Finite Group Databasebeta

← C31×#8418 #21465177

Subgroups — C31×#8418 #21465177

Order 7936.

772 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3520C211yesmaximal, characteristic
2C31×#1986C211yesmaximal, characteristic, complemented
3C31×#1762C211yesmaximal, characteristic, complemented
4C31×#1531C211yesmaximal, characteristic, complemented
5C31×#1524C211yesmaximal, characteristic, complemented
6C31×#1509C211yesmaximal, characteristic, complemented
7C31×#1490C211yesmaximal, characteristic, complemented
8C31×#556C411yescharacteristic, complemented
9C31×#554—21no—
10C31×#552—21no—
11C31×#551—22no—
12D4×Q8×C31C411yescharacteristic, complemented
13C31×#548—21no—
14C31×#547C411yescharacteristic, complemented
15C31×#547C411yescharacteristic, complemented
16C31×#546—21no—
17C31×#542—21no—
18C31×#539—22no—
19C31×#533C2211yescharacteristic
20C31×#529C2211yescharacteristic
21C31×#525—21no—
22C2×C31×C22⋊Q8—21no—
23C31×#520—21no—
24C2×C4×D4×C31C2211yescharacteristic
25C31×#432C2211yescharacteristic
26C2×C31×#102—21no—
27C4×M16×C31C2211yescharacteristic
28C2×C31×#97C2211yescharacteristic
29C4×C4⋊C4×C31C2211yescharacteristic
30C31×#340—21no—
31C31×#339—22no—
32C31×#330—21no—
33C31×#330—22no—
34C31×#329—21no—
35C31×#328—21no—
36C31×#327—21no—
37C31×D4⋊C8—21no—
38C31×D4⋊C8—21no—
39C2×#41×C31—21no—
40C2×#41×C31—21no—
41C22×Q8×C31—21no—
42C22×D4×C31D411yescharacteristic
43C23×C4×C31—21no—
44C2×M16×C31—21no—
45C22×C8×C31—21no—
46C31×C4⋊Q8—21no—
47C31×C4⋊Q8C2×C411yescharacteristic
48C31×C4⋊Q8C2×C411yescharacteristic
49C31×#126—22no—
50C31×#126—22no—
51C31×#126—21no—
52C31×#125—22no—
53C31×#125D411yescharacteristic
54C31×#125D412yes—
55C31×#124—22no—
56C31×#124—21no—
57C31×#124D411yescharacteristic
58C31×#123—22no—
59C31×#123—22no—
60C31×#123—21no—
61C31×#123—22no—
62C31×#123—22no—
63C31×C22⋊Q8—22no—
64C31×C22⋊Q8—22no—
65C31×C22⋊Q8—22no—
66C31×C22⋊Q8—21no—
67C31×C22⋊Q8—21no—
68C31×C22⋊Q8—21no—
69C31×C22⋊Q8—21no—
70C31×C22⋊Q8—21no—
71C31×C22⋊Q8—22no—
72C31×C22⋊Q8—21no—
73C31×C22⋊Q8D411yescharacteristic
74C31×#121—22no—
75C31×#121—21no—
76C31×#121—21no—
77C31×#121D411yescharacteristic
78C31×#120—22no—
79C4×Q8×C31—21no—
80C4×Q8×C31—21no—
81C4×Q8×C31—21no—
82C4×Q8×C31C2×C411yescharacteristic
83C4×D4×C31—22no—
84C4×D4×C31—22no—
85C4×D4×C31—22no—
86C4×D4×C31—21no—
87C4×D4×C31—21no—
88C4×D4×C31—21no—
89C4×D4×C31—21no—
90C4×D4×C31C2×C411yescharacteristic
91C4×D4×C31C2×C411yescharacteristic
92C4×D4×C31C2×C411yescharacteristic
93C4×D4×C31D411yescharacteristic
94C4×D4×C31D411yescharacteristic
95C31×#117—22no—
96C31×#117—21no—
97C31×#117—21no—
98C31×#117D412yes—
99C2×C4⋊C4×C31—22no—
100C2×C4⋊C4×C31—21no—
