Finite Group Databasebeta

← C31×#18642 #21475401

Subgroups — C31×#18642 #21475401

Order 7936.

1002 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3503C211yesmaximal, characteristic, complemented
2C31×#3208C211yesmaximal, characteristic, complemented
3C2×C31×#486C211yesmaximal, characteristic, complemented
4C31×#3038C211yesmaximal, characteristic
5C31×#1627C214yesmaximal, complemented
6C31×#1626C214yesmaximal, complemented
7C31×#1549C211yesmaximal, characteristic, complemented
8C31×#1493C211yesmaximal, characteristic, complemented
9C31×#1476C211yesmaximal, characteristic, complemented
10C2×C31×AGL(1,ℤ/8)—21no—
11Q82×C31—21no—
12C31×#557C2212yescomplemented
13C31×#554—21no—
14C31×#554—22no—
15D4×Q8×C31—22no—
16D4×Q8×C31C2212yescomplemented
17C31×#544—22no—
18C31×#544—21no—
19C31×#533—21no—
20C31×#533C2211yescharacteristic
21C2×C31×C4⋊Q8—21no—
22C2×C31×C4⋊Q8C2211yescharacteristic
23C2×C31×#127C2211yescharacteristic
24C31×#520—21no—
25C4×#41×C31C2211yescharacteristic
26C31×#492—21no—
27C31×#486—22no—
28C31×#486—21no—
29C31×#486C2214yes—
30C31×#460—21no—
31C31×#460C2212yes—
32C31×#460C2214yes—
33C31×#459—21no—
34C31×#459C2212yes—
35C31×#450—22no—
36C31×#431C2211yescharacteristic
37C31×#423C2211yescharacteristic
38C31×#418—21no—
39C2×C31×#102—22no—
40C2×C4×C8×C31C2211yescharacteristic
41C31×#332C2214yescomplemented
42C31×Q8⋊C8C2212yescomplemented
43C31×Q8⋊C8C2214yescomplemented
44C31×D4⋊C8C2212yescomplemented
45C2×#41×C31D411yescharacteristic
46C22×Q8×C31—22no—
47C22×Q8×C31—21no—
48C22×D4×C31—21no—
49C22×D4×C31—21no—
50C22×D4×C31—21no—
51C31×AGL(1,ℤ/8)—44no—
52C2×SD16×C31—22no—
53C2×SD16×C31—24no—
54C2×D8×C31—22no—
55C2×M16×C31—21no—
56C22×C8×C31—21no—
57C31×C4⋊Q8—22no—
58C31×C4⋊Q8—22no—
59C31×C4⋊Q8—24no—
60C31×C4⋊Q8—22no—
61C31×C4⋊Q8—22no—
62C31×C4⋊Q8—21no—
63C31×C4⋊Q8—21no—
64C31×C4⋊Q8D411yescharacteristic
65C31×C4⋊Q8D412yes—
66C31×C4⋊Q8D411yescharacteristic
67C31×C4⋊Q8C2312yes—
68C31×#127—21no—
69C31×#127—21no—
70C31×#127—21no—
71C31×#127C2312yes—
72C31×#125—22no—
73C31×#125—24no—
74C31×#125—22no—
75C31×#125—21no—
76C31×C22⋊Q8—22no—
77C31×C22⋊Q8—24no—
78C31×C22⋊Q8—24no—
79C31×C22⋊Q8—21no—
80C31×C22⋊Q8—22no—
81C31×C22⋊Q8—24no—
82C31×C22⋊Q8D412yes—
83C4×Q8×C31—22no—
84C4×Q8×C31—22no—
85C4×Q8×C31—22no—
86C4×Q8×C31—21no—
87C4×Q8×C31D411yescharacteristic
88C4×Q8×C31C2311yescharacteristic
89C4×Q8×C31C2312yes—
90C4×D4×C31—24no—
91C4×D4×C31—22no—
92C4×D4×C31—21no—
93C4×D4×C31—21no—
94C4×D4×C31D411yescharacteristic
95C4×D4×C31C2311yescharacteristic
