Finite Group Databasebeta

← C31×#24156 #21480915

Subgroups — C31×#24156 #21480915

Order 7936.

1394 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#532C211yesmaximal, characteristic
2C2×C31×#477C211yesmaximal, characteristic, complemented
3C2×C31×#460C211yesmaximal, characteristic, complemented
4C4×C31×C22⋊Q8C211yesmaximal, characteristic, complemented
5C31×#2033C212yesmaximal, complemented
6C31×#2000C212yesmaximal, complemented
7C31×#1987C211yesmaximal, characteristic, complemented
8C31×#1982C211yesmaximal, characteristic, complemented
9C31×#1877C212yesmaximal, complemented
10C31×#1862C212yesmaximal, complemented
11C31×#1797C211yesmaximal, characteristic, complemented
12C22×#41×C31—21no—
13C22×SD16×C31C2212yes—
14C31×#532—24no—
15C31×#532—24no—
16C2×C31×C4⋊Q8C2211yescharacteristic
17C2×C31×#127C2211yescharacteristic
18C2×C31×#124—21no—
19C2×C31×C22⋊Q8C2211yescharacteristic
20C2×C31×#121—21no—
21C2×C31×#121C2212yes—
22C2×C4×Q8×C31C2211yescharacteristic
23C2×C4×D4×C31—21no—
24C2×C4×D4×C31C2212yes—
25C22×C42×C31C2211yescharacteristic
26C31×#477—24no—
27C31×#460—24no—
28C2×C31×#107C2212yes—
29C2×C31×C4⋊C8C2211yescharacteristic
30C2×C31×C4⋊C8C2211yescharacteristic
31C2×C31×#103C2212yes—
32C2×C31×#102C2212yes—
33C2×C31×#102C2212yes—
34C2×C31×#102C2212yes—
35C2×C31×C22⋊C8C2212yes—
36C31×#391C2211yescharacteristic
37C31×#389C2211yescharacteristic
38C31×#387—21no—
39C31×#385—21no—
40C31×#380C2211yescharacteristic
41C4×C4⋊C4×C31—21no—
42C4×C4⋊C4×C31C2211yescharacteristic
43C4×C22⋊C4×C31—21no—
44C31×#340C2212yes—
45C31×#340C2212yes—
46C31×#336C2212yes—
47C2×#41×C31—28no—
48C2×#41×C31—44no—
49C2×#41×C31—44no—
50C22×Q8×C31—21no—
51C22×Q8×C31C2311yescharacteristic
52C22×D4×C31—22no—
53C22×D4×C31—21no—
54C22×D4×C31—21no—
55C22×D4×C31C2312yes—
56C23×C4×C31—22no—
57C23×C4×C31D411yescharacteristic
58C23×C4×C31D411yescharacteristic
59C23×C4×C31C2311yescharacteristic
60C2×SD16×C31—24no—
61C2×SD16×C31—24no—
62C2×SD16×C31—42no—
63C22×C8×C31C2312yes—
64C22×C8×C31C2312yes—
65C31×C4⋊Q8—22no—
66C31×C4⋊Q8D414yes—
67C31×#127—22no—
68C31×#127D414yes—
69C31×#124—28no—
70C31×C22⋊Q8—22no—
71C31×C22⋊Q8—22no—
72C31×#121—28no—
73C31×#121—24no—
74C31×#121—24no—
75C4×Q8×C31—22no—
76C4×Q8×C31—22no—
77C4×D4×C31—28no—
78C4×D4×C31—24no—
79C4×D4×C31—24no—
80C2×C4⋊C4×C31—21no—
81C2×C4⋊C4×C31—21no—
82C2×C4⋊C4×C31—21no—
83C2×C4⋊C4×C31—21no—
84C2×C4⋊C4×C31C2311yescharacteristic
85C2×C4⋊C4×C31C2311yescharacteristic
86C2×C4⋊C4×C31C2312yes—
87C2×C4⋊C4×C31C2311yescharacteristic
88C2×C22⋊C4×C31—21no—
89C2×C22⋊C4×C31—22no—
90C2×C22⋊C4×C31—22no—
91C2×C22⋊C4×C31—21no—
92C2×C42×C31—22no—
93C2×C42×C31—22no—
94C2×C42×C31—21no—
95C2×C42×C31—21no—
96C2×C42×C31D411yescharacteristic
97C2×C42×C31D411yescharacteristic
98C2×C42×C31C2311yescharacteristic
99C2×C42×C31C2311yescharacteristic