101C2×C4⋊C4×C31—21no—
102C2×C4⋊C4×C31—21no—
103C2×C4⋊C4×C31D411yescharacteristic
104C2×C22⋊C4×C31—22no—
105C2×C22⋊C4×C31—21no—
106C2×C42×C31—21no—
107C2×C42×C31C2311yescharacteristic
108C31×C4⋊C8—21no—
109C31×#102—24no—
110C31×C22⋊C8—21no—
111C31×#97—21no—
112C31×#97—21no—
113C31×#97—21no—
114C31×#97—21no—
115C4×C8×C31—21no—
116C31×#95—21no—
117C24×C31—21no—
118#41×C31—42no—
119#41×C31—42no—
120#41×C31—82no—
121C2×Q8×C31—22no—
122C2×Q8×C31—21no—
123C2×Q8×C31—21no—
124C2×Q8×C31—82no—
125C2×Q8×C31C22⋊C411yescharacteristic
126C2×D4×C31—22no—
127C2×D4×C31—22no—
128C2×D4×C31—21no—
129C2×D4×C31—22no—
130C2×D4×C31—21no—
131C2×D4×C31—41no—
132C2×D4×C31C22⋊C411yescharacteristic
133C2×D4×C31C22⋊C411yescharacteristic
134C2×D4×C31C22⋊C411yescharacteristic
135C22×C4×C31—22no—
136C22×C4×C31—22no—
137C22×C4×C31—22no—
138C22×C4×C31—21no—
139C22×C4×C31—21no—
140C22×C4×C31—21no—
141C22×C4×C31—21no—
142C22×C4×C31—21no—
143C22×C4×C31—21no—
144C22×C4×C31—41no—
145C22×C4×C31—82no—
146C22×C4×C31C2×D411yescharacteristic
147C22×C4×C31C2×D411yescharacteristic
148C22×C4×C31C2×D411yescharacteristic
149M16×C31—22no—
150M16×C31—41no—
151C2×C8×C31—22no—
152C2×C8×C31—41no—
153C2×C8×C31—41no—
154C2×C8×C31—41no—
155C2×C8×C31—41no—
156C4⋊C4×C31—22no—
157C4⋊C4×C31—22no—
158C4⋊C4×C31—22no—
159C4⋊C4×C31—22no—
160C4⋊C4×C31—21no—
161C4⋊C4×C31—21no—
162C4⋊C4×C31—21no—
163C4⋊C4×C31—21no—
164C4⋊C4×C31—22no—
165C4⋊C4×C31—21no—
166C4⋊C4×C31—21no—
167C4⋊C4×C31—21no—
168C4⋊C4×C31—22no—
169C4⋊C4×C31—22no—
170C4⋊C4×C31—41no—
171C4⋊C4×C31C22⋊C411yescharacteristic
172C4⋊C4×C31C22⋊C411yescharacteristic
173C4⋊C4×C31C22⋊C411yescharacteristic
174C4⋊C4×C31C22⋊C411yescharacteristic
175C4⋊C4×C31C22⋊C412yes—
176C4⋊C4×C31C22⋊C412yes—
177C22⋊C4×C31—22no—
178C22⋊C4×C31—22no—
179C22⋊C4×C31—22no—
180C22⋊C4×C31—22no—
181C22⋊C4×C31—22no—
182C22⋊C4×C31—21no—
183C22⋊C4×C31—22no—
184C22⋊C4×C31—21no—
185C22⋊C4×C31—21no—
186C22⋊C4×C31—21no—
187C22⋊C4×C31—21no—
188C22⋊C4×C31—21no—
189C22⋊C4×C31—21no—
190C22⋊C4×C31—21no—
191C22⋊C4×C31—21no—
192C42×C31—22no—
193C42×C31—21no—
194C42×C31—22no—
195C42×C31—42no—
196C42×C31C22×C411yescharacteristic
197C42×C31C2×D411yescharacteristic
198C42×C31C2×D411yescharacteristic
199C42×C31C2×D411yescharacteristic
200#8418C3111yesmaximal, characteristic, direct product factor, Sylow
201C23×C31—21no—
202C23×C31—21no—
203C23×C31—21no—
204C23×C31—42no—
205C23×C31—41no—
206C23×C31—41no—
207C23×C31—81no—
208C23×C31#12011yescharacteristic
209Q8×C31—41no—
210Q8×C31—42no—
211Q8×C31—41no—
212Q8×C31—41no—
213Q8×C31—82no—
214D4×C31—21no—
215D4×C31—21no—
216D4×C31—21no—
217D4×C31—42no—
218D4×C31—41no—
219D4×C31—44no—
220D4×C31—41no—
221D4×C31—41no—
