Finite Group Databasebeta

← C31×#23998 #21480757

Subgroups — C31×#23998 #21480757

Order 7936.

1266 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3508C211yesmaximal, characteristic, complemented
2C31×#3019C211yesmaximal, characteristic, complemented
3C2×C31×#453C211yesmaximal, characteristic, complemented
4C31×#2883C211yesmaximal, characteristic, complemented
5C31×#2031C212yesmaximal, complemented
6C31×#2008C212yesmaximal, complemented
7C31×#1911C211yesmaximal, characteristic, complemented
8C31×#1906C212yesmaximal, complemented
9C31×#1904C212yesmaximal, complemented
10C31×#1887C211yesmaximal, characteristic, complemented
11C31×#1829C211yesmaximal, characteristic, complemented
12C2×C31×H(5,𝔽2)C2211yescharacteristic
13C31×#575C2212yes—
14C2×C31×AGL(1,ℤ/8)C2212yes—
15C31×#559—22no—
16C31×#550—22no—
17C31×#546—24no—
18C31×#540—22no—
19C31×#535C2211yescharacteristic
20C31×#535C2211yescharacteristic, complemented
21C31×#533C2211yescharacteristic, complemented
22C31×#532—22no—
23C2×C31×#127C2211yescharacteristic
24C31×#518C2211yescharacteristic
25C31×#518C2211yescharacteristic
26C31×#518C2211yescharacteristic, complemented
27C4×#41×C31C2211yescharacteristic, complemented
28C31×#455—22no—
29C31×#453—22no—
30C31×#453—22no—
31C31×#453—22no—
32C31×#421C2212yes—
33C31×#421C2212yescomplemented
34C2×C31×#104C2212yes—
35C31×#418C2212yescomplemented
36C2×C31×#100C2211yescharacteristic
37C2×C31×#100C2211yescharacteristic
38C31×#410C2212yes—
39C31×#410C2212yes—
40C2×C31×#99—24no—
41C31×#344C2212yescomplemented
42C31×#344C2212yescomplemented
43C31×#339C2212yescomplemented
44C31×H(3,ℤ/4)C2212yescomplemented
45C31×H(5,𝔽2)—22no—
46C31×H(5,𝔽2)—22no—
47C31×H(5,𝔽2)—42no—
48C2×#41×C31—24no—
49C2×#41×C31—22no—
50C2×#41×C31C2311yescharacteristic
51C2×#41×C31C2312yes—
52C22×D4×C31—22no—
53C22×D4×C31—22no—
54C22×D4×C31—22no—
55C22×D4×C31C2311yescharacteristic
56C22×D4×C31C2311yescharacteristic
57C31×#137—22no—
58C31×#137—22no—
59C31×AGL(1,ℤ/8)—24no—
60C31×AGL(1,ℤ/8)—22no—
61C31×AGL(1,ℤ/8)—22no—
62C31×AGL(1,ℤ/8)—42no—
63C31×#135—42no—
64C2×SD16×C31—22no—
65C2×D8×C31—22no—
66C2×M16×C31C2312yes—
67C2×M16×C31C2312yes—
68C31×C4⋊Q8—21no—
69C31×#127—22no—
70C31×#127—21no—
71C31×#127D412yes—
72C31×#127D412yes—
73C31×#127D412yes—
74C31×#125—21no—
75C31×#124—24no—
76C31×#124—21no—
77C31×#124D412yes—
78C31×#123—24no—
79C31×#123—24no—
80C31×C22⋊Q8—22no—
81C31×#121—24no—
82C31×#121—24no—
83C31×#121—24no—
84C31×#121—22no—
85C31×#121—22no—
86C31×#120—22no—
87C31×#120—22no—
88C4×Q8×C31—21no—
89C4×Q8×C31—22no—
90C4×D4×C31—24no—
91C4×D4×C31—21no—
92C4×D4×C31—22no—
93C4×D4×C31—22no—
94C4×D4×C31—22no—
95C4×D4×C31—22no—
96C4×D4×C31—22no—
97C4×D4×C31—42no—
98C31×#117—21no—
99C31×#117C2312yes—
