Finite Group Databasebeta

← C31×#24751 #21481510

Subgroups — C31×#24751 #21481510

Order 7936.

1076 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#465C212yesmaximal
2C2×C31×#461C211yesmaximal, characteristic, complemented
3C22×C31×C4⋊C8C211yesmaximal, characteristic, complemented
4C31×#2857C211yesmaximal, characteristic, complemented
5C4×C31×#121C211yesmaximal, characteristic
6C31×#2066C212yesmaximal, complemented
7C31×#2055C212yesmaximal, complemented
8C31×#1981C211yesmaximal, characteristic, complemented
9C31×#1948C211yesmaximal, characteristic, complemented
10C31×#1892C212yesmaximal, complemented
11C31×#1802C211yesmaximal, characteristic, complemented
12C22×SD16×C31C2212yes—
13C23×C8×C31C2212yes—
14C2×C31×C4⋊Q8C2211yescharacteristic
15C2×C31×C22⋊Q8C2212yes—
16C2×C31×#121C2211yescharacteristic
17C2×C4×D4×C31—21no—
18C2×C4×D4×C31C2211yescharacteristic
19C22×C42×C31C2211yescharacteristic
20C31×#465—28no—
21C31×#461—24no—
22C2×C31×#106C2212yes—
23C2×C31×C4⋊C8—24no—
24C2×C31×C4⋊C8C2212yes—
25C2×C31×C4⋊C8C2211yescharacteristic
26C2×C31×C4⋊C8C2211yescharacteristic
27C2×C31×#103C2212yes—
28C2×C31×#103C2212yes—
29C2×C31×#103C2212yes—
30C2×C31×#102C2212yes—
31C31×#391C2212yes—
32C31×#390C2211yescharacteristic
33C31×#389—21no—
34C31×#389—21no—
35C31×#389C2211yescharacteristic
36C31×#388—21no—
37C31×#386—21no—
38C31×#384C2211yescharacteristic
39C31×#380—21no—
40C31×#380—21no—
41C31×#380C2211yescharacteristic
42C4×C4⋊C4×C31C2211yescharacteristic
43C4×C22⋊C4×C31—21no—
44C31×#340C2212yes—
45C31×#340C2212yes—
46C22×Q8×C31C2312yes—
47C22×D4×C31—21no—
48C22×D4×C31C2311yescharacteristic
49C23×C4×C31—21no—
50C23×C4×C31D411yescharacteristic
51C23×C4×C31D411yescharacteristic
52C23×C4×C31C2311yescharacteristic
53C2×SD16×C31—24no—
54C2×SD16×C31—24no—
55C2×SD16×C31—42no—
56C22×C8×C31—24no—
57C22×C8×C31—24no—
58C22×C8×C31—42no—
59C22×C8×C31C2312yes—
60C22×C8×C31C2312yes—
61C31×C4⋊Q8—22no—
62C31×C4⋊Q8—22no—
63C31×C22⋊Q8—24no—
64C31×C22⋊Q8—24no—
65C31×#121—22no—
66C31×#121D414yes—
67C4×D4×C31—22no—
68C4×D4×C31—22no—
69C4×D4×C31—44no—
70C2×C4⋊C4×C31—21no—
71C2×C4⋊C4×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×C31C2312yes—
77C2×C4⋊C4×C31C2311yescharacteristic
78C2×C4⋊C4×C31C2311yescharacteristic
79C2×C4⋊C4×C31C2311yescharacteristic
80C2×C22⋊C4×C31—21no—
81C2×C22⋊C4×C31—21no—
82C2×C22⋊C4×C31—22no—
83C2×C22⋊C4×C31—21no—
84C2×C22⋊C4×C31—21no—
85C2×C22⋊C4×C31—21no—
86C2×C42×C31—22no—
87C2×C42×C31—22no—
88C2×C42×C31—21no—
89C2×C42×C31D411yescharacteristic
90C2×C42×C31D411yescharacteristic
91C2×C42×C31C2311yescharacteristic
92C2×C42×C31C2311yescharacteristic
93C31×#106—24no—
94C31×C4⋊C8—24no—
95C31×C4⋊C8—22no—
96C31×C4⋊C8D414yescomplemented
97C31×#103—24no—
