Finite Group Databasebeta

← C31×#24048 #21480807

Subgroups — C31×#24048 #21480807

Order 7936.

1492 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#546C211yesmaximal, characteristic
2C2×C31×#481C211yesmaximal, characteristic, complemented
3C2×C31×#417C211yesmaximal, characteristic, complemented
4C31×#2486C211yesmaximal, characteristic, complemented
5C31×#2042C212yesmaximal, complemented
6C31×#2035C212yesmaximal, complemented
7C31×#1895C211yesmaximal, characteristic, complemented
8C31×#1870C211yesmaximal, characteristic, complemented
9C31×#1864C212yesmaximal, complemented
10C31×#1788C211yesmaximal, characteristic, complemented
11C31×#1786C212yesmaximal, complemented
12C22×#41×C31C2211yescharacteristic
13C23×D4×C31C2211yescharacteristic
14C22×M16×C31C2211yescharacteristic, complemented
15C31×#546—24no—
16C31×#546—24no—
17C2×C31×#124C2211yescharacteristic
18C2×C31×#123—21no—
19C2×C31×#123—21no—
20C2×C31×C22⋊Q8C2211yescharacteristic
21C2×C31×#121—21no—
22C2×C31×#121C2211yescharacteristic
23C2×C31×#120—21no—
24C2×C4×D4×C31C2211yescharacteristic
25C2×C4×D4×C31C2211yescharacteristic
26C22×C4⋊C4×C31C2211yescharacteristic, complemented
27C22×C22⋊C4×C31—21no—
28C31×#481—24no—
29C2×C31×#106C2212yescomplemented
30C31×#417—24no—
31C2×C31×#103C2212yescomplemented
32C2×C31×#103C2212yescomplemented
33C2×C31×#102C2212yes—
34C2×C31×#102C2212yes—
35C2×C31×#102C2212yes—
36C2×C31×#102C2212yes—
37C2×C31×C22⋊C8C2212yes—
38C2×C31×C22⋊C8C2211yescharacteristic, complemented
39C31×#398—21no—
40C31×#397—21no—
41C31×#389C2212yescomplemented
42C31×#381—21no—
43C31×#380C2212yes—
44C31×#380C2211yescharacteristic, complemented
45C31×#340C2212yescomplemented
46C31×#336C2212yescomplemented
47C25×C31—21no—
48C2×#41×C31—22no—
49C2×#41×C31—24no—
50C2×#41×C31—42no—
51C2×#41×C31D414yes—
52C22×Q8×C31C2311yescharacteristic
53C22×D4×C31—21no—
54C22×D4×C31—21no—
55C22×D4×C31—22no—
56C22×D4×C31—22no—
57C22×D4×C31—24no—
58C22×D4×C31—21no—
59C22×D4×C31—42no—
60C22×D4×C31D411yescharacteristic
61C22×D4×C31C2311yescharacteristic
62C22×D4×C31C2311yescharacteristic
63C23×C4×C31—21no—
64C23×C4×C31—21no—
65C23×C4×C31—21no—
66C23×C4×C31C2311yescharacteristic
67C23×C4×C31C2311yescharacteristic
68C2×M16×C31—22no—
69C2×M16×C31—42no—
70C22×C8×C31C2312yes—
71C22×C8×C31C2312yes—
72C31×#124—22no—
73C31×#124—22no—
74C31×#123—28no—
75C31×#123—44no—
76C31×C22⋊Q8—22no—
77C31×C22⋊Q8—22no—
78C31×#121—28no—
79C31×#121—22no—
80C31×#121D414yes—
81C31×#120—28no—
82C4×D4×C31—22no—
83C4×D4×C31—22no—
84C4×D4×C31—22no—
85C4×D4×C31—22no—
86C2×C4⋊C4×C31—21no—
87C2×C4⋊C4×C31—22no—
88C2×C4⋊C4×C31—22no—
89C2×C4⋊C4×C31—21no—
90C2×C4⋊C4×C31—22no—
91C2×C4⋊C4×C31D411yescharacteristic
92C2×C4⋊C4×C31C2312yes—
93C2×C4⋊C4×C31C2312yes—
94C2×C4⋊C4×C31C2311yescharacteristic
95C2×C22⋊C4×C31—28no—
96C2×C22⋊C4×C31—21no—
97C2×C22⋊C4×C31—21no—
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×#106—24no—
107C31×#103—24no—
108C31×#103—42no—
109C31×#102—24no—
110C31×#102—24no—
111C31×#102—24no—
112C31×#102—42no—
