Finite Group Databasebeta

← C31×#8447 #21465206

Subgroups — C31×#8447 #21465206

Order 7936.

1016 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×Q8×C31C211yesmaximal, characteristic
2C31×#1990C211yesmaximal, characteristic, complemented
3C31×#1803C211yesmaximal, characteristic, complemented
4C31×#1802C211yesmaximal, characteristic
5C2×C31×#332C211yesmaximal, characteristic, complemented
6C2×C31×Q8⋊C8C211yesmaximal, characteristic, complemented
7C2×C31×D4⋊C8C211yesmaximal, characteristic
8C23×Q8×C31—21no—
9D4×Q8×C31—22no—
10D4×Q8×C31—22no—
11D4×Q8×C31—22no—
12D4×Q8×C31C414yescomplemented
13C2×C31×C4⋊Q8—21no—
14C2×C31×C4⋊Q8C2211yescharacteristic
15C2×C31×C22⋊Q8—21no—
16C2×C31×C22⋊Q8—22no—
17C2×C4×Q8×C31C2211yescharacteristic
18C2×C4×D4×C31—21no—
19C2×C4×D4×C31C2211yescharacteristic
20C2×C31×C4⋊C8C2211yescharacteristic
21C2×C31×C4⋊C8C2211yescharacteristic
22C2×C31×#103—21no—
23C2×C31×#102—21no—
24C2×C4×C8×C31C2211yescharacteristic
25C31×#384C2211yescharacteristic
26C31×#340—21no—
27C31×#340—21no—
28C31×#340—22no—
29C31×#332—22no—
30C31×Q8⋊C8—22no—
31C31×D4⋊C8—22no—
32C22×Q8×C31—22no—
33C22×Q8×C31—24no—
34C22×Q8×C31—24no—
35C22×Q8×C31—21no—
36C22×Q8×C31—42no—
37C22×Q8×C31—42no—
38C22×Q8×C31—42no—
39C22×Q8×C31—42no—
40C22×Q8×C31D411yescharacteristic
41C22×D4×C31D411yescharacteristic
42C23×C4×C31—22no—
43C23×C4×C31—21no—
44C22×C8×C31—21no—
45C22×C8×C31—21no—
46C22×C8×C31—21no—
47C31×C4⋊Q8—24no—
48C31×C4⋊Q8—24no—
49C31×C4⋊Q8—21no—
50C31×C4⋊Q8—21no—
51C31×C4⋊Q8C2×C412yes—
52C31×C4⋊Q8D412yes—
53C31×C22⋊Q8—24no—
54C31×C22⋊Q8—28no—
55C31×C22⋊Q8—24no—
56C31×C22⋊Q8—28no—
57C4×Q8×C31—21no—
58C4×Q8×C31—21no—
59C4×Q8×C31C2×C412yes—
60C4×Q8×C31D412yes—
61C4×D4×C31—24no—
62C4×D4×C31—24no—
63C4×D4×C31—21no—
64C4×D4×C31—21no—
65C4×D4×C31C2×C412yes—
66C4×D4×C31D412yes—
67C2×C4⋊C4×C31—22no—
68C2×C4⋊C4×C31—21no—
69C2×C4⋊C4×C31—21no—
70C2×C4⋊C4×C31—21no—
71C2×C4⋊C4×C31—21no—
72C2×C4⋊C4×C31—21no—
73C2×C4⋊C4×C31D411yescharacteristic
74C2×C4⋊C4×C31D411yescharacteristic
75C2×C4⋊C4×C31D412yes—
76C2×C22⋊C4×C31—22no—
77C2×C22⋊C4×C31—21no—
78C2×C42×C31—21no—
79C2×C42×C31C2311yescharacteristic
80C31×C4⋊C8—21no—
81C31×C4⋊C8—21no—
82C31×C4⋊C8—21no—
83C31×C4⋊C8—21no—
84C31×#103—42no—
85C31×#102—42no—
86C4×C8×C31—21no—
87C4×C8×C31—21no—
88C24×C31—21no—
89C2×Q8×C31—28no—
90C2×Q8×C31—22no—
91C2×Q8×C31—21no—
92C2×Q8×C31—22no—
93C2×Q8×C31—21no—
94C2×Q8×C31—21no—
95C2×Q8×C31—44no—
96C2×Q8×C31—44no—
97C2×Q8×C31—48no—
98C2×Q8×C31—48no—
99C2×Q8×C31—48no—
100C2×Q8×C31—48no—
101C2×Q8×C31—41no—
102C2×Q8×C31—41no—
103C2×Q8×C31—41no—
104C2×Q8×C31—41no—
105C2×Q8×C31C22⋊C412yes—
106C2×Q8×C31D812yes—
107C2×Q8×C31SD1612yes—
108C2×D4×C31—21no—
109C2×D4×C31—21no—
110C2×D4×C31—21no—
111C2×D4×C31C22⋊C412yes—
112C2×D4×C31SD1612yes—
113C2×D4×C31Q1612yes—
114C22×C4×C31—21no—
115C22×C4×C31—28no—
116C22×C4×C31—21no—
117C22×C4×C31—22no—
118C22×C4×C31—21no—
119C22×C4×C31—22no—
120C22×C4×C31—22no—
121C22×C4×C31—44no—
122C22×C4×C31—44no—
123C22×C4×C31—41no—
124C22×C4×C31—41no—
125C22×C4×C31—41no—
126C22×C4×C31—41no—
127C22×C4×C31—41no—
128C22×C4×C31C2×D411yescharacteristic
