Finite Group Databasebeta

← C31×#23984 #21480743

Subgroups — C31×#23984 #21480743

Order 7936.

2810 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3490C211yesmaximal, characteristic, complemented
2C31×#2846C212yesmaximal, complemented
3C2×C31×#379C211yesmaximal, characteristic, complemented
4C31×#2029C212yesmaximal, complemented
5C31×#2021C212yesmaximal, complemented
6C31×#1898C212yesmaximal, complemented
7C31×#1889C214yesmaximal, complemented
8C31×#1819C211yesmaximal, characteristic, complemented
9C26×C31C2211yescharacteristic
10C31×#551—24no—
11C31×#546—24no—
12C31×#543C2211yescharacteristic, complemented
13C31×#538—22no—
14C31×#534C2211yescharacteristic, complemented
15C31×#525—22no—
16C2×C31×#120—24no—
17C2×C31×#120C2214yes—
18C2×C31×#120C2212yes—
19C31×#518C2212yescomplemented
20C22×C22⋊C4×C31—22no—
21C22×C22⋊C4×C31C2211yescharacteristic
22C22×C22⋊C4×C31C2211yescharacteristic
23C2×C31×#99C2214yescomplemented
24C31×#379—24no—
25C31×#379—22no—
26C31×#379C2214yes—
27C31×#379C2212yes—
28C31×#379C2212yes—
29C31×#379C2212yes—
30C31×#342C2212yescomplemented
31C31×#342C2212yescomplemented
32C31×#342C2214yescomplemented
33C25×C31—21no—
34C25×C31—24no—
35C25×C31—21no—
36C25×C31—24no—
37C25×C31—48no—
38C25×C31—42no—
39C25×C31C2312yes—
40C25×C31C2311yescharacteristic
41C2×#41×C31—22no—
42C22×D4×C31—24no—
43C22×D4×C31D414yescomplemented
44C22×D4×C31C2312yes—
45C23×C4×C31—21no—
46C23×C4×C31D412yescomplemented
47C31×#126—24no—
48C31×#124—24no—
49C31×#123—24no—
50C31×#123—24no—
51C31×#123—22no—
52C31×C22⋊Q8—22no—
53C31×C22⋊Q8—22no—
54C31×C22⋊Q8—24no—
55C31×#121—22no—
56C31×#121—24no—
57C31×#121—22no—
58C31×#120—22no—
59C31×#120—22no—
60C31×#120—24no—
61C31×#120—22no—
62C31×#120—28no—
63C31×#120—44no—
64C31×#120—416no—
65C4×D4×C31—24no—
66C4×D4×C31—24no—
67C4×D4×C31—24no—
68C31×#117—22no—
69C31×#117—22no—
70C2×C22⋊C4×C31—24no—
71C2×C22⋊C4×C31—22no—
72C2×C22⋊C4×C31—21no—
73C2×C22⋊C4×C31—24no—
74C2×C22⋊C4×C31—24no—
75C2×C22⋊C4×C31—24no—
76C2×C22⋊C4×C31—28no—
77C2×C22⋊C4×C31—24no—
78C2×C22⋊C4×C31—42no—
79C2×C22⋊C4×C31—48no—
80C2×C22⋊C4×C31D412yescomplemented
81C2×C22⋊C4×C31D414yescomplemented
82C2×C22⋊C4×C31C2312yes—
83C2×C22⋊C4×C31C2314yes—
84C2×C22⋊C4×C31C2311yescharacteristic
85C2×C22⋊C4×C31C2311yescharacteristic
86C2×C22⋊C4×C31C2312yes—
87C31×#99—44no—
88C24×C31—28no—
89C24×C31—24no—
90C24×C31—22no—
91C24×C31—24no—
92C24×C31—22no—
93C24×C31—22no—
94C24×C31—216no—
95C24×C31—24no—
96C24×C31—22no—
97C24×C31—24no—
98C24×C31—48no—
99C24×C31—44no—
100C24×C31—416no—
101C24×C31—42no—
102C24×C31—44no—
103C24×C31—48no—
104C24×C31—416no—
105C24×C31—432no—
106C24×C31—42no—
107C24×C31—48no—
108C24×C31—432no—
109C24×C31—42no—
110C24×C31—42no—
111C24×C31C2×D411yescharacteristic
112C24×C31C2×D414yes—
113C24×C31C2×D412yes—
114C24×C31C2×D411yescharacteristic
115C24×C31#4114yescomplemented
116C24×C31C2411yescharacteristic
117#41×C31—84no—
118C2×Q8×C31—22no—
119C2×D4×C31—24no—
120C2×D4×C31—22no—
121C2×D4×C31—22no—
122C2×D4×C31—24no—
123C2×D4×C31—24no—
124C2×D4×C31—22no—
125C2×D4×C31—24no—
126C2×D4×C31—48no—
127C2×D4×C31—48no—