98C31×#103—24no—
99C31×#103—42no—
100C31×#102—24no—
101C31×#95—21no—
102C31×#95—21no—
103C31×#95—21no—
104C31×#95—22no—
105C31×#95—22no—
106C24×C31—21no—
107C24×C31C2×D411yescharacteristic
108C2×Q8×C31—22no—
109C2×Q8×C31—22no—
110C2×Q8×C31—22no—
111C2×Q8×C31—22no—
112C2×Q8×C31—22no—
113C2×Q8×C31—22no—
114C2×D4×C31—24no—
115C2×D4×C31—21no—
116C2×D4×C31—21no—
117C2×D4×C31—21no—
118C2×D4×C31—21no—
119C2×D4×C31—21no—
120C2×D4×C31—42no—
121C2×D4×C31—42no—
122C2×D4×C31C2×D412yes—
123C22×C4×C31—22no—
124C22×C4×C31—22no—
125C22×C4×C31—21no—
126C22×C4×C31—21no—
127C22×C4×C31—21no—
128C22×C4×C31—21no—
129C22×C4×C31—22no—
130C22×C4×C31—21no—
131C22×C4×C31—22no—
132C22×C4×C31—21no—
133C22×C4×C31—21no—
134C22×C4×C31—22no—
135C22×C4×C31—22no—
136C22×C4×C31—41no—
137C22×C4×C31—42no—
138C22×C4×C31—42no—
139C22×C4×C31—42no—
140C22×C4×C31SD1614yescomplemented
141C22×C4×C31C2×D411yescharacteristic
142C22×C4×C31C2×D411yescharacteristic
143C22×C4×C31C2×D411yescharacteristic
144C22×C4×C31C2×D412yes—
145C22×C4×C31C2×D411yescharacteristic
146C22×C4×C31C2×D411yescharacteristic
147C22×C4×C31#4112yes—
148C22×C4×C31C2411yescharacteristic
149SD16×C31—48no—
150C2×C8×C31—22no—
151C2×C8×C31—22no—
152C2×C8×C31—22no—
153C2×C8×C31—22no—
154C2×C8×C31—22no—
155C2×C8×C31—48no—
156C2×C8×C31C2×D414yes—
157C4⋊C4×C31—21no—
158C4⋊C4×C31—22no—
159C4⋊C4×C31—21no—
160C4⋊C4×C31—21no—
161C4⋊C4×C31—22no—
162C4⋊C4×C31—21no—
163C4⋊C4×C31—21no—
164C4⋊C4×C31—42no—
165C4⋊C4×C31—44no—
166C4⋊C4×C31—42no—
167C4⋊C4×C31—42no—
168C4⋊C4×C31—42no—
169C4⋊C4×C31—42no—
170C4⋊C4×C31C2×D412yes—
171C22⋊C4×C31—24no—
172C22⋊C4×C31—42no—
173C22⋊C4×C31—44no—
174C22⋊C4×C31—42no—
175C22⋊C4×C31—42no—
176C22⋊C4×C31—42no—
177C42×C31—22no—
178C42×C31—22no—
179C42×C31—21no—
180C42×C31—21no—
181C42×C31—42no—
182C42×C31C2×D412yes—
183C42×C31#4112yes—
184#24751C3111yesmaximal, characteristic, direct product factor, Sylow
185C23×C31—21no—
186C23×C31—21no—
187C23×C31—21no—
188C23×C31—21no—
189C23×C31—22no—
190C23×C31—41no—
191C23×C31—41no—
192C23×C31—41no—
193C23×C31—41no—
194C23×C31—41no—
195C23×C31—41no—
196C23×C31#12112yes—
197C23×C31C2×SD1612yes—
198C23×C31#13712yes—
199C23×C31C22×D411yescharacteristic
200Q8×C31—42no—
201Q8×C31—42no—
202Q8×C31—42no—
203Q8×C31—42no—
204D4×C31—22no—
205D4×C31—22no—
206D4×C31—48no—
207D4×C31—41no—
208D4×C31—41no—
209C2×C4×C31—21no—
210C2×C4×C31—21no—
211C2×C4×C31—21no—
212C2×C4×C31—21no—
213C2×C4×C31—21no—
214C2×C4×C31—21no—
215C2×C4×C31—22no—
216C2×C4×C31—22no—
217C2×C4×C31—22no—
218C2×C4×C31—22no—
219C2×C4×C31—28no—
220C2×C4×C31—21no—
221C2×C4×C31—22no—
222C2×C4×C31—21no—
223C2×C4×C31—21no—
224C2×C4×C31—44no—
225C2×C4×C31—41no—
226C2×C4×C31—41no—
227C2×C4×C31—42no—
228C2×C4×C31—41no—