100C31×#117C2311yescharacteristic
101C31×#117C2311yescharacteristic
102C31×#117C2311yescharacteristic
103C2×C22⋊C4×C31—22no—
104C2×C22⋊C4×C31—22no—
105C2×C22⋊C4×C31D414yes—
106C2×C42×C31—21no—
107C2×C42×C31C2311yescharacteristic
108C31×#104—22no—
109C31×#104—22no—
110C31×#104—24no—
111C31×#104—42no—
112C31×#102—42no—
113C31×#100—22no—
114C31×#100—22no—
115C31×#99—24no—
116C31×#99—24no—
117C24×C31—22no—
118C24×C31—21no—
119C24×C31C2×D412yes—
120#41×C31—22no—
121#41×C31—22no—
122#41×C31—22no—
123#41×C31—22no—
124#41×C31—22no—
125#41×C31—22no—
126#41×C31—44no—
127#41×C31—44no—
128#41×C31—88no—
129C2×Q8×C31—22no—
130C2×Q8×C31C2×D411yescharacteristic
131C2×Q8×C31#4112yescomplemented
132C2×D4×C31—24no—
133C2×D4×C31—24no—
134C2×D4×C31—22no—
135C2×D4×C31—21no—
136C2×D4×C31—24no—
137C2×D4×C31—24no—
138C2×D4×C31—24no—
139C2×D4×C31—21no—
140C2×D4×C31—22no—
141C2×D4×C31—22no—
142C2×D4×C31—22no—
143C2×D4×C31—22no—
144C2×D4×C31—44no—
145C2×D4×C31—44no—
146C2×D4×C31—44no—
147C2×D4×C31—44no—
148C2×D4×C31—84no—
149C2×D4×C31C2×D412yes—
150C2×D4×C31C2×D411yescharacteristic
151C2×D4×C31C2×D412yes—
152C2×D4×C31C2×D412yes—
153C2×D4×C31#4112yescomplemented
154C22×C4×C31—22no—
155C22×C4×C31—22no—
156C22×C4×C31—21no—
157C22×C4×C31—24no—
158C22×C4×C31—24no—
159C22×C4×C31—21no—
160C22×C4×C31—22no—
161C22×C4×C31—42no—
162C22×C4×C31C2411yescharacteristic
163Q16×C31—42no—
164SD16×C31—44no—
165SD16×C31—42no—
166SD16×C31—42no—
167D8×C31—44no—
168D8×C31—42no—
169M16×C31—22no—
170M16×C31—22no—
171M16×C31—22no—
172M16×C31—22no—
173C2×C8×C31—22no—
174C2×C8×C31—22no—
175C4⋊C4×C31—22no—
176C4⋊C4×C31—22no—
177C4⋊C4×C31—22no—
178C4⋊C4×C31—22no—
179C4⋊C4×C31—21no—
180C4⋊C4×C31—21no—
181C4⋊C4×C31—21no—
182C4⋊C4×C31—22no—
183C22⋊C4×C31—24no—
184C22⋊C4×C31—22no—
185C22⋊C4×C31—24no—
186C22⋊C4×C31—24no—
187C22⋊C4×C31—24no—
188C22⋊C4×C31—21no—
189C22⋊C4×C31—22no—
190C22⋊C4×C31—42no—
191C22⋊C4×C31C2×D412yes—
192C22⋊C4×C31C2×D412yes—
193C42×C31—22no—
194C42×C31—21no—
195C42×C31—21no—
196C42×C31—21no—
197C42×C31—22no—
198C42×C31C2×D411yescharacteristic
199C42×C31C2×D411yescharacteristic
200C42×C31C2×D412yes—
201#23998C3111yesmaximal, characteristic, direct product factor, Sylow
202C23×C31—22no—
203C23×C31—22no—
204C23×C31—22no—
205C23×C31—21no—
206C23×C31—22no—
207C23×C31—44no—
208C23×C31—41no—
209C23×C31—41no—
210C23×C31—82no—
211C23×C31—82no—
212C23×C31—162no—
213C23×C31#12014yes—
214C23×C31C22×D411yescharacteristic
215Q8×C31—22no—
216Q8×C31—22no—
217Q8×C31—22no—
218Q8×C31—44no—
219D4×C31—24no—
220D4×C31—24no—
221D4×C31—22no—
222D4×C31—24no—
223D4×C31—22no—
224D4×C31—22no—
225D4×C31—48no—
226D4×C31—48no—
227D4×C31—42no—
228D4×C31—41no—
229D4×C31—42no—
230D4×C31—41no—
231D4×C31—42no—
232D4×C31—84no—
