Finite Group Databasebeta

← C31×#8482 #21465241

Subgroups — C31×#8482 #21465241

Order 7936.

768 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3538C211yesmaximal, characteristic
2C31×#1996C211yesmaximal, characteristic, complemented
3C31×#1901C211yesmaximal, characteristic, complemented
4C31×#1894C211yesmaximal, characteristic, complemented
5C31×#1871C211yesmaximal, characteristic
6C31×#1513C211yesmaximal, characteristic
7C31×#1508C211yesmaximal, characteristic, complemented
8C22×M16×C31C2211yescharacteristic
9C31×#563—22no—
10C31×#552—21no—
11C31×#552—21no—
12C31×#551—22no—
13D4×Q8×C31—21no—
14D4×Q8×C31—21no—
15C31×#544—21no—
16C31×#544C412yescomplemented
17C31×#541—21no—
18C31×#539—21no—
19C31×#539C412yescomplemented
20C31×#536—21no—
21C31×#525C2211yescharacteristic
22C2×C31×C22⋊Q8C2211yescharacteristic
23C2×C31×C22⋊Q8C2211yescharacteristic
24C2×C31×#104—21no—
25C2×C31×#103—21no—
26C2×C31×#101—21no—
27C31×#407C2211yescharacteristic
28C31×#380C2211yescharacteristic
29C4×C22⋊C4×C31C2211yescharacteristic
30C31×#343—21no—
31C31×#342—22no—
32C31×#340—21no—
33C31×#339—21no—
34C31×#328—22no—
35C31×#328—21no—
36C31×#327—21no—
37C2×#41×C31D411yescharacteristic
38C22×Q8×C31—21no—
39C22×Q8×C31D411yescharacteristic
40C22×Q8×C31D411yescharacteristic
41C23×C4×C31C2311yescharacteristic
42C2×M16×C31—21no—
43C2×M16×C31—21no—
44C2×M16×C31—21no—
45C2×M16×C31—42no—
46C22×C8×C31—21no—
47C31×C4⋊Q8—21no—
48C31×C4⋊Q8—22no—
49C31×C4⋊Q8—22no—
50C31×C4⋊Q8—22no—
51C31×#126—22no—
52C31×#126—22no—
53C31×#126—22no—
54C31×#126—22no—
55C31×#125—21no—
56C31×#125—22no—
57C31×#125—22no—
58C31×#125—22no—
59C31×#125—22no—
60C31×#124—22no—
61C31×#123—22no—
62C31×#123—22no—
63C31×#123—22no—
64C31×#123—22no—
65C31×#123—21no—
66C31×C22⋊Q8—21no—
67C31×C22⋊Q8—21no—
68C31×C22⋊Q8—22no—
69C31×C22⋊Q8—22no—
70C31×C22⋊Q8—22no—
71C31×C22⋊Q8—22no—
72C31×C22⋊Q8—21no—
73C31×C22⋊Q8—21no—
74C31×C22⋊Q8—21no—
75C31×C22⋊Q8—21no—
76C31×C22⋊Q8—21no—
77C31×C22⋊Q8—21no—
78C31×C22⋊Q8C2×C412yes—
79C31×C22⋊Q8C2×C411yescharacteristic
80C31×C22⋊Q8D412yes—
81C31×C22⋊Q8D411yescharacteristic
82C31×#121C2×C411yescharacteristic
83C31×#121D411yescharacteristic
84C31×#120—21no—
85C4×Q8×C31—21no—
86C4×Q8×C31—21no—
87C4×D4×C31—22no—
88C4×D4×C31—22no—
89C4×D4×C31—21no—
90C4×D4×C31—21no—
91C31×#117—21no—
92C31×#117—22no—
93C31×#117—22no—
94C31×#117D411yescharacteristic
95C2×C4⋊C4×C31—21no—
96C2×C4⋊C4×C31—21no—
97C2×C4⋊C4×C31C2×C412yes—
98C2×C4⋊C4×C31D411yescharacteristic
99C2×C4⋊C4×C31D411yescharacteristic
100C2×C22⋊C4×C31—21no—
101C2×C22⋊C4×C31—21no—
102C2×C22⋊C4×C31—21no—
103C2×C22⋊C4×C31D412yes—
104C2×C42×C31—21no—
105C31×#104—42no—
106C31×#103—42no—
107C31×#101—42no—
108C31×C22⋊C8—21no—
109C31×C22⋊C8—21no—
110C31×#95—21no—
111C31×#95—21no—
112C24×C31C2×D411yescharacteristic
113#41×C31—41no—
114#41×C31—41no—
115C2×Q8×C31—22no—
116C2×Q8×C31—21no—
117C2×Q8×C31—21no—
118C2×Q8×C31—21no—
119C2×Q8×C31—41no—
120C2×Q8×C31—41no—
121C2×Q8×C31—41no—
122C2×Q8×C31—41no—
123C2×Q8×C31—81no—
124C2×Q8×C31—81no—
125C2×Q8×C31C22⋊C412yes—
126C2×Q8×C31C22⋊C411yescharacteristic