222D4×C31—82no—
223D4×C31#10412yes—
224C2×C4×C31—21no—
225C2×C4×C31—21no—
226C2×C4×C31—21no—
227C2×C4×C31—21no—
228C2×C4×C31—21no—
229C2×C4×C31—22no—
230C2×C4×C31—21no—
231C2×C4×C31—21no—
232C2×C4×C31—22no—
233C2×C4×C31—21no—
234C2×C4×C31—21no—
235C2×C4×C31—41no—
236C2×C4×C31—42no—
237C2×C4×C31—42no—
238C2×C4×C31—44no—
239C2×C4×C31—41no—
240C2×C4×C31—41no—
241C2×C4×C31—42no—
242C2×C4×C31—81no—
243C2×C4×C31—81no—
244C2×C4×C31—81no—
245C2×C4×C31—82no—
246C2×C4×C31—82no—
247C2×C4×C31—81no—
248C2×C4×C31—81no—
249C2×C4×C31C2×C22⋊C411yescharacteristic
250C2×C4×C31C2×C22⋊C411yescharacteristic
251C2×C4×C31C2×C22⋊C411yescharacteristic
252C2×C4×C31#12011yescharacteristic
253C2×C4×C31#12011yescharacteristic
254C2×C4×C31#12011yescharacteristic
255C8×C31—42no—
256C8×C31—42no—
257C8×C31—81no—
258#3520C2×C3111yescharacteristic
259#1986C2×C3111yescharacteristic, complemented
260#1762C2×C3111yescharacteristic, complemented
261#1531C2×C3111yescharacteristic, complemented
262#1524C2×C3111yescharacteristic, complemented
263#1509C2×C3111yescharacteristic, complemented
264#1490C2×C3111yescharacteristic, complemented
265C22×C31—21no—
266C22×C31—21no—
267C22×C31—22no—
268C22×C31—41no—
269C22×C31—41no—
270C22×C31—41no—
271C22×C31—41no—
272C22×C31—41no—
273C22×C31—41no—
274C22×C31—41no—
275C22×C31—41no—
276C22×C31—82no—
277C22×C31—82no—
278C22×C31#37911yescharacteristic
279C22×C31#45411yescharacteristic
280C22×C31#45611yescharacteristic
281C4×C31—21no—
282C4×C31—21no—
283C4×C31—21no—
284C4×C31—21no—
285C4×C31—21no—
286C4×C31—41no—
287C4×C31—41no—
288C4×C31—41no—
289C4×C31—42no—
290C4×C31—41no—
291C4×C31—41no—
292C4×C31—42no—
293C4×C31—82no—
294C4×C31C2×#10411yescharacteristic
295C4×C31#42111yescharacteristic
296#556C4×C3111yescharacteristic, complemented
297#554—21no—
298#552—21no—
299#551—22no—
300D4×Q8C4×C3111yescharacteristic, complemented
301#548—21no—
302#547C4×C3111yescharacteristic, complemented
303#547C4×C3111yescharacteristic, complemented
304#546—21no—
305#542—21no—
306#539—22no—
307#533C22×C3111yescharacteristic
308#529C22×C3111yescharacteristic
309#525—21no—
310C2×C22⋊Q8—21no—
311#520—21no—
312C2×C4×D4C22×C3111yescharacteristic
313#432C22×C3111yescharacteristic
314C2×#102—21no—
315C4×M16C22×C3111yescharacteristic
316C2×#97C22×C3111yescharacteristic
317C4×C4⋊C4C22×C3111yescharacteristic
318#340—21no—
319#339—22no—
320#330—21no—
321#330—22no—
322#329—21no—
323#328—21no—
324#327—21no—
325D4⋊C8—21no—
326D4⋊C8—21no—
327C2×C31—21no—
328C2×C31—21no—
329C2×C31—42no—
330C2×C31—81no—
331C2×C31—81no—
332C2×C31#178911yescharacteristic
333C2×C31#179311yescharacteristic
334C2×C31#179511yescharacteristic
