Finite Group Databasebeta

← C31×#24753 #21481512

Subgroups — C31×#24753 #21481512

Order 7936.

1132 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#465C212yesmaximal, complemented
2C2×C31×#460C211yesmaximal, characteristic, complemented
3C22×C31×C4⋊C8C211yesmaximal, characteristic, complemented
4C31×#2856C211yesmaximal, characteristic
5C4×C31×C22⋊Q8C211yesmaximal, characteristic, complemented
6C31×#2065C212yesmaximal, complemented
7C31×#2055C212yesmaximal, complemented
8C31×#1982C211yesmaximal, characteristic, complemented
9C31×#1948C211yesmaximal, characteristic, complemented
10C31×#1893C212yesmaximal, complemented
11C31×#1803C211yesmaximal, characteristic, complemented
12C22×SD16×C31C2212yes—
13C23×C8×C31C2212yes—
14C2×C31×#127C2211yescharacteristic
15C2×C31×C22⋊Q8C2211yescharacteristic
16C2×C31×#121—21no—
17C2×C31×#121—21no—
18C2×C31×#121C2212yes—
19C2×C4×Q8×C31C2211yescharacteristic
20C22×C42×C31C2211yescharacteristic
21C31×#465—28no—
22C31×#460—24no—
23C2×C31×#106C2212yes—
24C2×C31×C4⋊C8—24no—
25C2×C31×C4⋊C8C2212yes—
26C2×C31×C4⋊C8C2211yescharacteristic
27C2×C31×C4⋊C8C2211yescharacteristic
28C2×C31×#103C2212yes—
29C2×C31×#102C2212yes—
30C2×C31×#102C2212yes—
31C2×C31×#102C2212yes—
32C31×#391C2211yescharacteristic
33C31×#390—21no—
34C31×#390—21no—
35C31×#390C2212yes—
36C31×#389C2211yescharacteristic
37C31×#387—21no—
38C31×#385—21no—
39C31×#384C2211yescharacteristic
40C31×#380C2211yescharacteristic
41C4×C4⋊C4×C31—21no—
42C4×C4⋊C4×C31C2211yescharacteristic
43C4×C22⋊C4×C31—21no—
44C31×#340C2212yes—
45C31×#340C2212yes—
46C22×Q8×C31C2311yescharacteristic
47C22×D4×C31—22no—
48C22×D4×C31—21no—
49C22×D4×C31—21no—
50C22×D4×C31C2312yes—
51C23×C4×C31D411yescharacteristic
52C23×C4×C31D411yescharacteristic
53C23×C4×C31C2311yescharacteristic
54C2×SD16×C31—24no—
55C2×SD16×C31—24no—
56C2×SD16×C31—42no—
57C22×C8×C31—24no—
58C22×C8×C31—24no—
59C22×C8×C31—42no—
60C22×C8×C31C2312yes—
61C22×C8×C31C2312yes—
62C31×#127—22no—
63C31×#127—22no—
64C31×C22⋊Q8—22no—
65C31×C22⋊Q8D414yescomplemented
66C31×#121—24no—
67C31×#121—24no—
68C31×#121—44no—
69C31×#121—44no—
70C4×Q8×C31—22no—
71C4×Q8×C31—22no—
72C2×C4⋊C4×C31—21no—
73C2×C4⋊C4×C31—21no—
74C2×C4⋊C4×C31—21no—
75C2×C4⋊C4×C31—21no—
76C2×C4⋊C4×C31—21no—
77C2×C4⋊C4×C31C2311yescharacteristic
78C2×C4⋊C4×C31C2311yescharacteristic
79C2×C4⋊C4×C31C2312yes—
80C2×C4⋊C4×C31C2311yescharacteristic
81C2×C22⋊C4×C31—21no—
82C2×C22⋊C4×C31—21no—
83C2×C22⋊C4×C31—22no—
84C2×C22⋊C4×C31—22no—
85C2×C22⋊C4×C31—22no—
86C2×C42×C31—22no—
87C2×C42×C31—22no—
88C2×C42×C31—21no—
89C2×C42×C31—21no—
90C2×C42×C31D411yescharacteristic
91C2×C42×C31D411yescharacteristic
92C2×C42×C31C2311yescharacteristic
93C2×C42×C31C2311yescharacteristic
94C31×#106—24no—
95C31×C4⋊C8—24no—
96C31×C4⋊C8—22no—
97C31×C4⋊C8D414yescomplemented
98C31×#103—24no—
99C31×#102—24no—