233D4×C31—84no—
234D4×C31—84no—
235C2×C4×C31—21no—
236C2×C4×C31—21no—
237C2×C4×C31—22no—
238C2×C4×C31—21no—
239C2×C4×C31—22no—
240C2×C4×C31—22no—
241C2×C4×C31—22no—
242C2×C4×C31—42no—
243C2×C4×C31—44no—
244C2×C4×C31—44no—
245C2×C4×C31—42no—
246C2×C4×C31—42no—
247C2×C4×C31—42no—
248C2×C4×C31—42no—
249C2×C4×C31—84no—
250C2×C4×C31—84no—
251C2×C4×C31—162no—
252C2×C4×C31C22×D411yescharacteristic
253C2×C4×C31C22×D411yescharacteristic
254C2×C4×C31C2×#4112yes—
255C2×C4×C31H(5,𝔽2)12yes—
256C8×C31—42no—
257C8×C31—42no—
258#3508C2×C3111yescharacteristic, complemented
259#3019C2×C3111yescharacteristic, complemented
260C2×#453C2×C3111yescharacteristic, complemented
261#2883C2×C3111yescharacteristic, complemented
262#2031C2×C3112yescomplemented
263#2008C2×C3112yescomplemented
264#1911C2×C3111yescharacteristic, complemented
265#1906C2×C3112yescomplemented
266#1904C2×C3112yescomplemented
267#1887C2×C3111yescharacteristic, complemented
268#1829C2×C3111yescharacteristic, complemented
269C22×C31—24no—
270C22×C31—41no—
271C22×C31—44no—
272C22×C31—42no—
273C22×C31—44no—
274C22×C31—42no—
275C22×C31—41no—
276C22×C31—42no—
277C22×C31—82no—
278C22×C31—82no—
279C22×C31—82no—
280C22×C31—161no—
281C22×C31—161no—
282C22×C31—164no—
283C22×C31C2×#12011yescharacteristic
284C22×C31#52511yescharacteristic
285C22×C31#53511yescharacteristic
286C4×C31—22no—
287C4×C31—42no—
288C4×C31—41no—
289C4×C31—42no—
290C4×C31—42no—
291C4×C31—42no—
292C4×C31—41no—
293C4×C31—82no—
294C4×C31#54612yes—
295C4×C31#54612yes—
296C2×H(5,𝔽2)C22×C3111yescharacteristic
297#575C22×C3112yes—
298C2×AGL(1,ℤ/8)C22×C3112yes—
299#559—22no—
300#550—22no—
301#546—24no—
302#540—22no—
303#535C22×C3111yescharacteristic
304#535C22×C3111yescharacteristic, complemented
305#533C22×C3111yescharacteristic, complemented
306#532—22no—
307C2×#127C22×C3111yescharacteristic
308#518C22×C3111yescharacteristic
309#518C22×C3111yescharacteristic
310#518C22×C3111yescharacteristic, complemented
311C4×#41C22×C3111yescharacteristic, complemented
312#455—22no—
313#453—22no—
314#453—22no—
315#453—22no—
316#421C22×C3112yes—
317#421C22×C3112yescomplemented
318C2×#104C22×C3112yes—
319#418C22×C3112yescomplemented
320C2×#100C22×C3111yescharacteristic
321C2×#100C22×C3111yescharacteristic
322#410C22×C3112yes—
323#410C22×C3112yes—
324C2×#99—24no—
325#344C22×C3112yescomplemented
326#344C22×C3112yescomplemented
327#339C22×C3112yescomplemented
328H(3,ℤ/4)C22×C3112yescomplemented
329C2×C31—21no—
330C2×C31—21no—
331C2×C31—21no—
332C2×C31—44no—
333C2×C31—81no—
334C2×C31—82no—
335C2×C31—82no—
336C2×C31#240411yescharacteristic
337H(5,𝔽2)—22no—
338H(5,𝔽2)—22no—
339H(5,𝔽2)—42no—
340C2×#41—24no—
341C2×#41—22no—
342C2×#41C23×C3111yescharacteristic
343C2×#41C23×C3112yes—
