C4C2×C4×C3111yescharacteristic
281C2×C4⋊C4D4×C3112yes—
282C2×C42D4×C3112yes—
283C2×C42C23×C3111yescharacteristic
284C4⋊C8—21no—
285C4⋊C8—21no—
286C4⋊C8—21no—
287C4⋊C8—21no—
288C22⋊C8—21no—
289C22⋊C8—21no—
290#97—21no—
291#97—21no—
292#97—21no—
293#97—21no—
294C31#846111yescharacteristic, direct product factor, Sylow
295#41—44no—
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8—21no—
300C2×Q8—42no—
301C2×Q8C22⋊C4×C3112yes—
302C2×D4—22no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4C22⋊C4×C3112yes—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4C22⋊C4×C3112yes—
309C22×C4C22⋊C4×C3112yes—
310C22×C4C22×C4×C3111yescharacteristic
311C22×C4C2×D4×C3111yescharacteristic
312C22×C4C2×D4×C3111yescharacteristic
313C2×C8—22no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—42no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—41no—
320C4⋊C4—24no—
321C4⋊C4—24no—
322C4⋊C4—24no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—22no—
332C4⋊C4—21no—
333C4⋊C4C22⋊C4×C3112yes—
334C4⋊C4C22⋊C4×C3112yes—
335C22⋊C4—24no—
336C22⋊C4—24no—
337C22⋊C4—24no—
338C22⋊C4—22no—
339C22⋊C4—22no—
340C22⋊C4—22no—
341C42—24no—
342C42—22no—
343C42—22no—
344C42—22no—
345C42C2×D4×C3111yescharacteristic
346C42C2×D4×C3111yescharacteristic
347C42C2×D4×C3111yescharacteristic
348C42C2×D4×C3111yescharacteristic
349C23—22no—
350C23C2×C22⋊C4×C3111yescharacteristic
351Q8—42no—
352Q8—42no—
353Q8—42no—
354Q8—41no—
355Q8—41no—
356Q8—81no—
357D4—42no—
358D4—44no—
359D4—41no—
360D4—41no—
361D4—81no—
362C2×C4—22no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—24no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—41no—
372C2×C4—44no—
373C2×C4—42no—
374C2×C4—84no—
375C2×C4C2×C22⋊C4×C3111yescharacteristic
376C2×C4C2×C22⋊C4×C3111yescharacteristic
377C2×C4C31×#12011yescharacteristic
378C2×C4C31×#12011yescharacteristic
379C2×C4C31×#12011yescharacteristic
380C2×C4C31×#12011yescharacteristic
381C8—42no—
382C8—81no—
383C8—81no—
384C8—81no—
385C22—21no—
386C22—21no—
387C22—42no—
388C22—42no—
389C22—42no—
390C22C31×#37911yescharacteristic
391C22C31×#41312yes—
392C4—21no—
393C4—21no—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—42no—
398C4—42no—
399C4—42no—
400C4—42no—
401C4—42no—
402C4C31×#45411yescharacteristic
403C4C31×#45611yescharacteristic
404C2—22no—
405C2—82no—
406C2C31×#178511yescharacteristic
407C2C31×#179311yescharacteristic
408C2C31×#179411yescharacteristic
subgroupG/H
center, socleC22×C31#379
commutatorC22×C4C22×C4×C31
Frattini, cosocleC2×C42C23×C31
prime radicalC31×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#1985C2semidirect
3. C31×#1552C2semidirect
5. C31×#1531C2semidirect
7. C31×#1467C2semidirect
144. #8461C31direct (N × H)
182. #1985C2×C31semidirect
183. #1552C2×C31semidirect
185. #1531C2×C31semidirect
187. #1467C2×C31semidirect
294. C31#8461direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35072
C31×#19852
C31×#15522
C31×#15512
C31×#15312
C31×#15302
C31×#14672
#846131