Finite Group Databasebeta

← C31×#23591 #21480350

Subgroups — C31×#23591 #21480350

Order 7936.

1182 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3460C211yesmaximal, characteristic, complemented
2C4×C31×AGL(1,ℤ/8)C212yesmaximal, complemented
3C31×#2328C211yesmaximal, characteristic, complemented
4C31×#2061C211yesmaximal, characteristic, complemented
5C31×#2040C211yesmaximal, characteristic, complemented
6C31×#2037C212yesmaximal, complemented
7C31×#2027C211yesmaximal, characteristic, complemented
8C31×#2006C211yesmaximal, characteristic, complemented
9C31×#2002C212yesmaximal, complemented
10C31×#1759C212yesmaximal, complemented
11C4×C31×#100C211yesmaximal, characteristic, complemented
12C2×C31×AGL(1,ℤ/8)C2212yes—
13C31×#544C2211yescharacteristic
14C31×#542—22no—
15C31×#540—22no—
16C31×#538—22no—
17C31×#537C2211yescharacteristic
18C31×#535C2212yes—
19C31×#532—22no—
20C2×C31×#127C2211yescharacteristic
21C31×#529—22no—
22C31×#525—22no—
23C4×#41×C31C2212yes—
24C2×C4×D4×C31C2211yescharacteristic
25C2×C4×D4×C31C2211yescharacteristic
26C31×#442—22no—
27C31×#440—22no—
28C4×SD16×C31—22no—
29C4×D8×C31—22no—
30C31×#428C2212yes—
31C2×C31×#104C2212yes—
32C2×C31×#104C2212yes—
33C31×#418C2212yes—
34C31×#417C2212yes—
35C2×C31×#100C2212yes—
36C4×M16×C31C2212yes—
37C31×#390C2211yescharacteristic
38C31×#390C2211yescharacteristic
39C31×#386—22no—
40C31×#386—22no—
41C31×#382C2211yescharacteristic
42C31×#376C2211yescharacteristic
43C31×#343C2211yescharacteristic
44C31×#343C2211yescharacteristic
45C31×#339C2212yes—
46C31×#339C2212yes—
47C2×#41×C31—22no—
48C2×#41×C31C2312yes—
49C22×D4×C31—22no—
50C22×D4×C31C2311yescharacteristic
51C22×D4×C31C2311yescharacteristic
52C23×C4×C31—21no—
53C23×C4×C31D412yes—
54C31×AGL(1,ℤ/8)—24no—
55C31×AGL(1,ℤ/8)—24no—
56C2×SD16×C31—22no—
57C2×D8×C31—22no—
58C2×M16×C31C2312yes—
59C2×M16×C31C2312yes—
60C31×C4⋊Q8—22no—
61C31×#127—22no—
62C31×#127—22no—
63C31×#127—22no—
64C31×#126—24no—
65C31×#125—22no—
66C31×#124—24no—
67C31×#124—22no—
68C31×#123—24no—
69C31×#123—24no—
70C31×C22⋊Q8—22no—
71C31×C22⋊Q8—22no—
72C31×C22⋊Q8—22no—
73C31×#121—22no—
74C31×#121—24no—
75C31×#121—22no—
76C31×#121—22no—
77C31×#121—22no—
78C31×#121—22no—
79C31×#120—24no—
80C4×Q8×C31D412yescomplemented
81C4×Q8×C31D412yescomplemented
82C4×D4×C31—24no—
83C4×D4×C31—22no—
84C4×D4×C31—22no—
85C4×D4×C31—22no—
86C4×D4×C31—22no—
87C4×D4×C31—22no—
88C4×D4×C31—22no—
89C4×D4×C31D412yescomplemented
90C4×D4×C31D412yescomplemented
91C31×#117—22no—
92C31×#117—22no—
93C31×#117C2312yes—
94C2×C4⋊C4×C31C2311yescharacteristic
