Finite Group Databasebeta

← C31×#24054 #21480813

Subgroups — C31×#24054 #21480813

Order 7936.

1358 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#547C211yesmaximal, characteristic
2C2×C31×#484C211yesmaximal, characteristic, complemented
3C2×C31×#417C211yesmaximal, characteristic, complemented
4C31×#2486C211yesmaximal, characteristic, complemented
5C31×#2043C212yesmaximal, complemented
6C31×#2036C212yesmaximal, complemented
7C31×#1894C211yesmaximal, characteristic, complemented
8C31×#1870C211yesmaximal, characteristic, complemented
9C31×#1862C212yesmaximal, complemented
10C31×#1788C211yesmaximal, characteristic, complemented
11C31×#1788C212yesmaximal, complemented
12C22×#41×C31C2211yescharacteristic
13C22×#41×C31C2211yescharacteristic
14C22×M16×C31C2211yescharacteristic
15C31×#547—24no—
16C31×#547—24no—
17C2×C31×C4⋊Q8C2211yescharacteristic
18C2×C31×#123—21no—
19C2×C31×#123—21no—
20C2×C31×#123—21no—
21C2×C31×C22⋊Q8C2211yescharacteristic
22C2×C31×C22⋊Q8C2211yescharacteristic
23C2×C31×#121—21no—
24C2×C4×D4×C31C2211yescharacteristic
25C2×C4×D4×C31C2211yescharacteristic
26C22×C4⋊C4×C31—21no—
27C22×C4⋊C4×C31C2211yescharacteristic
28C31×#484—24no—
29C2×C31×#107C2212yes—
30C31×#417—24no—
31C2×C31×#103C2212yes—
32C2×C31×#103C2212yes—
33C2×C31×#103C2212yes—
34C2×C31×#103C2212yes—
35C2×C31×#102C2212yes—
36C2×C31×#102C2212yes—
37C2×C31×C22⋊C8C2211yescharacteristic
38C2×C31×C22⋊C8C2212yes—
39C31×#398—21no—
40C31×#397—21no—
41C31×#389C2212yes—
42C31×#381—21no—
43C31×#380C2211yescharacteristic
44C31×#380C2212yes—
45C31×#340C2212yes—
46C31×#336C2212yes—
47C2×#41×C31—22no—
48C2×#41×C31—22no—
49C2×#41×C31—22no—
50C2×#41×C31—24no—
51C2×#41×C31—24no—
52C2×#41×C31—42no—
53C2×#41×C31—42no—
54C2×#41×C31D414yes—
55C22×Q8×C31C2311yescharacteristic
56C22×Q8×C31C2311yescharacteristic
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×M16×C31—22no—
68C2×M16×C31—42no—
69C22×C8×C31C2312yes—
70C22×C8×C31C2312yes—
71C31×C4⋊Q8—22no—
72C31×C4⋊Q8—22no—
73C31×#123—28no—
74C31×#123—28no—
75C31×#123—44no—
76C31×C22⋊Q8—22no—
77C31×C22⋊Q8—22no—
78C31×C22⋊Q8—22no—
79C31×C22⋊Q8D414yes—
80C31×#121—28no—
81C4×D4×C31—22no—
82C4×D4×C31—22no—
83C4×D4×C31—22no—
84C4×D4×C31—22no—
85C2×C4⋊C4×C31—28no—
86C2×C4⋊C4×C31—21no—
87C2×C4⋊C4×C31—21no—
88C2×C4⋊C4×C31—21no—
89C2×C4⋊C4×C31—21no—
90C2×C4⋊C4×C31—22no—
91C2×C4⋊C4×C31—22no—
92C2×C4⋊C4×C31—21no—
93C2×C4⋊C4×C31—22no—
94C2×C4⋊C4×C31D411yescharacteristic
95C2×C4⋊C4×C31C2312yes—
96C2×C4⋊C4×C31C2312yes—
97C2×C4⋊C4×C31C2311yescharacteristic
98C2×C22⋊C4×C31—21no—
99C2×C22⋊C4×C31—21no—
100C2×C22⋊C4×C31—21no—
101C2×C22⋊C4×C31—22no—
102C2×C22⋊C4×C31—21no—
103C2×C22⋊C4×C31D411yescharacteristic