100C31×#102—24no—
101C31×#102—42no—
102C31×#95—21no—
103C31×#95—21no—
104C31×#95—21no—
105C31×#95—21no—
106C31×#95—21no—
107C24×C31—22no—
108C24×C31C2×D411yescharacteristic
109C2×Q8×C31—21no—
110C2×Q8×C31—21no—
111C2×Q8×C31—21no—
112C2×Q8×C31—21no—
113C2×Q8×C31—21no—
114C2×Q8×C31C2×D412yes—
115C2×D4×C31—22no—
116C2×D4×C31—22no—
117C2×D4×C31—22no—
118C2×D4×C31—22no—
119C2×D4×C31—22no—
120C2×D4×C31—22no—
121C2×D4×C31—42no—
122C2×D4×C31—42no—
123C2×D4×C31—44no—
124C2×D4×C31—44no—
125C2×D4×C31—44no—
126C2×D4×C31—42no—
127C2×D4×C31—42no—
128C2×D4×C31—42no—
129C2×D4×C31—42no—
130C22×C4×C31—22no—
131C22×C4×C31—22no—
132C22×C4×C31—21no—
133C22×C4×C31—21no—
134C22×C4×C31—21no—
135C22×C4×C31—21no—
136C22×C4×C31—22no—
137C22×C4×C31—21no—
138C22×C4×C31—21no—
139C22×C4×C31—22no—
140C22×C4×C31—21no—
141C22×C4×C31—21no—
142C22×C4×C31—22no—
143C22×C4×C31—41no—
144C22×C4×C31SD1614yescomplemented
145C22×C4×C31C2×D411yescharacteristic
146C22×C4×C31C2×D411yescharacteristic
147C22×C4×C31C2×D411yescharacteristic
148C22×C4×C31C2×D412yes—
149C22×C4×C31C2×D411yescharacteristic
150C22×C4×C31C2×D411yescharacteristic
151C22×C4×C31#4112yes—
152C22×C4×C31C2411yescharacteristic
153SD16×C31—48no—
154C2×C8×C31—22no—
155C2×C8×C31—22no—
156C2×C8×C31—22no—
157C2×C8×C31—22no—
158C2×C8×C31—22no—
159C2×C8×C31—48no—
160C2×C8×C31C2×D414yes—
161C4⋊C4×C31—24no—
162C4⋊C4×C31—21no—
163C4⋊C4×C31—21no—
164C4⋊C4×C31—22no—
165C4⋊C4×C31—21no—
166C4⋊C4×C31—22no—
167C4⋊C4×C31—21no—
168C4⋊C4×C31—21no—
169C4⋊C4×C31—42no—
170C4⋊C4×C31—42no—
171C4⋊C4×C31—42no—
172C4⋊C4×C31—42no—
173C4⋊C4×C31C2×D412yes—
174C22⋊C4×C31—24no—
175C22⋊C4×C31—44no—
176C22⋊C4×C31—42no—
177C22⋊C4×C31—44no—
178C22⋊C4×C31—44no—
179C42×C31—22no—
180C42×C31—22no—
181C42×C31—21no—
182C42×C31—21no—
183C42×C31—42no—
184C42×C31—42no—
185C42×C31C2×D412yes—
186C42×C31#4112yes—
187#24753C3111yesmaximal, characteristic, direct product factor, Sylow
188C23×C31—21no—
189C23×C31—21no—
190C23×C31—21no—
191C23×C31—21no—
192C23×C31—42no—
193C23×C31—42no—
194C23×C31—42no—
195C23×C31—42no—
196C23×C31—42no—
197C23×C31—42no—
198C23×C31—42no—
199C23×C31#12112yes—
200C23×C31C2×SD1612yes—
201C23×C31AGL(1,ℤ/8)12yes—
202C23×C31C22×D411yescharacteristic
203Q8×C31—22no—
204Q8×C31—22no—
205Q8×C31—41no—
206Q8×C31—41no—
207D4×C31—42no—
208D4×C31—42no—
209D4×C31—42no—
210D4×C31—42no—
211D4×C31—84no—
212D4×C31—88no—
213D4×C31—84no—
214C2×C4×C31—21no—
215C2×C4×C31—21no—
216C2×C4×C31—21no—
217C2×C4×C31—21no—
218C2×C4×C31—21no—
219C2×C4×C31—21no—
220C2×C4×C31—22no—
221C2×C4×C31—22no—
222C2×C4×C31—22no—
223C2×C4×C31—22no—
224C2×C4×C31—28no—
225C2×C4×C31—21no—
226C2×C4×C31—22no—
227C2×C4×C31—22no—
228C2×C4×C31—21no—
229C2×C4×C31—21no—
230C2×C4×C31—44no—
231C2×C4×C31—41no—
232C2×C4×C31—41no—
233C2×C4×C31—41no—
234C2×C4×C31—41no—
235C2×C4×C31—41no—
236C2×C4×C31—42no—