127C2×D4×C31—21no—
128C2×D4×C31—21no—
129C2×D4×C31—21no—
130C2×D4×C31C22⋊C411yescharacteristic
131C22×C4×C31—21no—
132C22×C4×C31—22no—
133C22×C4×C31—21no—
134C22×C4×C31—21no—
135C22×C4×C31—41no—
136C22×C4×C31—41no—
137C22×C4×C31—41no—
138C22×C4×C31—41no—
139C22×C4×C31C22⋊C412yes—
140C22×C4×C31C22⋊C412yes—
141C22×C4×C31C22×C411yescharacteristic
142C22×C4×C31C2×D411yescharacteristic
143C22×C4×C31C2×D411yescharacteristic
144C22×C4×C31C2×D411yescharacteristic
145C22×C4×C31C2×D411yescharacteristic
146C22×C4×C31C2×D411yescharacteristic
147M16×C31—22no—
148M16×C31—22no—
149M16×C31—41no—
150M16×C31—41no—
151M16×C31—41no—
152M16×C31—41no—
153C2×C8×C31—22no—
154C2×C8×C31—41no—
155C2×C8×C31—41no—
156C2×C8×C31—41no—
157C4⋊C4×C31—22no—
158C4⋊C4×C31—22no—
159C4⋊C4×C31—22no—
160C4⋊C4×C31—22no—
161C4⋊C4×C31—22no—
162C4⋊C4×C31—22no—
163C4⋊C4×C31—22no—
164C4⋊C4×C31—22no—
165C4⋊C4×C31—22no—
166C4⋊C4×C31—22no—
167C4⋊C4×C31—22no—
168C4⋊C4×C31—21no—
169C4⋊C4×C31—22no—
170C4⋊C4×C31—21no—
171C4⋊C4×C31—21no—
172C4⋊C4×C31—21no—
173C4⋊C4×C31—21no—
174C4⋊C4×C31—41no—
175C4⋊C4×C31—41no—
176C4⋊C4×C31C22⋊C412yes—
177C4⋊C4×C31C22⋊C411yescharacteristic
178C4⋊C4×C31C22⋊C411yescharacteristic
179C22⋊C4×C31—22no—
180C22⋊C4×C31—22no—
181C22⋊C4×C31—22no—
182C22⋊C4×C31—22no—
183C22⋊C4×C31—21no—
184C22⋊C4×C31—22no—
185C22⋊C4×C31—22no—
186C22⋊C4×C31—22no—
187C22⋊C4×C31—21no—
188C22⋊C4×C31—21no—
189C22⋊C4×C31—21no—
190C22⋊C4×C31—21no—
191C22⋊C4×C31—22no—
192C22⋊C4×C31—41no—
193C22⋊C4×C31—42no—
194C42×C31—22no—
195C42×C31—22no—
196C42×C31—22no—
197C42×C31—21no—
198C42×C31—41no—
199C42×C31—41no—
200#8482C3111yesmaximal, characteristic, direct product factor, Sylow
201C23×C31—21no—
202C23×C31—21no—
203C23×C31—21no—
204C23×C31—41no—
205C23×C31—41no—
206C23×C31C2×C22⋊C411yescharacteristic
207C23×C31#12011yescharacteristic
208C23×C31#12011yescharacteristic
209Q8×C31—42no—
210Q8×C31—41no—
211Q8×C31—41no—
212Q8×C31—41no—
213Q8×C31—41no—
214Q8×C31—41no—
215Q8×C31—81no—
216Q8×C31—82no—
217D4×C31—41no—
218D4×C31—42no—
219D4×C31—41no—
220D4×C31—41no—
221D4×C31—81no—
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—22no—
232C2×C4×C31—21no—
233C2×C4×C31—22no—
234C2×C4×C31—22no—
235C2×C4×C31—41no—
236C2×C4×C31—41no—
237C2×C4×C31—41no—
238C2×C4×C31—41no—
239C2×C4×C31—41no—
240C2×C4×C31—41no—
241C2×C4×C31—41no—
242C2×C4×C31—42no—
243C2×C4×C31—42no—
244C2×C4×C31—42no—
245C2×C4×C31—81no—
246C2×C4×C31—82no—
247C2×C4×C31—81no—
248C2×C4×C31—81no—
249C2×C4×C31—81no—
250C2×C4×C31—81no—
251C2×C4×C31—81no—
252C2×C4×C31C2×C22⋊C411yescharacteristic
253C2×C4×C31C2×C22⋊C411yescharacteristic
254C2×C4×C31#12011yescharacteristic
255C2×C4×C31#12011yescharacteristic
256C8×C31—42no—
257C8×C31—81no—
258#3538C2×C3111yescharacteristic
259#1996C2×C3111yescharacteristic, complemented
260#1901C2×C3111yescharacteristic, complemented
261#1894C2×C3111yescharacteristic, complemented
262#1871C2×C3111yescharacteristic
263#1513C2×C3111yescharacteristic
264#1508C2×C3111yescharacteristic, complemented
265C22×C31—21no—
266C22×C31—21no—
267C22×C31—21no—
268C22×C31—21no—
