Finite Group Databasebeta

← C31×#8423 #21465182

Subgroups — C31×#8423 #21465182

Order 7936.

772 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3511C211yesmaximal, characteristic
2C31×#1981C211yesmaximal, characteristic, complemented
3C31×#1806C211yesmaximal, characteristic, complemented
4C31×#1548C211yesmaximal, characteristic, complemented
5C31×#1547C211yesmaximal, characteristic, complemented
6C31×#1502C211yesmaximal, characteristic, complemented
7C31×#1477C211yesmaximal, characteristic, complemented
8C31×#556—21no—
9C31×#555C412yescomplemented
10C31×#554C412yescomplemented
11C31×#553—22no—
12C31×#548—22no—
13C31×#547—21no—
14C31×#542—22no—
15C31×#537—21no—
16C31×#533—21no—
17C31×#532—21no—
18C31×#532C2211yescharacteristic
19C2×C31×C4⋊Q8C2211yescharacteristic
20C31×#520—21no—
21C4×#41×C31C2211yescharacteristic
22C31×#431C2211yescharacteristic
23C31×#423C2211yescharacteristic
24C2×C31×C4⋊C8C2211yescharacteristic
25C2×C31×#104—21no—
26C2×C31×#103—22no—
27C4×C4⋊C4×C31C2211yescharacteristic
28C31×#339—21no—
29C31×#333—22no—
30C31×#332—21no—
31C31×#331—21no—
32C31×#327—22no—
33C31×Q8⋊C8—21no—
34C31×D4⋊C8—21no—
35C2×#41×C31—22no—
36C2×#41×C31—21no—
37C2×#41×C31D411yescharacteristic
38C22×Q8×C31D412yes—
39C2×M16×C31—21no—
40C22×C8×C31—21no—
41C31×C4⋊Q8—21no—
42C31×C4⋊Q8—21no—
43C31×C4⋊Q8C2×C412yes—
44C31×C4⋊Q8D412yes—
45C31×C4⋊Q8D411yescharacteristic
46C31×#127D411yescharacteristic
47C31×#126—24no—
48C31×#125—22no—
49C31×#124—22no—
50C31×#124—24no—
51C31×#124—21no—
52C31×#123—24no—
53C31×C22⋊Q8—22no—
54C31×C22⋊Q8—22no—
55C31×C22⋊Q8—22no—
56C31×C22⋊Q8—24no—
57C31×#121—22no—
58C31×#121—24no—
59C31×#121—22no—
60C31×#121—22no—
61C31×#121—21no—
62C31×#121D412yes—
63C4×Q8×C31—22no—
64C4×Q8×C31—22no—
65C4×Q8×C31C2×C411yescharacteristic
66C4×Q8×C31D411yescharacteristic
67C4×D4×C31—22no—
68C4×D4×C31—24no—
69C4×D4×C31—22no—
70C4×D4×C31—21no—
71C4×D4×C31—21no—
72C4×D4×C31—21no—
73C4×D4×C31C2×C411yescharacteristic
74C4×D4×C31C2×C412yes—
75C4×D4×C31D411yescharacteristic
76C31×#117—22no—
77C31×#117—22no—
78C31×#117—21no—
79C31×#117D411yescharacteristic
80C2×C4⋊C4×C31—21no—
81C2×C4⋊C4×C31—21no—
82C2×C4⋊C4×C31—22no—
83C2×C4⋊C4×C31—21no—
84C2×C42×C31—21no—
85C2×C42×C31—21no—
86C2×C42×C31C2311yescharacteristic
87C31×C4⋊C8—21no—
88C31×C4⋊C8—21no—
89C31×C4⋊C8—21no—
90C31×C4⋊C8—21no—
91C31×C4⋊C8—21no—
92C31×#104—42no—
93C31×#103—44no—
94C31×C22⋊C8—21no—
95C4×C8×C31—21no—
96C31×#95—21no—
97#41×C31—21no—
98#41×C31—21no—
99#41×C31—21no—
100#41×C31—21no—
101#41×C31—41no—
102#41×C31—41no—
103#41×C31—41no—
104#41×C31—41no—
105#41×C31—82no—
106#41×C31—82no—
107C2×Q8×C31—21no—
108C2×Q8×C31—22no—
109C2×Q8×C31—42no—
110C2×Q8×C31—42no—
111C2×Q8×C31C22⋊C411yescharacteristic
112C2×Q8×C31C22⋊C414yes—
113C2×D4×C31—21no—
114C2×D4×C31—22no—
115C2×D4×C31—24no—
116C2×D4×C31—21no—
117C2×D4×C31—21no—
118C2×D4×C31C22⋊C411yescharacteristic
119C2×D4×C31C22⋊C412yes—
120C22×C4×C31—24no—
121C22×C4×C31—22no—