237C2×C4×C31—41no—
238C2×C4×C31—42no—
239C2×C4×C31—41no—
240C2×C4×C31—42no—
241C2×C4×C31—41no—
242C2×C4×C31—41no—
243C2×C4×C31—41no—
244C2×C4×C31#12112yes—
245C2×C4×C31#12112yes—
246C2×C4×C31#12112yes—
247C2×C4×C31C2×SD1612yes—
248C2×C4×C31C2×SD1612yes—
249C2×C4×C31#13512yes—
250C2×C4×C31C22×D411yescharacteristic
251C2×C4×C31C22×D411yescharacteristic
252C2×C4×C31C2×#4111yescharacteristic
253C2×C4×C31C2×#4111yescharacteristic
254C2×C4×C31H(5,𝔽2)11yescharacteristic
255C2×C4×C312−1+411yescharacteristic
256C8×C31—24no—
257C8×C31—42no—
258C2×#465C2×C3112yescomplemented
259C2×#460C2×C3111yescharacteristic, complemented
260C22×C4⋊C8C2×C3111yescharacteristic, complemented
261#2856C2×C3111yescharacteristic
262C4×C22⋊Q8C2×C3111yescharacteristic, complemented
263#2065C2×C3112yescomplemented
264#2055C2×C3112yescomplemented
265#1982C2×C3111yescharacteristic, complemented
266#1948C2×C3111yescharacteristic, complemented
267#1893C2×C3112yescomplemented
268#1803C2×C3111yescharacteristic, complemented
269C22×C31—22no—
270C22×C31—22no—
271C22×C31—22no—
272C22×C31—22no—
273C22×C31—22no—
274C22×C31—22no—
275C22×C31—82no—
276C22×C31—82no—
277C22×C31—82no—
278C22×C31—82no—
279C22×C31—82no—
280C22×C31—82no—
281C22×C31—82no—
282C22×C31#46014yescomplemented
283C22×C31C2×#12111yescharacteristic
284C22×C31C2×#12111yescharacteristic
285C22×C31#53711yescharacteristic
286C22×C31C22×SD1611yescharacteristic
287C22×C31C2×#13511yescharacteristic
288C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
289C22×C31#57511yescharacteristic
290C4×C31—22no—
291C4×C31—22no—
292C4×C31—22no—
293C4×C31—42no—
294C4×C31—42no—
295C4×C31—41no—
296C4×C31—81no—
297C4×C31—82no—
298C4×C31—81no—
299C4×C31#46514yescomplemented
300C4×C31D4211yescharacteristic
301C4×C31#54711yescharacteristic
302C4×C31#54811yescharacteristic
303C4×C31#55311yescharacteristic
304C22×SD16C22×C3112yes—
305C23×C8C22×C3112yes—
306C2×#127C22×C3111yescharacteristic
307C2×C22⋊Q8C22×C3111yescharacteristic
308C2×#121—21no—
309C2×#121—21no—
310C2×#121C22×C3112yes—
311C2×C4×Q8C22×C3111yescharacteristic
312C22×C42C22×C3111yescharacteristic
313#465—28no—
314#460—24no—
315C2×#106C22×C3112yes—
316C2×C4⋊C8—24no—
317C2×C4⋊C8C22×C3112yes—
318C2×C4⋊C8C22×C3111yescharacteristic
319C2×C4⋊C8C22×C3111yescharacteristic
320C2×#103C22×C3112yes—
321C2×#102C22×C3112yes—
322C2×#102C22×C3112yes—
323C2×#102C22×C3112yes—
324#391C22×C3111yescharacteristic
325#390—21no—
326#390—21no—
327#390C22×C3112yes—
328#389C22×C3111yescharacteristic
329#387—21no—
330#385—21no—
331#384C22×C3111yescharacteristic
332#380C22×C3111yescharacteristic
333C4×C4⋊C4—21no—
334C4×C4⋊C4C22×C3111yescharacteristic
335C4×C22⋊C4—21no—
336#340C22×C3112yes—
337#340C22×C3112yes—
338C2×C31—24no—
339C2×C31—162no—
340C2×C31#241611yescharacteristic
341C2×C31C2×#46011yescharacteristic
