Finite Group Databasebeta

← C31×#18533 #21475292

Subgroups — C31×#18533 #21475292

Order 7936.

1200 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#547C211yesmaximal, characteristic, complemented
2C31×#3141C211yesmaximal, characteristic, complemented
3C31×#3109C211yesmaximal, characteristic, complemented
4C31×#3109C211yesmaximal, characteristic, complemented
5C31×#1635C214yesmaximal, complemented
6C31×#1625C214yesmaximal, complemented
7C31×#1556C211yesmaximal, characteristic, complemented
8C31×#1490C211yesmaximal, characteristic, complemented
9C31×#1490C211yesmaximal, characteristic, complemented
10C22×#41×C31—21no—
11C31×#547—22no—
12C31×#547—22no—
13C31×#547—22no—
14C31×#547C2214yes—
15C31×#532C2211yescharacteristic
16C2×C31×C4⋊Q8C2211yescharacteristic
17C2×C31×#123—22no—
18C2×C31×C22⋊Q8—21no—
19C2×C31×#121—21no—
20C2×C4×D4×C31C2211yescharacteristic
21C2×C4×D4×C31C2211yescharacteristic
22C22×C4⋊C4×C31—21no—
23C31×#489—22no—
24C31×#487C2212yescomplemented
25C31×#486C2212yescomplemented
26C31×#481—21no—
27C31×#481—21no—
28C31×#476C2212yescomplemented
29C31×#476C2212yescomplemented
30C31×#459C2212yescomplemented
31C31×#459C2212yescomplemented
32C31×#447—21no—
33C31×#447—21no—
34C31×#442—22no—
35C31×#442—22no—
36C31×#432C2211yescharacteristic
37C31×#432C2211yescharacteristic
38C31×#417—22no—
39C4×M16×C31C2211yescharacteristic
40C31×#332C2214yescomplemented
41C31×D4⋊C8C2214yescomplemented
42C31×D4⋊C8C2214yescomplemented
43C2×#41×C31—24no—
44C2×#41×C31—24no—
45C2×#41×C31—21no—
46C2×#41×C31—48no—
47C22×Q8×C31—21no—
48C22×D4×C31—22no—
49C22×D4×C31D411yescharacteristic
50C22×D4×C31D411yescharacteristic
51C23×C4×C31—22no—
52C23×C4×C31—21no—
53C23×C4×C31—21no—
54C2×D8×C31—22no—
55C2×D8×C31—22no—
56C2×M16×C31—21no—
57C31×C4⋊Q8—21no—
58C31×C4⋊Q8—21no—
59C31×C4⋊Q8D412yes—
60C31×C4⋊Q8C2311yescharacteristic
61C31×C4⋊Q8C2312yes—
62C31×#127C2311yescharacteristic
63C31×#124—21no—
64C31×#123—28no—
65C31×#123—28no—
66C31×C22⋊Q8—24no—
67C31×C22⋊Q8—24no—
68C31×#121—24no—
69C31×#121—24no—
70C31×#121—21no—
71C31×#121—21no—
72C4×D4×C31—21no—
73C4×D4×C31—21no—
74C4×D4×C31—21no—
75C4×D4×C31—21no—
76C4×D4×C31—21no—
77C4×D4×C31—21no—
78C4×D4×C31D412yes—
79C4×D4×C31D412yes—
80C4×D4×C31C2312yes—
81C4×D4×C31C2312yes—
82C2×C4⋊C4×C31—22no—
83C2×C4⋊C4×C31—24no—
84C2×C4⋊C4×C31—21no—
85C2×C4⋊C4×C31—24no—
86C2×C4⋊C4×C31D412yes—
87C2×C4⋊C4×C31D411yescharacteristic
88C2×C4⋊C4×C31D411yescharacteristic
89C2×C22⋊C4×C31—22no—
90C2×C22⋊C4×C31—21no—
91C2×C22⋊C4×C31—21no—
92C2×C42×C31C2311yescharacteristic
93C31×#106—22no—
94C31×#106—22no—
95C31×C4⋊C8C2312yes—
96C31×C4⋊C8C2312yes—
97C31×#103—24no—
98C31×#102—22no—
99C31×#102—22no—
100C31×#102—24no—
101C31×#102—22no—
102C31×#102—22no—
103C31×C22⋊C8—21no—
104C31×C22⋊C8—21no—
105C31×#97—21no—
106C31×#97—21no—
107C31×#97—21no—
108C4×C8×C31C2312yes—
109C24×C31—21no—
110C24×C31—21no—
111#41×C31—44no—
112#41×C31—416no—
113#41×C31—48no—
114#41×C31—48no—
115C2×Q8×C31—22no—
116C2×Q8×C31—22no—
117C2×Q8×C31—21no—
118C2×Q8×C31—44no—
