Finite Group Databasebeta

← C31×#23961 #21480720

Subgroups — C31×#23961 #21480720

Order 7936.

1494 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#547C211yesmaximal, characteristic
2C2×C31×#448C211yesmaximal, characteristic, complemented
3C22×C31×#103C211yesmaximal, characteristic, complemented
4C31×#2409C211yesmaximal, characteristic, complemented
5C31×#2024C212yesmaximal, complemented
6C31×#2000C212yesmaximal, complemented
7C31×#1893C211yesmaximal, characteristic, complemented
8C31×#1877C212yesmaximal, complemented
9C31×#1872C212yesmaximal, complemented
10C31×#1869C211yesmaximal, characteristic, complemented
11C31×#1823C211yesmaximal, characteristic, complemented
12C22×#41×C31—21no—
13C23×Q8×C31C2211yescharacteristic
14C22×SD16×C31C2212yes—
15C23×C8×C31C2211yescharacteristic
16C31×#547—24no—
17C31×#547—24no—
18C2×C31×C4⋊Q8C2211yescharacteristic
19C2×C31×#123—21no—
20C2×C31×#123—21no—
21C2×C31×C22⋊Q8—21no—
22C2×C31×C22⋊Q8C2212yes—
23C2×C31×#121C2211yescharacteristic
24C2×C31×#121C2211yescharacteristic
25C2×C4×D4×C31C2211yescharacteristic
26C2×C4×D4×C31C2211yescharacteristic
27C22×C4⋊C4×C31C2211yescharacteristic
28C22×C4⋊C4×C31C2211yescharacteristic
29C31×#448—24no—
30C2×C31×#103—24no—
31C2×C31×#103C2212yes—
32C2×C31×#103C2212yes—
33C2×C31×#103C2212yes—
34C2×C31×#102C2212yes—
35C2×C31×C22⋊C8C2211yescharacteristic
36C2×C31×C22⋊C8C2212yes—
37C31×#395—21no—
38C31×#393—21no—
39C31×#391C2212yes—
40C31×#386—21no—
41C31×#385—21no—
42C31×#380C2211yescharacteristic
43C31×#340C2212yes—
44C31×#340C2212yes—
45C31×#340C2212yes—
46C31×#336C2212yes—
47C2×#41×C31—28no—
48C2×#41×C31—44no—
49C2×#41×C31—44no—
50C22×Q8×C31—21no—
51C22×Q8×C31—21no—
52C22×Q8×C31—24no—
53C22×Q8×C31—22no—
54C22×Q8×C31—22no—
55C22×Q8×C31—42no—
56C22×Q8×C31C2312yes—
57C22×D4×C31—21no—
58C22×D4×C31—21no—
59C22×D4×C31D411yescharacteristic
60C22×D4×C31C2311yescharacteristic
61C23×C4×C31—21no—
62C23×C4×C31—21no—
63C23×C4×C31—21no—
64C23×C4×C31—21no—
65C23×C4×C31C2311yescharacteristic
66C23×C4×C31C2311yescharacteristic
67C2×SD16×C31—24no—
68C2×SD16×C31—24no—
69C2×SD16×C31—42no—
70C22×C8×C31—22no—
71C22×C8×C31—42no—
72C22×C8×C31C2312yes—
73C22×C8×C31C2312yes—
74C31×C4⋊Q8—22no—
75C31×C4⋊Q8—22no—
76C31×#123—28no—
77C31×#123—28no—
78C31×C22⋊Q8—28no—
79C31×C22⋊Q8—24no—
80C31×C22⋊Q8—42no—
81C31×#121—22no—
82C31×#121—22no—
83C31×#121—22no—
84C31×#121D414yes—
85C4×D4×C31—22no—
86C4×D4×C31—22no—
87C4×D4×C31—22no—
88C4×D4×C31—22no—
89C2×C4⋊C4×C31—21no—
90C2×C4⋊C4×C31—21no—
91C2×C4⋊C4×C31—22no—
92C2×C4⋊C4×C31—21no—
93C2×C4⋊C4×C31—22no—
94C2×C4⋊C4×C31—22no—
95C2×C4⋊C4×C31—22no—
96C2×C4⋊C4×C31D411yescharacteristic
97C2×C4⋊C4×C31D414yes—
98C2×C4⋊C4×C31C2312yes—
99C2×C4⋊C4×C31C2311yescharacteristic
100C2×C4⋊C4×C31C2311yescharacteristic
101C2×C4⋊C4×C31C2311yescharacteristic
102C2×C22⋊C4×C31—21no—
103C2×C22⋊C4×C31—21no—