100C31×#107—24no—
101C31×C4⋊C8—22no—
102C31×C4⋊C8—22no—
103C31×#103—24no—
104C31×#102—24no—
105C31×#102—24no—
106C31×#102—42no—
107C31×C22⋊C8—42no—
108C31×#95—21no—
109C31×#95—21no—
110C31×#95—21no—
111C31×#95—21no—
112C31×#95—21no—
113C24×C31—22no—
114C24×C31C2×D411yescharacteristic
115#41×C31—432no—
116C2×Q8×C31—24no—
117C2×Q8×C31—21no—
118C2×Q8×C31—21no—
119C2×Q8×C31—21no—
120C2×Q8×C31—21no—
121C2×Q8×C31—21no—
122C2×Q8×C31—42no—
123C2×Q8×C31—42no—
124C2×Q8×C31#4112yescomplemented
125C2×D4×C31—28no—
126C2×D4×C31—24no—
127C2×D4×C31—24no—
128C2×D4×C31—22no—
129C2×D4×C31—22no—
130C2×D4×C31—22no—
131C2×D4×C31—22no—
132C2×D4×C31—22no—
133C2×D4×C31—42no—
134C2×D4×C31—42no—
135C2×D4×C31—44no—
136C2×D4×C31—44no—
137C2×D4×C31—42no—
138C2×D4×C31—42no—
139C2×D4×C31C2×D414yes—
140C22×C4×C31—22no—
141C22×C4×C31—22no—
142C22×C4×C31—22no—
143C22×C4×C31—28no—
144C22×C4×C31—21no—
145C22×C4×C31—21no—
146C22×C4×C31—22no—
147C22×C4×C31—22no—
148C22×C4×C31—22no—
149C22×C4×C31—22no—
150C22×C4×C31—21no—
151C22×C4×C31—21no—
152C22×C4×C31—21no—
153C22×C4×C31—21no—
154C22×C4×C31—21no—
155C22×C4×C31—21no—
156C22×C4×C31—21no—
157C22×C4×C31—21no—
158C22×C4×C31—44no—
159C22×C4×C31—44no—
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—48no—
167C2×C8×C31—22no—
168C2×C8×C31—22no—
169C2×C8×C31—22no—
170C2×C8×C31—22no—
171C2×C8×C31—22no—
172C2×C8×C31—22no—
173C4⋊C4×C31—24no—
174C4⋊C4×C31—24no—
175C4⋊C4×C31—22no—
176C4⋊C4×C31—21no—
177C4⋊C4×C31—21no—
178C4⋊C4×C31—21no—
179C4⋊C4×C31—21no—
180C4⋊C4×C31—21no—
181C4⋊C4×C31—42no—
182C4⋊C4×C31—42no—
183C4⋊C4×C31C2×D414yes—
184C4⋊C4×C31#4112yescomplemented
185C22⋊C4×C31—28no—
186C22⋊C4×C31—28no—
187C22⋊C4×C31—24no—
188C22⋊C4×C31—42no—
189C42×C31—22no—
190C42×C31—22no—
191C42×C31—21no—
192C42×C31—21no—
193C42×C31—42no—
194C42×C31—42no—
195C42×C31C2×D412yes—
196C42×C31C2×D412yes—
197#24156C3111yesmaximal, characteristic, direct product factor, Sylow
198C23×C31—21no—
199C23×C31—21no—
200C23×C31—21no—
201C23×C31—21no—
202C23×C31—21no—
203C23×C31—21no—
204C23×C31—24no—
205C23×C31—42no—
206C23×C31—42no—
207C23×C31—42no—
208C23×C31—42no—
209C23×C31—42no—
210C23×C31—42no—
211C23×C31AGL(1,ℤ/8)12yes—
212C23×C31C22×D411yescharacteristic
213Q8×C31—22no—
214Q8×C31—22no—
215Q8×C31—48no—
216Q8×C31—41no—
217Q8×C31—41no—
218D4×C31—24no—
219D4×C31—24no—
220D4×C31—48no—
221D4×C31—416no—
222D4×C31—42no—
223D4×C31—42no—
224D4×C31—48no—
225C2×C4×C31—28no—
226C2×C4×C31—21no—
227C2×C4×C31—21no—
228C2×C4×C31—21no—
229C2×C4×C31—21no—
230C2×C4×C31—21no—
231C2×C4×C31—21no—
232C2×C4×C31—24no—
233C2×C4×C31—22no—
234C2×C4×C31—22no—
235C2×C4×C31—21no—
236C2×C4×C31—21no—
237C2×C4×C31—21no—
238C2×C4×C31—21no—
239C2×C4×C31—416no—
240C2×C4×C31—41no—
241C2×C4×C31—41no—
242C2×C4×C31—41no—
243C2×C4×C31—41no—
244C2×C4×C31—42no—
