Finite Group Databasebeta

← C31×#24559 #21481318

Subgroups — C31×#24559 #21481318

Order 7936.

1156 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#483C211yesmaximal, characteristic, complemented
2C2×C31×#407C211yesmaximal, characteristic
3C31×#2848C211yesmaximal, characteristic
4C2×C31×#380C211yesmaximal, characteristic
5C31×#2077C212yesmaximal, complemented
6C31×#2036C212yesmaximal, complemented
7C31×#1894C212yesmaximal, complemented
8C31×#1872C211yesmaximal, characteristic, complemented
9C31×#1852C211yesmaximal, characteristic, complemented
10C31×#1823C211yesmaximal, characteristic, complemented
11C31×#1809C212yesmaximal, complemented
12C24×C4×C31C2211yescharacteristic
13C22×M16×C31C2212yes—
14C2×C31×C22⋊Q8C2211yescharacteristic
15C2×C31×C22⋊Q8C2211yescharacteristic
16C2×C31×C22⋊Q8C2211yescharacteristic
17C22×C4⋊C4×C31C2211yescharacteristic
18C22×C4⋊C4×C31C2211yescharacteristic
19C22×C22⋊C4×C31—21no—
20C31×#483—24no—
21C2×C31×#107C2212yescomplemented
22C2×C31×#106C2212yescomplemented
23C2×C31×#103C2212yescomplemented
24C2×C31×#103C2212yescomplemented
25C31×#407—24no—
26C2×C31×C22⋊C8C2212yes—
27C2×C31×C22⋊C8C2211yescharacteristic
28C2×C31×C22⋊C8C2211yescharacteristic
29C31×#385—21no—
30C31×#385—21no—
31C31×#385—21no—
32C31×#380—21no—
33C31×#380—21no—
34C31×#380C2211yescharacteristic
35C31×#380C2212yes—
36C31×#380C2211yescharacteristic
37C31×#380C2212yes—
38C31×#380C2211yescharacteristic
39C31×#379—21no—
40C2×C31×#95—21no—
41C31×#340C2212yescomplemented
42C31×#340C2212yescomplemented
43C31×#340C2212yescomplemented
44C31×#340C2212yescomplemented
45C25×C31D411yescharacteristic
46C22×Q8×C31C2311yescharacteristic
47C23×C4×C31—22no—
48C23×C4×C31—22no—
49C23×C4×C31—24no—
50C23×C4×C31—21no—
51C23×C4×C31—21no—
52C23×C4×C31—42no—
53C23×C4×C31D411yescharacteristic
54C23×C4×C31D411yescharacteristic
55C23×C4×C31D411yescharacteristic
56C23×C4×C31C2311yescharacteristic
57C23×C4×C31C2311yescharacteristic
58C23×C4×C31C2311yescharacteristic
59C2×M16×C31—24no—
60C2×M16×C31—24no—
61C2×M16×C31—42no—
62C22×C8×C31C2312yes—
63C22×C8×C31C2312yes—
64C31×C22⋊Q8—22no—
65C31×C22⋊Q8—22no—
66C31×C22⋊Q8—24no—
67C31×C22⋊Q8—42no—
68C2×C4⋊C4×C31—22no—
69C2×C4⋊C4×C31—22no—
70C2×C4⋊C4×C31—21no—
71C2×C4⋊C4×C31—21no—
72C2×C4⋊C4×C31—21no—
73C2×C4⋊C4×C31—21no—
74C2×C4⋊C4×C31—21no—
75C2×C4⋊C4×C31—42no—
76C2×C4⋊C4×C31C2312yes—
77C2×C4⋊C4×C31C2312yes—
78C2×C4⋊C4×C31C2311yescharacteristic
79C2×C4⋊C4×C31C2311yescharacteristic
80C2×C4⋊C4×C31C2311yescharacteristic
81C2×C22⋊C4×C31—22no—
82C2×C22⋊C4×C31—22no—
83C2×C22⋊C4×C31—21no—
84C2×C22⋊C4×C31—21no—
85C2×C22⋊C4×C31—21no—
86C2×C22⋊C4×C31—21no—
87C2×C22⋊C4×C31—21no—
88C2×C22⋊C4×C31—21no—
89C2×C22⋊C4×C31—44no—