104C2×C22⋊C4×C31—21no—
105C2×C22⋊C4×C31—22no—
106C2×C22⋊C4×C31—21no—
107C2×C22⋊C4×C31D411yescharacteristic
108C2×C22⋊C4×C31D411yescharacteristic
109C2×C42×C31C2311yescharacteristic
110C31×#103—24no—
111C31×#103—24no—
112C31×#103—24no—
113C31×#102—42no—
114C31×C22⋊C8—22no—
115C31×C22⋊C8—42no—
116C31×#95—21no—
117C31×#95—22no—
118C31×#95—22no—
119C24×C31—21no—
120C24×C31C2×D411yescharacteristic
121C24×C31C2×D411yescharacteristic
122#41×C31—432no—
123C2×Q8×C31—24no—
124C2×Q8×C31—22no—
125C2×Q8×C31—28no—
126C2×Q8×C31—22no—
127C2×Q8×C31—22no—
128C2×Q8×C31—22no—
129C2×Q8×C31—22no—
130C2×Q8×C31—44no—
131C2×Q8×C31—48no—
132C2×Q8×C31—44no—
133C2×Q8×C31—42no—
134C2×Q8×C31—42no—
135C2×Q8×C31—42no—
136C2×Q8×C31#4114yescomplemented
137C2×D4×C31—22no—
138C2×D4×C31—22no—
139C2×D4×C31—24no—
140C2×D4×C31—24no—
141C2×D4×C31—21no—
142C2×D4×C31—22no—
143C2×D4×C31—22no—
144C2×D4×C31—42no—
145C2×D4×C31—42no—
146C2×D4×C31—42no—
147C2×D4×C31—42no—
148C2×D4×C31SD1614yescomplemented
149C2×D4×C31C2×D412yes—
150C22×C4×C31—22no—
151C22×C4×C31—22no—
152C22×C4×C31—24no—
153C22×C4×C31—24no—
154C22×C4×C31—24no—
155C22×C4×C31—22no—
156C22×C4×C31—21no—
157C22×C4×C31—21no—
158C22×C4×C31—21no—
159C22×C4×C31—21no—
160C22×C4×C31—21no—
161C22×C4×C31—22no—
162C22×C4×C31—22no—
163C22×C4×C31—22no—
164C22×C4×C31—44no—
165C22×C4×C31—42no—
166C22×C4×C31—42no—
167C22×C4×C31—42no—
168C22×C4×C31—42no—
169C22×C4×C31—42no—
170C22×C4×C31—42no—
171C22×C4×C31—42no—
172C22×C4×C31C2×D411yescharacteristic
173C22×C4×C31C2×D411yescharacteristic
174C22×C4×C31C2×D411yescharacteristic
175C22×C4×C31C2×D412yes—
176C22×C4×C31C2×D412yes—
177C22×C4×C31C2×D411yescharacteristic
178C22×C4×C31C2411yescharacteristic
179SD16×C31—48no—
180C2×C8×C31—22no—
181C2×C8×C31—22no—
182C2×C8×C31—22no—
183C2×C8×C31—22no—
184C2×C8×C31—22no—
185C2×C8×C31—22no—
186C2×C8×C31—44no—
187C4⋊C4×C31—24no—
188C4⋊C4×C31—24no—
189C4⋊C4×C31—24no—
190C4⋊C4×C31—22no—
191C4⋊C4×C31—21no—
192C4⋊C4×C31—21no—
193C4⋊C4×C31—21no—
194C4⋊C4×C31—22no—
195C4⋊C4×C31—22no—
196C4⋊C4×C31—44no—
197C4⋊C4×C31C2×D412yes—
198C4⋊C4×C31C2×D412yes—
199C4⋊C4×C31C2×D412yes—
200C22⋊C4×C31—24no—
201C22⋊C4×C31—24no—
202C22⋊C4×C31—22no—
203C22⋊C4×C31—24no—
204C22⋊C4×C31—22no—
205C22⋊C4×C31—44no—
206C22⋊C4×C31—42no—
207C42×C31—21no—
208C42×C31—21no—
209#23961C3111yesmaximal, characteristic, direct product factor, Sylow
210C23×C31—22no—
211C23×C31—22no—
212C23×C31—21no—
213C23×C31—22no—
214C23×C31—21no—
215C23×C31—22no—
216C23×C31—22no—
217C23×C31—42no—
218C23×C31—41no—
219C23×C31—41no—
220C23×C31—42no—
221C23×C31#12012yes—
222C23×C31C2×SD1612yes—
223C23×C31C2×SD1612yes—
224C23×C31AGL(1,ℤ/8)12yes—
225C23×C31C22×D411yescharacteristic
226Q8×C31—24no—
227Q8×C31—24no—
228Q8×C31—48no—
229Q8×C31—48no—
230Q8×C31—42no—
231Q8×C31—42no—
232D4×C31—28no—
233D4×C31—24no—
234D4×C31—48no—
235D4×C31—48no—