269C22×C31—21no—
270C22×C31—21no—
271C22×C31—21no—
272C22×C31—21no—
273C22×C31—41no—
274C22×C31—41no—
275C22×C31—41no—
276C22×C31—81no—
277C22×C31—81no—
278C22×C31#37911yescharacteristic
279C22×C31#45411yescharacteristic
280C22×C31#45611yescharacteristic
281C4×C31—21no—
282C4×C31—21no—
283C4×C31—21no—
284C4×C31—42no—
285C4×C31—41no—
286C4×C31—42no—
287C4×C31—41no—
288C4×C31—41no—
289C4×C31—42no—
290C4×C31—42no—
291C4×C31—82no—
292C4×C31—161no—
293C4×C31B(4,𝔽2)11yescharacteristic
294C4×C31#45811yescharacteristic
295C22×M16C22×C3111yescharacteristic
296#563—22no—
297#552—21no—
298#552—21no—
299#551—22no—
300D4×Q8—21no—
301D4×Q8—21no—
302#544—21no—
303#544C4×C3112yescomplemented
304#541—21no—
305#539—21no—
306#539C4×C3112yescomplemented
307#536—21no—
308#525C22×C3111yescharacteristic
309C2×C22⋊Q8C22×C3111yescharacteristic
310C2×C22⋊Q8C22×C3111yescharacteristic
311C2×#104—21no—
312C2×#103—21no—
313C2×#101—21no—
314#407C22×C3111yescharacteristic
315#380C22×C3111yescharacteristic
316C4×C22⋊C4C22×C3111yescharacteristic
317#343—21no—
318#342—22no—
319#340—21no—
320#339—21no—
321#328—22no—
322#328—21no—
323#327—21no—
324C2×C31—21no—
325C2×C31—21no—
326C2×C31—41no—
327C2×C31—41no—
328C2×C31—81no—
329C2×C31#179011yescharacteristic
330C2×C31#179311yescharacteristic
331C2×C31#179611yescharacteristic
332C2×#41D4×C3111yescharacteristic
333C22×Q8—21no—
334C22×Q8D4×C3111yescharacteristic
335C22×Q8D4×C3111yescharacteristic
336C23×C4C23×C3111yescharacteristic
337C2×M16—21no—
338C2×M16—21no—
339C2×M16—21no—
340C2×M16—42no—
341C22×C8—21no—
342C4⋊Q8—21no—
343C4⋊Q8—22no—
344C4⋊Q8—22no—
345C4⋊Q8—22no—
346#126—22no—
347#126—22no—
348#126—22no—
349#126—22no—
350#125—21no—
351#125—22no—
352#125—22no—
353#125—22no—
354#125—22no—
355#124—22no—
356#123—22no—
357#123—22no—
358#123—22no—
359#123—22no—
360#123—21no—
361C22⋊Q8—21no—
362C22⋊Q8—21no—
363C22⋊Q8—22no—
364C22⋊Q8—22no—
365C22⋊Q8—22no—
366C22⋊Q8—22no—
367C22⋊Q8—21no—
368C22⋊Q8—21no—
369C22⋊Q8—21no—
370C22⋊Q8—21no—
371C22⋊Q8—21no—
372C22⋊Q8—21no—
373C22⋊Q8C2×C4×C3112yes—
374C22⋊Q8C2×C4×C3111yescharacteristic
375C22⋊Q8D4×C3112yes—
376C22⋊Q8D4×C3111yescharacteristic
377#121C2×C4×C3111yescharacteristic
378#121D4×C3111yescharacteristic
379#120—21no—
380C4×Q8—21no—
381C4×Q8—21no—
382C4×D4—22no—
383C4×D4—22no—
384C4×D4—21no—
385C4×D4—21no—
386#117—21no—
387#117—22no—
388#117—22no—
389#117D4×C3111yescharacteristic
390C2×C4⋊C4—21no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4C2×C4×C3112yes—
393C2×C4⋊C4D4×C3111yescharacteristic
394C2×C4⋊C4D4×C3111yescharacteristic
395C2×C22⋊C4—21no—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—21no—
398C2×C22⋊C4D4×C3112yes—
399C2×C42—21no—
400#104—42no—
401#103—42no—
402#101—42no—
403C22⋊C8—21no—
404C22⋊C8—21no—
405#95—21no—
406#95—21no—
407C31#848211yescharacteristic, direct product factor, Sylow
408C24C2×D4×C3111yescharacteristic
409#41—41no—
410#41—41no—
411C2×Q8—22no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—41no—
416C2×Q8—41no—
417C2×Q8—41no—
418C2×Q8—41no—
419C2×Q8—81no—
420C2×Q8—81no—
