Finite Group Databasebeta

← C31×#25641 #21482400

Subgroups — C31×#25641 #21482400

Order 7936.

1172 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#559C211yesmaximal, characteristic
2C2×C31×#497C211yesmaximal, characteristic, complemented
3C2×C31×#461C211yesmaximal, characteristic, complemented
4C31×#2568C211yesmaximal, characteristic, complemented
5C31×#2015C212yesmaximal, complemented
6C31×#2014C212yesmaximal
7C31×#1985C211yesmaximal, characteristic, complemented
8C31×#1983C211yesmaximal, characteristic, complemented
9C31×#1878C212yesmaximal, complemented
10C31×#1877C212yesmaximal, complemented
11C31×#1762C211yesmaximal, characteristic, complemented
12C22×Q16×C31C2212yes—
13C22×SD16×C31C2212yes—
14C22×SD16×C31C2212yes—
15C31×#559—24no—
16C31×#559—24no—
17C2×C31×C4⋊Q8C2211yescharacteristic
18C2×C31×#127C2211yescharacteristic
19C2×C31×#124—21no—
20C2×C31×#124C2211yescharacteristic
21C2×C31×#124C2211yescharacteristic
22C2×C31×#123—21no—
23C2×C31×#123—21no—
24C2×C4×Q8×C31C2211yescharacteristic
25C2×C4×Q8×C31C2211yescharacteristic
26C2×C4×D4×C31—21no—
27C2×C4×D4×C31C2211yescharacteristic
28C2×C4×D4×C31C2211yescharacteristic
29C31×#497—24no—
30C31×#461—24no—
31C2×C31×C4⋊C8C2211yescharacteristic
32C2×C31×#103C2212yes—
33C2×C31×#103C2212yes—
34C2×C31×#102C2212yes—
35C2×C31×#102C2212yes—
36C2×C31×#97C2211yescharacteristic
37C31×#400—21no—
38C31×#400—21no—
39C31×#391—21no—
40C31×#391C2212yes—
41C31×#391C2212yes—
42C31×#382—21no—
43C4×C4⋊C4×C31C2211yescharacteristic
44C31×#340C2212yes—
45C31×#336C2212yes—
46C31×#336C2212yes—
47C22×Q8×C31D411yescharacteristic
48C22×Q8×C31C2312yes—
49C22×Q8×C31C2311yescharacteristic
50C22×D4×C31—21no—
51C22×D4×C31—21no—
52C22×D4×C31C2311yescharacteristic
53C22×D4×C31C2311yescharacteristic
54C23×C4×C31—21no—
55C23×C4×C31—21no—
56C2×Q16×C31—24no—
57C2×Q16×C31—24no—
58C2×Q16×C31—42no—
59C2×SD16×C31—24no—
60C2×SD16×C31—24no—
61C2×SD16×C31—24no—
62C2×SD16×C31—42no—
63C2×SD16×C31—42no—
64C2×SD16×C31—42no—
65C22×C8×C31C2312yes—
66C22×C8×C31C2312yes—
67C31×C4⋊Q8—22no—
68C31×C4⋊Q8—22no—
69C31×#127—22no—
70C31×#127—22no—
71C31×#124—28no—
72C31×#124—22no—
73C31×#124—22no—
74C31×#124—22no—
75C31×#124D414yes—
76C31×#123—28no—
77C31×#123—28no—
78C4×Q8×C31—22no—
79C4×Q8×C31—22no—
80C4×Q8×C31—22no—
81C4×Q8×C31—22no—
82C4×D4×C31—28no—
83C4×D4×C31—22no—
84C4×D4×C31—22no—
85C4×D4×C31—22no—
86C4×D4×C31D414yes—
87C2×C4⋊C4×C31—21no—
88C2×C4⋊C4×C31—21no—
89C2×C4⋊C4×C31—21no—
90C2×C4⋊C4×C31D411yescharacteristic
91C2×C4⋊C4×C31D411yescharacteristic
92C2×C4⋊C4×C31D411yescharacteristic
93C2×C4⋊C4×C31C2312yes—
94C2×C4⋊C4×C31C2311yescharacteristic
95C2×C22⋊C4×C31—21no—
96C2×C22⋊C4×C31—21no—
97C2×C22⋊C4×C31—21no—
98C2×C22⋊C4×C31—21no—
99C2×C22⋊C4×C31—21no—
100C2×C22⋊C4×C31—21no—
101C2×C42×C31—21no—
102C2×C42×C31—21no—
103C2×C42×C31C2311yescharacteristic
104C2×C42×C31C2311yescharacteristic
105C2×C42×C31C2311yescharacteristic
106C31×C4⋊C8—22no—
107C31×#103—24no—
108C31×#103—42no—
109C31×#102—24no—
110C31×#102—42no—
111C31×#97—22no—
112C31×#95—21no—
113C31×#95—22no—
114C31×#95—22no—
115C31×#95—22no—
116C31×#95—22no—
117C24×C31—21no—
118C24×C31—21no—
119C2×Q8×C31—22no—
120C2×Q8×C31—22no—
121C2×Q8×C31—21no—