119C2×D4×C31—24no—
120C2×D4×C31—28no—
121C2×D4×C31—21no—
122C2×D4×C31—21no—
123C2×D4×C31—21no—
124C2×D4×C31—21no—
125C2×D4×C31—21no—
126C2×D4×C31—42no—
127C2×D4×C31—48no—
128C2×D4×C31—41no—
129C2×D4×C31C2×D412yes—
130C2×D4×C31C2×D412yes—
131C22×C4×C31—28no—
132C22×C4×C31—21no—
133C22×C4×C31—21no—
134C22×C4×C31—22no—
135C22×C4×C31—22no—
136C22×C4×C31—22no—
137C22×C4×C31—22no—
138C22×C4×C31—22no—
139C22×C4×C31—44no—
140C22×C4×C31—44no—
141C22×C4×C31—48no—
142C22×C4×C31—41no—
143C22×C4×C31C2×D411yescharacteristic
144C22×C4×C31C2×D411yescharacteristic
145C22×C4×C31C2×D411yescharacteristic
146D8×C31—44no—
147D8×C31—44no—
148M16×C31—42no—
149C2×C8×C31—22no—
150C2×C8×C31—22no—
151C2×C8×C31—22no—
152C4⋊C4×C31—28no—
153C4⋊C4×C31—21no—
154C4⋊C4×C31—22no—
155C4⋊C4×C31—21no—
156C4⋊C4×C31—22no—
157C4⋊C4×C31—22no—
158C4⋊C4×C31—21no—
159C4⋊C4×C31—21no—
160C4⋊C4×C31C2×D414yes—
161C4⋊C4×C31C2×D412yes—
162C4⋊C4×C31C2×D412yes—
163C22⋊C4×C31—28no—
164C22⋊C4×C31—22no—
165C22⋊C4×C31—22no—
166C22⋊C4×C31—22no—
167C22⋊C4×C31—22no—
168C22⋊C4×C31—41no—
169C22⋊C4×C31—41no—
170C42×C31C2×D411yescharacteristic
171C42×C31C2×D411yescharacteristic
172C42×C31C2×D411yescharacteristic
173C42×C31C2411yescharacteristic
174#18533C3111yesmaximal, characteristic, direct product factor, Sylow
175C23×C31—24no—
176C23×C31—22no—
177C23×C31—22no—
178C23×C31—42no—
179C23×C31—42no—
180C23×C31—42no—
181C23×C31—42no—
182C23×C31—42no—
183C23×C31—41no—
184C23×C31#12011yescharacteristic
185Q8×C31—44no—
186Q8×C31—42no—
187Q8×C31—44no—
188D4×C31—24no—
189D4×C31—22no—
190D4×C31—22no—
191D4×C31—22no—
192D4×C31—42no—
193D4×C31—42no—
194D4×C31—416no—
195D4×C31—42no—
196D4×C31—42no—
197D4×C31—82no—
198D4×C31AGL(1,ℤ/8)14yes—
199C2×C4×C31—22no—
200C2×C4×C31—22no—
201C2×C4×C31—22no—
202C2×C4×C31—22no—
203C2×C4×C31—22no—
204C2×C4×C31—24no—
205C2×C4×C31—44no—
206C2×C4×C31—44no—
207C2×C4×C31—42no—
208C2×C4×C31—42no—
209C2×C4×C31—44no—
210C2×C4×C31—416no—
211C2×C4×C31—44no—
212C2×C4×C31—44no—
213C2×C4×C31—41no—
214C2×C4×C31—82no—
215C2×C4×C31#12011yescharacteristic
216C2×C4×C31#12011yescharacteristic
217C2×C4×C31#12011yescharacteristic
218C2×C4×C31C22×D411yescharacteristic
219C2×C4×C31C22×D411yescharacteristic
220C2×C4×C31C22×D411yescharacteristic
221C8×C31—42no—
222C8×C31—42no—
223C8×C31—42no—
224C2×#547C2×C3111yescharacteristic, complemented
225#3141C2×C3111yescharacteristic, complemented
226#3109C2×C3111yescharacteristic, complemented
227#3109C2×C3111yescharacteristic, complemented
228#1635C2×C3114yescomplemented
229#1625C2×C3114yescomplemented
230#1556C2×C3111yescharacteristic, complemented
231#1490C2×C3111yescharacteristic, complemented
232#1490C2×C3111yescharacteristic, complemented
233C22×C31—21no—
234C22×C31—21no—
235C22×C31—24no—
236C22×C31—48no—
237C22×C31—42no—
238C22×C31—42no—
239C22×C31—42no—
240C22×C31—42no—
241C22×C31—42no—
242C22×C31—84no—
243C22×C31—81no—
244C22×C31—81no—
245C22×C31—81no—
246C22×C31#45412yes—
247C22×C31C2×#12011yescharacteristic
248C4×C31—21no—
249C4×C31—21no—
250C4×C31—21no—