128C2×D4×C31—416no—
129C22×C4×C31—22no—
130C22×C4×C31—24no—
131C22×C4×C31—22no—
132C22×C4×C31—22no—
133C22×C4×C31—24no—
134C22×C4×C31—44no—
135C22×C4×C31—44no—
136C22×C4×C31—42no—
137C22×C4×C31C2×D412yes—
138C22×C4×C31C2×D412yes—
139C22×C4×C31C2×D414yes—
140C22×C4×C31C2×D412yes—
141C4⋊C4×C31—24no—
142C4⋊C4×C31—22no—
143C4⋊C4×C31—22no—
144C4⋊C4×C31—24no—
145C22⋊C4×C31—24no—
146C22⋊C4×C31—24no—
147C22⋊C4×C31—24no—
148C22⋊C4×C31—24no—
149C22⋊C4×C31—24no—
150C22⋊C4×C31—22no—
151C22⋊C4×C31—22no—
152C22⋊C4×C31—22no—
153C22⋊C4×C31—22no—
154C22⋊C4×C31—24no—
155C22⋊C4×C31—24no—
156C22⋊C4×C31—24no—
157C22⋊C4×C31—48no—
158C22⋊C4×C31—48no—
159C22⋊C4×C31—48no—
160C22⋊C4×C31—42no—
161C22⋊C4×C31—416no—
162C22⋊C4×C31—48no—
163C42×C31—24no—
164#23984C3111yesmaximal, characteristic, direct product factor, Sylow
165C23×C31—21no—
166C23×C31—21no—
167C23×C31—24no—
168C23×C31—28no—
169C23×C31—216no—
170C23×C31—216no—
171C23×C31—21no—
172C23×C31—24no—
173C23×C31—21no—
174C23×C31—28no—
175C23×C31—28no—
176C23×C31—28no—
177C23×C31—22no—
178C23×C31—24no—
179C23×C31—42no—
180C23×C31—44no—
181C23×C31—48no—
182C23×C31—42no—
183C23×C31—48no—
184C23×C31—416no—
185C23×C31—48no—
186C23×C31—48no—
187C23×C31—48no—
188C23×C31—44no—
189C23×C31—48no—
190C23×C31—48no—
191C23×C31—416no—
192C23×C31—464no—
193C23×C31—48no—
194C23×C31—416no—
195C23×C31—48no—
196C23×C31—416no—
197C23×C31—44no—
198C23×C31—464no—
199C23×C31—416no—
200C23×C31—42no—
201C23×C31—42no—
202C23×C31—44no—
203C23×C31—48no—
204C23×C31#12012yes—
205C23×C31#12012yes—
206C23×C31C22×D411yescharacteristic
207C23×C31C22×D412yes—
208C23×C31C2×#4112yes—
209C23×C31H(5,𝔽2)12yes—
210Q8×C31—82no—
211D4×C31—42no—
212D4×C31—42no—
213D4×C31—44no—
214D4×C31—48no—
215D4×C31—84no—
216D4×C31—82no—
217D4×C31—816no—
218C2×C4×C31—22no—
219C2×C4×C31—22no—
220C2×C4×C31—24no—
221C2×C4×C31—22no—
222C2×C4×C31—24no—
223C2×C4×C31—44no—
224C2×C4×C31—42no—
225C2×C4×C31—42no—
226C2×C4×C31—44no—
227C2×C4×C31—44no—
228C2×C4×C31—48no—
229C2×C4×C31—42no—
230C2×C4×C31—44no—
231C2×C4×C31—82no—
232C2×C4×C31—84no—
233C2×C4×C31—84no—
234#3490C2×C3111yescharacteristic, complemented
235#2846C2×C3112yescomplemented
236C2×#379C2×C3111yescharacteristic, complemented
237#2029C2×C3112yescomplemented
238#2021C2×C3112yescomplemented
239#1898C2×C3112yescomplemented
240#1889C2×C3114yescomplemented
241#1819C2×C3111yescharacteristic, complemented
242C22×C31—24no—
243C22×C31—24no—
244C22×C31—22no—
245C22×C31—22no—
246C22×C31—24no—
247C22×C31—22no—
248C22×C31—216no—
249C22×C31—28no—
250C22×C31—22no—
251C22×C31—24no—
252C22×C31—42no—
253C22×C31—42no—
254C22×C31—416no—
255C22×C31—48no—
256C22×C31—42no—
257C22×C31—416no—
258C22×C31—48no—
259C22×C31—42no—
260C22×C31—44no—
261C22×C31—432no—
262C22×C31—432no—
263C22×C31—44no—
264C22×C31—48no—
265C22×C31—42no—
266C22×C31—44no—
267C22×C31—84no—
268C22×C31B(4,𝔽2)14yes—
269C22×C31C2×#12011yescharacteristic
270C22×C31#52511yescharacteristic
271C22×C31#53511yescharacteristic
272C22×C31#54614yes—
273C4×C31—42no—
274C4×C31—42no—
275C4×C31—42no—
276C4×C31—44no—
277C4×C31—82no—
278C4×C31—84no—
279C26C22×C3111yescharacteristic