96C31×#117—22no—
97C31×#117—21no—
98C31×#117—21no—
99C31×#117D411yescharacteristic
100C2×C4⋊C4×C31—22no—
101C2×C4⋊C4×C31—22no—
102C2×C4⋊C4×C31—21no—
103C2×C4⋊C4×C31D412yes—
104C2×C42×C31—21no—
105C2×C42×C31C2311yescharacteristic
106C31×C4⋊C8—21no—
107C31×C4⋊C8C2312yes—
108C31×C4⋊C8C2312yes—
109C31×#102—28no—
110C31×#102—22no—
111C31×#102—22no—
112C31×#102—24no—
113C31×C22⋊C8—21no—
114C4×C8×C31—21no—
115C4×C8×C31—21no—
116C4×C8×C31C2312yes—
117C24×C31—41no—
118#41×C31—22no—
119#41×C31SD1614yescomplemented
120C2×Q8×C31—24no—
121C2×Q8×C31—21no—
122C2×Q8×C31—22no—
123C2×Q8×C31—22no—
124C2×Q8×C31—82no—
125C2×Q8×C31—84no—
126C2×Q8×C31C2×D412yes—
127C2×Q8×C31C2×D411yescharacteristic
128C2×D4×C31—21no—
129C2×D4×C31—22no—
130C2×D4×C31—22no—
131C2×D4×C31—22no—
132C2×D4×C31—44no—
133C2×D4×C31—41no—
134C2×D4×C31—41no—
135C2×D4×C31—41no—
136C2×D4×C31—42no—
137C2×D4×C31—42no—
138C2×D4×C31—82no—
139C2×D4×C31C2×D411yescharacteristic
140C22×C4×C31—21no—
141C22×C4×C31—24no—
142C22×C4×C31—22no—
143C22×C4×C31—21no—
144C22×C4×C31—21no—
145C22×C4×C31C2×D411yescharacteristic
146C22×C4×C31C2×D411yescharacteristic
147C22×C4×C31C2×D411yescharacteristic
148SD16×C31—44no—
149SD16×C31—84no—
150D8×C31—44no—
151M16×C31—42no—
152C2×C8×C31—22no—
153C2×C8×C31—22no—
154C2×C8×C31—22no—
155C2×C8×C31—42no—
156C4⋊C4×C31—24no—
157C4⋊C4×C31—24no—
158C4⋊C4×C31—24no—
159C4⋊C4×C31—24no—
160C4⋊C4×C31—21no—
161C4⋊C4×C31—22no—
162C4⋊C4×C31—21no—
163C4⋊C4×C31—22no—
164C4⋊C4×C31—21no—
165C4⋊C4×C31—21no—
166C4⋊C4×C31—22no—
167C4⋊C4×C31—22no—
168C4⋊C4×C31—21no—
169C4⋊C4×C31—22no—
170C4⋊C4×C31C2×D414yes—
171C4⋊C4×C31C2×D411yescharacteristic
172C4⋊C4×C31C2×D411yescharacteristic
173C4⋊C4×C31C2×D412yes—
174C22⋊C4×C31—24no—
175C22⋊C4×C31—22no—
176C22⋊C4×C31—21no—
177C22⋊C4×C31—21no—
178C22⋊C4×C31—21no—
179C42×C31—24no—
180C42×C31—21no—
181C42×C31—21no—
182C42×C31—21no—
183C42×C31—22no—
184C42×C31C2×D411yescharacteristic
185C42×C31C2×D411yescharacteristic
186C42×C31C2×D411yescharacteristic
187C42×C31C2411yescharacteristic
188#18642C3111yesmaximal, characteristic, direct product factor, Sylow
189C23×C31—21no—
190C23×C31—42no—
191C23×C31—82no—
192C23×C31—82no—
193C23×C31—82no—
194C23×C31#12011yescharacteristic
195Q8×C31—21no—
196Q8×C31—24no—
197Q8×C31—44no—
198Q8×C31—42no—
199Q8×C31—82no—
200Q8×C31—84no—
201Q8×C31C2×SD1612yes—
202D4×C31—24no—
203D4×C31—21no—
204D4×C31—44no—
205D4×C31—42no—
206D4×C31—42no—
207D4×C31—44no—
208D4×C31—82no—
209D4×C31—82no—
210D4×C31—82no—
211D4×C31C2×SD1612yes—
212C2×C4×C31—21no—