90C31×#107—42no—
91C31×#106—24no—
92C31×#103—24no—
93C31×#103—42no—
94C31×C22⋊C8—22no—
95C31×C22⋊C8—24no—
96C31×C22⋊C8D414yes—
97C31×#95—22no—
98C31×#95—22no—
99C31×#95—21no—
100C31×#95—21no—
101C31×#95—21no—
102C24×C31—22no—
103C24×C31—22no—
104C24×C31—22no—
105C24×C31—22no—
106C24×C31—22no—
107C24×C31—41no—
108C24×C31—41no—
109C24×C31C2×D411yescharacteristic
110C24×C31C2×D411yescharacteristic
111C24×C31C2×D411yescharacteristic
112C2×Q8×C31—21no—
113C2×Q8×C31—21no—
114C2×Q8×C31—22no—
115C2×Q8×C31—22no—
116C22×C4×C31—21no—
117C22×C4×C31—21no—
118C22×C4×C31—22no—
119C22×C4×C31—22no—
120C22×C4×C31—22no—
121C22×C4×C31—22no—
122C22×C4×C31—22no—
123C22×C4×C31—22no—
124C22×C4×C31—22no—
125C22×C4×C31—22no—
126C22×C4×C31—22no—
127C22×C4×C31—21no—
128C22×C4×C31—21no—
129C22×C4×C31—22no—
130C22×C4×C31—22no—
131C22×C4×C31—21no—
132C22×C4×C31—21no—
133C22×C4×C31—21no—
134C22×C4×C31—22no—
135C22×C4×C31—22no—
136C22×C4×C31—21no—
137C22×C4×C31—41no—
138C22×C4×C31—41no—
139C22×C4×C31—41no—
140C22×C4×C31—48no—
141C22×C4×C31—48no—
142C22×C4×C31—42no—
143C22×C4×C31—42no—
144C22×C4×C31—42no—
145C22×C4×C31—42no—
146C22×C4×C31—42no—
147C22×C4×C31—42no—
148C22×C4×C31C2×D411yescharacteristic
149C22×C4×C31C2×D411yescharacteristic
150C22×C4×C31C2×D412yes—
151C22×C4×C31C2×D411yescharacteristic
152C22×C4×C31#4112yes—
153C22×C4×C31#4114yes—
154C22×C4×C31C2411yescharacteristic
155M16×C31—48no—
156C2×C8×C31—22no—
157C2×C8×C31—22no—
158C2×C8×C31—22no—
159C2×C8×C31—22no—
160C2×C8×C31—22no—
161C2×C8×C31C2×D414yes—
162C4⋊C4×C31—22no—
163C4⋊C4×C31—21no—
164C4⋊C4×C31—22no—
165C4⋊C4×C31—21no—
166C4⋊C4×C31—22no—
167C4⋊C4×C31—22no—
168C4⋊C4×C31—22no—
169C4⋊C4×C31—22no—
170C4⋊C4×C31—42no—
171C4⋊C4×C31—42no—
172C4⋊C4×C31—42no—
173C4⋊C4×C31—42no—
174C4⋊C4×C31—42no—
175C22⋊C4×C31—42no—
176C22⋊C4×C31—44no—
177C22⋊C4×C31—44no—
178C22⋊C4×C31—42no—
179C22⋊C4×C31—42no—
180C22⋊C4×C31—42no—
181C22⋊C4×C31—42no—
182C22⋊C4×C31—42no—
183#24559C3111yesmaximal, characteristic, direct product factor, Sylow
184C23×C31—21no—
185C23×C31—21no—
186C23×C31—21no—
187C23×C31—21no—
188C23×C31—28no—
189C23×C31—21no—
190C23×C31—21no—
191C23×C31—21no—
192C23×C31—21no—
193C23×C31—21no—
194C23×C31—21no—
195C23×C31—21no—
196C23×C31—21no—
197C23×C31—21no—
198C23×C31—21no—
199C23×C31—21no—
200C23×C31—21no—
201C23×C31—44no—
202C23×C31—44no—
203C23×C31—44no—
204C23×C31—44no—
205C23×C31—44no—
206C23×C31—44no—
207C23×C31#12112yes—
208C23×C31#12312yes—
209C23×C31AGL(1,ℤ/8)12yes—
210C23×C31#13712yes—
211C23×C31#13712yes—
212C23×C31C22×D411yescharacteristic
213Q8×C31—42no—
214Q8×C31—42no—
215C2×C4×C31—21no—
216C2×C4×C31—21no—
217C2×C4×C31—21no—
218C2×C4×C31—21no—
219C2×C4×C31—24no—