129C22×C4×C31C2×D411yescharacteristic
130C22×C4×C31C2×D411yescharacteristic
131C2×C8×C31—22no—
132C2×C8×C31—41no—
133C2×C8×C31—41no—
134C2×C8×C31—41no—
135C2×C8×C31—41no—
136C2×C8×C31—41no—
137C2×C8×C31—41no—
138C2×C8×C31—41no—
139C2×C8×C31—41no—
140C4⋊C4×C31—28no—
141C4⋊C4×C31—22no—
142C4⋊C4×C31—22no—
143C4⋊C4×C31—22no—
144C4⋊C4×C31—22no—
145C4⋊C4×C31—21no—
146C4⋊C4×C31—21no—
147C4⋊C4×C31—22no—
148C4⋊C4×C31—42no—
149C4⋊C4×C31—41no—
150C4⋊C4×C31—41no—
151C4⋊C4×C31—41no—
152C4⋊C4×C31—41no—
153C4⋊C4×C31C22⋊C412yes—
154C4⋊C4×C31C22⋊C412yes—
155C4⋊C4×C31C22⋊C414yes—
156C22⋊C4×C31—28no—
157C22⋊C4×C31—22no—
158C22⋊C4×C31—22no—
159C42×C31—22no—
160C42×C31—22no—
161C42×C31C22×C411yescharacteristic
162C42×C31C2×D411yescharacteristic
163C42×C31C2×D411yescharacteristic
164C42×C31C2×D411yescharacteristic
165#8447C3111yesmaximal, characteristic, direct product factor, Sylow
166C23×C31—22no—
167C23×C31—22no—
168C23×C31—22no—
169C23×C31—41no—
170C23×C31—41no—
171C23×C31—41no—
172C23×C31—41no—
173C23×C31#12011yescharacteristic
174Q8×C31—22no—
175Q8×C31—22no—
176Q8×C31—22no—
177Q8×C31—416no—
178Q8×C31—42no—
179Q8×C31—42no—
180Q8×C31—42no—
181Q8×C31—42no—
182Q8×C31#10214yescomplemented
183D4×C31—22no—
184D4×C31—22no—
185D4×C31—22no—
186D4×C31#10314yescomplemented
187C2×C4×C31—22no—
188C2×C4×C31—22no—
189C2×C4×C31—22no—
190C2×C4×C31—21no—
191C2×C4×C31—21no—
192C2×C4×C31—22no—
193C2×C4×C31—24no—
194C2×C4×C31—22no—
195C2×C4×C31—416no—
196C2×C4×C31—42no—
197C2×C4×C31—41no—
198C2×C4×C31—42no—
199C2×C4×C31—41no—
200C2×C4×C31—41no—
201C2×C4×C31—41no—
202C2×C4×C31—42no—
203C2×C4×C31—42no—
204C2×C4×C31—42no—
205C2×C4×C31—42no—
206C2×C4×C31—81no—
207C2×C4×C31—81no—
208C2×C4×C31—81no—
209C2×C4×C31C2×C22⋊C411yescharacteristic
210C2×C4×C31C2×C22⋊C411yescharacteristic
211C2×C4×C31C2×C22⋊C411yescharacteristic
212C2×C4×C31#12011yescharacteristic
213C2×C4×C31#12011yescharacteristic
214C2×C4×C31#12011yescharacteristic
215C2×C4×C31C2×D811yescharacteristic
216C2×C4×C31C2×SD1611yescharacteristic
217C2×C4×C31C2×SD1611yescharacteristic
218C2×C4×C31C2×Q1611yescharacteristic
219C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
220C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
221C2×C4×C31#13711yescharacteristic
222C2×C4×C31#13711yescharacteristic
223C8×C31—42no—
224C8×C31—81no—
225C8×C31—81no—
226C2×D4×Q8C2×C3111yescharacteristic
227#1990C2×C3111yescharacteristic, complemented
228#1803C2×C3111yescharacteristic, complemented
229#1802C2×C3111yescharacteristic
230C2×#332C2×C3111yescharacteristic, complemented
231C2×Q8⋊C8C2×C3111yescharacteristic, complemented
232C2×D4⋊C8C2×C3111yescharacteristic
233C22×C31—24no—
234C22×C31—42no—
235C22×C31—42no—
236C22×C31—42no—
237C22×C31—42no—
238C22×C31—42no—
239C22×C31—42no—
240C22×C31#37911yescharacteristic
241C22×C31#44711yescharacteristic
242C22×C31#44811yescharacteristic
243C22×C31#45011yescharacteristic
244C22×C31#45111yescharacteristic
245C22×C31#45411yescharacteristic
246C22×C31#45611yescharacteristic
247C4×C31—21no—
248C4×C31—21no—
249C4×C31—24no—
250C4×C31—42no—
251C4×C31—44no—