236D4×C31—41no—
237D4×C31—41no—
238C2×C4×C31—21no—
239C2×C4×C31—22no—
240C2×C4×C31—21no—
241C2×C4×C31—22no—
242C2×C4×C31—22no—
243C2×C4×C31—24no—
244C2×C4×C31—21no—
245C2×C4×C31—21no—
246C2×C4×C31—22no—
247C2×C4×C31—22no—
248C2×C4×C31—22no—
249C2×C4×C31—24no—
250C2×C4×C31—22no—
251C2×C4×C31—22no—
252C2×C4×C31—48no—
253C2×C4×C31—48no—
254C2×C4×C31—48no—
255C2×C4×C31—42no—
256C2×C4×C31—42no—
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—48no—
264C2×C4×C31—41no—
265C2×C4×C31—41no—
266C2×C4×C31—42no—
267C2×C4×C31—42no—
268C2×C4×C31—42no—
269C2×C4×C31—42no—
270C2×C4×C31#12012yes—
271C2×C4×C31C2×SD1612yes—
272C2×C4×C31C22×D411yescharacteristic
273C2×C4×C31C22×D411yescharacteristic
274C2×C4×C31C2×#4112yes—
275C2×C4×C31H(5,𝔽2)12yes—
276C8×C31—42no—
277C8×C31—42no—
278C2×#547C2×C3111yescharacteristic
279C2×#448C2×C3111yescharacteristic, complemented
280C22×#103C2×C3111yescharacteristic, complemented
281#2409C2×C3111yescharacteristic, complemented
282#2024C2×C3112yescomplemented
283#2000C2×C3112yescomplemented
284#1893C2×C3111yescharacteristic, complemented
285#1877C2×C3112yescomplemented
286#1872C2×C3112yescomplemented
287#1869C2×C3111yescharacteristic, complemented
288#1823C2×C3111yescharacteristic, complemented
289C22×C31—22no—
290C22×C31—22no—
291C22×C31—24no—
292C22×C31—24no—
293C22×C31—22no—
294C22×C31—24no—
295C22×C31—41no—
296C22×C31—41no—
297C22×C31—42no—
298C22×C31—42no—
299C22×C31—42no—
300C22×C31—42no—
301C22×C31—82no—
302C22×C31—82no—
303C22×C31#45014yes—
304C22×C31C2×#12011yescharacteristic
305C22×C31#54612yes—
306C22×C31C22×SD1611yescharacteristic
307C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
308C22×C31#57812yes—
309C4×C31—22no—
310C4×C31—22no—
311C4×C31—41no—
312C4×C31—44no—
313C4×C31—42no—
314C4×C31—42no—
315C4×C31—42no—
316C4×C31—41no—
317C4×C31—82no—
318C4×C31#52511yescharacteristic
319C4×C31#53511yescharacteristic
320C4×C31#54612yes—
321C22×#41—21no—
322C23×Q8C22×C3111yescharacteristic
323C22×SD16C22×C3112yes—
324C23×C8C22×C3111yescharacteristic
325#547—24no—
326#547—24no—
327C2×C4⋊Q8C22×C3111yescharacteristic
328C2×#123—21no—
329C2×#123—21no—
330C2×C22⋊Q8—21no—
331C2×C22⋊Q8C22×C3112yes—
332C2×#121C22×C3111yescharacteristic
333C2×#121C22×C3111yescharacteristic
334C2×C4×D4C22×C3111yescharacteristic
335C2×C4×D4C22×C3111yescharacteristic
336C22×C4⋊C4C22×C3111yescharacteristic
337C22×C4⋊C4C22×C3111yescharacteristic
338#448—24no—
339C2×#103—24no—
340C2×#103C22×C3112yes—
341C2×#103C22×C3112yes—
342C2×#103C22×C3112yes—
343C2×#102C22×C3112yes—
344C2×C22⋊C8C22×C3111yescharacteristic
345C2×C22⋊C8C22×C3112yes—
346#395—21no—
347#393—21no—
348#391C22×C3112yes—
349#386—21no—
350#385—21no—
351#380C22×C3111yescharacteristic
352#340C22×C3112yes—
353#340C22×C3112yes—
354#340C22×C3112yes—
355#336C22×C3112yes—
356C2×C31—24no—
357C2×C31—42no—
