Finite Group Databasebeta

← C31×#8412 #21465171

Subgroups — C31×#8412 #21465171

Order 7936.

768 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
1C31×#3514C211yesmaximal, characteristic
2C31×#1759C211yesmaximal, characteristic, complemented
3C31×#1529C211yesmaximal, characteristic, complemented
4C31×#1505C211yesmaximal, characteristic, complemented
5C31×#1500C211yesmaximal, characteristic, complemented
6C31×#1488C211yesmaximal, characteristic, complemented
7C31×#1484C211yesmaximal, characteristic, complemented
8C31×#559C411yescharacteristic, complemented
9C31×#556C411yescharacteristic, complemented
10C31×#555—21no—
11C31×#551—22no—
12D4×Q8×C31C411yescharacteristic, complemented
13C31×#548—21no—
14C31×#546—21no—
15C31×#546—21no—
16D42×C31C411yescharacteristic, complemented
17C31×#542—21no—
18C31×#538—22no—
19C31×#532C2211yescharacteristic
20C31×#529—21no—
21C31×#529C2211yescharacteristic
22C2×C31×#124—21no—
23C31×#518—21no—
24C4×#41×C31C2211yescharacteristic
25C31×#432C2211yescharacteristic, complemented
26C31×#431C2211yescharacteristic, complemented
27C2×C31×#104—21no—
28C4×M16×C31C2211yescharacteristic, complemented
29C31×#376C2211yescharacteristic, complemented
30C31×#339—21no—
31C31×#330—21no—
32C31×#329—21no—
33C31×#328—21no—
34C31×#327—21no—
35C31×#327—22no—
36C31×Q8⋊C8—21no—
37C31×D4⋊C8—22no—
38C31×D4⋊C8—21no—
39C2×#41×C31—21no—
40C2×#41×C31D411yescharacteristic
41C22×Q8×C31—21no—
42C22×D4×C31—21no—
43C22×D4×C31—21no—
44C2×M16×C31—21no—
45C31×C4⋊Q8—21no—
46C31×C4⋊Q8C2×C411yescharacteristic
47C31×#127C2×C411yescharacteristic
48C31×#126—22no—
49C31×#126—22no—
50C31×#126—21no—
51C31×#125D411yescharacteristic
52C31×#124—22no—
53C31×#124—22no—
54C31×#124—22no—
55C31×#124—21no—
56C31×#124—21no—
57C31×#124—21no—
58C31×#124—21no—
59C31×#124D411yescharacteristic
60C31×#123—22no—
61C31×#123—22no—
62C31×#123—22no—
63C31×#123—21no—
64C31×C22⋊Q8—22no—
65C31×C22⋊Q8—21no—
66C31×C22⋊Q8—21no—
67C31×C22⋊Q8—22no—
68C31×C22⋊Q8D411yescharacteristic
69C31×#121—22no—
70C31×#121—22no—
71C31×#121—21no—
72C31×#121—21no—
73C31×#121—21no—
74C31×#121D411yescharacteristic
75C31×#121D412yes—
76C31×#120—22no—
77C31×#120—22no—
78C4×Q8×C31—21no—
79C4×Q8×C31C2×C411yescharacteristic
80C4×Q8×C31C2×C411yescharacteristic
81C4×Q8×C31D411yescharacteristic
82C4×D4×C31—22no—
83C4×D4×C31—22no—
84C4×D4×C31—21no—
85C4×D4×C31—21no—
86C4×D4×C31—21no—
87C4×D4×C31—21no—
88C4×D4×C31—21no—
89C4×D4×C31—21no—
90C4×D4×C31—21no—
91C4×D4×C31C2×C411yescharacteristic
92C4×D4×C31C2×C411yescharacteristic
93C4×D4×C31D411yescharacteristic
94C4×D4×C31D412yes—
95C31×#117—22no—
96C31×#117—21no—
97C31×#117—21no—