245C2×C4×C31—42no—
246C2×C4×C31—41no—
247C2×C4×C31—41no—
248C2×C4×C31—41no—
249C2×C4×C31—41no—
250C2×C4×C31—44no—
251C2×C4×C31—41no—
252C2×C4×C31—41no—
253C2×C4×C31—41no—
254C2×C4×C31—41no—
255C2×C4×C31—41no—
256C2×C4×C31—41no—
257C2×C4×C31#12012yes—
258C2×C4×C31#12012yes—
259C2×C4×C31#13512yes—
260C2×C4×C31AGL(1,ℤ/8)12yes—
261C2×C4×C31#13712yes—
262C2×C4×C31C22×D411yescharacteristic
263C2×C4×C31C22×D411yescharacteristic
264C2×C4×C31C2×#4111yescharacteristic
265C2×C4×C31C2×#4111yescharacteristic
266C2×C4×C31H(5,𝔽2)11yescharacteristic
267C2×C4×C312−1+411yescharacteristic
268C8×C31—42no—
269C8×C31—42no—
270C2×#532C2×C3111yescharacteristic
271C2×#477C2×C3111yescharacteristic, complemented
272C2×#460C2×C3111yescharacteristic, complemented
273C4×C22⋊Q8C2×C3111yescharacteristic, complemented
274#2033C2×C3112yescomplemented
275#2000C2×C3112yescomplemented
276#1987C2×C3111yescharacteristic, complemented
277#1982C2×C3111yescharacteristic, complemented
278#1877C2×C3112yescomplemented
279#1862C2×C3112yescomplemented
280#1797C2×C3111yescharacteristic, complemented
281C22×C31—22no—
282C22×C31—22no—
283C22×C31—22no—
284C22×C31—41no—
285C22×C31—41no—
286C22×C31—41no—
287C22×C31—41no—
288C22×C31—44no—
289C22×C31—44no—
290C22×C31—44no—
291C22×C31—82no—
292C22×C31—82no—
293C22×C31—82no—
294C22×C31—82no—
295C22×C31C2×#12011yescharacteristic
296C22×C31#52511yescharacteristic
297C22×C31#53711yescharacteristic
298C22×C31C2×#13511yescharacteristic
299C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
300C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
301C22×C31C2×#13711yescharacteristic
302C4×C31—22no—
303C4×C31—22no—
304C4×C31—42no—
305C4×C31—41no—
306C4×C31—42no—
307C4×C31—41no—
308C4×C31—44no—
309C4×C31—81no—
310C4×C31—81no—
311C4×C31#45214yes—
312C4×C31#54611yescharacteristic
313C4×C31#54711yescharacteristic
314C4×C31#54811yescharacteristic
315C4×C31#55011yescharacteristic
316C22×#41—21no—
317C22×SD16C22×C3112yes—
318#532—24no—
319#532—24no—
320C2×C4⋊Q8C22×C3111yescharacteristic
321C2×#127C22×C3111yescharacteristic
322C2×#124—21no—
323C2×C22⋊Q8C22×C3111yescharacteristic
324C2×#121—21no—
325C2×#121C22×C3112yes—
326C2×C4×Q8C22×C3111yescharacteristic
327C2×C4×D4—21no—
328C2×C4×D4C22×C3112yes—
329C22×C42C22×C3111yescharacteristic
330#477—24no—
331#460—24no—
332C2×#107C22×C3112yes—
333C2×C4⋊C8C22×C3111yescharacteristic
334C2×C4⋊C8C22×C3111yescharacteristic
335C2×#103C22×C3112yes—
336C2×#102C22×C3112yes—
337C2×#102C22×C3112yes—
338C2×#102C22×C3112yes—
339C2×C22⋊C8C22×C3112yes—
340#391C22×C3111yescharacteristic
341#389C22×C3111yescharacteristic
342#387—21no—
343#385—21no—
344#380C22×C3111yescharacteristic
345C4×C4⋊C4—21no—
346C4×C4⋊C4C22×C3111yescharacteristic
347C4×C22⋊C4—21no—
348#340C22×C3112yes—
349#340C22×C3112yes—
350#336C22×C3112yes—