358C2×C31—82no—
359C2×C31#240411yescharacteristic
360C2×C31C2×#45011yescharacteristic
361C2×C31#301111yescharacteristic
362C2×C31#329812yes—
363C2×C31#330812yes—
364C2×#41—28no—
365C2×#41—44no—
366C2×#41—44no—
367C22×Q8—21no—
368C22×Q8—21no—
369C22×Q8—24no—
370C22×Q8—22no—
371C22×Q8—22no—
372C22×Q8—42no—
373C22×Q8C23×C3112yes—
374C22×D4—21no—
375C22×D4—21no—
376C22×D4D4×C3111yescharacteristic
377C22×D4C23×C3111yescharacteristic
378C23×C4—21no—
379C23×C4—21no—
380C23×C4—21no—
381C23×C4—21no—
382C23×C4C23×C3111yescharacteristic
383C23×C4C23×C3111yescharacteristic
384C2×SD16—24no—
385C2×SD16—24no—
386C2×SD16—42no—
387C22×C8—22no—
388C22×C8—42no—
389C22×C8C23×C3112yes—
390C22×C8C23×C3112yes—
391C4⋊Q8—22no—
392C4⋊Q8—22no—
393#123—28no—
394#123—28no—
395C22⋊Q8—28no—
396C22⋊Q8—24no—
397C22⋊Q8—42no—
398#121—22no—
399#121—22no—
400#121—22no—
401#121D4×C3114yes—
402C4×D4—22no—
403C4×D4—22no—
404C4×D4—22no—
405C4×D4—22no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—22no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—22no—
411C2×C4⋊C4—22no—
412C2×C4⋊C4—22no—
413C2×C4⋊C4D4×C3111yescharacteristic
414C2×C4⋊C4D4×C3114yes—
415C2×C4⋊C4C23×C3112yes—
416C2×C4⋊C4C23×C3111yescharacteristic
417C2×C4⋊C4C23×C3111yescharacteristic
418C2×C4⋊C4C23×C3111yescharacteristic
419C2×C22⋊C4—21no—
420C2×C22⋊C4—21no—
421C2×C22⋊C4—21no—
422C2×C22⋊C4—22no—
423C2×C22⋊C4—21no—
424C2×C22⋊C4D4×C3111yescharacteristic
425C2×C22⋊C4D4×C3111yescharacteristic
426C2×C42C23×C3111yescharacteristic
427#103—24no—
428#103—24no—
429#103—24no—
430#102—42no—
431C22⋊C8—22no—
432C22⋊C8—42no—
433#95—21no—
434#95—22no—
435#95—22no—
436C31#2396111yescharacteristic, direct product factor, Sylow
437C24—21no—
438C24C2×D4×C3111yescharacteristic
439C24C2×D4×C3111yescharacteristic
440#41—432no—
441C2×Q8—24no—
442C2×Q8—22no—
443C2×Q8—28no—
444C2×Q8—22no—
445C2×Q8—22no—
446C2×Q8—22no—
447C2×Q8—22no—
448C2×Q8—44no—
449C2×Q8—48no—
450C2×Q8—44no—
451C2×Q8—42no—
452C2×Q8—42no—
453C2×Q8—42no—
454C2×Q8#41×C3114yescomplemented
455C2×D4—22no—
456C2×D4—22no—
457C2×D4—24no—
458C2×D4—24no—
459C2×D4—21no—
460C2×D4—22no—
461C2×D4—22no—
462C2×D4—42no—
463C2×D4—42no—
464C2×D4—42no—
465C2×D4—42no—
466C2×D4SD16×C3114yescomplemented
467C2×D4C2×D4×C3112yes—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—22no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—44no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4C2×D4×C3111yescharacteristic
491C22×C4C2×D4×C3111yescharacteristic
492C22×C4C2×D4×C3111yescharacteristic
493C22×C4C2×D4×C3112yes—
494C22×C4C2×D4×C3112yes—
495C22×C4C2×D4×C3111yescharacteristic
496C22×C4C24×C3111yescharacteristic
497SD16—48no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—44no—
505C4⋊C4—24no—
506C4⋊C4—24no—
507C4⋊C4—24no—
508C4⋊C4—22no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—44no—