95C2×C4⋊C4×C31C2311yescharacteristic
96C2×C22⋊C4×C31—21no—
97C2×C22⋊C4×C31—22no—
98C2×C22⋊C4×C31—22no—
99C2×C22⋊C4×C31D412yes—
100C2×C42×C31—22no—
101C2×C42×C31C2311yescharacteristic
102C2×C42×C31C2311yescharacteristic
103C2×C42×C31C2311yescharacteristic
104C31×#107—22no—
105C31×#106—22no—
106C31×#104—24no—
107C31×#104—24no—
108C31×#103—22no—
109C31×#102—22no—
110C31×#102—22no—
111C31×#102—22no—
112C31×#100—24no—
113C31×#97—22no—
114C4×C8×C31—22no—
115C31×#95—21no—
116C31×#95—21no—
117C31×#95—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—84no—
125C2×Q8×C31—22no—
126C2×Q8×C31C2×D412yes—
127C2×D4×C31—22no—
128C2×D4×C31—24no—
129C2×D4×C31—24no—
130C2×D4×C31—22no—
131C2×D4×C31—28no—
132C2×D4×C31—22no—
133C2×D4×C31—22no—
134C2×D4×C31—22no—
135C2×D4×C31—22no—
136C2×D4×C31—21no—
137C2×D4×C31—21no—
138C2×D4×C31—21no—
139C2×D4×C31—82no—
140C2×D4×C31—82no—
141C2×D4×C31C2×D412yes—
142C2×D4×C31#4112yescomplemented
143C22×C4×C31—21no—
144C22×C4×C31—22no—
145C22×C4×C31—22no—
146C22×C4×C31—22no—
147C22×C4×C31—22no—
148C22×C4×C31—22no—
149C22×C4×C31—21no—
150C22×C4×C31—21no—
151C22×C4×C31—42no—
152C22×C4×C31—82no—
153C22×C4×C31—82no—
154C22×C4×C31C2×D412yes—
155C22×C4×C31C2×D411yescharacteristic
156C22×C4×C31C2×D411yescharacteristic
157C22×C4×C31C2411yescharacteristic
158SD16×C31—28no—
159D8×C31—28no—
160M16×C31—22no—
161M16×C31—22no—
162M16×C31—22no—
163M16×C31#4114yescomplemented
164C2×C8×C31—22no—
165C2×C8×C31—22no—
166C4⋊C4×C31—22no—
167C4⋊C4×C31—22no—
168C4⋊C4×C31—24no—
169C4⋊C4×C31—22no—
170C4⋊C4×C31—22no—
171C4⋊C4×C31—22no—
172C4⋊C4×C31—22no—
173C4⋊C4×C31C2×D412yes—
174C4⋊C4×C31C2×D412yes—
175C22⋊C4×C31—24no—
176C22⋊C4×C31—24no—
177C22⋊C4×C31—24no—
178C22⋊C4×C31—22no—
179C22⋊C4×C31—24no—
180C22⋊C4×C31—22no—
181C22⋊C4×C31—22no—
182C22⋊C4×C31—41no—
183C22⋊C4×C31—41no—
184C22⋊C4×C31—82no—
185C22⋊C4×C31—82no—
186C42×C31—22no—
187C42×C31—22no—
188C42×C31—22no—
189C42×C31—21no—
190C42×C31—22no—
191C42×C31C2×D412yes—
192C42×C31C2×D412yes—
193C42×C31#4112yescomplemented
194#23591C3111yesmaximal, characteristic, direct product factor, Sylow
195C23×C31—22no—
196C23×C31—22no—
197C23×C31—22no—
198C23×C31—22no—
199C23×C31—22no—
200C23×C31—42no—
201C23×C31—42no—
202C23×C31—42no—
203C23×C31—42no—
204C23×C31—41no—
205C23×C31—41no—
206C23×C31C22×D411yescharacteristic
207Q8×C31—22no—
208Q8×C31—22no—
209Q8×C31—82no—
210D4×C31—22no—
211D4×C31—24no—
212D4×C31—22no—
213D4×C31—42no—
214D4×C31—42no—
215D4×C31—42no—
216D4×C31—42no—
217D4×C31—42no—
218D4×C31—84no—
219D4×C31—88no—
220D4×C31—82no—
221D4×C31—84no—
222C2×C4×C31—22no—
223C2×C4×C31—22no—