351C2×C31—42no—
352C2×C31—84no—
353C2×C31#240711yescharacteristic
354C2×C31C2×#45211yescharacteristic
355C2×C31#300311yescharacteristic
356C2×C31#310611yescharacteristic
357C2×C31#311211yescharacteristic
358C2×C31#324711yescharacteristic
359C2×C31#325411yescharacteristic
360C2×#41—28no—
361C2×#41—44no—
362C2×#41—44no—
363C22×Q8—21no—
364C22×Q8C23×C3111yescharacteristic
365C22×D4—22no—
366C22×D4—21no—
367C22×D4—21no—
368C22×D4C23×C3112yes—
369C23×C4—22no—
370C23×C4D4×C3111yescharacteristic
371C23×C4D4×C3111yescharacteristic
372C23×C4C23×C3111yescharacteristic
373C2×SD16—24no—
374C2×SD16—24no—
375C2×SD16—42no—
376C22×C8C23×C3112yes—
377C22×C8C23×C3112yes—
378C4⋊Q8—22no—
379C4⋊Q8D4×C3114yes—
380#127—22no—
381#127D4×C3114yes—
382#124—28no—
383C22⋊Q8—22no—
384C22⋊Q8—22no—
385#121—28no—
386#121—24no—
387#121—24no—
388C4×Q8—22no—
389C4×Q8—22no—
390C4×D4—28no—
391C4×D4—24no—
392C4×D4—24no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4C23×C3111yescharacteristic
398C2×C4⋊C4C23×C3111yescharacteristic
399C2×C4⋊C4C23×C3112yes—
400C2×C4⋊C4C23×C3111yescharacteristic
401C2×C22⋊C4—21no—
402C2×C22⋊C4—22no—
403C2×C22⋊C4—22no—
404C2×C22⋊C4—21no—
405C2×C42—22no—
406C2×C42—22no—
407C2×C42—21no—
408C2×C42—21no—
409C2×C42D4×C3111yescharacteristic
410C2×C42D4×C3111yescharacteristic
411C2×C42C23×C3111yescharacteristic
412C2×C42C23×C3111yescharacteristic
413#107—24no—
414C4⋊C8—22no—
415C4⋊C8—22no—
416#103—24no—
417#102—24no—
418#102—24no—
419#102—42no—
420C22⋊C8—42no—
421#95—21no—
422#95—21no—
423#95—21no—
424#95—21no—
425#95—21no—
426C31#2415611yescharacteristic, direct product factor, Sylow
427C24—22no—
428C24C2×D4×C3111yescharacteristic
429#41—432no—
430C2×Q8—24no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—42no—
437C2×Q8—42no—
438C2×Q8#41×C3112yescomplemented
439C2×D4—28no—
440C2×D4—24no—
441C2×D4—24no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—42no—
448C2×D4—42no—
449C2×D4—44no—
450C2×D4—44no—
451C2×D4—42no—
452C2×D4—42no—
453C2×D4C2×D4×C3114yes—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—28no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4C2×D4×C3111yescharacteristic
475C22×C4C2×D4×C3111yescharacteristic
476C22×C4C2×D4×C3111yescharacteristic
477C22×C4C2×D4×C3111yescharacteristic
478C22×C4C2×D4×C3111yescharacteristic
479C22×C4C24×C3111yescharacteristic
480SD16—48no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4C2×D4×C3114yes—
498C4⋊C4#41×C3112yescomplemented
499C22⋊C4—28no—
500C22⋊C4—28no—
501C22⋊C4—24no—
502C22⋊C4—42no—
503C42—22no—
504C42—22no—
505C42—21no—
506C42—21no—
507C42—42no—
508C42—42no—
509C42C2×D4×C3112yes—
510C42C2×D4×C3112yes—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—24no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23C31×AGL(1,ℤ/8)12yes—
525C23C22×D4×C3111yescharacteristic