113C31×C22⋊C8—22no—
114C31×C22⋊C8—42no—
115C31×#95—21no—
116C31×#95—22no—
117C31×#95—22no—
118C24×C31—21no—
119C24×C31—24no—
120C24×C31—21no—
121C24×C31—44no—
122C24×C31—42no—
123C24×C31—42no—
124C24×C31—82no—
125C24×C31C2×D411yescharacteristic
126C24×C31C2×D411yescharacteristic
127#41×C31—28no—
128#41×C31—44no—
129#41×C31—48no—
130C2×Q8×C31—21no—
131C2×Q8×C31—22no—
132C2×Q8×C31—22no—
133C2×Q8×C31C2×D412yes—
134C2×D4×C31—24no—
135C2×D4×C31—22no—
136C2×D4×C31—24no—
137C2×D4×C31—22no—
138C2×D4×C31—24no—
139C2×D4×C31—21no—
140C2×D4×C31—21no—
141C2×D4×C31—22no—
142C2×D4×C31—22no—
143C2×D4×C31—22no—
144C2×D4×C31—41no—
145C2×D4×C31—44no—
146C2×D4×C31—48no—
147C2×D4×C31—44no—
148C2×D4×C31—44no—
149C2×D4×C31—44no—
150C2×D4×C31—82no—
151C2×D4×C31C2×D412yes—
152C2×D4×C31C2×D412yes—
153C2×D4×C31#4114yes—
154C22×C4×C31—22no—
155C22×C4×C31—22no—
156C22×C4×C31—21no—
157C22×C4×C31—21no—
158C22×C4×C31—24no—
159C22×C4×C31—24no—
160C22×C4×C31—21no—
161C22×C4×C31—21no—
162C22×C4×C31—22no—
163C22×C4×C31—22no—
164C22×C4×C31—22no—
165C22×C4×C31—22no—
166C22×C4×C31—44no—
167C22×C4×C31—42no—
168C22×C4×C31—42no—
169C22×C4×C31—42no—
170C22×C4×C31—82no—
171C22×C4×C31C2×D411yescharacteristic
172C22×C4×C31C2×D411yescharacteristic
173C22×C4×C31C2×D411yescharacteristic
174C22×C4×C31C2×D411yescharacteristic
175C22×C4×C31C2×D412yes—
176C22×C4×C31C2×D412yes—
177C22×C4×C31C2411yescharacteristic
178M16×C31—44no—
179C2×C8×C31—22no—
180C2×C8×C31—22no—
181C2×C8×C31—22no—
182C2×C8×C31—22no—
183C2×C8×C31—22no—
184C2×C8×C31—22no—
185C4⋊C4×C31—24no—
186C4⋊C4×C31—22no—
187C4⋊C4×C31—21no—
188C4⋊C4×C31—22no—
189C4⋊C4×C31—22no—
190C4⋊C4×C31—22no—
191C4⋊C4×C31—22no—
192C4⋊C4×C31—44no—
193C4⋊C4×C31—42no—
194C4⋊C4×C31C2×D412yes—
195C22⋊C4×C31—24no—
196C22⋊C4×C31—24no—
197C22⋊C4×C31—24no—
198C22⋊C4×C31—24no—
199C22⋊C4×C31—22no—
200C22⋊C4×C31—22no—
201C22⋊C4×C31—24no—
202C22⋊C4×C31—42no—
203C22⋊C4×C31—44no—
204C42×C31—21no—
205C42×C31—21no—
206#24048C3111yesmaximal, characteristic, direct product factor, Sylow
207C23×C31—21no—
208C23×C31—22no—
209C23×C31—22no—
210C23×C31—22no—
211C23×C31—24no—
212C23×C31—21no—
213C23×C31—21no—
214C23×C31—21no—
215C23×C31—22no—
216C23×C31—22no—
217C23×C31—22no—
218C23×C31—48no—
219C23×C31—48no—
220C23×C31—41no—
221C23×C31—41no—
222C23×C31—42no—
223C23×C31—42no—
224C23×C31—41no—
225C23×C31—41no—
226C23×C31—42no—
227C23×C31—88no—
228C23×C31—84no—
229C23×C31—88no—
230C23×C31#12012yes—
231C23×C31AGL(1,ℤ/8)12yes—
232C23×C31C22×D411yescharacteristic
233Q8×C31—22no—
234Q8×C31—22no—
235Q8×C31—42no—
236D4×C31—22no—
237D4×C31—24no—
238D4×C31—24no—
239D4×C31—44no—
240D4×C31—48no—
241D4×C31—48no—
242D4×C31—41no—
243D4×C31—41no—
244D4×C31—84no—
245D4×C31#12314yescomplemented
246C2×C4×C31—22no—
247C2×C4×C31—22no—
248C2×C4×C31—22no—
249C2×C4×C31—24no—
250C2×C4×C31—21no—
251C2×C4×C31—21no—
252C2×C4×C31—22no—
253C2×C4×C31—22no—
254C2×C4×C31—21no—
255C2×C4×C31—22no—
256C2×C4×C31—21no—
257C2×C4×C31—21no—
258C2×C4×C31—22no—
259C2×C4×C31—48no—
260C2×C4×C31—42no—
261C2×C4×C31—42no—