122C2×Q8×C31—22no—
123C2×Q8×C31—22no—
124C2×Q8×C31—22no—
125C2×Q8×C31—22no—
126C2×Q8×C31—22no—
127C2×Q8×C31—22no—
128C2×Q8×C31—22no—
129C2×Q8×C31—22no—
130C2×Q8×C31—41no—
131C2×Q8×C31C2×D412yes—
132C2×D4×C31—24no—
133C2×D4×C31—24no—
134C2×D4×C31—21no—
135C2×D4×C31—21no—
136C2×D4×C31—22no—
137C2×D4×C31—22no—
138C2×D4×C31—22no—
139C2×D4×C31—22no—
140C2×D4×C31—42no—
141C2×D4×C31—42no—
142C2×D4×C31—82no—
143C2×D4×C31C2×D412yes—
144C2×D4×C31C2×D412yes—
145C22×C4×C31—21no—
146C22×C4×C31—21no—
147C22×C4×C31—24no—
148C22×C4×C31—24no—
149C22×C4×C31—22no—
150C22×C4×C31—22no—
151C22×C4×C31—82no—
152C22×C4×C31—82no—
153C22×C4×C31C2×D411yescharacteristic
154C22×C4×C31C2×D411yescharacteristic
155C22×C4×C31C2×D411yescharacteristic
156C22×C4×C31C2×D411yescharacteristic
157C22×C4×C31C2×D411yescharacteristic
158C22×C4×C31C2×D411yescharacteristic
159C22×C4×C31C2411yescharacteristic
160Q16×C31—48no—
161SD16×C31—48no—
162SD16×C31—48no—
163C2×C8×C31—22no—
164C2×C8×C31—22no—
165C2×C8×C31—22no—
166C2×C8×C31—22no—
167C2×C8×C31—22no—
168C2×C8×C31#4114yes—
169C4⋊C4×C31—24no—
170C4⋊C4×C31—22no—
171C4⋊C4×C31—22no—
172C4⋊C4×C31—22no—
173C4⋊C4×C31—21no—
174C4⋊C4×C31—22no—
175C4⋊C4×C31—22no—
176C4⋊C4×C31—42no—
177C4⋊C4×C31—42no—
178C4⋊C4×C31C2×D412yes—
179C22⋊C4×C31—24no—
180C22⋊C4×C31—24no—
181C22⋊C4×C31—24no—
182C22⋊C4×C31—24no—
183C22⋊C4×C31—24no—
184C22⋊C4×C31—24no—
185C42×C31—24no—
186C42×C31—21no—
187C42×C31—21no—
188C42×C31—21no—
189C42×C31—21no—
190C42×C31—42no—
191C42×C31C2×D412yes—
192C42×C31C2×D412yes—
193#25641C3111yesmaximal, characteristic, direct product factor, Sylow
194C23×C31—22no—
195C23×C31—22no—
196C23×C31—42no—
197C23×C31—42no—
198C23×C31—41no—
199C23×C31—41no—
200C23×C31—41no—
201C23×C31—41no—
202C23×C31—42no—
203C23×C31—42no—
204C23×C31C22×D411yescharacteristic
205Q8×C31—24no—
206Q8×C31—44no—
207Q8×C31—41no—
208Q8×C31—41no—
209Q8×C31—42no—
210Q8×C31—42no—
211Q8×C31—42no—
212Q8×C31—42no—
213D4×C31—22no—
214D4×C31—22no—
215D4×C31—24no—
216D4×C31—48no—
217D4×C31—41no—
218D4×C31—41no—
219D4×C31—42no—
220D4×C31—84no—
221C2×C4×C31—21no—
222C2×C4×C31—21no—
223C2×C4×C31—21no—
224C2×C4×C31—21no—
225C2×C4×C31—22no—
226C2×C4×C31—22no—
227C2×C4×C31—21no—
228C2×C4×C31—21no—
229C2×C4×C31—22no—
230C2×C4×C31—21no—
231C2×C4×C31—21no—
232C2×C4×C31—21no—
233C2×C4×C31—22no—
234C2×C4×C31—22no—
235C2×C4×C31—42no—
236C2×C4×C31—42no—
237C2×C4×C31—42no—
238C2×C4×C31—42no—
239C2×C4×C31—42no—
240C2×C4×C31—42no—
241C2×C4×C31—42no—
242C2×C4×C31—42no—
243C2×C4×C31—84no—
244C2×C4×C31—84no—
245C2×C4×C31#12012yes—
246C2×C4×C31#12012yes—
247C2×C4×C31#13712yes—
248C2×C4×C31C22×D411yescharacteristic
249C2×C4×C31C22×D411yescharacteristic
250C2×C4×C31C2×#4112yes—
251C2×C4×C31H(5,𝔽2)12yes—
252C8×C31—24no—
253C8×C31—42no—
254C2×#559C2×C3111yescharacteristic
255C2×#497C2×C3111yescharacteristic, complemented
256C2×#461C2×C3111yescharacteristic, complemented
257#2568C2×C3111yescharacteristic, complemented
258#2015C2×C3112yescomplemented
259#2014C2×C3112yes—
260#1985C2×C3111yescharacteristic, complemented
261#1983C2×C3111yescharacteristic, complemented
262#1878C2×C3112yescomplemented
263#1877C2×C3112yescomplemented
264#1762C2×C3111yescharacteristic, complemented
