Finite Group Databasebeta

← C31×#8348 #21465107

Subgroups — C31×#8348 #21465107

Order 7936.

726 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3491C211yesmaximal, characteristic
2C31×#1750C211yesmaximal, characteristic, complemented
3C31×#1532C212yesmaximal, complemented
4C31×#1494C212yesmaximal, complemented
5C31×#1464C211yesmaximal, characteristic, complemented
6C31×#552—22no—
7C31×#552—22no—
8C31×#551—22no—
9C31×#551—22no—
10C31×#543C411yescharacteristic
11C31×#539—21no—
12C31×#539C412yes—
13C31×#534C411yescharacteristic
14C31×#529—21no—
15C31×#529C2212yes—
16C2×C31×#123—21no—
17C31×#518—21no—
18C2×C31×#117C2211yescharacteristic
19C31×#432C2212yescomplemented
20C2×C31×C22⋊C8—21no—
21C2×C31×#97C2211yescharacteristic, complemented
22C2×C4×C8×C31C2211yescharacteristic, complemented
23C31×#336—21no—
24C31×#330—22no—
25C31×#329—22no—
26C31×Q8⋊C8—22no—
27C31×D4⋊C8—22no—
28C31×#324—21no—
29C31×#323—21no—
30C22×D4×C31—21no—
31C23×C4×C31D411yescharacteristic
32C22×C8×C31—21no—
33C22×C8×C31—21no—
34C31×#126—22no—
35C31×#126—22no—
36C31×#126—22no—
37C31×#126—22no—
38C31×#125—21no—
39C31×#125—22no—
40C31×#125—22no—
41C31×#125D412yes—
42C31×#124—21no—
43C31×#124D412yes—
44C31×#123—22no—
45C31×#123—22no—
46C31×#123—22no—
47C31×#123—22no—
48C31×#123—22no—
49C31×#123—22no—
50C31×#123—22no—
51C31×C22⋊Q8—22no—
52C31×C22⋊Q8—22no—
53C31×C22⋊Q8—22no—
54C31×C22⋊Q8—21no—
55C31×C22⋊Q8—22no—
56C31×C22⋊Q8C2×C412yes—
57C31×#121—21no—
58C31×#121C2×C412yes—
59C31×#120—22no—
60C4×Q8×C31—21no—
61C4×Q8×C31D412yes—
62C4×D4×C31—22no—
63C4×D4×C31—22no—
64C4×D4×C31—21no—
65C4×D4×C31—22no—
66C4×D4×C31—22no—
67C4×D4×C31D412yes—
68C31×#117—22no—
69C31×#117—22no—
70C31×#117—22no—
71C31×#117—22no—
72C31×#117—22no—
73C2×C4⋊C4×C31—21no—
74C2×C4⋊C4×C31—21no—
75C2×C4⋊C4×C31C2×C411yescharacteristic
76C2×C4⋊C4×C31D411yescharacteristic
77C2×C22⋊C4×C31—22no—
78C2×C22⋊C4×C31—21no—
79C2×C22⋊C4×C31—21no—
80C2×C22⋊C4×C31—21no—
81C2×C22⋊C4×C31C2×C411yescharacteristic
82C2×C22⋊C4×C31D411yescharacteristic
83C2×C42×C31D411yescharacteristic
84C2×C42×C31C2311yescharacteristic
85C31×C4⋊C8—22no—
86C31×C22⋊C8—24no—
87C31×C22⋊C8—22no—
88C31×#97—22no—
89C31×#97—22no—
90C4×C8×C31—22no—
91C24×C31—21no—
92C24×C31C22⋊C411yescharacteristic
93C2×Q8×C31—21no—
94C2×Q8×C31C22⋊C412yes—
95C2×D4×C31—22no—
96C2×D4×C31—22no—
97C2×D4×C31—21no—
98C2×D4×C31—42no—
99C2×D4×C31—42no—
100C2×D4×C31C22⋊C412yes—
101C22×C4×C31—21no—
102C22×C4×C31—22no—
103C22×C4×C31—21no—
104C22×C4×C31—21no—
105C22×C4×C31—22no—
106C22×C4×C31—21no—
107C22×C4×C31—21no—
108C22×C4×C31—22no—
109C22×C4×C31—41no—
110C22×C4×C31C22⋊C411yescharacteristic
111C22×C4×C31C22⋊C411yescharacteristic
112C22×C4×C31C22⋊C411yescharacteristic
113C22×C4×C31C22×C411yescharacteristic
114C22×C4×C31C2×D411yescharacteristic
115C22×C4×C31C2×D411yescharacteristic
116C2×C8×C31—22no—
117C2×C8×C31—22no—
118C2×C8×C31—41no—
119C2×C8×C31—41no—
120C2×C8×C31—41no—
121C2×C8×C31—41no—
122C2×C8×C31—42no—
123C4⋊C4×C31—22no—
124C4⋊C4×C31—22no—
125C4⋊C4×C31—22no—
126C4⋊C4×C31—22no—
127C4⋊C4×C31—21no—
128C4⋊C4×C31—21no—
129C4⋊C4×C31—22no—
130C4⋊C4×C31—22no—
131C4⋊C4×C31—22no—
132C4⋊C4×C31—22no—
133C4⋊C4×C31—22no—
134C4⋊C4×C31—22no—
135C4⋊C4×C31C22⋊C412yes—
136C4⋊C4×C31C22⋊C412yes—
137C22⋊C4×C31—22no—
138C22⋊C4×C31—22no—