122C22×C4×C31—21no—
123C22×C4×C31—22no—
124C22×C4×C31—21no—
125C22×C4×C31—21no—
126C22×C4×C31—41no—
127C22×C4×C31C2×D411yescharacteristic
128C22×C4×C31C2×D411yescharacteristic
129C22×C4×C31C2×D411yescharacteristic
130M16×C31—41no—
131M16×C31—41no—
132C2×C8×C31—42no—
133C2×C8×C31—41no—
134C2×C8×C31—41no—
135C2×C8×C31—41no—
136C4⋊C4×C31—24no—
137C4⋊C4×C31—24no—
138C4⋊C4×C31—22no—
139C4⋊C4×C31—22no—
140C4⋊C4×C31—21no—
141C4⋊C4×C31—22no—
142C4⋊C4×C31—22no—
143C4⋊C4×C31—21no—
144C4⋊C4×C31—21no—
145C4⋊C4×C31—22no—
146C4⋊C4×C31—41no—
147C4⋊C4×C31C22⋊C411yescharacteristic
148C4⋊C4×C31C22⋊C411yescharacteristic
149C4⋊C4×C31C22⋊C412yes—
150C22⋊C4×C31—24no—
151C22⋊C4×C31—24no—
152C22⋊C4×C31—24no—
153C22⋊C4×C31—21no—
154C22⋊C4×C31—21no—
155C22⋊C4×C31—22no—
156C22⋊C4×C31—21no—
157C22⋊C4×C31—22no—
158C42×C31—24no—
159C42×C31—21no—
160C42×C31—21no—
161C42×C31—21no—
162C42×C31—41no—
163C42×C31—41no—
164C42×C31C22×C411yescharacteristic
165C42×C31C2×D411yescharacteristic
166C42×C31C2×D411yescharacteristic
167C42×C31C2×D411yescharacteristic
168#8423C3111yesmaximal, characteristic, direct product factor, Sylow
169C23×C31—21no—
170C23×C31—22no—
171C23×C31#12011yescharacteristic
172Q8×C31—21no—
173Q8×C31—21no—
174Q8×C31—44no—
175Q8×C31—41no—
176Q8×C31—41no—
177Q8×C31—42no—
178Q8×C31—42no—
179D4×C31—21no—
180D4×C31—21no—
181D4×C31—42no—
182D4×C31—41no—
183D4×C31—42no—
184D4×C31—41no—
185D4×C31—42no—
186D4×C31—41no—
187D4×C31—41no—
188D4×C31—41no—
189D4×C31—41no—
190D4×C31—84no—
191C2×C4×C31—21no—
192C2×C4×C31—21no—
193C2×C4×C31—21no—
194C2×C4×C31—21no—
195C2×C4×C31—22no—
196C2×C4×C31—21no—
197C2×C4×C31—24no—
198C2×C4×C31—41no—
199C2×C4×C31—42no—
200C2×C4×C31—42no—
201C2×C4×C31—42no—
202C2×C4×C31—41no—
203C2×C4×C31—41no—
204C2×C4×C31—41no—
205C2×C4×C31—41no—
206C2×C4×C31—42no—
207C2×C4×C31—82no—
208C2×C4×C31—84no—
209C2×C4×C31—81no—
210C2×C4×C31—81no—
211C2×C4×C31C2×C22⋊C411yescharacteristic
212C2×C4×C31C2×C22⋊C411yescharacteristic
213C2×C4×C31C2×C22⋊C411yescharacteristic
214C2×C4×C31#12011yescharacteristic
215C2×C4×C31#12011yescharacteristic
216C2×C4×C31#12011yescharacteristic
217C2×C4×C31#13511yescharacteristic
218C2×C4×C31#13511yescharacteristic
219C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
220C2×C4×C31#13711yescharacteristic
221C8×C31—81no—
222C8×C31—81no—
223C8×C31—81no—
224#3511C2×C3111yescharacteristic
225#1981C2×C3111yescharacteristic, complemented
226#1806C2×C3111yescharacteristic, complemented
227#1548C2×C3111yescharacteristic, complemented
228#1547C2×C3111yescharacteristic, complemented
229#1502C2×C3111yescharacteristic, complemented
230#1477C2×C3111yescharacteristic, complemented
231C22×C31—21no—
232C22×C31—21no—
233C22×C31—41no—
234C22×C31—42no—
235C22×C31—41no—
236C22×C31—42no—
237C22×C31—41no—
238C22×C31—42no—
239C22×C31#37911yescharacteristic
240C22×C31#44911yescharacteristic
241C22×C31#45211yescharacteristic
242C4×C31—21no—
243C4×C31—21no—
244C4×C31—21no—
245C4×C31—41no—