220C2×C4×C31—24no—
221C2×C4×C31—24no—
222C2×C4×C31—24no—
223C2×C4×C31—21no—
224C2×C4×C31—21no—
225C2×C4×C31—21no—
226C2×C4×C31—44no—
227C2×C4×C31—44no—
228C2×C4×C31—44no—
229C2×C4×C31—42no—
230C2×C4×C31—42no—
231C2×C4×C31—41no—
232C2×C4×C31—41no—
233C2×C4×C31—42no—
234C2×C4×C31—41no—
235C2×C4×C31—41no—
236C2×C4×C31—41no—
237C2×C4×C31—41no—
238C2×C4×C31—42no—
239C2×C4×C31—42no—
240C2×C4×C31—42no—
241C2×C4×C31—41no—
242C2×C4×C31—41no—
243C2×C4×C31—41no—
244C2×C4×C31—41no—
245C2×C4×C31—42no—
246C2×C4×C31—41no—
247C2×C4×C31—41no—
248C2×C4×C31—42no—
249C2×C4×C31—84no—
250C2×C4×C31—84no—
251C2×C4×C31#12112yes—
252C2×C4×C31#12312yes—
253C2×C4×C31C22×D411yescharacteristic
254C2×C4×C31C2×#4112yes—
255C2×C4×C31C2×#4111yescharacteristic
256C2×C4×C31H(5,𝔽2)12yes—
257C8×C31—24no—
258C8×C31—42no—
259C2×#483C2×C3111yescharacteristic, complemented
260C2×#407C2×C3111yescharacteristic
261#2848C2×C3111yescharacteristic
262C2×#380C2×C3111yescharacteristic
263#2077C2×C3112yescomplemented
264#2036C2×C3112yescomplemented
265#1894C2×C3112yescomplemented
266#1872C2×C3111yescharacteristic, complemented
267#1852C2×C3111yescharacteristic, complemented
268#1823C2×C3111yescharacteristic, complemented
269#1809C2×C3112yescomplemented
270C22×C31—22no—
271C22×C31—22no—
272C22×C31—22no—
273C22×C31—22no—
274C22×C31—22no—
275C22×C31—22no—
276C22×C31—22no—
277C22×C31—22no—
278C22×C31—22no—
279C22×C31—22no—
280C22×C31—22no—
281C22×C31—22no—
282C22×C31—416no—
283C22×C31—41no—
284C22×C31—41no—
285C22×C31—41no—
286C22×C31—41no—
287C22×C31—41no—
288C22×C31—41no—
289C22×C31—41no—
290C22×C31—41no—
291C22×C31#46814yes—
292C22×C31C2×#12111yescharacteristic
293C22×C31C2×#12311yescharacteristic
294C22×C31#53411yescharacteristic
295C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
296C22×C31C2×#13711yescharacteristic
297C22×C31C2×#13711yescharacteristic
298C22×C31#57511yescharacteristic
299C4×C31—22no—
300C4×C31—22no—
301C4×C31—41no—
302C4×C31—81no—
303C4×C31—81no—
304C4×C31—81no—
305C4×C31—82no—
306C4×C31—82no—
307C4×C31—81no—
308C4×C31#54612yes—
309C4×C31#55112yes—
310C24×C4C22×C3111yescharacteristic
311C22×M16C22×C3112yes—
312C2×C22⋊Q8C22×C3111yescharacteristic
313C2×C22⋊Q8C22×C3111yescharacteristic
314C2×C22⋊Q8C22×C3111yescharacteristic
315C22×C4⋊C4C22×C3111yescharacteristic
316C22×C4⋊C4C22×C3111yescharacteristic
317C22×C22⋊C4—21no—
318#483—24no—
319C2×#107C22×C3112yescomplemented
320C2×#106C22×C3112yescomplemented
321C2×#103C22×C3112yescomplemented
322C2×#103C22×C3112yescomplemented
323#407—24no—
324C2×C22⋊C8C22×C3112yes—
325C2×C22⋊C8C22×C3111yescharacteristic
326C2×C22⋊C8C22×C3111yescharacteristic
327#385—21no—
328#385—21no—
329#385—21no—
330#380—21no—