213C2×C4×C31—22no—
214C2×C4×C31—24no—
215C2×C4×C31—22no—
216C2×C4×C31—22no—
217C2×C4×C31—21no—
218C2×C4×C31—21no—
219C2×C4×C31—22no—
220C2×C4×C31—24no—
221C2×C4×C31—42no—
222C2×C4×C31—42no—
223C2×C4×C31—82no—
224C2×C4×C31—84no—
225C2×C4×C31#12011yescharacteristic
226C2×C4×C31#12011yescharacteristic
227C2×C4×C31#12011yescharacteristic
228C2×C4×C31C2×SD1612yes—
229C2×C4×C31#13712yes—
230C2×C4×C31C22×D411yescharacteristic
231C2×C4×C31C22×D411yescharacteristic
232C2×C4×C31C22×D411yescharacteristic
233C8×C31—42no—
234C8×C31—42no—
235C8×C31—42no—
236#3503C2×C3111yescharacteristic, complemented
237#3208C2×C3111yescharacteristic, complemented
238C2×#486C2×C3111yescharacteristic, complemented
239#3038C2×C3111yescharacteristic
240#1627C2×C3114yescomplemented
241#1626C2×C3114yescomplemented
242#1549C2×C3111yescharacteristic, complemented
243#1493C2×C3111yescharacteristic, complemented
244#1476C2×C3111yescharacteristic, complemented
245C22×C31—21no—
246C22×C31—21no—
247C22×C31—22no—
248C22×C31—41no—
249C22×C31—41no—
250C22×C31—82no—
251C22×C31—82no—
252C22×C31—82no—
253C22×C31—164no—
254C22×C31#44812yes—
255C22×C31C2×#12011yescharacteristic
256C4×C31—22no—
257C4×C31—21no—
258C4×C31—21no—
259C4×C31—41no—
260C4×C31—42no—
261C4×C31—44no—
262C4×C31—41no—
263C4×C31—42no—
264C4×C31#44812yes—
265C4×C31#45412yes—
266C4×C31C22×SD1611yescharacteristic
267C4×C31C2×#13711yescharacteristic
268C4×C31#57612yes—
269C2×AGL(1,ℤ/8)—21no—
270Q82—21no—
271#557C22×C3112yescomplemented
272#554—21no—
273#554—22no—
274D4×Q8—22no—
275D4×Q8C22×C3112yescomplemented
276#544—22no—
277#544—21no—
278#533—21no—
279#533C22×C3111yescharacteristic
280C2×C4⋊Q8—21no—
281C2×C4⋊Q8C22×C3111yescharacteristic
282C2×#127C22×C3111yescharacteristic
283#520—21no—
284C4×#41C22×C3111yescharacteristic
285#492—21no—
286#486—22no—
287#486—21no—
288#486C22×C3114yes—
289#460—21no—
290#460C22×C3112yes—
291#460C22×C3114yes—
292#459—21no—
293#459C22×C3112yes—
294#450—22no—
295#431C22×C3111yescharacteristic
296#423C22×C3111yescharacteristic
297#418—21no—
298C2×#102—22no—
299C2×C4×C8C22×C3111yescharacteristic
300#332C22×C3114yescomplemented
301Q8⋊C8C22×C3112yescomplemented
302Q8⋊C8C22×C3114yescomplemented
303D4⋊C8C22×C3112yescomplemented
304C2×C31—22no—
305C2×C31—42no—
306C2×C31—162no—
307C2×C31C2×#44811yescharacteristic
308C2×C31#300811yescharacteristic
309C2×C31C2×#45411yescharacteristic
310C2×#41D4×C3111yescharacteristic
311C22×Q8—22no—
312C22×Q8—21no—
313C22×D4—21no—
314C22×D4—21no—
315C22×D4—21no—