98C31×#117—21no—
99C31×#117D411yescharacteristic
100C2×C22⋊C4×C31—22no—
101C2×C22⋊C4×C31—22no—
102C2×C22⋊C4×C31—22no—
103C2×C42×C31—21no—
104C2×C42×C31—21no—
105C2×C42×C31C2311yescharacteristic
106C31×C4⋊C8—21no—
107C31×C4⋊C8—21no—
108C31×#104—24no—
109C31×C22⋊C8—21no—
110C31×C22⋊C8—21no—
111C31×#97—21no—
112C31×#97—21no—
113C4×C8×C31—21no—
114C4×C8×C31—21no—
115C31×#95—21no—
116C31×#95—21no—
117C24×C31—22no—
118#41×C31—22no—
119#41×C31—41no—
120#41×C31—82no—
121C2×Q8×C31—22no—
122C2×Q8×C31—21no—
123C2×Q8×C31—82no—
124C2×Q8×C31C22⋊C411yescharacteristic
125C2×Q8×C31C22⋊C411yescharacteristic
126C2×D4×C31—22no—
127C2×D4×C31—21no—
128C2×D4×C31—22no—
129C2×D4×C31—21no—
130C2×D4×C31—22no—
131C2×D4×C31—21no—
132C2×D4×C31—21no—
133C2×D4×C31—42no—
134C2×D4×C31—42no—
135C2×D4×C31—82no—
136C2×D4×C31C22⋊C411yescharacteristic
137C2×D4×C31C22⋊C411yescharacteristic
138C2×D4×C31C22⋊C412yes—
139C22×C4×C31—22no—
140C22×C4×C31—22no—
141C22×C4×C31—21no—
142C22×C4×C31—21no—
143C22×C4×C31—21no—
144C22×C4×C31—21no—
145C22×C4×C31—22no—
146C22×C4×C31—41no—
147C22×C4×C31C2×D411yescharacteristic
148C22×C4×C31C2×D411yescharacteristic
149C22×C4×C31C2×D411yescharacteristic
150M16×C31—22no—
151M16×C31—41no—
152C2×C8×C31—41no—
153C2×C8×C31—41no—
154C2×C8×C31—41no—
155C4⋊C4×C31—22no—
156C4⋊C4×C31—22no—
157C4⋊C4×C31—21no—
158C4⋊C4×C31—21no—
159C4⋊C4×C31—21no—
160C4⋊C4×C31—21no—
161C4⋊C4×C31—21no—
162C4⋊C4×C31—21no—
163C4⋊C4×C31—21no—
164C4⋊C4×C31C22⋊C411yescharacteristic
165C4⋊C4×C31C22⋊C411yescharacteristic
166C4⋊C4×C31C22⋊C411yescharacteristic
167C4⋊C4×C31C22⋊C411yescharacteristic
168C4⋊C4×C31C22⋊C412yes—
169C22⋊C4×C31—22no—
170C22⋊C4×C31—22no—
171C22⋊C4×C31—22no—
172C22⋊C4×C31—22no—
173C22⋊C4×C31—22no—
174C22⋊C4×C31—22no—
175C22⋊C4×C31—22no—
176C22⋊C4×C31—21no—
177C22⋊C4×C31—21no—
178C22⋊C4×C31—21no—
179C22⋊C4×C31—21no—
180C22⋊C4×C31—21no—
181C22⋊C4×C31—21no—
182C22⋊C4×C31—21no—
183C22⋊C4×C31—21no—
184C22⋊C4×C31—22no—
185C22⋊C4×C31—22no—
186C42×C31—22no—
187C42×C31—21no—
188C42×C31—21no—
189C42×C31—21no—
190C42×C31—21no—
191C42×C31—22no—
192C42×C31—41no—
193C42×C31C22×C411yescharacteristic
194C42×C31C2×D411yescharacteristic
195C42×C31C2×D411yescharacteristic
196C42×C31C2×D411yescharacteristic
197#8412C3111yesmaximal, characteristic, direct product factor, Sylow
198C23×C31—21no—
199C23×C31—21no—
200C23×C31—22no—
201C23×C31—44no—
202C23×C31—82no—
203C23×C31—82no—
204C23×C31#12011yescharacteristic
205Q8×C31—21no—
206Q8×C31—21no—
207Q8×C31—41no—
208Q8×C31—42no—
209Q8×C31—82no—
210D4×C31—21no—
211D4×C31—41no—
212D4×C31—42no—
213D4×C31—42no—
214D4×C31—42no—
215D4×C31—44no—
216D4×C31—41no—
217D4×C31—41no—