331#380—21no—
332#380C22×C3111yescharacteristic
333#380C22×C3112yes—
334#380C22×C3111yescharacteristic
335#380C22×C3112yes—
336#380C22×C3111yescharacteristic
337#379—21no—
338C2×#95—21no—
339#340C22×C3112yescomplemented
340#340C22×C3112yescomplemented
341#340C22×C3112yescomplemented
342#340C22×C3112yescomplemented
343C2×C31—24no—
344C2×C31—42no—
345C2×C31—42no—
346C2×C31#241211yescharacteristic
347C2×C31C2×#46811yescharacteristic
348C2×C31#305611yescharacteristic
349C2×C31#309211yescharacteristic
350C2×C31#309511yescharacteristic
351C2×C31#319411yescharacteristic
352C2×C31#319511yescharacteristic
353C25D4×C3111yescharacteristic
354C22×Q8C23×C3111yescharacteristic
355C23×C4—22no—
356C23×C4—22no—
357C23×C4—24no—
358C23×C4—21no—
359C23×C4—21no—
360C23×C4—42no—
361C23×C4D4×C3111yescharacteristic
362C23×C4D4×C3111yescharacteristic
363C23×C4D4×C3111yescharacteristic
364C23×C4C23×C3111yescharacteristic
365C23×C4C23×C3111yescharacteristic
366C23×C4C23×C3111yescharacteristic
367C2×M16—24no—
368C2×M16—24no—
369C2×M16—42no—
370C22×C8C23×C3112yes—
371C22×C8C23×C3112yes—
372C22⋊Q8—22no—
373C22⋊Q8—22no—
374C22⋊Q8—24no—
375C22⋊Q8—42no—
376C2×C4⋊C4—22no—
377C2×C4⋊C4—22no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—42no—
384C2×C4⋊C4C23×C3112yes—
385C2×C4⋊C4C23×C3112yes—
386C2×C4⋊C4C23×C3111yescharacteristic
387C2×C4⋊C4C23×C3111yescharacteristic
388C2×C4⋊C4C23×C3111yescharacteristic
389C2×C22⋊C4—22no—
390C2×C22⋊C4—22no—
391C2×C22⋊C4—21no—
392C2×C22⋊C4—21no—
393C2×C22⋊C4—21no—
394C2×C22⋊C4—21no—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—44no—
398#107—42no—
399#106—24no—
400#103—24no—
401#103—42no—
402C22⋊C8—22no—
403C22⋊C8—24no—
404C22⋊C8D4×C3114yes—
405#95—22no—
406#95—22no—
407#95—21no—
408#95—21no—
409#95—21no—
410C31#2455911yescharacteristic, direct product factor, Sylow
411C24—22no—
412C24—22no—
413C24—22no—
414C24—22no—
415C24—22no—
416C24—41no—
417C24—41no—
418C24C2×D4×C3111yescharacteristic
419C24C2×D4×C3111yescharacteristic
420C24C2×D4×C3111yescharacteristic
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—22no—
424C2×Q8—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4C2×D4×C3111yescharacteristic
458C22×C4C2×D4×C3111yescharacteristic
459C22×C4C2×D4×C3112yes—
460C22×C4C2×D4×C3111yescharacteristic
461C22×C4#41×C3112yes—
462C22×C4#41×C3114yes—
463C22×C4C24×C3111yescharacteristic
464M16—48no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8C2×D4×C3114yes—
471C4⋊C4—22no—
472C4⋊C4—21no—
473C4⋊C4—22no—
474C4⋊C4—21no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C22⋊C4—42no—
485C22⋊C4—44no—
486C22⋊C4—44no—
487C22⋊C4—42no—
488C22⋊C4—42no—
489C22⋊C4—42no—
490C22⋊C4—42no—