342C2×C31#303511yescharacteristic
343C2×C31C2×#46511yescharacteristic
344C2×C31#305511yescharacteristic
345C2×C31#324611yescharacteristic
346C2×C31#324711yescharacteristic
347C22×Q8C23×C3111yescharacteristic
348C22×D4—22no—
349C22×D4—21no—
350C22×D4—21no—
351C22×D4C23×C3112yes—
352C23×C4D4×C3111yescharacteristic
353C23×C4D4×C3111yescharacteristic
354C23×C4C23×C3111yescharacteristic
355C2×SD16—24no—
356C2×SD16—24no—
357C2×SD16—42no—
358C22×C8—24no—
359C22×C8—24no—
360C22×C8—42no—
361C22×C8C23×C3112yes—
362C22×C8C23×C3112yes—
363#127—22no—
364#127—22no—
365C22⋊Q8—22no—
366C22⋊Q8D4×C3114yescomplemented
367#121—24no—
368#121—24no—
369#121—44no—
370#121—44no—
371C4×Q8—22no—
372C4×Q8—22no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4C23×C3111yescharacteristic
379C2×C4⋊C4C23×C3112yes—
380C2×C4⋊C4C23×C3111yescharacteristic
381C2×C4⋊C4C23×C3111yescharacteristic
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—22no—
385C2×C22⋊C4—22no—
386C2×C22⋊C4—22no—
387C2×C42—22no—
388C2×C42—22no—
389C2×C42—21no—
390C2×C42—21no—
391C2×C42D4×C3111yescharacteristic
392C2×C42D4×C3111yescharacteristic
393C2×C42C23×C3111yescharacteristic
394C2×C42C23×C3111yescharacteristic
395#106—24no—
396C4⋊C8—24no—
397C4⋊C8—22no—
398C4⋊C8D4×C3114yescomplemented
399#103—24no—
400#102—24no—
401#102—24no—
402#102—42no—
403#95—21no—
404#95—21no—
405#95—21no—
406#95—21no—
407#95—21no—
408C31#2475311yescharacteristic, direct product factor, Sylow
409C24—22no—
410C24C2×D4×C3111yescharacteristic
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8C2×D4×C3112yes—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—42no—
424C2×D4—42no—
425C2×D4—44no—
426C2×D4—44no—
427C2×D4—44no—
428C2×D4—42no—
429C2×D4—42no—
430C2×D4—42no—
431C2×D4—42no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—41no—
446C22×C4SD16×C3114yescomplemented
447C22×C4C2×D4×C3111yescharacteristic
448C22×C4C2×D4×C3111yescharacteristic
449C22×C4C2×D4×C3111yescharacteristic
450C22×C4C2×D4×C3112yes—
451C22×C4C2×D4×C3111yescharacteristic
452C22×C4C2×D4×C3111yescharacteristic
453C22×C4#41×C3112yes—
454C22×C4C24×C3111yescharacteristic
455SD16—48no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—48no—
462C2×C8C2×D4×C3114yes—
463C4⋊C4—24no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—22no—
467C4⋊C4—21no—
468C4⋊C4—22no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4C2×D4×C3112yes—
476C22⋊C4—24no—
477C22⋊C4—44no—
478C22⋊C4—42no—
479C22⋊C4—44no—
480C22⋊C4—44no—
481C42—22no—
482C42—22no—
483C42—21no—
484C42—21no—
485C42—42no—
486C42—42no—
487C42C2×D4×C3112yes—
488C42#41×C3112yes—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23C31×#12112yes—
501C23C2×SD16×C3112yes—
502C23C31×AGL(1,ℤ/8)12yes—
503C23C22×D4×C3111yescharacteristic
504Q8—22no—
505Q8—22no—
506Q8—41no—
507Q8—41no—
508D4—42no—
509D4—42no—
510D4—42no—
511D4—42no—