104C2×C22⋊C4×C31D411yescharacteristic
105C2×C42×C31C2311yescharacteristic
106C31×#107—24no—
107C31×#103—24no—
108C31×#103—24no—
109C31×#103—42no—
110C31×#103—42no—
111C31×#102—24no—
112C31×#102—24no—
113C31×C22⋊C8—22no—
114C31×C22⋊C8—42no—
115C31×#95—21no—
116C31×#95—22no—
117C31×#95—22no—
118C24×C31—21no—
119C24×C31C2×D411yescharacteristic
120C24×C31C2×D411yescharacteristic
121#41×C31—28no—
122#41×C31—44no—
123#41×C31—48no—
124#41×C31—48no—
125#41×C31—44no—
126#41×C31—44no—
127C2×Q8×C31—21no—
128C2×Q8×C31—21no—
129C2×Q8×C31—22no—
130C2×Q8×C31—22no—
131C2×Q8×C31—22no—
132C2×Q8×C31—22no—
133C2×Q8×C31C2×D412yes—
134C2×Q8×C31C2×D412yes—
135C2×D4×C31—24no—
136C2×D4×C31—22no—
137C2×D4×C31—24no—
138C2×D4×C31—22no—
139C2×D4×C31—21no—
140C2×D4×C31—22no—
141C2×D4×C31—41no—
142C2×D4×C31—44no—
143C2×D4×C31—44no—
144C2×D4×C31C2×D412yes—
145C2×D4×C31#4114yes—
146C22×C4×C31—22no—
147C22×C4×C31—22no—
148C22×C4×C31—24no—
149C22×C4×C31—24no—
150C22×C4×C31—21no—
151C22×C4×C31—21no—
152C22×C4×C31—21no—
153C22×C4×C31—24no—
154C22×C4×C31—21no—
155C22×C4×C31—21no—
156C22×C4×C31—22no—
157C22×C4×C31—22no—
158C22×C4×C31—22no—
159C22×C4×C31—22no—
160C22×C4×C31—44no—
161C22×C4×C31—44no—
162C22×C4×C31—42no—
163C22×C4×C31—42no—
164C22×C4×C31—42no—
165C22×C4×C31—42no—
166C22×C4×C31—42no—
167C22×C4×C31C2×D411yescharacteristic
168C22×C4×C31C2×D411yescharacteristic
169C22×C4×C31C2×D411yescharacteristic
170C22×C4×C31C2×D412yes—
171C22×C4×C31C2×D412yes—
172C22×C4×C31C2×D411yescharacteristic
173C22×C4×C31C2411yescharacteristic
174M16×C31—44no—
175C2×C8×C31—22no—
176C2×C8×C31—22no—
177C2×C8×C31—22no—
178C2×C8×C31—22no—
179C2×C8×C31—22no—
180C2×C8×C31—22no—
181C4⋊C4×C31—24no—
182C4⋊C4×C31—24no—
183C4⋊C4×C31—24no—
184C4⋊C4×C31—22no—
185C4⋊C4×C31—24no—
186C4⋊C4×C31—21no—
187C4⋊C4×C31—22no—
188C4⋊C4×C31—22no—
189C4⋊C4×C31—22no—
190C4⋊C4×C31—22no—
191C4⋊C4×C31—42no—
192C4⋊C4×C31—44no—
193C4⋊C4×C31C2×D412yes—
194C22⋊C4×C31—24no—
195C22⋊C4×C31—24no—
196C22⋊C4×C31—24no—
197C22⋊C4×C31—22no—
198C22⋊C4×C31—22no—
199C22⋊C4×C31—42no—
200C22⋊C4×C31—44no—
201C42×C31—21no—
202C42×C31—21no—
203#24054C3111yesmaximal, characteristic, direct product factor, Sylow
204C23×C31—22no—
205C23×C31—22no—
206C23×C31—24no—
207C23×C31—21no—
208C23×C31—21no—
209C23×C31—22no—
210C23×C31—21no—
211C23×C31—22no—
212C23×C31—21no—
213C23×C31—21no—
214C23×C31—22no—
215C23×C31—41no—
216C23×C31—41no—
217C23×C31—42no—
218C23×C31#12012yes—
219C23×C31C22×D411yescharacteristic
220Q8×C31—22no—
221Q8×C31—22no—
222Q8×C31—24no—
223Q8×C31—41no—
224Q8×C31—41no—
225Q8×C31—42no—
226D4×C31—22no—
227D4×C31—24no—
228D4×C31—44no—
229D4×C31—48no—
230D4×C31—48no—
231D4×C31#12314yescomplemented
