Finite Group Databasebeta

← C31×#18160 #21474919

Subgroups — C31×#18160 #21474919

Order 7936.

976 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3461C211yesmaximal, characteristic, complemented
2C31×#3114C212yesmaximal, complemented
3C31×#2536C211yesmaximal, characteristic, complemented
4C31×#1611C211yesmaximal, characteristic, complemented
5C31×#1610C211yesmaximal, characteristic, complemented
6C31×#1609C212yesmaximal, complemented
7C31×#1599C211yesmaximal, characteristic, complemented
8C31×#1598C211yesmaximal, characteristic
9C31×#1597C212yesmaximal, complemented
10C31×#1512C212yesmaximal, complemented
11C31×#1467C211yesmaximal, characteristic, complemented
12C31×#544C2211yescharacteristic
13C31×#542—22no—
14C31×#541—22no—
15C31×#539—21no—
16C31×#539C2212yes—
17C31×#537C2211yescharacteristic
18C31×#536—21no—
19C31×#533—22no—
20C31×#529—22no—
21C31×#529—21no—
22C31×#529C2212yes—
23C2×C31×C22⋊Q8—21no—
24C4×#41×C31—21no—
25C2×C4×Q8×C31C2211yescharacteristic
26C2×C4×D4×C31C2211yescharacteristic
27C31×#483C2212yes—
28C31×#480C2212yes—
29C31×#478—22no—
30C31×#463—22no—
31C31×#450C2212yes—
32C31×#447C2212yes—
33C31×#442—22no—
34C31×#440—22no—
35C31×#432C2212yes—
36C31×#398C2211yescharacteristic
37C31×#398C2211yescharacteristic
38C31×#394C2211yescharacteristic
39C31×#394C2211yescharacteristic
40C31×#388—21no—
41C31×#388—21no—
42C31×#382C2211yescharacteristic
43C31×#380—21no—
44C31×#328C2212yes—
45C31×#328C2212yes—
46C31×#327C2212yes—
47C31×#327C2212yes—
48C31×#324C2212yes—
49C31×#324C2211yescharacteristic
50C31×#324C2211yescharacteristic
51C2×#41×C31—21no—
52C22×Q8×C31D411yescharacteristic
53C22×D4×C31C2311yescharacteristic
54C23×C4×C31—21no—
55C2×SD16×C31—22no—
56C2×D8×C31—22no—
57C31×C4⋊Q8—22no—
58C31×C4⋊Q8—22no—
59C31×#126—22no—
60C31×#126—22no—
61C31×#126—22no—
62C31×#126—22no—
63C31×#125—21no—
64C31×#125—22no—
65C31×#125—22no—
66C31×#125—22no—
67C31×#125D412yes—
68C31×#124—22no—
69C31×#124—21no—
70C31×#124D412yes—
71C31×#123—22no—
72C31×#123—22no—
73C31×#123—22no—
74C31×#123—22no—
75C31×C22⋊Q8—22no—
76C31×C22⋊Q8—22no—
77C31×C22⋊Q8—22no—
78C31×C22⋊Q8—22no—
79C31×C22⋊Q8—21no—
80C31×C22⋊Q8—22no—
81C31×C22⋊Q8—44no—
82C31×C22⋊Q8C2312yes—
83C31×#121—22no—
84C31×#121—21no—
85C31×#121C2312yes—
86C4×Q8×C31—21no—
87C4×Q8×C31—21no—
88C4×Q8×C31—21no—
89C4×Q8×C31—22no—
90C4×Q8×C31—22no—
91C4×Q8×C31D412yes—
92C4×D4×C31—21no—
93C4×D4×C31—21no—
94C4×D4×C31—21no—
95C4×D4×C31—22no—
96C4×D4×C31—22no—
97C4×D4×C31—22no—
98C4×D4×C31—22no—
99C4×D4×C31D412yes—
100C31×#117—22no—
101C31×#117—22no—
102C31×#117—21no—
103C31×#117—22no—