280#551—24no—
281#546—24no—
282#543C22×C3111yescharacteristic, complemented
283#538—22no—
284#534C22×C3111yescharacteristic, complemented
285#525—22no—
286C2×#120—24no—
287C2×#120C22×C3114yes—
288C2×#120C22×C3112yes—
289#518C22×C3112yescomplemented
290C22×C22⋊C4—22no—
291C22×C22⋊C4C22×C3111yescharacteristic
292C22×C22⋊C4C22×C3111yescharacteristic
293C2×#99C22×C3114yescomplemented
294#379—24no—
295#379—22no—
296#379C22×C3112yes—
297#379C22×C3114yes—
298#379C22×C3112yes—
299#379C22×C3112yes—
300#342C22×C3112yescomplemented
301#342C22×C3112yescomplemented
302#342C22×C3114yescomplemented
303C2×C31—21no—
304C2×C31—24no—
305C2×C31—21no—
306C2×C31—24no—
307C2×C31—48no—
308C2×C31—42no—
309C2×C31—82no—
310C2×C31#240411yescharacteristic
311C2×C31C2×B(4,𝔽2)12yes—
312C25—21no—
313C25—24no—
314C25—21no—
315C25—24no—
316C25—48no—
317C25—42no—
318C25C23×C3112yes—
319C25C23×C3111yescharacteristic
320C2×#41—22no—
321C22×D4—24no—
322C22×D4D4×C3114yescomplemented
323C22×D4C23×C3112yes—
324C23×C4—21no—
325C23×C4D4×C3112yescomplemented
326#126—24no—
327#124—24no—
328#123—24no—
329#123—24no—
330#123—22no—
331C22⋊Q8—22no—
332C22⋊Q8—22no—
333C22⋊Q8—24no—
334#121—22no—
335#121—24no—
336#121—22no—
337#120—22no—
338#120—22no—
339#120—24no—
340#120—22no—
341#120—28no—
342#120—44no—
343#120—416no—
344C4×D4—24no—
345C4×D4—24no—
346C4×D4—24no—
347#117—22no—
348#117—22no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—24no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4—24no—
355C2×C22⋊C4—28no—
356C2×C22⋊C4—24no—
357C2×C22⋊C4—42no—
358C2×C22⋊C4—48no—
359C2×C22⋊C4D4×C3112yescomplemented
360C2×C22⋊C4D4×C3114yescomplemented
361C2×C22⋊C4C23×C3112yes—
362C2×C22⋊C4C23×C3114yes—
363C2×C22⋊C4C23×C3111yescharacteristic
364C2×C22⋊C4C23×C3111yescharacteristic
365C2×C22⋊C4C23×C3112yes—
366#99—44no—
367C31#2398411yescharacteristic, direct product factor, Sylow
368C24—28no—
369C24—24no—
370C24—22no—
371C24—24no—
372C24—22no—
373C24—22no—
374C24—216no—
375C24—24no—
376C24—22no—
377C24—24no—
378C24—44no—
379C24—48no—
380C24—416no—
381C24—44no—
382C24—42no—
383C24—48no—
384C24—416no—
385C24—432no—
386C24—48no—
387C24—42no—
388C24—432no—
389C24—42no—
390C24—42no—
391C24C2×D4×C3111yescharacteristic
392C24C2×D4×C3114yes—
393C24C2×D4×C3112yes—
394C24C2×D4×C3111yescharacteristic
395C24#41×C3114yescomplemented
396C24C24×C3111yescharacteristic
397#41—84no—
398C2×Q8—22no—
399C2×D4—24no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—24no—
403C2×D4—24no—
404C2×D4—22no—
405C2×D4—24no—
406C2×D4—48no—
407C2×D4—48no—
408C2×D4—416no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—42no—
417C22×C4C2×D4×C3112yes—
418C22×C4C2×D4×C3112yes—
419C22×C4C2×D4×C3114yes—
420C22×C4C2×D4×C3112yes—
421C4⋊C4—24no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—24no—
425C22⋊C4—24no—
426C22⋊C4—24no—
427C22⋊C4—24no—
428C22⋊C4—24no—
429C22⋊C4—24no—
430C22⋊C4—22no—
431C22⋊C4—22no—
432C22⋊C4—22no—
433C22⋊C4—22no—
434C22⋊C4—24no—
435C22⋊C4—24no—
436C22⋊C4—24no—
437C22⋊C4—48no—
438C22⋊C4—48no—
439C22⋊C4—48no—
440C22⋊C4—42no—
441C22⋊C4—416no—
442C22⋊C4—48no—