251C4×C31—21no—
252C4×C31—42no—
253C4×C31—44no—
254C4×C31—42no—
255C4×C31—81no—
256C4×C31C2×AGL(1,ℤ/8)12yes—
257C4×C31#57512yes—
258C22×#41—21no—
259#547—22no—
260#547—22no—
261#547—22no—
262#547C22×C3114yes—
263#532C22×C3111yescharacteristic
264C2×C4⋊Q8C22×C3111yescharacteristic
265C2×#123—22no—
266C2×C22⋊Q8—21no—
267C2×#121—21no—
268C2×C4×D4C22×C3111yescharacteristic
269C2×C4×D4C22×C3111yescharacteristic
270C22×C4⋊C4—21no—
271#489—22no—
272#487C22×C3112yescomplemented
273#486C22×C3112yescomplemented
274#481—21no—
275#481—21no—
276#476C22×C3112yescomplemented
277#476C22×C3112yescomplemented
278#459C22×C3112yescomplemented
279#459C22×C3112yescomplemented
280#447—21no—
281#447—21no—
282#442—22no—
283#442—22no—
284#432C22×C3111yescharacteristic
285#432C22×C3111yescharacteristic
286#417—22no—
287C4×M16C22×C3111yescharacteristic
288#332C22×C3114yescomplemented
289D4⋊C8C22×C3114yescomplemented
290D4⋊C8C22×C3114yescomplemented
291C2×C31—22no—
292C2×C31—44no—
293C2×C31—82no—
294C2×C31—161no—
295C2×C31#300311yescharacteristic
296C2×C31#300511yescharacteristic
297C2×C31C2×#45411yescharacteristic
298C2×#41—24no—
299C2×#41—24no—
300C2×#41—21no—
301C2×#41—48no—
302C22×Q8—21no—
303C22×D4—22no—
304C22×D4D4×C3111yescharacteristic
305C22×D4D4×C3111yescharacteristic
306C23×C4—22no—
307C23×C4—21no—
308C23×C4—21no—
309C2×D8—22no—
310C2×D8—22no—
311C2×M16—21no—
312C4⋊Q8—21no—
313C4⋊Q8—21no—
314C4⋊Q8D4×C3112yes—
315C4⋊Q8C23×C3111yescharacteristic
316C4⋊Q8C23×C3112yes—
317#127C23×C3111yescharacteristic
318#124—21no—
319#123—28no—
320#123—28no—
321C22⋊Q8—24no—
322C22⋊Q8—24no—
323#121—24no—
324#121—24no—
325#121—21no—
326#121—21no—
327C4×D4—21no—
328C4×D4—21no—
329C4×D4—21no—
330C4×D4—21no—
331C4×D4—21no—
332C4×D4—21no—
333C4×D4D4×C3112yes—
334C4×D4D4×C3112yes—
335C4×D4C23×C3112yes—
336C4×D4C23×C3112yes—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—24no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—24no—
341C2×C4⋊C4D4×C3111yescharacteristic
342C2×C4⋊C4D4×C3111yescharacteristic
343C2×C4⋊C4D4×C3112yes—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—21no—
347C2×C42C23×C3111yescharacteristic
348#106—22no—
349#106—22no—
350C4⋊C8C23×C3112yes—
351C4⋊C8C23×C3112yes—
352#103—24no—
353#102—22no—
354#102—22no—
355#102—24no—
356#102—22no—
357#102—22no—
358C22⋊C8—21no—
359C22⋊C8—21no—
360#97—21no—
361#97—21no—
362#97—21no—
363C4×C8C23×C3112yes—
364C31#1853311yescharacteristic, direct product factor, Sylow
365C24—21no—
366C24—21no—
367#41—44no—
368#41—416no—
369#41—48no—
370#41—48no—
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8—21no—
374C2×Q8—44no—
375C2×D4—24no—
376C2×D4—28no—
377C2×D4—21no—
378C2×D4—21no—
379C2×D4—21no—
380C2×D4—21no—
381C2×D4—21no—
382C2×D4—42no—
383C2×D4—48no—
384C2×D4—41no—
385C2×D4C2×D4×C3112yes—
386C2×D4C2×D4×C3112yes—
387C22×C4—28no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—48no—
398C22×C4—41no—
399C22×C4C2×D4×C3111yescharacteristic
400C22×C4C2×D4×C3111yescharacteristic
401C22×C4C2×D4×C3111yescharacteristic