512D4—84no—
513D4—88no—
514D4—84no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—28no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—44no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4C31×#12112yes—
546C2×C4C31×#12112yes—
547C2×C4C31×#12112yes—
548C2×C4C2×SD16×C3112yes—
549C2×C4C2×SD16×C3112yes—
550C2×C4C31×#13512yes—
551C2×C4C22×D4×C3111yescharacteristic
552C2×C4C22×D4×C3111yescharacteristic
553C2×C4C2×#41×C3111yescharacteristic
554C2×C4C2×#41×C3111yescharacteristic
555C2×C4C31×H(5,𝔽2)11yescharacteristic
556C2×C4C31×2−1+411yescharacteristic
557C8—24no—
558C8—42no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—82no—
566C22—82no—
567C22—82no—
568C22—82no—
569C22—82no—
570C22—82no—
571C22—82no—
572C22C31×#46014yescomplemented
573C22C2×C31×#12111yescharacteristic
574C22C2×C31×#12111yescharacteristic
575C22C31×#53711yescharacteristic
576C22C22×SD16×C3111yescharacteristic
577C22C2×C31×#13511yescharacteristic
578C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
579C22C31×#57511yescharacteristic
580C4—22no—
581C4—22no—
582C4—22no—
583C4—42no—
584C4—42no—
585C4—41no—
586C4—81no—
587C4—82no—
588C4—81no—
589C4C31×#46514yescomplemented
590C4D42×C3111yescharacteristic
591C4C31×#54711yescharacteristic
592C4C31×#54811yescharacteristic
593C4C31×#55311yescharacteristic
594C2—24no—
595C2—162no—
596C2C31×#241611yescharacteristic
597C2C2×C31×#46011yescharacteristic
598C2C31×#303511yescharacteristic
599C2C2×C31×#46511yescharacteristic
600C2C31×#305511yescharacteristic
601C2C31×#324611yescharacteristic
602C2C31×#324711yescharacteristic
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#2856C2
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
1. C2×C31×#465C2semidirect
2. C2×C31×#460C2semidirect
3. C22×C31×C4⋊C8C2semidirect
5. C4×C31×C22⋊Q8C2semidirect
6. C31×#2065C2semidirect
7. C31×#2055C2semidirect
8. C31×#1982C2semidirect
9. C31×#1948C2semidirect
10. C31×#1893C2semidirect
11. C31×#1803C2semidirect
65. C31×C22⋊Q8D4semidirect
97. C31×C4⋊C8D4semidirect
144. C22×C4×C31SD16semidirect
187. #24753C31direct (N × H)
258. C2×#465C2×C31semidirect
259. C2×#460C2×C31semidirect
260. C22×C4⋊C8C2×C31semidirect
262. C4×C22⋊Q8C2×C31semidirect
263. #2065C2×C31semidirect
264. #2055C2×C31semidirect
265. #1982C2×C31semidirect
266. #1948C2×C31semidirect
267. #1893C2×C31semidirect
268. #1803C2×C31semidirect
282. C22×C31#460semidirect
299. C4×C31#465semidirect
366. C22⋊Q8D4×C31semidirect
398. C4⋊C8D4×C31semidirect
408. C31#24753direct (N × H)
446. C22×C4SD16×C31semidirect
572. C22C31×#460semidirect
589. C4C31×#465semidirect

Maximal subgroups

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

subgroupindex
C2×C31×#4652
C2×C31×#4602
C22×C31×C4⋊C82
C31×#28562
C4×C31×C22⋊Q82
C31×#20652
C31×#20552
C31×#19822
C31×#19482
C31×#18932
C31×#18032
#2475331