104C2×C4⋊C4×C31—21no—
105C2×C4⋊C4×C31—21no—
106C2×C4⋊C4×C31—21no—
107C2×C4⋊C4×C31D411yescharacteristic
108C2×C4⋊C4×C31C2311yescharacteristic
109C2×C4⋊C4×C31C2311yescharacteristic
110C2×C4⋊C4×C31C2311yescharacteristic
111C2×C22⋊C4×C31—21no—
112C2×C22⋊C4×C31—21no—
113C2×C22⋊C4×C31—21no—
114C2×C22⋊C4×C31—21no—
115C2×C22⋊C4×C31—21no—
116C2×C42×C31—22no—
117C2×C42×C31D411yescharacteristic
118C2×C42×C31D411yescharacteristic
119C2×C42×C31C2311yescharacteristic
120C31×#107—22no—
121C31×#106—22no—
122C31×C4⋊C8—22no—
123C31×#103—22no—
124C31×#102—22no—
125C31×#102—22no—
126C31×#102—22no—
127C31×C22⋊C8C2312yes—
128C31×C22⋊C8C2312yes—
129C31×#97—22no—
130C31×#95—21no—
131C31×#95—21no—
132C31×#95—21no—
133C31×#95C2311yescharacteristic
134C31×#95C2311yescharacteristic
135C24×C31—21no—
136#41×C31—42no—
137#41×C31—42no—
138C2×Q8×C31—21no—
139C2×Q8×C31—21no—
140C2×Q8×C31—22no—
141C2×Q8×C31—22no—
142C2×Q8×C31—41no—
143C2×Q8×C31C2×D412yes—
144C2×D4×C31—21no—
145C2×D4×C31—22no—
146C2×D4×C31—21no—
147C2×D4×C31—22no—
148C2×D4×C31—42no—
149C2×D4×C31C2×D412yes—
150C22×C4×C31—22no—
151C22×C4×C31—22no—
152C22×C4×C31—22no—
153C22×C4×C31—21no—
154C22×C4×C31—21no—
155C22×C4×C31—21no—
156C22×C4×C31—42no—
157C22×C4×C31—42no—
158C22×C4×C31—42no—
159C22×C4×C31C2×D411yescharacteristic
160C22×C4×C31C2×D411yescharacteristic
161C22×C4×C31C2×D411yescharacteristic
162C22×C4×C31C2×D411yescharacteristic
163C22×C4×C31C2×D411yescharacteristic
164C22×C4×C31C2×D411yescharacteristic
165C22×C4×C31C2411yescharacteristic
166SD16×C31—44no—
167D8×C31—44no—
168C2×C8×C31—22no—
169C2×C8×C31—22no—
170C4⋊C4×C31—21no—
171C4⋊C4×C31—22no—
172C4⋊C4×C31—21no—
173C4⋊C4×C31—22no—
174C4⋊C4×C31—22no—
175C4⋊C4×C31—21no—
176C4⋊C4×C31—21no—
177C4⋊C4×C31—22no—
178C4⋊C4×C31—22no—
179C4⋊C4×C31—22no—
180C4⋊C4×C31—22no—
181C4⋊C4×C31—22no—
182C4⋊C4×C31—22no—
183C4⋊C4×C31—22no—
184C4⋊C4×C31—22no—
185C4⋊C4×C31—22no—
186C4⋊C4×C31—42no—
187C4⋊C4×C31—42no—
188C4⋊C4×C31—42no—
189C4⋊C4×C31C2×D412yes—
190C4⋊C4×C31C2×D412yes—
191C22⋊C4×C31—22no—
192C22⋊C4×C31—22no—
193C22⋊C4×C31—22no—
194C22⋊C4×C31—22no—
195C22⋊C4×C31—22no—
196C22⋊C4×C31—22no—
197C22⋊C4×C31—22no—
198C22⋊C4×C31—22no—
199C22⋊C4×C31—22no—
200C22⋊C4×C31—42no—
201C22⋊C4×C31—42no—
202C22⋊C4×C31—42no—
203C22⋊C4×C31—41no—
204C22⋊C4×C31—41no—
205C22⋊C4×C31—42no—
206C42×C31—22no—
207C42×C31—22no—
208C42×C31—21no—
209C42×C31—22no—
210C42×C31—21no—
211C42×C31—21no—
212C42×C31—21no—
213C42×C31C2×D412yes—
214C42×C31C2×D412yes—
215#18160C3111yesmaximal, characteristic, direct product factor, Sylow
216C23×C31—22no—
217C23×C31—22no—
218C23×C31—41no—