139C22⋊C4×C31—22no—
140C22⋊C4×C31—22no—
141C22⋊C4×C31—22no—
142C22⋊C4×C31—22no—
143C22⋊C4×C31—22no—
144C22⋊C4×C31—22no—
145C22⋊C4×C31—22no—
146C22⋊C4×C31—22no—
147C22⋊C4×C31—22no—
148C22⋊C4×C31—22no—
149C42×C31—22no—
150C42×C31—21no—
151C42×C31—21no—
152C42×C31—22no—
153C42×C31C2×D412yes—
154C42×C31C2×D412yes—
155#8348C3111yesmaximal, characteristic, direct product factor, Sylow
156C23×C31—21no—
157C23×C31—21no—
158C23×C31—22no—
159C23×C31—41no—
160C23×C31—41no—
161C23×C31—42no—
162C23×C31—81no—
163C23×C31—81no—
164C23×C31#10412yes—
165C23×C31C2×C22⋊C411yescharacteristic
166Q8×C31—42no—
167Q8×C31—81no—
168D4×C31—42no—
169D4×C31—44no—
170D4×C31—42no—
171D4×C31—42no—
172D4×C31—81no—
173C2×C4×C31—21no—
174C2×C4×C31—21no—
175C2×C4×C31—21no—
176C2×C4×C31—21no—
177C2×C4×C31—22no—
178C2×C4×C31—22no—
179C2×C4×C31—22no—
180C2×C4×C31—21no—
181C2×C4×C31—21no—
182C2×C4×C31—21no—
183C2×C4×C31—41no—
184C2×C4×C31—41no—
185C2×C4×C31—42no—
186C2×C4×C31—42no—
187C2×C4×C31—42no—
188C2×C4×C31—41no—
189C2×C4×C31—41no—
190C2×C4×C31—82no—
191C2×C4×C31—81no—
192C2×C4×C31—82no—
193C2×C4×C31—81no—
194C2×C4×C31C2×C22⋊C412yes—
195C2×C4×C31#12012yes—
196C2×C4×C31#12011yescharacteristic
197C2×C4×C31#12011yescharacteristic
198C8×C31—42no—
199C8×C31—42no—
200C8×C31—82no—
201#3491C2×C3111yescharacteristic
202#1750C2×C3111yescharacteristic, complemented
203#1532C2×C3112yescomplemented
204#1494C2×C3112yescomplemented
205#1464C2×C3111yescharacteristic, complemented
206C22×C31—21no—
207C22×C31—21no—
208C22×C31—22no—
209C22×C31—42no—
210C22×C31—41no—
211C22×C31—41no—
212C22×C31—42no—
213C22×C31—42no—
214C22×C31—42no—
215C22×C31—81no—
216C22×C31—162no—
217C22×C31#37911yescharacteristic
218C22×C31C2×#10411yescharacteristic
219C22×C31#42111yescharacteristic
220C4×C31—22no—
221C4×C31—21no—
222C4×C31—21no—
223C4×C31—22no—
224C4×C31—41no—
225C4×C31—42no—
226C4×C31—41no—
227C4×C31—42no—
228C4×C31—41no—
229C4×C31—42no—
230#552—22no—
231#552—22no—
232#551—22no—
233#551—22no—
234#543C4×C3111yescharacteristic
235#539—21no—
236#539C4×C3112yes—
237#534C4×C3111yescharacteristic
238#529—21no—
239#529C22×C3112yes—
240C2×#123—21no—
241#518—21no—
242C2×#117C22×C3111yescharacteristic
243#432C22×C3112yescomplemented
244C2×C22⋊C8—21no—
245C2×#97C22×C3111yescharacteristic, complemented
246C2×C4×C8C22×C3111yescharacteristic, complemented
247#336—21no—
248#330—22no—
249#329—22no—
250Q8⋊C8—22no—
251D4⋊C8—22no—
252#324—21no—
253#323—21no—
254C2×C31—21no—
255C2×C31—21no—
256C2×C31—42no—
257C2×C31—82no—
258C2×C31#178911yescharacteristic
259C2×C31#179111yescharacteristic
260C2×C31#179411yescharacteristic
261C22×D4—21no—
262C23×C4D4×C3111yescharacteristic
263C22×C8—21no—
264C22×C8—21no—
265#126—22no—
266#126—22no—
267#126—22no—
268#126—22no—
269#125—21no—
270#125—22no—
271#125—22no—
272#125D4×C3112yes—
273#124—21no—
274#124D4×C3112yes—
275#123—22no—
276#123—22no—
277#123—22no—
278#123—22no—
279#123—22no—
280#123—22no—
281#123—22no—
282C22⋊Q8—22no—
283C22⋊Q8—22no—
284C22⋊Q8—22no—
285C22⋊Q8—21no—
286C22⋊Q8—22no—
287C22⋊Q8C2×C4×C3112yes—
288#121—21no—
289#121C2×C4×C3112yes—
290#120—22no—
291C4×Q8—21no—
292C4×Q8D4×C3112yes—
293C4×D4—22no—
294C4×D4—22no—
295C4×D4—21no—
296C4×D4—22no—
297C4×D4—22no—
298C4×D4D4×C3112yes—
299#117—22no—
300#117—22no—
301#117—22no—