218D4×C31—81no—
219D4×C31—82no—
220D4×C31—82no—
221D4×C31#10412yes—
222C2×C4×C31—21no—
223C2×C4×C31—21no—
224C2×C4×C31—21no—
225C2×C4×C31—21no—
226C2×C4×C31—21no—
227C2×C4×C31—21no—
228C2×C4×C31—21no—
229C2×C4×C31—21no—
230C2×C4×C31—22no—
231C2×C4×C31—41no—
232C2×C4×C31—41no—
233C2×C4×C31—42no—
234C2×C4×C31—42no—
235C2×C4×C31—41no—
236C2×C4×C31—41no—
237C2×C4×C31—42no—
238C2×C4×C31—81no—
239C2×C4×C31—82no—
240C2×C4×C31—82no—
241C2×C4×C31—82no—
242C2×C4×C31—81no—
243C2×C4×C31—81no—
244C2×C4×C31C2×C22⋊C411yescharacteristic
245C2×C4×C31C2×C22⋊C411yescharacteristic
246C2×C4×C31C2×C22⋊C411yescharacteristic
247C2×C4×C31#12011yescharacteristic
248C2×C4×C31#12011yescharacteristic
249C2×C4×C31#12011yescharacteristic
250C8×C31—42no—
251C8×C31—81no—
252C8×C31—81no—
253#3514C2×C3111yescharacteristic
254#1759C2×C3111yescharacteristic, complemented
255#1529C2×C3111yescharacteristic, complemented
256#1505C2×C3111yescharacteristic, complemented
257#1500C2×C3111yescharacteristic, complemented
258#1488C2×C3111yescharacteristic, complemented
259#1484C2×C3111yescharacteristic, complemented
260C22×C31—21no—
261C22×C31—21no—
262C22×C31—21no—
263C22×C31—22no—
264C22×C31—41no—
265C22×C31—41no—
266C22×C31—42no—
267C22×C31—41no—
268C22×C31—42no—
269C22×C31—41no—
270C22×C31—41no—
271C22×C31—42no—
272C22×C31—164no—
273C22×C31#37911yescharacteristic
274C4×C31—21no—
275C4×C31—21no—
276C4×C31—21no—
277C4×C31—21no—
278C4×C31—21no—
279C4×C31—41no—
280C4×C31—41no—
281C4×C31—41no—
282C4×C31—42no—
283C4×C31—41no—
284C4×C31—41no—
285C4×C31—41no—
286C4×C31—82no—
287C4×C31C2×#10411yescharacteristic
288C4×C31#42111yescharacteristic
289#559C4×C3111yescharacteristic, complemented
290#556C4×C3111yescharacteristic, complemented
291#555—21no—
292#551—22no—
293D4×Q8C4×C3111yescharacteristic, complemented
294#548—21no—
295#546—21no—
296#546—21no—
297D42C4×C3111yescharacteristic, complemented
298#542—21no—
299#538—22no—
300#532C22×C3111yescharacteristic
301#529—21no—
302#529C22×C3111yescharacteristic
303C2×#124—21no—
304#518—21no—
305C4×#41C22×C3111yescharacteristic
306#432C22×C3111yescharacteristic, complemented
307#431C22×C3111yescharacteristic, complemented
308C2×#104—21no—
309C4×M16C22×C3111yescharacteristic, complemented
310#376C22×C3111yescharacteristic, complemented
311#339—21no—
312#330—21no—
313#329—21no—
314#328—21no—
315#327—21no—
316#327—22no—
317Q8⋊C8—21no—
318D4⋊C8—22no—
319D4⋊C8—21no—
320C2×C31—41no—
321C2×C31—42no—
322C2×C31—82no—
323C2×C31—81no—
324C2×C31#178911yescharacteristic
325C2×C31#179411yescharacteristic
326C2×C31#179511yescharacteristic
327C2×#41—21no—
328C2×#41D4×C3111yescharacteristic