265C22×C31—42no—
266C22×C31—42no—
267C22×C31—42no—
268C22×C31—42no—
269C22×C31—42no—
270C22×C31—42no—
271C22×C31—82no—
272C22×C31—82no—
273C22×C31—82no—
274C22×C31—82no—
275C22×C31C2×#12011yescharacteristic
276C22×C31#55012yes—
277C22×C31C2×#13711yescharacteristic
278C22×C31#57511yescharacteristic
279C22×C31#57712yes—
280C4×C31—22no—
281C4×C31—22no—
282C4×C31—22no—
283C4×C31—42no—
284C4×C31—41no—
285C4×C31—41no—
286C4×C31—41no—
287C4×C31—42no—
288C4×C31—82no—
289C4×C31—82no—
290C4×C31#53211yescharacteristic
291C4×C31#53511yescharacteristic
292C4×C31#54612yes—
293C22×Q16C22×C3112yes—
294C22×SD16C22×C3112yes—
295C22×SD16C22×C3112yes—
296#559—24no—
297#559—24no—
298C2×C4⋊Q8C22×C3111yescharacteristic
299C2×#127C22×C3111yescharacteristic
300C2×#124—21no—
301C2×#124C22×C3111yescharacteristic
302C2×#124C22×C3111yescharacteristic
303C2×#123—21no—
304C2×#123—21no—
305C2×C4×Q8C22×C3111yescharacteristic
306C2×C4×Q8C22×C3111yescharacteristic
307C2×C4×D4—21no—
308C2×C4×D4C22×C3111yescharacteristic
309C2×C4×D4C22×C3111yescharacteristic
310#497—24no—
311#461—24no—
312C2×C4⋊C8C22×C3111yescharacteristic
313C2×#103C22×C3112yes—
314C2×#103C22×C3112yes—
315C2×#102C22×C3112yes—
316C2×#102C22×C3112yes—
317C2×#97C22×C3111yescharacteristic
318#400—21no—
319#400—21no—
320#391—21no—
321#391C22×C3112yes—
322#391C22×C3112yes—
323#382—21no—
324C4×C4⋊C4C22×C3111yescharacteristic
325#340C22×C3112yes—
326#336C22×C3112yes—
327#336C22×C3112yes—
328C2×C31—82no—
329C2×C31—82no—
330C2×C31#243111yescharacteristic
331C2×C31#300711yescharacteristic
332C2×C31#300911yescharacteristic
333C2×C31#340012yes—
334C2×C31#340112yes—
335C22×Q8D4×C3111yescharacteristic
336C22×Q8C23×C3112yes—
337C22×Q8C23×C3111yescharacteristic
338C22×D4—21no—
339C22×D4—21no—
340C22×D4C23×C3111yescharacteristic
341C22×D4C23×C3111yescharacteristic
342C23×C4—21no—
343C23×C4—21no—
344C2×Q16—24no—
345C2×Q16—24no—
346C2×Q16—42no—
347C2×SD16—24no—
348C2×SD16—24no—
349C2×SD16—24no—
350C2×SD16—42no—
351C2×SD16—42no—
352C2×SD16—42no—
353C22×C8C23×C3112yes—
354C22×C8C23×C3112yes—
355C4⋊Q8—22no—
356C4⋊Q8—22no—
357#127—22no—
358#127—22no—
359#124—28no—
360#124—22no—
361#124—22no—
362#124—22no—
363#124D4×C3114yes—
364#123—28no—
365#123—28no—
366C4×Q8—22no—
367C4×Q8—22no—
368C4×Q8—22no—
369C4×Q8—22no—
370C4×D4—28no—
371C4×D4—22no—
372C4×D4—22no—
373C4×D4—22no—
374C4×D4D4×C3114yes—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4D4×C3111yescharacteristic
379C2×C4⋊C4D4×C3111yescharacteristic
380C2×C4⋊C4D4×C3111yescharacteristic
381C2×C4⋊C4C23×C3112yes—
382C2×C4⋊C4C23×C3111yescharacteristic
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C22⋊C4—21no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—21no—
389C2×C42—21no—
390C2×C42—21no—
391C2×C42C23×C3111yescharacteristic
392C2×C42C23×C3111yescharacteristic
393C2×C42C23×C3111yescharacteristic
394C4⋊C8—22no—
395#103—24no—
396#103—42no—
397#102—24no—
398#102—42no—
399#97—22no—
400#95—21no—
401#95—22no—
402#95—22no—
403#95—22no—
404#95—22no—
405C31#2564111yescharacteristic, direct product factor, Sylow
406C24—21no—
407C24—21no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—21no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—41no—