219C23×C31—41no—
220C23×C31—41no—
221C23×C31—41no—
222C23×C31—41no—
223C23×C31—41no—
224C23×C31C22×D411yescharacteristic
225Q8×C31—44no—
226Q8×C31—42no—
227Q8×C31—42no—
228Q8×C31—81no—
229D4×C31—24no—
230D4×C31—42no—
231D4×C31—42no—
232D4×C31—42no—
233D4×C31—42no—
234D4×C31—42no—
235D4×C31—81no—
236C2×C4×C31—21no—
237C2×C4×C31—21no—
238C2×C4×C31—21no—
239C2×C4×C31—21no—
240C2×C4×C31—21no—
241C2×C4×C31—22no—
242C2×C4×C31—22no—
243C2×C4×C31—21no—
244C2×C4×C31—21no—
245C2×C4×C31—22no—
246C2×C4×C31—21no—
247C2×C4×C31—21no—
248C2×C4×C31—41no—
249C2×C4×C31—41no—
250C2×C4×C31—44no—
251C2×C4×C31—44no—
252C2×C4×C31—41no—
253C2×C4×C31—41no—
254C2×C4×C31—41no—
255C2×C4×C31—41no—
256C2×C4×C31—41no—
257C2×C4×C31—41no—
258C2×C4×C31—41no—
259C2×C4×C31—41no—
260C2×C4×C31—41no—
261C2×C4×C31—41no—
262C2×C4×C31—41no—
263C2×C4×C31—84no—
264C2×C4×C31—82no—
265C2×C4×C31#12011yescharacteristic
266C2×C4×C31#12011yescharacteristic
267C2×C4×C31#12012yes—
268C2×C4×C31AGL(1,ℤ/8)12yes—
269C2×C4×C31C22×D412yes—
270C8×C31—42no—
271C8×C31—42no—
272#3461C2×C3111yescharacteristic, complemented
273#3114C2×C3112yescomplemented
274#2536C2×C3111yescharacteristic, complemented
275#1611C2×C3111yescharacteristic, complemented
276#1610C2×C3111yescharacteristic, complemented
277#1609C2×C3112yescomplemented
278#1599C2×C3111yescharacteristic, complemented
279#1598C2×C3111yescharacteristic
280#1597C2×C3112yescomplemented
281#1512C2×C3112yescomplemented
282#1467C2×C3111yescharacteristic, complemented
283C22×C31—21no—
284C22×C31—21no—
285C22×C31—42no—
286C22×C31—42no—
287C22×C31—42no—
288C22×C31—42no—
289C22×C31—42no—
290C22×C31—42no—
291C22×C31—81no—
292C22×C31—81no—
293C22×C31—81no—
294C22×C31—81no—
295C22×C31C2×#12011yescharacteristic
296C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
297C22×C31#57511yescharacteristic
298C4×C31—22no—
299C4×C31—22no—
300C4×C31—21no—
301C4×C31—21no—
302C4×C31—22no—
303C4×C31—41no—
304C4×C31—42no—
305C4×C31—42no—
306C4×C31—41no—
307C4×C31—41no—
308C4×C31—42no—
309C4×C31—81no—
310C4×C31—81no—
311C4×C31—81no—
312#544C22×C3111yescharacteristic
313#542—22no—
314#541—22no—
315#539—21no—
316#539C22×C3112yes—
317#537C22×C3111yescharacteristic
318#536—21no—
319#533—22no—
320#529—22no—
321#529—21no—
322#529C22×C3112yes—
323C2×C22⋊Q8—21no—
324C4×#41—21no—
325C2×C4×Q8C22×C3111yescharacteristic
326C2×C4×D4C22×C3111yescharacteristic
327#483C22×C3112yes—
328#480C22×C3112yes—
329#478—22no—
330#463—22no—
331#450C22×C3112yes—
332#447C22×C3112yes—
333#442—22no—
334#440—22no—
335#432C22×C3112yes—
336#398C22×C3111yescharacteristic