515C4⋊C4C2×D4×C3112yes—
516C4⋊C4C2×D4×C3112yes—
517C4⋊C4C2×D4×C3112yes—
518C22⋊C4—24no—
519C22⋊C4—24no—
520C22⋊C4—22no—
521C22⋊C4—24no—
522C22⋊C4—22no—
523C22⋊C4—44no—
524C22⋊C4—42no—
525C42—21no—
526C42—21no—
527C23—22no—
528C23—22no—
529C23—21no—
530C23—22no—
531C23—21no—
532C23—22no—
533C23—22no—
534C23—42no—
535C23—41no—
536C23—41no—
537C23—42no—
538C23C31×#12012yes—
539C23C2×SD16×C3112yes—
540C23C2×SD16×C3112yes—
541C23C31×AGL(1,ℤ/8)12yes—
542C23C22×D4×C3111yescharacteristic
543Q8—24no—
544Q8—24no—
545Q8—48no—
546Q8—48no—
547Q8—42no—
548Q8—42no—
549D4—28no—
550D4—24no—
551D4—48no—
552D4—48no—
553D4—41no—
554D4—41no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—48no—
570C2×C4—48no—
571C2×C4—48no—
572C2×C4—48no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4C31×#12012yes—
588C2×C4C2×SD16×C3112yes—
589C2×C4C22×D4×C3111yescharacteristic
590C2×C4C22×D4×C3111yescharacteristic
591C2×C4C2×#41×C3112yes—
592C2×C4C31×H(5,𝔽2)12yes—
593C8—42no—
594C8—42no—
595C22—22no—
596C22—22no—
597C22—24no—
598C22—24no—
599C22—22no—
600C22—24no—
601C22—41no—
602C22—41no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—82no—
608C22—82no—
609C22C31×#45014yes—
610C22C2×C31×#12011yescharacteristic
611C22C31×#54612yes—
612C22C22×SD16×C3111yescharacteristic
613C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
614C22C31×#57812yes—
615C4—22no—
616C4—22no—
617C4—41no—
618C4—44no—
619C4—42no—
620C4—42no—
621C4—42no—
622C4—41no—
623C4—82no—
624C4C31×#52511yescharacteristic
625C4C31×#53511yescharacteristic
626C4C31×#54612yes—
627C2—24no—
628C2—42no—
629C2—82no—
630C2C31×#240411yescharacteristic
631C2C2×C31×#45011yescharacteristic
632C2C31×#301111yescharacteristic
633C2C31×#329812yes—
634C2C31×#330812yes—
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC2×C31×#547C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C2×C4×D4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×#448C2semidirect
3. C22×C31×#103C2semidirect
4. C31×#2409C2semidirect
5. C31×#2024C2semidirect
6. C31×#2000C2semidirect
7. C31×#1893C2semidirect
8. C31×#1877C2semidirect
9. C31×#1872C2semidirect
10. C31×#1869C2semidirect
11. C31×#1823C2semidirect
136. C2×Q8×C31#41semidirect
148. C2×D4×C31SD16semidirect
209. #23961C31direct (N × H)
279. C2×#448C2×C31semidirect
280. C22×#103C2×C31semidirect
281. #2409C2×C31semidirect
282. #2024C2×C31semidirect
283. #2000C2×C31semidirect
284. #1893C2×C31semidirect
285. #1877C2×C31semidirect
286. #1872C2×C31semidirect
287. #1869C2×C31semidirect
288. #1823C2×C31semidirect
436. C31#23961direct (N × H)
454. C2×Q8#41×C31semidirect
466. C2×D4SD16×C31semidirect

Maximal subgroups

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

subgroupindex
C2×C31×#5472
C2×C31×#4482
C22×C31×#1032
C31×#24092
C31×#20242
C31×#20002
C31×#18932
C31×#18772
C31×#18722
C31×#18692
C31×#18232
#2396131