232C2×C4×C31—21no—
233C2×C4×C31—22no—
234C2×C4×C31—21no—
235C2×C4×C31—22no—
236C2×C4×C31—22no—
237C2×C4×C31—24no—
238C2×C4×C31—21no—
239C2×C4×C31—22no—
240C2×C4×C31—21no—
241C2×C4×C31—22no—
242C2×C4×C31—22no—
243C2×C4×C31—22no—
244C2×C4×C31—22no—
245C2×C4×C31—48no—
246C2×C4×C31—48no—
247C2×C4×C31—48no—
248C2×C4×C31—42no—
249C2×C4×C31—42no—
250C2×C4×C31—41no—
251C2×C4×C31—41no—
252C2×C4×C31—41no—
253C2×C4×C31—41no—
254C2×C4×C31—42no—
255C2×C4×C31—42no—
256C2×C4×C31—42no—
257C2×C4×C31—42no—
258C2×C4×C31—42no—
259C2×C4×C31—42no—
260C2×C4×C31—42no—
261C2×C4×C31—84no—
262C2×C4×C31#12012yes—
263C2×C4×C31AGL(1,ℤ/8)12yes—
264C2×C4×C31C22×D411yescharacteristic
265C2×C4×C31C22×D411yescharacteristic
266C2×C4×C31C2×#4112yes—
267C2×C4×C31H(5,𝔽2)12yes—
268C8×C31—42no—
269C8×C31—42no—
270C2×#547C2×C3111yescharacteristic
271C2×#484C2×C3111yescharacteristic, complemented
272C2×#417C2×C3111yescharacteristic, complemented
273#2486C2×C3111yescharacteristic, complemented
274#2043C2×C3112yescomplemented
275#2036C2×C3112yescomplemented
276#1894C2×C3111yescharacteristic, complemented
277#1870C2×C3111yescharacteristic, complemented
278#1862C2×C3112yescomplemented
279#1788C2×C3112yescomplemented
280#1788C2×C3111yescharacteristic, complemented
281C22×C31—22no—
282C22×C31—24no—
283C22×C31—22no—
284C22×C31—22no—
285C22×C31—42no—
286C22×C31—44no—
287C22×C31—42no—
288C22×C31—42no—
289C22×C31—42no—
290C22×C31—44no—
291C22×C31—42no—
292C22×C31—41no—
293C22×C31—41no—
294C22×C31—41no—
295C22×C31—42no—
296C22×C31C2×#12011yescharacteristic
297C22×C31#54612yes—
298C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
299C22×C31#57511yescharacteristic
300C22×C31#57812yes—
301C4×C31—22no—
302C4×C31—22no—
303C4×C31—41no—
304C4×C31—42no—
305C4×C31—42no—
306C4×C31—42no—
307C4×C31—41no—
308C4×C31—82no—
309C4×C31—82no—
310C4×C31C2×#12311yescharacteristic
311C4×C31#53411yescharacteristic
312C4×C31#54612yes—
313C22×#41C22×C3111yescharacteristic
314C22×#41C22×C3111yescharacteristic
315C22×M16C22×C3111yescharacteristic
316#547—24no—
317#547—24no—
318C2×C4⋊Q8C22×C3111yescharacteristic
319C2×#123—21no—
320C2×#123—21no—
321C2×#123—21no—
322C2×C22⋊Q8C22×C3111yescharacteristic
323C2×C22⋊Q8C22×C3111yescharacteristic
324C2×#121—21no—
325C2×C4×D4C22×C3111yescharacteristic
326C2×C4×D4C22×C3111yescharacteristic
327C22×C4⋊C4—21no—
328C22×C4⋊C4C22×C3111yescharacteristic
329#484—24no—
330C2×#107C22×C3112yes—
331#417—24no—
332C2×#103C22×C3112yes—
333C2×#103C22×C3112yes—
334C2×#103C22×C3112yes—
335C2×#103C22×C3112yes—
336C2×#102C22×C3112yes—
337C2×#102C22×C3112yes—
338C2×C22⋊C8C22×C3111yescharacteristic
339C2×C22⋊C8C22×C3112yes—
340#398—21no—