421C2×Q8C22⋊C4×C3112yes—
422C2×Q8C22⋊C4×C3111yescharacteristic
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4C22⋊C4×C3111yescharacteristic
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4C22⋊C4×C3112yes—
436C22×C4C22⋊C4×C3112yes—
437C22×C4C22×C4×C3111yescharacteristic
438C22×C4C2×D4×C3111yescharacteristic
439C22×C4C2×D4×C3111yescharacteristic
440C22×C4C2×D4×C3111yescharacteristic
441C22×C4C2×D4×C3111yescharacteristic
442C22×C4C2×D4×C3111yescharacteristic
443M16—22no—
444M16—22no—
445M16—41no—
446M16—41no—
447M16—41no—
448M16—41no—
449C2×C8—22no—
450C2×C8—41no—
451C2×C8—41no—
452C2×C8—41no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—21no—
465C4⋊C4—22no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—41no—
471C4⋊C4—41no—
472C4⋊C4C22⋊C4×C3112yes—
473C4⋊C4C22⋊C4×C3111yescharacteristic
474C4⋊C4C22⋊C4×C3111yescharacteristic
475C22⋊C4—22no—
476C22⋊C4—22no—
477C22⋊C4—22no—
478C22⋊C4—22no—
479C22⋊C4—21no—
480C22⋊C4—22no—
481C22⋊C4—22no—
482C22⋊C4—22no—
483C22⋊C4—21no—
484C22⋊C4—21no—
485C22⋊C4—21no—
486C22⋊C4—21no—
487C22⋊C4—22no—
488C22⋊C4—41no—
489C22⋊C4—42no—
490C42—22no—
491C42—22no—
492C42—22no—
493C42—21no—
494C42—41no—
495C42—41no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—41no—
500C23—41no—
501C23C2×C22⋊C4×C3111yescharacteristic
502C23C31×#12011yescharacteristic
503C23C31×#12011yescharacteristic
504Q8—42no—
505Q8—41no—
506Q8—41no—
507Q8—41no—
508Q8—41no—
509Q8—41no—
510Q8—81no—
511Q8—82no—
512D4—41no—
513D4—42no—
514D4—41no—
515D4—41no—
516D4—81no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—81no—
541C2×C4—82no—
542C2×C4—81no—
543C2×C4—81no—
544C2×C4—81no—
545C2×C4—81no—
546C2×C4—81no—
547C2×C4C2×C22⋊C4×C3111yescharacteristic
548C2×C4C2×C22⋊C4×C3111yescharacteristic
549C2×C4C31×#12011yescharacteristic
550C2×C4C31×#12011yescharacteristic
551C8—42no—
552C8—81no—
553C22—21no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—81no—
565C22—81no—
566C22C31×#37911yescharacteristic
567C22C31×#45411yescharacteristic
568C22C31×#45611yescharacteristic
569C4—21no—
570C4—21no—
571C4—21no—
572C4—42no—
573C4—41no—
574C4—41no—
575C4—42no—
576C4—42no—
577C4—42no—
578C4—41no—
579C4—82no—
580C4—161no—
581C4C31×B(4,𝔽2)11yescharacteristic
582C4C31×#45811yescharacteristic
583C2—21no—
584C2—21no—
585C2—41no—
586C2—41no—
587C2—81no—
588C2C31×#179011yescharacteristic
589C2C31×#179311yescharacteristic
590C2C31×#179611yescharacteristic
subgroupG/H
center, socleC22×C31#379
commutatorC22×C4C22×C4×C31
Frattini, cosocleC23×C4C23×C31
prime radicalC31×#525C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#1996C2semidirect
3. C31×#1901C2semidirect
4. C31×#1894C2semidirect
7. C31×#1508C2semidirect
16. C31×#544C4semidirect
19. C31×#539C4semidirect
200. #8482C31direct (N × H)
259. #1996C2×C31semidirect
260. #1901C2×C31semidirect
261. #1894C2×C31semidirect
264. #1508C2×C31semidirect
303. #544C4×C31semidirect
306. #539C4×C31semidirect
407. C31#8482direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35382
C31×#19962
C31×#19012
C31×#18942
C31×#18712
C31×#15132
C31×#15082
#848231