246C4×C31—44no—
247C4×C31—41no—
248C4×C31—41no—
249C4×C31—42no—
250C4×C31—82no—
251C4×C31#41611yescharacteristic
252C4×C31#41711yescharacteristic
253C4×C31#44911yescharacteristic
254C4×C31#45211yescharacteristic
255C4×C31#45311yescharacteristic
256C4×C31#45511yescharacteristic
257#556—21no—
258#555C4×C3112yescomplemented
259#554C4×C3112yescomplemented
260#553—22no—
261#548—22no—
262#547—21no—
263#542—22no—
264#537—21no—
265#533—21no—
266#532—21no—
267#532C22×C3111yescharacteristic
268C2×C4⋊Q8C22×C3111yescharacteristic
269#520—21no—
270C4×#41C22×C3111yescharacteristic
271#431C22×C3111yescharacteristic
272#423C22×C3111yescharacteristic
273C2×C4⋊C8C22×C3111yescharacteristic
274C2×#104—21no—
275C2×#103—22no—
276C4×C4⋊C4C22×C3111yescharacteristic
277#339—21no—
278#333—22no—
279#332—21no—
280#331—21no—
281#327—22no—
282Q8⋊C8—21no—
283D4⋊C8—21no—
284C2×C31—21no—
285C2×C31—21no—
286C2×C31—81no—
287C2×C31—82no—
288C2×C31#178811yescharacteristic
289C2×C31#179211yescharacteristic
290C2×C31#179611yescharacteristic
291C2×#41—22no—
292C2×#41—21no—
293C2×#41D4×C3111yescharacteristic
294C22×Q8D4×C3112yes—
295C2×M16—21no—
296C22×C8—21no—
297C4⋊Q8—21no—
298C4⋊Q8—21no—
299C4⋊Q8C2×C4×C3112yes—
300C4⋊Q8D4×C3112yes—
301C4⋊Q8D4×C3111yescharacteristic
302#127D4×C3111yescharacteristic
303#126—24no—
304#125—22no—
305#124—22no—
306#124—24no—
307#124—21no—
308#123—24no—
309C22⋊Q8—22no—
310C22⋊Q8—22no—
311C22⋊Q8—22no—
312C22⋊Q8—24no—
313#121—22no—
314#121—24no—
315#121—22no—
316#121—22no—
317#121—21no—
318#121D4×C3112yes—
319C4×Q8—22no—
320C4×Q8—22no—
321C4×Q8C2×C4×C3111yescharacteristic
322C4×Q8D4×C3111yescharacteristic
323C4×D4—22no—
324C4×D4—24no—
325C4×D4—22no—
326C4×D4—21no—
327C4×D4—21no—
328C4×D4—21no—
329C4×D4C2×C4×C3111yescharacteristic
330C4×D4C2×C4×C3112yes—
331C4×D4D4×C3111yescharacteristic
332#117—22no—
333#117—22no—
334#117—21no—
335#117D4×C3111yescharacteristic
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—21no—
340C2×C42—21no—
341C2×C42—21no—
342C2×C42C23×C3111yescharacteristic
343C4⋊C8—21no—
344C4⋊C8—21no—
345C4⋊C8—21no—
346C4⋊C8—21no—
347C4⋊C8—21no—
348#104—42no—
349#103—44no—
350C22⋊C8—21no—
351C4×C8—21no—
352#95—21no—
353C31#842311yescharacteristic, direct product factor, Sylow
354#41—21no—
355#41—21no—
356#41—21no—
357#41—21no—
358#41—41no—
359#41—41no—
360#41—41no—
361#41—41no—
362#41—82no—
363#41—82no—
364C2×Q8—21no—
365C2×Q8—22no—
366C2×Q8—42no—
367C2×Q8—42no—
368C2×Q8C22⋊C4×C3111yescharacteristic
369C2×Q8C22⋊C4×C3114yes—
370C2×D4—21no—
371C2×D4—22no—
372C2×D4—24no—
373C2×D4—21no—
374C2×D4—21no—
375C2×D4C22⋊C4×C3111yescharacteristic
376C2×D4C22⋊C4×C3112yes—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—41no—
384C22×C4C2×D4×C3111yescharacteristic
385C22×C4C2×D4×C3111yescharacteristic
386C22×C4C2×D4×C3111yescharacteristic
387M16—41no—
388M16—41no—
389C2×C8—42no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—41no—
393C4⋊C4—24no—