224C2×C4×C31—22no—
225C2×C4×C31—22no—
226C2×C4×C31—22no—
227C2×C4×C31—41no—
228C2×C4×C31—41no—
229C2×C4×C31—44no—
230C2×C4×C31—44no—
231C2×C4×C31—41no—
232C2×C4×C31—41no—
233C2×C4×C31—42no—
234C2×C4×C31—42no—
235C2×C4×C31—42no—
236C2×C4×C31—41no—
237C2×C4×C31—41no—
238C2×C4×C31—42no—
239C2×C4×C31—41no—
240C2×C4×C31—41no—
241C2×C4×C31—82no—
242C2×C4×C31—82no—
243C2×C4×C31—84no—
244C2×C4×C31#12011yescharacteristic
245C2×C4×C31#12012yes—
246C2×C4×C31#12011yescharacteristic
247C2×C4×C31C22×D412yes—
248C2×C4×C31C2×#4111yescharacteristic
249C2×C4×C31C2×#4112yes—
250C2×C4×C31C2×#4111yescharacteristic
251C8×C31—24no—
252C8×C31—42no—
253#3460C2×C3111yescharacteristic, complemented
254C4×AGL(1,ℤ/8)C2×C3112yescomplemented
255#2328C2×C3111yescharacteristic, complemented
256#2061C2×C3111yescharacteristic, complemented
257#2040C2×C3111yescharacteristic, complemented
258#2037C2×C3112yescomplemented
259#2027C2×C3111yescharacteristic, complemented
260#2006C2×C3111yescharacteristic, complemented
261#2002C2×C3112yescomplemented
262#1759C2×C3112yescomplemented
263C4×#100C2×C3111yescharacteristic, complemented
264C22×C31—21no—
265C22×C31—21no—
266C22×C31—44no—
267C22×C31—44no—
268C22×C31—42no—
269C22×C31—42no—
270C22×C31—42no—
271C22×C31—42no—
272C22×C31—42no—
273C22×C31—42no—
274C22×C31—42no—
275C22×C31—82no—
276C22×C31—82no—
277C22×C31—81no—
278C22×C31—81no—
279C22×C31C2×#12011yescharacteristic
280C22×C31#52511yescharacteristic
281C22×C31#53211yescharacteristic
282C4×C31—21no—
283C4×C31—21no—
284C4×C31—22no—
285C4×C31—42no—
286C4×C31—42no—
287C4×C31—42no—
288C4×C31—42no—
289C4×C31—42no—
290C4×C31—81no—
291C4×C31—81no—
292C4×C31#52512yes—
293C4×C31#53212yes—
294C2×AGL(1,ℤ/8)C22×C3112yes—
295#544C22×C3111yescharacteristic
296#542—22no—
297#540—22no—
298#538—22no—
299#537C22×C3111yescharacteristic
300#535C22×C3112yes—
301#532—22no—
302C2×#127C22×C3111yescharacteristic
303#529—22no—
304#525—22no—
305C4×#41C22×C3112yes—
306C2×C4×D4C22×C3111yescharacteristic
307C2×C4×D4C22×C3111yescharacteristic
308#442—22no—
309#440—22no—
310C4×SD16—22no—
311C4×D8—22no—
312#428C22×C3112yes—
313C2×#104C22×C3112yes—
314C2×#104C22×C3112yes—
315#418C22×C3112yes—
316#417C22×C3112yes—
317C2×#100C22×C3112yes—
318C4×M16C22×C3112yes—
319#390C22×C3111yescharacteristic
320#390C22×C3111yescharacteristic
321#386—22no—
322#386—22no—
323#382C22×C3111yescharacteristic
324#376C22×C3111yescharacteristic
325#343C22×C3111yescharacteristic
326#343C22×C3111yescharacteristic
327#339C22×C3112yes—
328#339C22×C3112yes—
329C2×C31—21no—
330C2×C31—21no—
331C2×C31—82no—