337#398C22×C3111yescharacteristic
338#394C22×C3111yescharacteristic
339#394C22×C3111yescharacteristic
340#388—21no—
341#388—21no—
342#382C22×C3111yescharacteristic
343#380—21no—
344#328C22×C3112yes—
345#328C22×C3112yes—
346#327C22×C3112yes—
347#327C22×C3112yes—
348#324C22×C3112yes—
349#324C22×C3111yescharacteristic
350#324C22×C3111yescharacteristic
351C2×C31—21no—
352C2×C31—21no—
353C2×C31—82no—
354C2×C31—82no—
355C2×C31#300611yescharacteristic
356C2×C31#302011yescharacteristic
357C2×C31#302511yescharacteristic
358C2×#41—21no—
359C22×Q8D4×C3111yescharacteristic
360C22×D4C23×C3111yescharacteristic
361C23×C4—21no—
362C2×SD16—22no—
363C2×D8—22no—
364C4⋊Q8—22no—
365C4⋊Q8—22no—
366#126—22no—
367#126—22no—
368#126—22no—
369#126—22no—
370#125—21no—
371#125—22no—
372#125—22no—
373#125—22no—
374#125D4×C3112yes—
375#124—22no—
376#124—21no—
377#124D4×C3112yes—
378#123—22no—
379#123—22no—
380#123—22no—
381#123—22no—
382C22⋊Q8—22no—
383C22⋊Q8—22no—
384C22⋊Q8—22no—
385C22⋊Q8—22no—
386C22⋊Q8—21no—
387C22⋊Q8—22no—
388C22⋊Q8—44no—
389C22⋊Q8C23×C3112yes—
390#121—22no—
391#121—21no—
392#121C23×C3112yes—
393C4×Q8—21no—
394C4×Q8—21no—
395C4×Q8—21no—
396C4×Q8—22no—
397C4×Q8—22no—
398C4×Q8D4×C3112yes—
399C4×D4—21no—
400C4×D4—21no—
401C4×D4—21no—
402C4×D4—22no—
403C4×D4—22no—
404C4×D4—22no—
405C4×D4—22no—
406C4×D4D4×C3112yes—
407#117—22no—
408#117—22no—
409#117—21no—
410#117—22no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4D4×C3111yescharacteristic
415C2×C4⋊C4C23×C3111yescharacteristic
416C2×C4⋊C4C23×C3111yescharacteristic
417C2×C4⋊C4C23×C3111yescharacteristic
418C2×C22⋊C4—21no—
419C2×C22⋊C4—21no—
420C2×C22⋊C4—21no—
421C2×C22⋊C4—21no—
422C2×C22⋊C4—21no—
423C2×C42—22no—
424C2×C42D4×C3111yescharacteristic
425C2×C42D4×C3111yescharacteristic
426C2×C42C23×C3111yescharacteristic
427#107—22no—
428#106—22no—
429C4⋊C8—22no—
430#103—22no—
431#102—22no—
432#102—22no—
433#102—22no—
434C22⋊C8C23×C3112yes—
435C22⋊C8C23×C3112yes—
436#97—22no—
437#95—21no—
438#95—21no—
439#95—21no—
440#95C23×C3111yescharacteristic
441#95C23×C3111yescharacteristic
442C31#1816011yescharacteristic, direct product factor, Sylow
443C24—21no—
444#41—42no—
445#41—42no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—22no—
449C2×Q8—22no—
450C2×Q8—41no—
451C2×Q8C2×D4×C3112yes—
452C2×D4—21no—
453C2×D4—22no—
454C2×D4—21no—
455C2×D4—22no—
456C2×D4—42no—
457C2×D4C2×D4×C3112yes—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4C2×D4×C3111yescharacteristic
468C22×C4C2×D4×C3111yescharacteristic
469C22×C4C2×D4×C3111yescharacteristic
470C22×C4C2×D4×C3111yescharacteristic