420C2×Q8C2×D4×C3112yes—
421C2×D4—24no—
422C2×D4—24no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—42no—
430C2×D4—42no—
431C2×D4—82no—
432C2×D4C2×D4×C3112yes—
433C2×D4C2×D4×C3112yes—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—82no—
441C22×C4—82no—
442C22×C4C2×D4×C3111yescharacteristic
443C22×C4C2×D4×C3111yescharacteristic
444C22×C4C2×D4×C3111yescharacteristic
445C22×C4C2×D4×C3111yescharacteristic
446C22×C4C2×D4×C3111yescharacteristic
447C22×C4C2×D4×C3111yescharacteristic
448C22×C4C24×C3111yescharacteristic
449Q16—48no—
450SD16—48no—
451SD16—48no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8#41×C3114yes—
458C4⋊C4—24no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—21no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4C2×D4×C3112yes—
468C22⋊C4—24no—
469C22⋊C4—24no—
470C22⋊C4—24no—
471C22⋊C4—24no—
472C22⋊C4—24no—
473C22⋊C4—24no—
474C42—24no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—42no—
480C42C2×D4×C3112yes—
481C42C2×D4×C3112yes—
482C23—22no—
483C23—22no—
484C23—42no—
485C23—42no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—42no—
491C23—42no—
492C23C22×D4×C3111yescharacteristic
493Q8—24no—
494Q8—44no—
495Q8—41no—
496Q8—41no—
497Q8—42no—
498Q8—42no—
499Q8—42no—
500Q8—42no—
501D4—22no—
502D4—22no—
503D4—24no—
504D4—48no—
505D4—41no—
506D4—41no—
507D4—42no—
508D4—84no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—84no—
532C2×C4—84no—
533C2×C4C31×#12012yes—
534C2×C4C31×#12012yes—
535C2×C4C31×#13712yes—
536C2×C4C22×D4×C3111yescharacteristic
537C2×C4C22×D4×C3111yescharacteristic
538C2×C4C2×#41×C3112yes—
539C2×C4C31×H(5,𝔽2)12yes—
540C8—24no—
541C8—42no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—42no—
546C22—42no—
547C22—42no—
548C22—82no—
549C22—82no—
550C22—82no—
551C22—82no—
552C22C2×C31×#12011yescharacteristic
553C22C31×#55012yes—
554C22C2×C31×#13711yescharacteristic
555C22C31×#57511yescharacteristic
556C22C31×#57712yes—
557C4—22no—
558C4—22no—
559C4—22no—
560C4—42no—
561C4—41no—
562C4—41no—
563C4—41no—
564C4—42no—
565C4—82no—
566C4—82no—
567C4C31×#53211yescharacteristic
568C4C31×#53511yescharacteristic
569C4C31×#54612yes—
570C2—82no—
571C2—82no—
572C2C31×#243111yescharacteristic
573C2C31×#300711yescharacteristic
574C2C31×#300911yescharacteristic
575C2C31×#340012yes—
576C2C31×#340112yes—
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC2×C31×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C2×C4×Q8×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×#497C2semidirect
3. C2×C31×#461C2semidirect
4. C31×#2568C2semidirect
5. C31×#2015C2semidirect
7. C31×#1985C2semidirect
8. C31×#1983C2semidirect
9. C31×#1878C2semidirect
10. C31×#1877C2semidirect
11. C31×#1762C2semidirect
193. #25641C31direct (N × H)
255. C2×#497C2×C31semidirect
256. C2×#461C2×C31semidirect
257. #2568C2×C31semidirect
258. #2015C2×C31semidirect
260. #1985C2×C31semidirect
261. #1983C2×C31semidirect
262. #1878C2×C31semidirect
263. #1877C2×C31semidirect
264. #1762C2×C31semidirect
405. C31#25641direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5592
C2×C31×#4972
C2×C31×#4612
C31×#25682
C31×#20152
C31×#20142
C31×#19852
C31×#19832
C31×#18782
C31×#18772
C31×#17622
#2564131