332C2×C31—82no—
333C2×C31—82no—
334C2×C31#238811yescharacteristic
335C2×C31#301911yescharacteristic
336C2×C31#302111yescharacteristic
337C2×#41—22no—
338C2×#41C23×C3112yes—
339C22×D4—22no—
340C22×D4C23×C3111yescharacteristic
341C22×D4C23×C3111yescharacteristic
342C23×C4—21no—
343C23×C4D4×C3112yes—
344AGL(1,ℤ/8)—24no—
345AGL(1,ℤ/8)—24no—
346C2×SD16—22no—
347C2×D8—22no—
348C2×M16C23×C3112yes—
349C2×M16C23×C3112yes—
350C4⋊Q8—22no—
351#127—22no—
352#127—22no—
353#127—22no—
354#126—24no—
355#125—22no—
356#124—24no—
357#124—22no—
358#123—24no—
359#123—24no—
360C22⋊Q8—22no—
361C22⋊Q8—22no—
362C22⋊Q8—22no—
363#121—22no—
364#121—24no—
365#121—22no—
366#121—22no—
367#121—22no—
368#121—22no—
369#120—24no—
370C4×Q8D4×C3112yescomplemented
371C4×Q8D4×C3112yescomplemented
372C4×D4—24no—
373C4×D4—22no—
374C4×D4—22no—
375C4×D4—22no—
376C4×D4—22no—
377C4×D4—22no—
378C4×D4—22no—
379C4×D4D4×C3112yescomplemented
380C4×D4D4×C3112yescomplemented
381#117—22no—
382#117—22no—
383#117C23×C3112yes—
384C2×C4⋊C4C23×C3111yescharacteristic
385C2×C4⋊C4C23×C3111yescharacteristic
386C2×C22⋊C4—21no—
387C2×C22⋊C4—22no—
388C2×C22⋊C4—22no—
389C2×C22⋊C4D4×C3112yes—
390C2×C42—22no—
391C2×C42C23×C3111yescharacteristic
392C2×C42C23×C3111yescharacteristic
393C2×C42C23×C3111yescharacteristic
394#107—22no—
395#106—22no—
396#104—24no—
397#104—24no—
398#103—22no—
399#102—22no—
400#102—22no—
401#102—22no—
402#100—24no—
403#97—22no—
404C4×C8—22no—
405#95—21no—
406#95—21no—
407#95—22no—
408C31#2359111yescharacteristic, direct product factor, Sylow
409C24—21no—
410C24C2×D4×C3112yes—
411#41—22no—
412#41—22no—
413#41—22no—
414#41—22no—
415#41—84no—
416C2×Q8—22no—
417C2×Q8C2×D4×C3112yes—
418C2×D4—22no—
419C2×D4—24no—
420C2×D4—24no—
421C2×D4—22no—
422C2×D4—28no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—82no—
431C2×D4—82no—
432C2×D4C2×D4×C3112yes—
433C2×D4#41×C3112yescomplemented
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—42no—
443C22×C4—82no—
444C22×C4—82no—
445C22×C4C2×D4×C3112yes—
446C22×C4C2×D4×C3111yescharacteristic
447C22×C4C2×D4×C3111yescharacteristic
448C22×C4C24×C3111yescharacteristic
449SD16—28no—
450D8—28no—
451M16—22no—
452M16—22no—
453M16—22no—
454M16#41×C3114yescomplemented
455C2×C8—22no—
456C2×C8—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—24no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4C2×D4×C3112yes—
465C4⋊C4C2×D4×C3112yes—
466C22⋊C4—24no—
467C22⋊C4—24no—
468C22⋊C4—24no—
469C22⋊C4—22no—
470C22⋊C4—24no—
471C22⋊C4—22no—
472C22⋊C4—22no—
473C22⋊C4—41no—
474C22⋊C4—41no—
475C22⋊C4—82no—
476C22⋊C4—82no—