262C2×C4×C31—41no—
263C2×C4×C31—41no—
264C2×C4×C31—42no—
265C2×C4×C31—42no—
266C2×C4×C31—42no—
267C2×C4×C31—42no—
268C2×C4×C31—42no—
269C2×C4×C31—42no—
270C2×C4×C31—42no—
271C2×C4×C31—84no—
272C2×C4×C31—84no—
273C2×C4×C31#12012yes—
274C2×C4×C31C22×D411yescharacteristic
275C2×C4×C31C22×D411yescharacteristic
276C2×C4×C31C2×#4112yes—
277C2×C4×C31H(5,𝔽2)12yes—
278C8×C31—42no—
279C8×C31—42no—
280C2×#546C2×C3111yescharacteristic
281C2×#481C2×C3111yescharacteristic, complemented
282C2×#417C2×C3111yescharacteristic, complemented
283#2486C2×C3111yescharacteristic, complemented
284#2042C2×C3112yescomplemented
285#2035C2×C3112yescomplemented
286#1895C2×C3111yescharacteristic, complemented
287#1870C2×C3111yescharacteristic, complemented
288#1864C2×C3112yescomplemented
289#1788C2×C3111yescharacteristic, complemented
290#1786C2×C3112yescomplemented
291C22×C31—22no—
292C22×C31—24no—
293C22×C31—22no—
294C22×C31—22no—
295C22×C31—22no—
296C22×C31—24no—
297C22×C31—41no—
298C22×C31—42no—
299C22×C31—42no—
300C22×C31—41no—
301C22×C31—42no—
302C22×C31—44no—
303C22×C31—42no—
304C22×C31—42no—
305C22×C31—44no—
306C22×C31—42no—
307C22×C31—42no—
308C22×C31—42no—
309C22×C31—816no—
310C22×C31—82no—
311C22×C31—82no—
312C22×C31C2×#12011yescharacteristic
313C22×C31#54612yes—
314C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
315C22×C31#57511yescharacteristic
316C22×C31#57712yes—
317C4×C31—22no—
318C4×C31—41no—
319C4×C31—42no—
320C4×C31—41no—
321C4×C31—41no—
322C4×C31—42no—
323C4×C31—82no—
324C4×C31—82no—
325C4×C31C2×#12311yescharacteristic
326C4×C31#53411yescharacteristic
327C4×C31#54612yes—
328C22×#41C22×C3111yescharacteristic
329C23×D4C22×C3111yescharacteristic
330C22×M16C22×C3111yescharacteristic, complemented
331#546—24no—
332#546—24no—
333C2×#124C22×C3111yescharacteristic
334C2×#123—21no—
335C2×#123—21no—
336C2×C22⋊Q8C22×C3111yescharacteristic
337C2×#121—21no—
338C2×#121C22×C3111yescharacteristic
339C2×#120—21no—
340C2×C4×D4C22×C3111yescharacteristic
341C2×C4×D4C22×C3111yescharacteristic
342C22×C4⋊C4C22×C3111yescharacteristic, complemented
343C22×C22⋊C4—21no—
344#481—24no—
345C2×#106C22×C3112yescomplemented
346#417—24no—
347C2×#103C22×C3112yescomplemented
348C2×#103C22×C3112yescomplemented
349C2×#102C22×C3112yes—
350C2×#102C22×C3112yes—
351C2×#102C22×C3112yes—
352C2×#102C22×C3112yes—
353C2×C22⋊C8C22×C3112yes—
354C2×C22⋊C8C22×C3111yescharacteristic, complemented
355#398—21no—
356#397—21no—
357#389C22×C3112yescomplemented
358#381—21no—
359#380C22×C3112yes—
360#380C22×C3111yescharacteristic, complemented
361#340C22×C3112yescomplemented
362#336C22×C3112yescomplemented
363C2×C31—42no—
364C2×C31—42no—
365C2×C31—44no—
366C2×C31—82no—
367C2×C31#240511yescharacteristic
368C2×C31#300611yescharacteristic
369C2×C31#300911yescharacteristic
370C2×C31#330612yes—
371C2×C31#332312yes—
372C25—21no—
373C2×#41—22no—
374C2×#41—24no—
375C2×#41—42no—
376C2×#41D4×C3114yes—
377C22×Q8C23×C3111yescharacteristic
378C22×D4—21no—
379C22×D4—21no—
380C22×D4—22no—
381C22×D4—22no—