443C42—24no—
444C23—24no—
445C23—21no—
446C23—21no—
447C23—28no—
448C23—216no—
449C23—216no—
450C23—28no—
451C23—21no—
452C23—24no—
453C23—21no—
454C23—28no—
455C23—28no—
456C23—22no—
457C23—24no—
458C23—42no—
459C23—44no—
460C23—48no—
461C23—42no—
462C23—48no—
463C23—416no—
464C23—48no—
465C23—48no—
466C23—48no—
467C23—44no—
468C23—48no—
469C23—48no—
470C23—416no—
471C23—464no—
472C23—48no—
473C23—416no—
474C23—48no—
475C23—416no—
476C23—44no—
477C23—464no—
478C23—416no—
479C23—48no—
480C23—42no—
481C23—42no—
482C23—44no—
483C23C31×#12012yes—
484C23C31×#12012yes—
485C23C22×D4×C3111yescharacteristic
486C23C22×D4×C3112yes—
487C23C2×#41×C3112yes—
488C23C31×H(5,𝔽2)12yes—
489Q8—82no—
490D4—48no—
491D4—42no—
492D4—42no—
493D4—44no—
494D4—84no—
495D4—82no—
496D4—816no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—48no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—82no—
511C2×C4—84no—
512C2×C4—84no—
513C22—24no—
514C22—24no—
515C22—22no—
516C22—22no—
517C22—24no—
518C22—22no—
519C22—216no—
520C22—28no—
521C22—22no—
522C22—24no—
523C22—42no—
524C22—42no—
525C22—416no—
526C22—48no—
527C22—42no—
528C22—416no—
529C22—48no—
530C22—42no—
531C22—44no—
532C22—432no—
533C22—432no—
534C22—44no—
535C22—48no—
536C22—42no—
537C22—44no—
538C22—84no—
539C22C31×B(4,𝔽2)14yes—
540C22C2×C31×#12011yescharacteristic
541C22C31×#52511yescharacteristic
542C22C31×#53511yescharacteristic
543C22C31×#54614yes—
544C4—42no—
545C4—42no—
546C4—42no—
547C4—44no—
548C4—82no—
549C4—84no—
550C2—21no—
551C2—24no—
552C2—21no—
553C2—24no—
554C2—48no—
555C2—42no—
556C2—82no—
557C2C31×#240411yescharacteristic
558C2C2×C31×B(4,𝔽2)12yes—
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC24C24×C31
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C25×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#3490C2semidirect
2. C31×#2846C2semidirect
3. C2×C31×#379C2semidirect
4. C31×#2029C2semidirect
5. C31×#2021C2semidirect
6. C31×#1898C2semidirect
7. C31×#1889C2semidirect
8. C31×#1819C2semidirect
12. C31×#543C22semidirect
14. C31×#534C22semidirect
19. C31×#518C22semidirect
23. C2×C31×#99C22semidirect
30. C31×#342C22semidirect
31. C31×#342C22semidirect
32. C31×#342C22semidirect
43. C22×D4×C31D4semidirect
46. C23×C4×C31D4semidirect
80. C2×C22⋊C4×C31D4semidirect
81. C2×C22⋊C4×C31D4semidirect
115. C24×C31#41semidirect
164. #23984C31direct (N × H)
234. #3490C2×C31semidirect
235. #2846C2×C31semidirect
236. C2×#379C2×C31semidirect
237. #2029C2×C31semidirect
238. #2021C2×C31semidirect
239. #1898C2×C31semidirect
240. #1889C2×C31semidirect
241. #1819C2×C31semidirect
282. #543C22×C31semidirect
284. #534C22×C31semidirect
289. #518C22×C31semidirect
293. C2×#99C22×C31semidirect
300. #342C22×C31semidirect
301. #342C22×C31semidirect
302. #342C22×C31semidirect
322. C22×D4D4×C31semidirect
325. C23×C4D4×C31semidirect
359. C2×C22⋊C4D4×C31semidirect
360. C2×C22⋊C4D4×C31semidirect
367. C31#23984direct (N × H)
395. C24#41×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#34902
C31×#28462
C2×C31×#3792
C31×#20292
C31×#20212
C31×#18982
C31×#18892
C31×#18192
#2398431