341#397—21no—
342#389C22×C3112yes—
343#381—21no—
344#380C22×C3111yescharacteristic
345#380C22×C3112yes—
346#340C22×C3112yes—
347#336C22×C3112yes—
348C2×C31—42no—
349C2×C31—44no—
350C2×C31—81no—
351C2×C31#240511yescharacteristic
352C2×C31#300611yescharacteristic
353C2×C31#301111yescharacteristic
354C2×C31#331512yes—
355C2×C31#333012yes—
356C2×#41—22no—
357C2×#41—22no—
358C2×#41—22no—
359C2×#41—24no—
360C2×#41—24no—
361C2×#41—42no—
362C2×#41—42no—
363C2×#41D4×C3114yes—
364C22×Q8C23×C3111yescharacteristic
365C22×Q8C23×C3111yescharacteristic
366C22×D4—21no—
367C22×D4—21no—
368C22×D4D4×C3111yescharacteristic
369C22×D4C23×C3111yescharacteristic
370C23×C4—21no—
371C23×C4—21no—
372C23×C4—21no—
373C23×C4—21no—
374C23×C4C23×C3111yescharacteristic
375C23×C4C23×C3111yescharacteristic
376C2×M16—22no—
377C2×M16—42no—
378C22×C8C23×C3112yes—
379C22×C8C23×C3112yes—
380C4⋊Q8—22no—
381C4⋊Q8—22no—
382#123—28no—
383#123—28no—
384#123—44no—
385C22⋊Q8—22no—
386C22⋊Q8—22no—
387C22⋊Q8—22no—
388C22⋊Q8D4×C3114yes—
389#121—28no—
390C4×D4—22no—
391C4×D4—22no—
392C4×D4—22no—
393C4×D4—22no—
394C2×C4⋊C4—28no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—22no—
400C2×C4⋊C4—22no—
401C2×C4⋊C4—21no—
402C2×C4⋊C4—22no—
403C2×C4⋊C4D4×C3111yescharacteristic
404C2×C4⋊C4C23×C3112yes—
405C2×C4⋊C4C23×C3112yes—
406C2×C4⋊C4C23×C3111yescharacteristic
407C2×C22⋊C4—21no—
408C2×C22⋊C4—21no—
409C2×C22⋊C4—21no—
410C2×C22⋊C4—21no—
411C2×C22⋊C4—22no—
412C2×C22⋊C4D4×C3111yescharacteristic
413C2×C22⋊C4D4×C3111yescharacteristic
414C2×C42C23×C3111yescharacteristic
415#107—24no—
416#103—24no—
417#103—24no—
418#103—42no—
419#103—42no—
420#102—24no—
421#102—24no—
422C22⋊C8—22no—
423C22⋊C8—42no—
424#95—22no—
425#95—21no—
426#95—22no—
427C31#2405411yescharacteristic, direct product factor, Sylow
428C24—21no—
429C24C2×D4×C3111yescharacteristic
430C24C2×D4×C3111yescharacteristic
431#41—28no—
432#41—44no—
433#41—48no—
434#41—48no—
435#41—44no—
436#41—44no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—22no—
440C2×Q8—22no—
441C2×Q8—22no—
442C2×Q8—22no—
443C2×Q8C2×D4×C3112yes—
444C2×Q8C2×D4×C3112yes—
445C2×D4—24no—
446C2×D4—22no—
447C2×D4—24no—
448C2×D4—22no—
449C2×D4—21no—
450C2×D4—22no—
451C2×D4—41no—
452C2×D4—44no—
453C2×D4—44no—
454C2×D4C2×D4×C3112yes—
455C2×D4#41×C3114yes—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—24no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4C2×D4×C3111yescharacteristic
478C22×C4C2×D4×C3111yescharacteristic
479C22×C4C2×D4×C3111yescharacteristic
480C22×C4C2×D4×C3112yes—
481C22×C4C2×D4×C3112yes—
482C22×C4C2×D4×C3111yescharacteristic
483C22×C4C24×C3111yescharacteristic
484M16—44no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—22no—