252C4×C31—42no—
253C4×C31—161no—
254C4×C31C2×#10211yescharacteristic
255C4×C31C2×#10311yescharacteristic
256C4×C31#41811yescharacteristic
257C4×C31#41911yescharacteristic
258C4×C31#44711yescharacteristic
259C4×C31#44811yescharacteristic
260C4×C31#45011yescharacteristic
261C4×C31#45111yescharacteristic
262C23×Q8—21no—
263D4×Q8—22no—
264D4×Q8—22no—
265D4×Q8—22no—
266D4×Q8C4×C3114yescomplemented
267C2×C4⋊Q8—21no—
268C2×C4⋊Q8C22×C3111yescharacteristic
269C2×C22⋊Q8—21no—
270C2×C22⋊Q8—22no—
271C2×C4×Q8C22×C3111yescharacteristic
272C2×C4×D4—21no—
273C2×C4×D4C22×C3111yescharacteristic
274C2×C4⋊C8C22×C3111yescharacteristic
275C2×C4⋊C8C22×C3111yescharacteristic
276C2×#103—21no—
277C2×#102—21no—
278C2×C4×C8C22×C3111yescharacteristic
279#384C22×C3111yescharacteristic
280#340—21no—
281#340—21no—
282#340—22no—
283#332—22no—
284Q8⋊C8—22no—
285D4⋊C8—22no—
286C2×C31—44no—
287C2×C31#162711yescharacteristic
288C2×C31#163711yescharacteristic
289C2×C31#164911yescharacteristic
290C2×C31#165111yescharacteristic
291C2×C31#178611yescharacteristic
292C2×C31#178711yescharacteristic
293C2×C31#179311yescharacteristic
294C22×Q8—22no—
295C22×Q8—24no—
296C22×Q8—24no—
297C22×Q8—21no—
298C22×Q8—42no—
299C22×Q8—42no—
300C22×Q8—42no—
301C22×Q8—42no—
302C22×Q8D4×C3111yescharacteristic
303C22×D4D4×C3111yescharacteristic
304C23×C4—22no—
305C23×C4—21no—
306C22×C8—21no—
307C22×C8—21no—
308C22×C8—21no—
309C4⋊Q8—24no—
310C4⋊Q8—24no—
311C4⋊Q8—21no—
312C4⋊Q8—21no—
313C4⋊Q8C2×C4×C3112yes—
314C4⋊Q8D4×C3112yes—
315C22⋊Q8—24no—
316C22⋊Q8—28no—
317C22⋊Q8—24no—
318C22⋊Q8—28no—
319C4×Q8—21no—
320C4×Q8—21no—
321C4×Q8C2×C4×C3112yes—
322C4×Q8D4×C3112yes—
323C4×D4—24no—
324C4×D4—24no—
325C4×D4—21no—
326C4×D4—21no—
327C4×D4C2×C4×C3112yes—
328C4×D4D4×C3112yes—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4D4×C3111yescharacteristic
336C2×C4⋊C4D4×C3111yescharacteristic
337C2×C4⋊C4D4×C3112yes—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—21no—
340C2×C42—21no—
341C2×C42C23×C3111yescharacteristic
342C4⋊C8—21no—
343C4⋊C8—21no—
344C4⋊C8—21no—
345C4⋊C8—21no—
346#103—42no—
347#102—42no—
348C4×C8—21no—
349C4×C8—21no—
350C31#844711yescharacteristic, direct product factor, Sylow
351C24—21no—
352C2×Q8—28no—
353C2×Q8—22no—
354C2×Q8—21no—
355C2×Q8—22no—
356C2×Q8—21no—
357C2×Q8—21no—
358C2×Q8—44no—
359C2×Q8—44no—
360C2×Q8—48no—
361C2×Q8—48no—
362C2×Q8—48no—
363C2×Q8—48no—
364C2×Q8—41no—
365C2×Q8—41no—
366C2×Q8—41no—
367C2×Q8—41no—
368C2×Q8C22⋊C4×C3112yes—
369C2×Q8D8×C3112yes—
370C2×Q8SD16×C3112yes—
371C2×D4—21no—
372C2×D4—21no—
373C2×D4—21no—
374C2×D4C22⋊C4×C3112yes—
375C2×D4SD16×C3112yes—
376C2×D4Q16×C3112yes—
377C22×C4—21no—
378C22×C4—28no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4C2×D4×C3111yescharacteristic
392C22×C4C2×D4×C3111yescharacteristic
393C22×C4C2×D4×C3111yescharacteristic
394C2×C8—22no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—41no—
400C2×C8—41no—
401C2×C8—41no—
402C2×C8—41no—
403C4⋊C4—28no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—21no—
409C4⋊C4—21no—
410C4⋊C4—22no—
411C4⋊C4—42no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4C22⋊C4×C3112yes—