302#117—22no—
303#117—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4C2×C4×C3111yescharacteristic
307C2×C4⋊C4D4×C3111yescharacteristic
308C2×C22⋊C4—22no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—21no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4C2×C4×C3111yescharacteristic
313C2×C22⋊C4D4×C3111yescharacteristic
314C2×C42D4×C3111yescharacteristic
315C2×C42C23×C3111yescharacteristic
316C4⋊C8—22no—
317C22⋊C8—24no—
318C22⋊C8—22no—
319#97—22no—
320#97—22no—
321C4×C8—22no—
322C31#834811yescharacteristic, direct product factor, Sylow
323C24—21no—
324C24C22⋊C4×C3111yescharacteristic
325C2×Q8—21no—
326C2×Q8C22⋊C4×C3112yes—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—21no—
330C2×D4—42no—
331C2×D4—42no—
332C2×D4C22⋊C4×C3112yes—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—41no—
342C22×C4C22⋊C4×C3111yescharacteristic
343C22×C4C22⋊C4×C3111yescharacteristic
344C22×C4C22⋊C4×C3111yescharacteristic
345C22×C4C22×C4×C3111yescharacteristic
346C22×C4C2×D4×C3111yescharacteristic
347C22×C4C2×D4×C3111yescharacteristic
348C2×C8—22no—
349C2×C8—22no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—42no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4C22⋊C4×C3112yes—
368C4⋊C4C22⋊C4×C3112yes—
369C22⋊C4—22no—
370C22⋊C4—22no—
371C22⋊C4—22no—
372C22⋊C4—22no—
373C22⋊C4—22no—
374C22⋊C4—22no—
375C22⋊C4—22no—
376C22⋊C4—22no—
377C22⋊C4—22no—
378C22⋊C4—22no—
379C22⋊C4—22no—
380C22⋊C4—22no—
381C42—22no—
382C42—21no—
383C42—21no—
384C42—22no—
385C42C2×D4×C3112yes—
386C42C2×D4×C3112yes—
387C23—21no—
388C23—21no—
389C23—22no—
390C23—41no—
391C23—41no—
392C23—42no—
393C23—81no—
394C23—81no—
395C23C31×#10412yes—
396C23C2×C22⋊C4×C3111yescharacteristic
397Q8—42no—
398Q8—81no—
399D4—42no—
400D4—44no—
401D4—42no—
402D4—42no—
403D4—81no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—21no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4C2×C22⋊C4×C3112yes—
426C2×C4C31×#12011yescharacteristic
427C2×C4C31×#12011yescharacteristic
428C2×C4C31×#12012yes—
429C8—42no—
430C8—42no—
431C8—82no—
432C22—21no—
433C22—21no—
434C22—22no—
435C22—42no—
436C22—41no—
437C22—41no—
438C22—42no—
439C22—42no—
440C22—42no—
441C22—81no—
442C22—162no—
443C22C31×#37911yescharacteristic
444C22C2×C31×#10411yescharacteristic
445C22C31×#42111yescharacteristic
446C4—21no—
447C4—21no—
448C4—22no—
449C4—22no—
450C4—41no—
451C4—42no—
452C4—41no—
453C4—42no—
454C4—41no—
455C4—42no—
456C2—21no—
457C2—21no—
458C2—42no—
459C2—82no—
460C2C31×#178911yescharacteristic
461C2C31×#179111yescharacteristic
462C2C31×#179411yescharacteristic
subgroupG/H
center, socleC22×C31#379
commutatorC22×C4C22×C4×C31
Frattini, cosocleC2×C42C23×C31
prime radicalC31×#534C4
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×#1750C2semidirect
3. C31×#1532C2semidirect
4. C31×#1494C2semidirect
5. C31×#1464C2semidirect
19. C31×#432C22semidirect
21. C2×C31×#97C22semidirect
22. C2×C4×C8×C31C22semidirect
155. #8348C31direct (N × H)
202. #1750C2×C31semidirect
203. #1532C2×C31semidirect
204. #1494C2×C31semidirect
205. #1464C2×C31semidirect
243. #432C22×C31semidirect
245. C2×#97C22×C31semidirect
246. C2×C4×C8C22×C31semidirect
322. C31#8348direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#34912
C31×#17502
C31×#15322
C31×#14942
C31×#14642
#834831