402D8—44no—
403D8—44no—
404M16—42no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—22no—
408C4⋊C4—28no—
409C4⋊C4—21no—
410C4⋊C4—22no—
411C4⋊C4—21no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4C2×D4×C3114yes—
417C4⋊C4C2×D4×C3112yes—
418C4⋊C4C2×D4×C3112yes—
419C22⋊C4—28no—
420C22⋊C4—22no—
421C22⋊C4—22no—
422C22⋊C4—22no—
423C22⋊C4—22no—
424C22⋊C4—41no—
425C22⋊C4—41no—
426C42C2×D4×C3111yescharacteristic
427C42C2×D4×C3111yescharacteristic
428C42C2×D4×C3111yescharacteristic
429C42C24×C3111yescharacteristic
430C23—24no—
431C23—22no—
432C23—22no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—41no—
439C23C31×#12011yescharacteristic
440Q8—44no—
441Q8—42no—
442Q8—44no—
443D4—24no—
444D4—22no—
445D4—22no—
446D4—22no—
447D4—42no—
448D4—42no—
449D4—416no—
450D4—42no—
451D4—42no—
452D4—82no—
453D4C31×AGL(1,ℤ/8)14yes—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—44no—
465C2×C4—416no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—41no—
469C2×C4—82no—
470C2×C4C31×#12011yescharacteristic
471C2×C4C31×#12011yescharacteristic
472C2×C4C31×#12011yescharacteristic
473C2×C4C22×D4×C3111yescharacteristic
474C2×C4C22×D4×C3111yescharacteristic
475C2×C4C22×D4×C3111yescharacteristic
476C8—42no—
477C8—42no—
478C8—42no—
479C22—21no—
480C22—21no—
481C22—24no—
482C22—48no—
483C22—42no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22—84no—
489C22—81no—
490C22—81no—
491C22—81no—
492C22C31×#45412yes—
493C22C2×C31×#12011yescharacteristic
494C4—21no—
495C4—21no—
496C4—21no—
497C4—21no—
498C4—42no—
499C4—44no—
500C4—42no—
501C4—81no—
502C4C2×C31×AGL(1,ℤ/8)12yes—
503C4C31×#57512yes—
504C2—22no—
505C2—44no—
506C2—82no—
507C2—161no—
508C2C31×#300311yescharacteristic
509C2C31×#300511yescharacteristic
510C2C2×C31×#45411yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC42C24×C31
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. C2×C31×#547C2semidirect
2. C31×#3141C2semidirect
3. C31×#3109C2semidirect
4. C31×#3109C2semidirect
5. C31×#1635C2semidirect
6. C31×#1625C2semidirect
7. C31×#1556C2semidirect
8. C31×#1490C2semidirect
9. C31×#1490C2semidirect
24. C31×#487C22semidirect
25. C31×#486C22semidirect
28. C31×#476C22semidirect
29. C31×#476C22semidirect
30. C31×#459C22semidirect
31. C31×#459C22semidirect
40. C31×#332C22semidirect
41. C31×D4⋊C8C22semidirect
42. C31×D4⋊C8C22semidirect
174. #18533C31direct (N × H)
224. C2×#547C2×C31semidirect
225. #3141C2×C31semidirect
226. #3109C2×C31semidirect
227. #3109C2×C31semidirect
228. #1635C2×C31semidirect
229. #1625C2×C31semidirect
230. #1556C2×C31semidirect
231. #1490C2×C31semidirect
232. #1490C2×C31semidirect
272. #487C22×C31semidirect
273. #486C22×C31semidirect
276. #476C22×C31semidirect
277. #476C22×C31semidirect
278. #459C22×C31semidirect
279. #459C22×C31semidirect
288. #332C22×C31semidirect
289. D4⋊C8C22×C31semidirect
290. D4⋊C8C22×C31semidirect
364. C31#18533direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5472
C31×#31412
C31×#31092
C31×#16352
C31×#16252
C31×#15562
C31×#14902
#1853331