417C4⋊C4C22⋊C4×C3112yes—
418C4⋊C4C22⋊C4×C3114yes—
419C22⋊C4—28no—
420C22⋊C4—22no—
421C22⋊C4—22no—
422C42—22no—
423C42—22no—
424C42C22×C4×C3111yescharacteristic
425C42C2×D4×C3111yescharacteristic
426C42C2×D4×C3111yescharacteristic
427C42C2×D4×C3111yescharacteristic
428C23—22no—
429C23—22no—
430C23—22no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23C31×#12011yescharacteristic
436Q8—22no—
437Q8—22no—
438Q8—22no—
439Q8—416no—
440Q8—42no—
441Q8—42no—
442Q8—42no—
443Q8—42no—
444Q8C31×#10214yescomplemented
445D4—22no—
446D4—22no—
447D4—22no—
448D4C31×#10314yescomplemented
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—24no—
456C2×C4—22no—
457C2×C4—416no—
458C2×C4—42no—
459C2×C4—41no—
460C2×C4—42no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4C2×C22⋊C4×C3111yescharacteristic
472C2×C4C2×C22⋊C4×C3111yescharacteristic
473C2×C4C2×C22⋊C4×C3111yescharacteristic
474C2×C4C31×#12011yescharacteristic
475C2×C4C31×#12011yescharacteristic
476C2×C4C31×#12011yescharacteristic
477C2×C4C2×D8×C3111yescharacteristic
478C2×C4C2×SD16×C3111yescharacteristic
479C2×C4C2×SD16×C3111yescharacteristic
480C2×C4C2×Q16×C3111yescharacteristic
481C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
482C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
483C2×C4C31×#13711yescharacteristic
484C2×C4C31×#13711yescharacteristic
485C8—42no—
486C8—81no—
487C8—81no—
488C22—24no—
489C22—42no—
490C22—42no—
491C22—42no—
492C22—42no—
493C22—42no—
494C22—42no—
495C22C31×#37911yescharacteristic
496C22C31×#44711yescharacteristic
497C22C31×#44811yescharacteristic
498C22C31×#45011yescharacteristic
499C22C31×#45111yescharacteristic
500C22C31×#45411yescharacteristic
501C22C31×#45611yescharacteristic
502C4—24no—
503C4—21no—
504C4—21no—
505C4—44no—
506C4—42no—
507C4—42no—
508C4—161no—
509C4C2×C31×#10211yescharacteristic
510C4C2×C31×#10311yescharacteristic
511C4C31×#41811yescharacteristic
512C4C31×#41911yescharacteristic
513C4C31×#44711yescharacteristic
514C4C31×#44811yescharacteristic
515C4C31×#45011yescharacteristic
516C4C31×#45111yescharacteristic
517C2—44no—
518C2C31×#162711yescharacteristic
519C2C31×#163711yescharacteristic
520C2C31×#164911yescharacteristic
521C2C31×#165111yescharacteristic
522C2C31×#178611yescharacteristic
523C2C31×#178711yescharacteristic
524C2C31×#179311yescharacteristic
subgroupG/H
center, socleC23×C31#120
commutatorC42C22×C4×C31
Frattini, cosocleC2×C42C23×C31
prime radicalC22×D4×C31D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#1990C2semidirect
3. C31×#1803C2semidirect
5. C2×C31×#332C2semidirect
6. C2×C31×Q8⋊C8C2semidirect
12. D4×Q8×C31C4semidirect
165. #8447C31direct (N × H)
182. Q8×C31#102semidirect
186. D4×C31#103semidirect
227. #1990C2×C31semidirect
228. #1803C2×C31semidirect
230. C2×#332C2×C31semidirect
231. C2×Q8⋊C8C2×C31semidirect
266. D4×Q8C4×C31semidirect
350. C31#8447direct (N × H)
444. Q8C31×#102semidirect
448. D4C31×#103semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Q8×C312
C31×#19902
C31×#18032
C31×#18022
C2×C31×#3322
C2×C31×Q8⋊C82
C2×C31×D4⋊C82
#844731