329C22×Q8—21no—
330C22×D4—21no—
331C22×D4—21no—
332C2×M16—21no—
333C4⋊Q8—21no—
334C4⋊Q8C2×C4×C3111yescharacteristic
335#127C2×C4×C3111yescharacteristic
336#126—22no—
337#126—22no—
338#126—21no—
339#125D4×C3111yescharacteristic
340#124—22no—
341#124—22no—
342#124—22no—
343#124—21no—
344#124—21no—
345#124—21no—
346#124—21no—
347#124D4×C3111yescharacteristic
348#123—22no—
349#123—22no—
350#123—22no—
351#123—21no—
352C22⋊Q8—22no—
353C22⋊Q8—21no—
354C22⋊Q8—21no—
355C22⋊Q8—22no—
356C22⋊Q8D4×C3111yescharacteristic
357#121—22no—
358#121—22no—
359#121—21no—
360#121—21no—
361#121—21no—
362#121D4×C3111yescharacteristic
363#121D4×C3112yes—
364#120—22no—
365#120—22no—
366C4×Q8—21no—
367C4×Q8C2×C4×C3111yescharacteristic
368C4×Q8C2×C4×C3111yescharacteristic
369C4×Q8D4×C3111yescharacteristic
370C4×D4—22no—
371C4×D4—22no—
372C4×D4—21no—
373C4×D4—21no—
374C4×D4—21no—
375C4×D4—21no—
376C4×D4—21no—
377C4×D4—21no—
378C4×D4—21no—
379C4×D4C2×C4×C3111yescharacteristic
380C4×D4C2×C4×C3111yescharacteristic
381C4×D4D4×C3111yescharacteristic
382C4×D4D4×C3112yes—
383#117—22no—
384#117—21no—
385#117—21no—
386#117—21no—
387#117D4×C3111yescharacteristic
388C2×C22⋊C4—22no—
389C2×C22⋊C4—22no—
390C2×C22⋊C4—22no—
391C2×C42—21no—
392C2×C42—21no—
393C2×C42C23×C3111yescharacteristic
394C4⋊C8—21no—
395C4⋊C8—21no—
396#104—24no—
397C22⋊C8—21no—
398C22⋊C8—21no—
399#97—21no—
400#97—21no—
401C4×C8—21no—
402C4×C8—21no—
403#95—21no—
404#95—21no—
405C31#841211yescharacteristic, direct product factor, Sylow
406C24—22no—
407#41—22no—
408#41—41no—
409#41—82no—
410C2×Q8—22no—
411C2×Q8—21no—
412C2×Q8—82no—
413C2×Q8C22⋊C4×C3111yescharacteristic
414C2×Q8C22⋊C4×C3111yescharacteristic
415C2×D4—22no—
416C2×D4—21no—
417C2×D4—22no—
418C2×D4—21no—
419C2×D4—22no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—82no—
425C2×D4C22⋊C4×C3111yescharacteristic
426C2×D4C22⋊C4×C3111yescharacteristic
427C2×D4C22⋊C4×C3112yes—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—41no—
436C22×C4C2×D4×C3111yescharacteristic
437C22×C4C2×D4×C3111yescharacteristic
438C22×C4C2×D4×C3111yescharacteristic
439M16—22no—
440M16—41no—
441C2×C8—41no—
442C2×C8—41no—
443C2×C8—41no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4C22⋊C4×C3111yescharacteristic
454C4⋊C4C22⋊C4×C3111yescharacteristic
455C4⋊C4C22⋊C4×C3111yescharacteristic
456C4⋊C4C22⋊C4×C3111yescharacteristic
457C4⋊C4C22⋊C4×C3112yes—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—22no—
461C22⋊C4—22no—
462C22⋊C4—22no—
463C22⋊C4—22no—
464C22⋊C4—22no—
465C22⋊C4—21no—
466C22⋊C4—21no—
467C22⋊C4—21no—
468C22⋊C4—21no—
469C22⋊C4—21no—
470C22⋊C4—21no—
471C22⋊C4—21no—
472C22⋊C4—21no—