394C4⋊C4—24no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—21no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—22no—
403C4⋊C4—41no—
404C4⋊C4C22⋊C4×C3111yescharacteristic
405C4⋊C4C22⋊C4×C3111yescharacteristic
406C4⋊C4C22⋊C4×C3112yes—
407C22⋊C4—24no—
408C22⋊C4—24no—
409C22⋊C4—24no—
410C22⋊C4—21no—
411C22⋊C4—21no—
412C22⋊C4—22no—
413C22⋊C4—21no—
414C22⋊C4—22no—
415C42—24no—
416C42—21no—
417C42—21no—
418C42—21no—
419C42—41no—
420C42—41no—
421C42C22×C4×C3111yescharacteristic
422C42C2×D4×C3111yescharacteristic
423C42C2×D4×C3111yescharacteristic
424C42C2×D4×C3111yescharacteristic
425C23—21no—
426C23—22no—
427C23C31×#12011yescharacteristic
428Q8—21no—
429Q8—21no—
430Q8—44no—
431Q8—41no—
432Q8—41no—
433Q8—42no—
434Q8—42no—
435D4—21no—
436D4—21no—
437D4—42no—
438D4—41no—
439D4—42no—
440D4—41no—
441D4—42no—
442D4—41no—
443D4—41no—
444D4—41no—
445D4—41no—
446D4—84no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—22no—
452C2×C4—21no—
453C2×C4—24no—
454C2×C4—41no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4—82no—
464C2×C4—84no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4C2×C22⋊C4×C3111yescharacteristic
468C2×C4C2×C22⋊C4×C3111yescharacteristic
469C2×C4C2×C22⋊C4×C3111yescharacteristic
470C2×C4C31×#12011yescharacteristic
471C2×C4C31×#12011yescharacteristic
472C2×C4C31×#12011yescharacteristic
473C2×C4C31×#13511yescharacteristic
474C2×C4C31×#13511yescharacteristic
475C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
476C2×C4C31×#13711yescharacteristic
477C8—81no—
478C8—81no—
479C8—81no—
480C22—21no—
481C22—21no—
482C22—41no—
483C22—42no—
484C22—41no—
485C22—42no—
486C22—41no—
487C22—42no—
488C22C31×#37911yescharacteristic
489C22C31×#44911yescharacteristic
490C22C31×#45211yescharacteristic
491C4—21no—
492C4—21no—
493C4—21no—
494C4—41no—
495C4—41no—
496C4—42no—
497C4—41no—
498C4—44no—
499C4—82no—
500C4C31×#41611yescharacteristic
501C4C31×#41711yescharacteristic
502C4C31×#44911yescharacteristic
503C4C31×#45211yescharacteristic
504C4C31×#45311yescharacteristic
505C4C31×#45511yescharacteristic
506C2—21no—
507C2—21no—
508C2—81no—
509C2—82no—
510C2C31×#178811yescharacteristic
511C2C31×#179211yescharacteristic
512C2C31×#179611yescharacteristic
subgroupG/H
center, socleC22×C31#379
commutatorC42C22×C4×C31
Frattini, cosocleC2×C42C23×C31
prime radicalC31×#3511C2
LCSG ▷ C42 ▷ 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×#1981C2semidirect
3. C31×#1806C2semidirect
4. C31×#1548C2semidirect
5. C31×#1547C2semidirect
6. C31×#1502C2semidirect
7. C31×#1477C2semidirect
9. C31×#555C4semidirect
10. C31×#554C4semidirect
168. #8423C31direct (N × H)
225. #1981C2×C31semidirect
226. #1806C2×C31semidirect
227. #1548C2×C31semidirect
228. #1547C2×C31semidirect
229. #1502C2×C31semidirect
230. #1477C2×C31semidirect
258. #555C4×C31semidirect
259. #554C4×C31semidirect
353. C31#8423direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35112
C31×#19812
C31×#18062
C31×#15482
C31×#15472
C31×#15022
C31×#14772
#842331