382C22×D4—24no—
383C22×D4—21no—
384C22×D4—42no—
385C22×D4D4×C3111yescharacteristic
386C22×D4C23×C3111yescharacteristic
387C22×D4C23×C3111yescharacteristic
388C23×C4—21no—
389C23×C4—21no—
390C23×C4—21no—
391C23×C4C23×C3111yescharacteristic
392C23×C4C23×C3111yescharacteristic
393C2×M16—22no—
394C2×M16—42no—
395C22×C8C23×C3112yes—
396C22×C8C23×C3112yes—
397#124—22no—
398#124—22no—
399#123—28no—
400#123—44no—
401C22⋊Q8—22no—
402C22⋊Q8—22no—
403#121—28no—
404#121—22no—
405#121D4×C3114yes—
406#120—28no—
407C4×D4—22no—
408C4×D4—22no—
409C4×D4—22no—
410C4×D4—22no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—22no—
413C2×C4⋊C4—22no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—22no—
416C2×C4⋊C4D4×C3111yescharacteristic
417C2×C4⋊C4C23×C3112yes—
418C2×C4⋊C4C23×C3112yes—
419C2×C4⋊C4C23×C3111yescharacteristic
420C2×C22⋊C4—28no—
421C2×C22⋊C4—21no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—21no—
424C2×C22⋊C4—21no—
425C2×C22⋊C4—21no—
426C2×C22⋊C4—22no—
427C2×C22⋊C4—21no—
428C2×C22⋊C4D4×C3111yescharacteristic
429C2×C22⋊C4D4×C3111yescharacteristic
430C2×C42C23×C3111yescharacteristic
431#106—24no—
432#103—24no—
433#103—42no—
434#102—24no—
435#102—24no—
436#102—24no—
437#102—42no—
438C22⋊C8—22no—
439C22⋊C8—42no—
440#95—22no—
441#95—21no—
442#95—22no—
443C31#2404811yescharacteristic, direct product factor, Sylow
444C24—21no—
445C24—24no—
446C24—21no—
447C24—44no—
448C24—42no—
449C24—42no—
450C24—82no—
451C24C2×D4×C3111yescharacteristic
452C24C2×D4×C3111yescharacteristic
453#41—28no—
454#41—44no—
455#41—48no—
456C2×Q8—21no—
457C2×Q8—22no—
458C2×Q8—22no—
459C2×Q8C2×D4×C3112yes—
460C2×D4—24no—
461C2×D4—22no—
462C2×D4—24no—
463C2×D4—22no—
464C2×D4—24no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—22no—
468C2×D4—22no—
469C2×D4—22no—
470C2×D4—41no—
471C2×D4—44no—
472C2×D4—48no—
473C2×D4—44no—
474C2×D4—44no—
475C2×D4—44no—
476C2×D4—82no—
477C2×D4C2×D4×C3112yes—
478C2×D4C2×D4×C3112yes—
479C2×D4#41×C3114yes—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—24no—
485C22×C4—24no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—44no—
493C22×C4—42no—
494C22×C4—42no—
495C22×C4—42no—
496C22×C4—82no—
497C22×C4C2×D4×C3111yescharacteristic
498C22×C4C2×D4×C3111yescharacteristic
499C22×C4C2×D4×C3111yescharacteristic
500C22×C4C2×D4×C3111yescharacteristic
501C22×C4C2×D4×C3112yes—
502C22×C4C2×D4×C3112yes—
503C22×C4C24×C3111yescharacteristic
504M16—44no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C4⋊C4—24no—
512C4⋊C4—22no—
513C4⋊C4—21no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—44no—
519C4⋊C4—42no—
520C4⋊C4C2×D4×C3112yes—
521C22⋊C4—24no—
522C22⋊C4—24no—
523C22⋊C4—24no—
524C22⋊C4—24no—
525C22⋊C4—22no—
526C22⋊C4—22no—
527C22⋊C4—24no—
528C22⋊C4—42no—
529C22⋊C4—44no—
530C42—21no—
531C42—21no—
532C23—21no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—24no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—48no—
544C23—48no—
545C23—41no—
546C23—41no—
547C23—42no—
548C23—42no—
549C23—41no—
550C23—41no—
551C23—42no—