473C22⋊C4—22no—
474C22⋊C4—22no—
475C42—22no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—22no—
481C42—41no—
482C42C22×C4×C3111yescharacteristic
483C42C2×D4×C3111yescharacteristic
484C42C2×D4×C3111yescharacteristic
485C42C2×D4×C3111yescharacteristic
486C23—21no—
487C23—21no—
488C23—22no—
489C23—44no—
490C23—82no—
491C23—82no—
492C23C31×#12011yescharacteristic
493Q8—21no—
494Q8—21no—
495Q8—41no—
496Q8—42no—
497Q8—82no—
498D4—21no—
499D4—41no—
500D4—42no—
501D4—42no—
502D4—42no—
503D4—44no—
504D4—41no—
505D4—41no—
506D4—81no—
507D4—82no—
508D4—82no—
509D4C31×#10412yes—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—81no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4C2×C22⋊C4×C3111yescharacteristic
533C2×C4C2×C22⋊C4×C3111yescharacteristic
534C2×C4C2×C22⋊C4×C3111yescharacteristic
535C2×C4C31×#12011yescharacteristic
536C2×C4C31×#12011yescharacteristic
537C2×C4C31×#12011yescharacteristic
538C8—42no—
539C8—81no—
540C8—81no—
541C22—21no—
542C22—21no—
543C22—22no—
544C22—21no—
545C22—41no—
546C22—41no—
547C22—42no—
548C22—41no—
549C22—41no—
550C22—42no—
551C22—42no—
552C22—41no—
553C22—164no—
554C22C31×#37911yescharacteristic
555C4—21no—
556C4—21no—
557C4—21no—
558C4—21no—
559C4—21no—
560C4—41no—
561C4—42no—
562C4—41no—
563C4—41no—
564C4—41no—
565C4—41no—
566C4—41no—
567C4—82no—
568C4C2×C31×#10411yescharacteristic
569C4C31×#42111yescharacteristic
570C2—41no—
571C2—42no—
572C2—81no—
573C2—82no—
574C2C31×#178911yescharacteristic
575C2C31×#179411yescharacteristic
576C2C31×#179511yescharacteristic
subgroupG/H
center, socleC22×C31#379
commutatorC42C22×C4×C31
Frattini, cosocleC2×C42C23×C31
prime radicalC31×#3514C2
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
2. C31×#1759C2semidirect
3. C31×#1529C2semidirect
4. C31×#1505C2semidirect
5. C31×#1500C2semidirect
6. C31×#1488C2semidirect
7. C31×#1484C2semidirect
8. C31×#559C4semidirect
9. C31×#556C4semidirect
12. D4×Q8×C31C4semidirect
16. D42×C31C4semidirect
25. C31×#432C22semidirect
26. C31×#431C22semidirect
28. C4×M16×C31C22semidirect
29. C31×#376C22semidirect
197. #8412C31direct (N × H)
254. #1759C2×C31semidirect
255. #1529C2×C31semidirect
256. #1505C2×C31semidirect
257. #1500C2×C31semidirect
258. #1488C2×C31semidirect
259. #1484C2×C31semidirect
289. #559C4×C31semidirect
290. #556C4×C31semidirect
293. D4×Q8C4×C31semidirect
297. D42C4×C31semidirect
306. #432C22×C31semidirect
307. #431C22×C31semidirect
309. C4×M16C22×C31semidirect
310. #376C22×C31semidirect
405. C31#8412direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35142
C31×#17592
C31×#15292
C31×#15052
C31×#15002
C31×#14882
C31×#14842
#841231