344C22×D4—22no—
345C22×D4—22no—
346C22×D4—22no—
347C22×D4C23×C3111yescharacteristic
348C22×D4C23×C3111yescharacteristic
349#137—22no—
350#137—22no—
351AGL(1,ℤ/8)—24no—
352AGL(1,ℤ/8)—22no—
353AGL(1,ℤ/8)—22no—
354AGL(1,ℤ/8)—42no—
355#135—42no—
356C2×SD16—22no—
357C2×D8—22no—
358C2×M16C23×C3112yes—
359C2×M16C23×C3112yes—
360C4⋊Q8—21no—
361#127—22no—
362#127—21no—
363#127D4×C3112yes—
364#127D4×C3112yes—
365#127D4×C3112yes—
366#125—21no—
367#124—24no—
368#124—21no—
369#124D4×C3112yes—
370#123—24no—
371#123—24no—
372C22⋊Q8—22no—
373#121—24no—
374#121—24no—
375#121—24no—
376#121—22no—
377#121—22no—
378#120—22no—
379#120—22no—
380C4×Q8—21no—
381C4×Q8—22no—
382C4×D4—24no—
383C4×D4—21no—
384C4×D4—22no—
385C4×D4—22no—
386C4×D4—22no—
387C4×D4—22no—
388C4×D4—22no—
389C4×D4—42no—
390#117—21no—
391#117C23×C3112yes—
392#117C23×C3111yescharacteristic
393#117C23×C3111yescharacteristic
394#117C23×C3111yescharacteristic
395C2×C22⋊C4—22no—
396C2×C22⋊C4—22no—
397C2×C22⋊C4D4×C3114yes—
398C2×C42—21no—
399C2×C42C23×C3111yescharacteristic
400#104—22no—
401#104—22no—
402#104—24no—
403#104—42no—
404#102—42no—
405#100—22no—
406#100—22no—
407#99—24no—
408#99—24no—
409C31#2399811yescharacteristic, direct product factor, Sylow
410C24—22no—
411C24—21no—
412C24C2×D4×C3112yes—
413#41—22no—
414#41—22no—
415#41—22no—
416#41—22no—
417#41—22no—
418#41—22no—
419#41—44no—
420#41—44no—
421#41—88no—
422C2×Q8—22no—
423C2×Q8C2×D4×C3111yescharacteristic
424C2×Q8#41×C3112yescomplemented
425C2×D4—24no—
426C2×D4—24no—
427C2×D4—22no—
428C2×D4—21no—
429C2×D4—24no—
430C2×D4—24no—
431C2×D4—24no—
432C2×D4—21no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—22no—
437C2×D4—44no—
438C2×D4—44no—
439C2×D4—44no—
440C2×D4—44no—
441C2×D4—84no—
442C2×D4C2×D4×C3112yes—
443C2×D4C2×D4×C3111yescharacteristic
444C2×D4C2×D4×C3112yes—
445C2×D4C2×D4×C3112yes—
446C2×D4#41×C3112yescomplemented
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—42no—
455C22×C4C24×C3111yescharacteristic
456Q16—42no—
457SD16—44no—
458SD16—42no—
459SD16—42no—
460D8—44no—
461D8—42no—
462M16—22no—
463M16—22no—
464M16—22no—
465M16—22no—
466C2×C8—22no—
467C2×C8—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—22no—
476C22⋊C4—24no—
477C22⋊C4—22no—
478C22⋊C4—24no—
479C22⋊C4—24no—
480C22⋊C4—24no—
481C22⋊C4—21no—
482C22⋊C4—22no—
483C22⋊C4—42no—
484C22⋊C4C2×D4×C3112yes—
485C22⋊C4C2×D4×C3112yes—
486C42—22no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—22no—
491C42C2×D4×C3111yescharacteristic
492C42C2×D4×C3111yescharacteristic
493C42C2×D4×C3112yes—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—21no—
498C23—22no—
499C23—44no—
500C23—41no—
501C23—41no—
502C23—82no—
503C23—82no—
504C23—162no—
505C23C31×#12014yes—