552C23—88no—
553C23—84no—
554C23—88no—
555C23C31×#12012yes—
556C23C31×AGL(1,ℤ/8)12yes—
557C23C22×D4×C3111yescharacteristic
558Q8—22no—
559Q8—22no—
560Q8—42no—
561D4—22no—
562D4—24no—
563D4—24no—
564D4—44no—
565D4—48no—
566D4—48no—
567D4—41no—
568D4—41no—
569D4—84no—
570D4C31×#12314yescomplemented
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—48no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—84no—
597C2×C4—84no—
598C2×C4C31×#12012yes—
599C2×C4C22×D4×C3111yescharacteristic
600C2×C4C22×D4×C3111yescharacteristic
601C2×C4C2×#41×C3112yes—
602C2×C4C31×H(5,𝔽2)12yes—
603C8—42no—
604C8—42no—
605C22—22no—
606C22—24no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—24no—
611C22—41no—
612C22—42no—
613C22—42no—
614C22—41no—
615C22—42no—
616C22—44no—
617C22—42no—
618C22—42no—
619C22—44no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—816no—
624C22—82no—
625C22—82no—
626C22C2×C31×#12011yescharacteristic
627C22C31×#54612yes—
628C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
629C22C31×#57511yescharacteristic
630C22C31×#57712yes—
631C4—22no—
632C4—41no—
633C4—42no—
634C4—41no—
635C4—41no—
636C4—42no—
637C4—82no—
638C4—82no—
639C4C2×C31×#12311yescharacteristic
640C4C31×#53411yescharacteristic
641C4C31×#54612yes—
642C2—42no—
643C2—42no—
644C2—44no—
645C2—82no—
646C2C31×#240511yescharacteristic
647C2C31×#300611yescharacteristic
648C2C31×#300911yescharacteristic
649C2C31×#330612yes—
650C2C31×#332312yes—
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC2×C31×#546C2
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×#481C2semidirect
3. C2×C31×#417C2semidirect
4. C31×#2486C2semidirect
5. C31×#2042C2semidirect
6. C31×#2035C2semidirect
7. C31×#1895C2semidirect
8. C31×#1870C2semidirect
9. C31×#1864C2semidirect
10. C31×#1788C2semidirect
11. C31×#1786C2semidirect
14. C22×M16×C31C22semidirect
26. C22×C4⋊C4×C31C22semidirect
29. C2×C31×#106C22semidirect
31. C2×C31×#103C22semidirect
32. C2×C31×#103C22semidirect
38. C2×C31×C22⋊C8C22semidirect
41. C31×#389C22semidirect
44. C31×#380C22semidirect
45. C31×#340C22semidirect
46. C31×#336C22semidirect
206. #24048C31direct (N × H)
245. D4×C31#123semidirect
281. C2×#481C2×C31semidirect
282. C2×#417C2×C31semidirect
283. #2486C2×C31semidirect
284. #2042C2×C31semidirect
285. #2035C2×C31semidirect
286. #1895C2×C31semidirect
287. #1870C2×C31semidirect
288. #1864C2×C31semidirect
289. #1788C2×C31semidirect
290. #1786C2×C31semidirect
330. C22×M16C22×C31semidirect
342. C22×C4⋊C4C22×C31semidirect
345. C2×#106C22×C31semidirect
347. C2×#103C22×C31semidirect
348. C2×#103C22×C31semidirect
354. C2×C22⋊C8C22×C31semidirect
357. #389C22×C31semidirect
360. #380C22×C31semidirect
361. #340C22×C31semidirect
362. #336C22×C31semidirect
443. C31#24048direct (N × H)
570. D4C31×#123semidirect

Maximal subgroups

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

subgroupindex
C2×C31×#5462
C2×C31×#4812
C2×C31×#4172
C31×#24862
C31×#20422
C31×#20352
C31×#18952
C31×#18702
C31×#18642
C31×#17882
C31×#17862
#2404831