495C4⋊C4—24no—
496C4⋊C4—21no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—42no—
502C4⋊C4—44no—
503C4⋊C4C2×D4×C3112yes—
504C22⋊C4—24no—
505C22⋊C4—24no—
506C22⋊C4—24no—
507C22⋊C4—22no—
508C22⋊C4—22no—
509C22⋊C4—42no—
510C22⋊C4—44no—
511C42—21no—
512C42—21no—
513C23—22no—
514C23—22no—
515C23—24no—
516C23—21no—
517C23—21no—
518C23—22no—
519C23—21no—
520C23—22no—
521C23—21no—
522C23—21no—
523C23—22no—
524C23—41no—
525C23—41no—
526C23—42no—
527C23C31×#12012yes—
528C23C22×D4×C3111yescharacteristic
529Q8—22no—
530Q8—22no—
531Q8—24no—
532Q8—41no—
533Q8—41no—
534Q8—42no—
535D4—22no—
536D4—24no—
537D4—44no—
538D4—48no—
539D4—48no—
540D4C31×#12314yescomplemented
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—48no—
555C2×C4—48no—
556C2×C4—48no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—84no—
571C2×C4C31×#12012yes—
572C2×C4C31×AGL(1,ℤ/8)12yes—
573C2×C4C22×D4×C3111yescharacteristic
574C2×C4C22×D4×C3111yescharacteristic
575C2×C4C2×#41×C3112yes—
576C2×C4C31×H(5,𝔽2)12yes—
577C8—42no—
578C8—42no—
579C22—22no—
580C22—24no—
581C22—22no—
582C22—22no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—44no—
587C22—42no—
588C22—44no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—42no—
593C22—42no—
594C22C2×C31×#12011yescharacteristic
595C22C31×#54612yes—
596C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
597C22C31×#57511yescharacteristic
598C22C31×#57812yes—
599C4—22no—
600C4—22no—
601C4—41no—
602C4—42no—
603C4—42no—
604C4—42no—
605C4—41no—
606C4—82no—
607C4—82no—
608C4C2×C31×#12311yescharacteristic
609C4C31×#53411yescharacteristic
610C4C31×#54612yes—
611C2—42no—
612C2—44no—
613C2—81no—
614C2C31×#240511yescharacteristic
615C2C31×#300611yescharacteristic
616C2C31×#301111yescharacteristic
617C2C31×#331512yes—
618C2C31×#333012yes—
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×#484C2semidirect
3. C2×C31×#417C2semidirect
4. C31×#2486C2semidirect
5. C31×#2043C2semidirect
6. C31×#2036C2semidirect
7. C31×#1894C2semidirect
8. C31×#1870C2semidirect
9. C31×#1862C2semidirect
10. C31×#1788C2semidirect
11. C31×#1788C2semidirect
203. #24054C31direct (N × H)
231. D4×C31#123semidirect
271. C2×#484C2×C31semidirect
272. C2×#417C2×C31semidirect
273. #2486C2×C31semidirect
274. #2043C2×C31semidirect
275. #2036C2×C31semidirect
276. #1894C2×C31semidirect
277. #1870C2×C31semidirect
278. #1862C2×C31semidirect
279. #1788C2×C31semidirect
280. #1788C2×C31semidirect
427. C31#24054direct (N × H)
540. D4C31×#123semidirect

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×#4842
C2×C31×#4172
C31×#24862
C31×#20432
C31×#20362
C31×#18942
C31×#18702
C31×#18622
C31×#17882
#2405431