477C42—22no—
478C42—22no—
479C42—22no—
480C42—21no—
481C42—22no—
482C42C2×D4×C3112yes—
483C42C2×D4×C3112yes—
484C42#41×C3112yescomplemented
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—41no—
495C23—41no—
496C23C22×D4×C3111yescharacteristic
497Q8—22no—
498Q8—22no—
499Q8—82no—
500D4—22no—
501D4—24no—
502D4—22no—
503D4—42no—
504D4—42no—
505D4—42no—
506D4—42no—
507D4—42no—
508D4—84no—
509D4—88no—
510D4—82no—
511D4—84no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—84no—
534C2×C4C31×#12011yescharacteristic
535C2×C4C31×#12012yes—
536C2×C4C31×#12011yescharacteristic
537C2×C4C22×D4×C3112yes—
538C2×C4C2×#41×C3111yescharacteristic
539C2×C4C2×#41×C3112yes—
540C2×C4C2×#41×C3111yescharacteristic
541C8—24no—
542C8—42no—
543C22—21no—
544C22—21no—
545C22—44no—
546C22—44no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—42no—
554C22—82no—
555C22—82no—
556C22—81no—
557C22—81no—
558C22C2×C31×#12011yescharacteristic
559C22C31×#52511yescharacteristic
560C22C31×#53211yescharacteristic
561C4—21no—
562C4—21no—
563C4—22no—
564C4—42no—
565C4—42no—
566C4—42no—
567C4—42no—
568C4—42no—
569C4—81no—
570C4—81no—
571C4C31×#52512yes—
572C4C31×#53212yes—
573C2—21no—
574C2—21no—
575C2—82no—
576C2—82no—
577C2—82no—
578C2C31×#238811yescharacteristic
579C2C31×#301911yescharacteristic
580C2C31×#302111yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#3460C2semidirect
2. C4×C31×AGL(1,ℤ/8)C2semidirect
3. C31×#2328C2semidirect
4. C31×#2061C2semidirect
5. C31×#2040C2semidirect
6. C31×#2037C2semidirect
7. C31×#2027C2semidirect
8. C31×#2006C2semidirect
9. C31×#2002C2semidirect
10. C31×#1759C2semidirect
11. C4×C31×#100C2semidirect
80. C4×Q8×C31D4semidirect
81. C4×Q8×C31D4semidirect
89. C4×D4×C31D4semidirect
90. C4×D4×C31D4semidirect
142. C2×D4×C31#41semidirect
163. M16×C31#41semidirect
193. C42×C31#41semidirect
194. #23591C31direct (N × H)
253. #3460C2×C31semidirect
254. C4×AGL(1,ℤ/8)C2×C31semidirect
255. #2328C2×C31semidirect
256. #2061C2×C31semidirect
257. #2040C2×C31semidirect
258. #2037C2×C31semidirect
259. #2027C2×C31semidirect
260. #2006C2×C31semidirect
261. #2002C2×C31semidirect
262. #1759C2×C31semidirect
263. C4×#100C2×C31semidirect
370. C4×Q8D4×C31semidirect
371. C4×Q8D4×C31semidirect
379. C4×D4D4×C31semidirect
380. C4×D4D4×C31semidirect
408. C31#23591direct (N × H)
433. C2×D4#41×C31semidirect
454. M16#41×C31semidirect
484. C42#41×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#34602
C4×C31×AGL(1,ℤ/8)2
C31×#23282
C31×#20612
C31×#20402
C31×#20372
C31×#20272
C31×#20062
C31×#20022
C31×#17592
C4×C31×#1002
#2359131