316AGL(1,ℤ/8)—44no—
317C2×SD16—22no—
318C2×SD16—24no—
319C2×D8—22no—
320C2×M16—21no—
321C22×C8—21no—
322C4⋊Q8—22no—
323C4⋊Q8—22no—
324C4⋊Q8—24no—
325C4⋊Q8—22no—
326C4⋊Q8—22no—
327C4⋊Q8—21no—
328C4⋊Q8—21no—
329C4⋊Q8D4×C3111yescharacteristic
330C4⋊Q8D4×C3112yes—
331C4⋊Q8D4×C3111yescharacteristic
332C4⋊Q8C23×C3112yes—
333#127—21no—
334#127—21no—
335#127—21no—
336#127C23×C3112yes—
337#125—22no—
338#125—24no—
339#125—22no—
340#125—21no—
341C22⋊Q8—22no—
342C22⋊Q8—24no—
343C22⋊Q8—24no—
344C22⋊Q8—21no—
345C22⋊Q8—22no—
346C22⋊Q8—24no—
347C22⋊Q8D4×C3112yes—
348C4×Q8—22no—
349C4×Q8—22no—
350C4×Q8—22no—
351C4×Q8—21no—
352C4×Q8D4×C3111yescharacteristic
353C4×Q8C23×C3111yescharacteristic
354C4×Q8C23×C3112yes—
355C4×D4—24no—
356C4×D4—22no—
357C4×D4—21no—
358C4×D4—21no—
359C4×D4D4×C3111yescharacteristic
360C4×D4C23×C3111yescharacteristic
361#117—22no—
362#117—21no—
363#117—21no—
364#117D4×C3111yescharacteristic
365C2×C4⋊C4—22no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4D4×C3112yes—
369C2×C42—21no—
370C2×C42C23×C3111yescharacteristic
371C4⋊C8—21no—
372C4⋊C8C23×C3112yes—
373C4⋊C8C23×C3112yes—
374#102—28no—
375#102—22no—
376#102—22no—
377#102—24no—
378C22⋊C8—21no—
379C4×C8—21no—
380C4×C8—21no—
381C4×C8C23×C3112yes—
382C31#1864211yescharacteristic, direct product factor, Sylow
383C24—41no—
384#41—22no—
385#41SD16×C3114yescomplemented
386C2×Q8—24no—
387C2×Q8—21no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—82no—
391C2×Q8—84no—
392C2×Q8C2×D4×C3112yes—
393C2×Q8C2×D4×C3111yescharacteristic
394C2×D4—21no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—44no—
399C2×D4—41no—
400C2×D4—41no—
401C2×D4—41no—
402C2×D4—42no—
403C2×D4—42no—
404C2×D4—82no—
405C2×D4C2×D4×C3111yescharacteristic
406C22×C4—21no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4C2×D4×C3111yescharacteristic
412C22×C4C2×D4×C3111yescharacteristic
413C22×C4C2×D4×C3111yescharacteristic
414SD16—44no—
415SD16—84no—
416D8—44no—
417M16—42no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—42no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—21no—
427C4⋊C4—22no—
428C4⋊C4—21no—
429C4⋊C4—22no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—21no—
435C4⋊C4—22no—
436C4⋊C4C2×D4×C3114yes—
437C4⋊C4C2×D4×C3111yescharacteristic
438C4⋊C4C2×D4×C3111yescharacteristic
439C4⋊C4C2×D4×C3112yes—
440C22⋊C4—24no—
441C22⋊C4—22no—
442C22⋊C4—21no—
443C22⋊C4—21no—
444C22⋊C4—21no—
445C42—24no—
446C42—21no—
447C42—21no—
448C42—21no—
449C42—22no—
450C42C2×D4×C3111yescharacteristic
451C42C2×D4×C3111yescharacteristic