506C23C22×D4×C3111yescharacteristic
507Q8—22no—
508Q8—22no—
509Q8—22no—
510Q8—44no—
511D4—24no—
512D4—24no—
513D4—22no—
514D4—24no—
515D4—22no—
516D4—22no—
517D4—48no—
518D4—48no—
519D4—42no—
520D4—41no—
521D4—42no—
522D4—41no—
523D4—42no—
524D4—84no—
525D4—84no—
526D4—84no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—84no—
542C2×C4—84no—
543C2×C4—162no—
544C2×C4C22×D4×C3111yescharacteristic
545C2×C4C22×D4×C3111yescharacteristic
546C2×C4C2×#41×C3112yes—
547C2×C4C31×H(5,𝔽2)12yes—
548C8—42no—
549C8—42no—
550C22—24no—
551C22—41no—
552C22—44no—
553C22—42no—
554C22—44no—
555C22—42no—
556C22—41no—
557C22—42no—
558C22—82no—
559C22—82no—
560C22—82no—
561C22—161no—
562C22—161no—
563C22—164no—
564C22C2×C31×#12011yescharacteristic
565C22C31×#52511yescharacteristic
566C22C31×#53511yescharacteristic
567C4—22no—
568C4—42no—
569C4—41no—
570C4—42no—
571C4—42no—
572C4—42no—
573C4—41no—
574C4—82no—
575C4C31×#54612yes—
576C4C31×#54612yes—
577C2—21no—
578C2—21no—
579C2—21no—
580C2—44no—
581C2—82no—
582C2—81no—
583C2—82no—
584C2C31×#240411yescharacteristic
subgroupG/H
center, socleC2×C31#2404
commutator, Frattini, cosocleC22×C4C24×C31
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C31 ◁ C22×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#3508C2semidirect
2. C31×#3019C2semidirect
3. C2×C31×#453C2semidirect
4. C31×#2883C2semidirect
5. C31×#2031C2semidirect
6. C31×#2008C2semidirect
7. C31×#1911C2semidirect
8. C31×#1906C2semidirect
9. C31×#1904C2semidirect
10. C31×#1887C2semidirect
11. C31×#1829C2semidirect
20. C31×#535C22semidirect
21. C31×#533C22semidirect
26. C31×#518C22semidirect
27. C4×#41×C31C22semidirect
33. C31×#421C22semidirect
35. C31×#418C22semidirect
41. C31×#344C22semidirect
42. C31×#344C22semidirect
43. C31×#339C22semidirect
44. C31×H(3,ℤ/4)C22semidirect
131. C2×Q8×C31#41semidirect
153. C2×D4×C31#41semidirect
201. #23998C31direct (N × H)
258. #3508C2×C31semidirect
259. #3019C2×C31semidirect
260. C2×#453C2×C31semidirect
261. #2883C2×C31semidirect
262. #2031C2×C31semidirect
263. #2008C2×C31semidirect
264. #1911C2×C31semidirect
265. #1906C2×C31semidirect
266. #1904C2×C31semidirect
267. #1887C2×C31semidirect
268. #1829C2×C31semidirect
304. #535C22×C31semidirect
305. #533C22×C31semidirect
310. #518C22×C31semidirect
311. C4×#41C22×C31semidirect
317. #421C22×C31semidirect
319. #418C22×C31semidirect
325. #344C22×C31semidirect
326. #344C22×C31semidirect
327. #339C22×C31semidirect
328. H(3,ℤ/4)C22×C31semidirect
409. C31#23998direct (N × H)
424. C2×Q8#41×C31semidirect
446. C2×D4#41×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#35082
C31×#30192
C2×C31×#4532
C31×#28832
C31×#20312
C31×#20082
C31×#19112
C31×#19062
C31×#19042
C31×#18872
C31×#18292
#2399831