526Q8—22no—
527Q8—22no—
528Q8—48no—
529Q8—41no—
530Q8—41no—
531D4—24no—
532D4—24no—
533D4—48no—
534D4—416no—
535D4—48no—
536D4—42no—
537D4—42no—
538C2×C4—28no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—416no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—44no—
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×C4C31×#12012yes—
571C2×C4C31×#12012yes—
572C2×C4C31×#13512yes—
573C2×C4C31×AGL(1,ℤ/8)12yes—
574C2×C4C31×#13712yes—
575C2×C4C22×D4×C3111yescharacteristic
576C2×C4C22×D4×C3111yescharacteristic
577C2×C4C2×#41×C3111yescharacteristic
578C2×C4C2×#41×C3111yescharacteristic
579C2×C4C31×H(5,𝔽2)11yescharacteristic
580C2×C4C31×2−1+411yescharacteristic
581C8—42no—
582C8—42no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—44no—
591C22—44no—
592C22—44no—
593C22—82no—
594C22—82no—
595C22—82no—
596C22—82no—
597C22C2×C31×#12011yescharacteristic
598C22C31×#52511yescharacteristic
599C22C31×#53711yescharacteristic
600C22C2×C31×#13511yescharacteristic
601C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
602C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
603C22C2×C31×#13711yescharacteristic
604C4—22no—
605C4—22no—
606C4—42no—
607C4—41no—
608C4—42no—
609C4—41no—
610C4—44no—
611C4—81no—
612C4—81no—
613C4C31×#45214yes—
614C4C31×#54611yescharacteristic
615C4C31×#54711yescharacteristic
616C4C31×#54811yescharacteristic
617C4C31×#55011yescharacteristic
618C2—42no—
619C2—84no—
620C2C31×#240711yescharacteristic
621C2C2×C31×#45211yescharacteristic
622C2C31×#300311yescharacteristic
623C2C31×#310611yescharacteristic
624C2C31×#311211yescharacteristic
625C2C31×#324711yescharacteristic
626C2C31×#325411yescharacteristic
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC2×C31×#532C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C22×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×#477C2semidirect
3. C2×C31×#460C2semidirect
4. C4×C31×C22⋊Q8C2semidirect
5. C31×#2033C2semidirect
6. C31×#2000C2semidirect
7. C31×#1987C2semidirect
8. C31×#1982C2semidirect
9. C31×#1877C2semidirect
10. C31×#1862C2semidirect
11. C31×#1797C2semidirect
124. C2×Q8×C31#41semidirect
184. C4⋊C4×C31#41semidirect
197. #24156C31direct (N × H)
271. C2×#477C2×C31semidirect
272. C2×#460C2×C31semidirect
273. C4×C22⋊Q8C2×C31semidirect
274. #2033C2×C31semidirect
275. #2000C2×C31semidirect
276. #1987C2×C31semidirect
277. #1982C2×C31semidirect
278. #1877C2×C31semidirect
279. #1862C2×C31semidirect
280. #1797C2×C31semidirect
426. C31#24156direct (N × H)
438. C2×Q8#41×C31semidirect
498. C4⋊C4#41×C31semidirect

Maximal subgroups

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

subgroupindex
C2×C31×#5322
C2×C31×#4772
C2×C31×#4602
C4×C31×C22⋊Q82
C31×#20332
C31×#20002
C31×#19872
C31×#19822
C31×#18772
C31×#18622
C31×#17972
#2415631