452C42C2×D4×C3111yescharacteristic
453C42C24×C3111yescharacteristic
454C23—21no—
455C23—42no—
456C23—82no—
457C23—82no—
458C23—82no—
459C23C31×#12011yescharacteristic
460Q8—21no—
461Q8—24no—
462Q8—42no—
463Q8—44no—
464Q8—82no—
465Q8—84no—
466Q8C2×SD16×C3112yes—
467D4—24no—
468D4—21no—
469D4—44no—
470D4—42no—
471D4—42no—
472D4—44no—
473D4—82no—
474D4—82no—
475D4—82no—
476D4C2×SD16×C3112yes—
477C2×C4—21no—
478C2×C4—24no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—82no—
489C2×C4—84no—
490C2×C4C31×#12011yescharacteristic
491C2×C4C31×#12011yescharacteristic
492C2×C4C31×#12011yescharacteristic
493C2×C4C2×SD16×C3112yes—
494C2×C4C31×#13712yes—
495C2×C4C22×D4×C3111yescharacteristic
496C2×C4C22×D4×C3111yescharacteristic
497C2×C4C22×D4×C3111yescharacteristic
498C8—42no—
499C8—42no—
500C8—42no—
501C22—21no—
502C22—21no—
503C22—22no—
504C22—41no—
505C22—41no—
506C22—82no—
507C22—82no—
508C22—82no—
509C22—164no—
510C22C31×#44812yes—
511C22C2×C31×#12011yescharacteristic
512C4—22no—
513C4—21no—
514C4—21no—
515C4—41no—
516C4—42no—
517C4—44no—
518C4—41no—
519C4—42no—
520C4C31×#44812yes—
521C4C31×#45412yes—
522C4C22×SD16×C3111yescharacteristic
523C4C2×C31×#13711yescharacteristic
524C4C31×#57612yes—
525C2—22no—
526C2—42no—
527C2—162no—
528C2C2×C31×#44811yescharacteristic
529C2C31×#300811yescharacteristic
530C2C2×C31×#45411yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC42C24×C31
prime radicalC31×#3038C2
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
1. C31×#3503C2semidirect
2. C31×#3208C2semidirect
3. C2×C31×#486C2semidirect
5. C31×#1627C2semidirect
6. C31×#1626C2semidirect
7. C31×#1549C2semidirect
8. C31×#1493C2semidirect
9. C31×#1476C2semidirect
12. C31×#557C22semidirect
16. D4×Q8×C31C22semidirect
41. C31×#332C22semidirect
42. C31×Q8⋊C8C22semidirect
43. C31×Q8⋊C8C22semidirect
44. C31×D4⋊C8C22semidirect
119. #41×C31SD16semidirect
188. #18642C31direct (N × H)
236. #3503C2×C31semidirect
237. #3208C2×C31semidirect
238. C2×#486C2×C31semidirect
240. #1627C2×C31semidirect
241. #1626C2×C31semidirect
242. #1549C2×C31semidirect
243. #1493C2×C31semidirect
244. #1476C2×C31semidirect
271. #557C22×C31semidirect
275. D4×Q8C22×C31semidirect
300. #332C22×C31semidirect
301. Q8⋊C8C22×C31semidirect
302. Q8⋊C8C22×C31semidirect
303. D4⋊C8C22×C31semidirect
382. C31#18642direct (N × H)
385. #41SD16×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#35032
C31×#32082
C2×C31×#4862
C31×#30382
C31×#16272
C31×#16262
C31×#15492
C31×#14932
C31×#14762
#1864231