335C2×#41—21no—
336C2×#41—21no—
337C22×Q8—21no—
338C22×D4D4×C3111yescharacteristic
339C23×C4—21no—
340C2×M16—21no—
341C22×C8—21no—
342C4⋊Q8—21no—
343C4⋊Q8C2×C4×C3111yescharacteristic
344C4⋊Q8C2×C4×C3111yescharacteristic
345#126—22no—
346#126—22no—
347#126—21no—
348#125—22no—
349#125D4×C3111yescharacteristic
350#125D4×C3112yes—
351#124—22no—
352#124—21no—
353#124D4×C3111yescharacteristic
354#123—22no—
355#123—22no—
356#123—21no—
357#123—22no—
358#123—22no—
359C22⋊Q8—22no—
360C22⋊Q8—22no—
361C22⋊Q8—22no—
362C22⋊Q8—21no—
363C22⋊Q8—21no—
364C22⋊Q8—21no—
365C22⋊Q8—21no—
366C22⋊Q8—21no—
367C22⋊Q8—22no—
368C22⋊Q8—21no—
369C22⋊Q8D4×C3111yescharacteristic
370#121—22no—
371#121—21no—
372#121—21no—
373#121D4×C3111yescharacteristic
374#120—22no—
375C4×Q8—21no—
376C4×Q8—21no—
377C4×Q8—21no—
378C4×Q8C2×C4×C3111yescharacteristic
379C4×D4—22no—
380C4×D4—22no—
381C4×D4—22no—
382C4×D4—21no—
383C4×D4—21no—
384C4×D4—21no—
385C4×D4—21no—
386C4×D4C2×C4×C3111yescharacteristic
387C4×D4C2×C4×C3111yescharacteristic
388C4×D4C2×C4×C3111yescharacteristic
389C4×D4D4×C3111yescharacteristic
390C4×D4D4×C3111yescharacteristic
391#117—22no—
392#117—21no—
393#117—21no—
394#117D4×C3112yes—
395C2×C4⋊C4—22no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4D4×C3111yescharacteristic
400C2×C22⋊C4—22no—
401C2×C22⋊C4—21no—
402C2×C42—21no—
403C2×C42C23×C3111yescharacteristic
404C4⋊C8—21no—
405#102—24no—
406C22⋊C8—21no—
407#97—21no—
408#97—21no—
409#97—21no—
410#97—21no—
411C4×C8—21no—
412#95—21no—
413C31#841811yescharacteristic, direct product factor, Sylow
414C24—21no—
415#41—42no—
416#41—42no—
417#41—82no—
418C2×Q8—22no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—82no—
422C2×Q8C22⋊C4×C3111yescharacteristic
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—21no—
426C2×D4—22no—
427C2×D4—21no—
428C2×D4—41no—
429C2×D4C22⋊C4×C3111yescharacteristic
430C2×D4C22⋊C4×C3111yescharacteristic
431C2×D4C22⋊C4×C3111yescharacteristic
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—41no—
442C22×C4—82no—
443C22×C4C2×D4×C3111yescharacteristic
444C22×C4C2×D4×C3111yescharacteristic
445C22×C4C2×D4×C3111yescharacteristic
446M16—22no—
447M16—41no—
448C2×C8—22no—
449C2×C8—41no—
450C2×C8—41no—
451C2×C8—41no—
452C2×C8—41no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—22no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—41no—
468C4⋊C4C22⋊C4×C3111yescharacteristic
469C4⋊C4C22⋊C4×C3111yescharacteristic
470C4⋊C4C22⋊C4×C3111yescharacteristic
471C4⋊C4C22⋊C4×C3111yescharacteristic
472C4⋊C4C22⋊C4×C3112yes—
473C4⋊C4C22⋊C4×C3112yes—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—22no—
477C22⋊C4—22no—
478C22⋊C4—22no—
479C22⋊C4—21no—