471C22×C4C2×D4×C3111yescharacteristic
472C22×C4C2×D4×C3111yescharacteristic
473C22×C4C24×C3111yescharacteristic
474SD16—44no—
475D8—44no—
476C2×C8—22no—
477C2×C8—22no—
478C4⋊C4—21no—
479C4⋊C4—22no—
480C4⋊C4—21no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4C2×D4×C3112yes—
498C4⋊C4C2×D4×C3112yes—
499C22⋊C4—22no—
500C22⋊C4—22no—
501C22⋊C4—22no—
502C22⋊C4—22no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—22no—
507C22⋊C4—22no—
508C22⋊C4—42no—
509C22⋊C4—42no—
510C22⋊C4—42no—
511C22⋊C4—41no—
512C22⋊C4—41no—
513C22⋊C4—42no—
514C42—22no—
515C42—22no—
516C42—21no—
517C42—22no—
518C42—21no—
519C42—21no—
520C42—21no—
521C42C2×D4×C3112yes—
522C42C2×D4×C3112yes—
523C23—22no—
524C23—22no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23C22×D4×C3111yescharacteristic
532Q8—44no—
533Q8—42no—
534Q8—42no—
535Q8—81no—
536D4—24no—
537D4—42no—
538D4—42no—
539D4—42no—
540D4—42no—
541D4—42no—
542D4—81no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—84no—
571C2×C4—82no—
572C2×C4C31×#12011yescharacteristic
573C2×C4C31×#12011yescharacteristic
574C2×C4C31×#12012yes—
575C2×C4C31×AGL(1,ℤ/8)12yes—
576C2×C4C22×D4×C3112yes—
577C8—42no—
578C8—42no—
579C22—21no—
580C22—21no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—81no—
588C22—81no—
589C22—81no—
590C22—81no—
591C22C2×C31×#12011yescharacteristic
592C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
593C22C31×#57511yescharacteristic
594C4—22no—
595C4—22no—
596C4—21no—
597C4—21no—
598C4—22no—
599C4—41no—
600C4—42no—
601C4—42no—
602C4—41no—
603C4—41no—
604C4—42no—
605C4—81no—
606C4—81no—
607C4—81no—
608C2—21no—
609C2—21no—
610C2—82no—
611C2—82no—
612C2C31×#300611yescharacteristic
613C2C31×#302011yescharacteristic
614C2C31×#302511yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#1598C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#3461C2semidirect
2. C31×#3114C2semidirect
3. C31×#2536C2semidirect
4. C31×#1611C2semidirect
5. C31×#1610C2semidirect
6. C31×#1609C2semidirect
7. C31×#1599C2semidirect
9. C31×#1597C2semidirect
10. C31×#1512C2semidirect
11. C31×#1467C2semidirect
215. #18160C31direct (N × H)
272. #3461C2×C31semidirect
273. #3114C2×C31semidirect
274. #2536C2×C31semidirect
275. #1611C2×C31semidirect
276. #1610C2×C31semidirect
277. #1609C2×C31semidirect
278. #1599C2×C31semidirect
280. #1597C2×C31semidirect
281. #1512C2×C31semidirect
282. #1467C2×C31semidirect
442. C31#18160direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#34612
C31×#31142
C31×#25362
C31×#16112
C31×#16102
C31×#16092
C31×#15992
C31×#15982
C31×#15972
C31×#15122
C31×#14672
#1816031