491C22⋊C4—42no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—28no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23C31×#12112yes—
516C23C31×#12312yes—
517C23C31×AGL(1,ℤ/8)12yes—
518C23C31×#13712yes—
519C23C31×#13712yes—
520C23C22×D4×C3111yescharacteristic
521Q8—42no—
522Q8—42no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—84no—
558C2×C4—84no—
559C2×C4C31×#12112yes—
560C2×C4C31×#12312yes—
561C2×C4C22×D4×C3111yescharacteristic
562C2×C4C2×#41×C3112yes—
563C2×C4C2×#41×C3111yescharacteristic
564C2×C4C31×H(5,𝔽2)12yes—
565C8—24no—
566C8—42no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—416no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22—41no—
587C22—41no—
588C22C31×#46814yes—
589C22C2×C31×#12111yescharacteristic
590C22C2×C31×#12311yescharacteristic
591C22C31×#53411yescharacteristic
592C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
593C22C2×C31×#13711yescharacteristic
594C22C2×C31×#13711yescharacteristic
595C22C31×#57511yescharacteristic
596C4—22no—
597C4—22no—
598C4—41no—
599C4—81no—
600C4—81no—
601C4—82no—
602C4—81no—
603C4—82no—
604C4—81no—
605C4C31×#54612yes—
606C4C31×#55112yes—
607C2—24no—
608C2—42no—
609C2—42no—
610C2C31×#241211yescharacteristic
611C2C2×C31×#46811yescharacteristic
612C2C31×#305611yescharacteristic
613C2C31×#309211yescharacteristic
614C2C31×#309511yescharacteristic
615C2C31×#319411yescharacteristic
616C2C31×#319511yescharacteristic
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC25×C31D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C24×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C31×#483C2semidirect
5. C31×#2077C2semidirect
6. C31×#2036C2semidirect
7. C31×#1894C2semidirect
8. C31×#1872C2semidirect
9. C31×#1852C2semidirect
10. C31×#1823C2semidirect
11. C31×#1809C2semidirect
21. C2×C31×#107C22semidirect
22. C2×C31×#106C22semidirect
23. C2×C31×#103C22semidirect
24. C2×C31×#103C22semidirect
41. C31×#340C22semidirect
42. C31×#340C22semidirect
43. C31×#340C22semidirect
44. C31×#340C22semidirect
183. #24559C31direct (N × H)
259. C2×#483C2×C31semidirect
263. #2077C2×C31semidirect
264. #2036C2×C31semidirect
265. #1894C2×C31semidirect
266. #1872C2×C31semidirect
267. #1852C2×C31semidirect
268. #1823C2×C31semidirect
269. #1809C2×C31semidirect
319. C2×#107C22×C31semidirect
320. C2×#106C22×C31semidirect
321. C2×#103C22×C31semidirect
322. C2×#103C22×C31semidirect
339. #340C22×C31semidirect
340. #340C22×C31semidirect
341. #340C22×C31semidirect
342. #340C22×C31semidirect
410. C31#24559direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#4832
C2×C31×#4072
C31×#28482
C2×C31×#3802
C31×#20772
C31×#20362
C31×#18942
C31×#18722
C31×#18522
C31×#18232
C31×#18092
#2455931