480C22⋊C4—22no—
481C22⋊C4—21no—
482C22⋊C4—21no—
483C22⋊C4—21no—
484C22⋊C4—21no—
485C22⋊C4—21no—
486C22⋊C4—21no—
487C22⋊C4—21no—
488C22⋊C4—21no—
489C42—22no—
490C42—21no—
491C42—22no—
492C42—42no—
493C42C22×C4×C3111yescharacteristic
494C42C2×D4×C3111yescharacteristic
495C42C2×D4×C3111yescharacteristic
496C42C2×D4×C3111yescharacteristic
497C23—21no—
498C23—21no—
499C23—21no—
500C23—42no—
501C23—41no—
502C23—41no—
503C23—81no—
504C23C31×#12011yescharacteristic
505Q8—41no—
506Q8—42no—
507Q8—41no—
508Q8—41no—
509Q8—82no—
510D4—21no—
511D4—21no—
512D4—21no—
513D4—42no—
514D4—41no—
515D4—44no—
516D4—41no—
517D4—41no—
518D4—82no—
519D4C31×#10412yes—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—41no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—42no—
538C2×C4—81no—
539C2×C4—81no—
540C2×C4—81no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—81no—
544C2×C4—81no—
545C2×C4C2×C22⋊C4×C3111yescharacteristic
546C2×C4C2×C22⋊C4×C3111yescharacteristic
547C2×C4C2×C22⋊C4×C3111yescharacteristic
548C2×C4C31×#12011yescharacteristic
549C2×C4C31×#12011yescharacteristic
550C2×C4C31×#12011yescharacteristic
551C8—42no—
552C8—42no—
553C8—81no—
554C22—21no—
555C22—21no—
556C22—22no—
557C22—41no—
558C22—41no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—82no—
566C22—82no—
567C22C31×#37911yescharacteristic
568C22C31×#45411yescharacteristic
569C22C31×#45611yescharacteristic
570C4—21no—
571C4—21no—
572C4—21no—
573C4—21no—
574C4—21no—
575C4—41no—
576C4—41no—
577C4—42no—
578C4—41no—
579C4—41no—
580C4—42no—
581C4—41no—
582C4—82no—
583C4C2×C31×#10411yescharacteristic
584C4C31×#42111yescharacteristic
585C2—21no—
586C2—21no—
587C2—42no—
588C2—81no—
589C2—81no—
590C2C31×#178911yescharacteristic
591C2C31×#179311yescharacteristic
592C2C31×#179511yescharacteristic
subgroupG/H
center, socleC22×C31#379
commutatorC42C22×C4×C31
Frattini, cosocleC2×C42C23×C31
prime radicalC31×#3520C2
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×#1986C2semidirect
3. C31×#1762C2semidirect
4. C31×#1531C2semidirect
5. C31×#1524C2semidirect
6. C31×#1509C2semidirect
7. C31×#1490C2semidirect
8. C31×#556C4semidirect
12. D4×Q8×C31C4semidirect
14. C31×#547C4semidirect
15. C31×#547C4semidirect
200. #8418C31direct (N × H)
259. #1986C2×C31semidirect
260. #1762C2×C31semidirect
261. #1531C2×C31semidirect
262. #1524C2×C31semidirect
263. #1509C2×C31semidirect
264. #1490C2×C31semidirect
296. #556C4×C31semidirect
300. D4×Q8C4×C31semidirect
302. #547C4×C31semidirect
303. #547C4×C31semidirect
413. C31#8418direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35202
C31×#19862
C31×#17622
C31×#15312
C31×#15242
C31×#15092
C31×#14902
#841831