229C2×C4×C31—42no—
230C2×C4×C31—42no—
231C2×C4×C31—41no—
232C2×C4×C31—42no—
233C2×C4×C31—41no—
234C2×C4×C31—41no—
235C2×C4×C31—42no—
236C2×C4×C31—41no—
237C2×C4×C31—41no—
238C2×C4×C31—42no—
239C2×C4×C31—84no—
240C2×C4×C31#12112yes—
241C2×C4×C31#12112yes—
242C2×C4×C31#12112yes—
243C2×C4×C31C2×SD1612yes—
244C2×C4×C31C2×SD1612yes—
245C2×C4×C31#13512yes—
246C2×C4×C31C22×D411yescharacteristic
247C2×C4×C31C22×D411yescharacteristic
248C2×C4×C31C2×#4111yescharacteristic
249C2×C4×C31C2×#4111yescharacteristic
250C2×C4×C31H(5,𝔽2)11yescharacteristic
251C2×C4×C312−1+411yescharacteristic
252C8×C31—24no—
253C8×C31—42no—
254C2×#465C2×C3112yes—
255C2×#461C2×C3111yescharacteristic, complemented
256C22×C4⋊C8C2×C3111yescharacteristic, complemented
257#2857C2×C3111yescharacteristic, complemented
258C4×#121C2×C3111yescharacteristic
259#2066C2×C3112yescomplemented
260#2055C2×C3112yescomplemented
261#1981C2×C3111yescharacteristic, complemented
262#1948C2×C3111yescharacteristic, complemented
263#1892C2×C3112yescomplemented
264#1802C2×C3111yescharacteristic, complemented
265C22×C31—22no—
266C22×C31—22no—
267C22×C31—22no—
268C22×C31—22no—
269C22×C31—22no—
270C22×C31—22no—
271C22×C31—42no—
272C22×C31—42no—
273C22×C31—42no—
274C22×C31—81no—
275C22×C31—81no—
276C22×C31—81no—
277C22×C31—81no—
278C22×C31#46114yescomplemented
279C22×C31C2×#12111yescharacteristic
280C22×C31C2×#12111yescharacteristic
281C22×C31#53711yescharacteristic
282C22×C31C22×SD1611yescharacteristic
283C22×C31C2×#13511yescharacteristic
284C22×C31C2×#13711yescharacteristic
285C22×C31#57511yescharacteristic
286C4×C31—22no—
287C4×C31—22no—
288C4×C31—22no—
289C4×C31—42no—
290C4×C31—41no—
291C4×C31—82no—
292C4×C31—81no—
293C4×C31—82no—
294C4×C31—81no—
295C4×C31#46514yescomplemented
296C4×C31D4211yescharacteristic
297C4×C31#54711yescharacteristic
298C4×C31#54811yescharacteristic
299C4×C31#55311yescharacteristic
300C22×SD16C22×C3112yes—
301C23×C8C22×C3112yes—
302C2×C4⋊Q8C22×C3111yescharacteristic
303C2×C22⋊Q8C22×C3112yes—
304C2×#121C22×C3111yescharacteristic
305C2×C4×D4—21no—
306C2×C4×D4C22×C3111yescharacteristic
307C22×C42C22×C3111yescharacteristic
308#465—28no—
309#461—24no—
310C2×#106C22×C3112yes—
311C2×C4⋊C8—24no—
312C2×C4⋊C8C22×C3112yes—
313C2×C4⋊C8C22×C3111yescharacteristic
314C2×C4⋊C8C22×C3111yescharacteristic
315C2×#103C22×C3112yes—
316C2×#103C22×C3112yes—
317C2×#103C22×C3112yes—
318C2×#102C22×C3112yes—
319#391C22×C3112yes—
320#390C22×C3111yescharacteristic
321#389—21no—
322#389—21no—
323#389C22×C3111yescharacteristic
324#388—21no—
325#386—21no—
326#384C22×C3111yescharacteristic
327#380—21no—
328#380—21no—
329#380C22×C3111yescharacteristic
330C4×C4⋊C4C22×C3111yescharacteristic
331C4×C22⋊C4—21no—
332#340C22×C3112yes—
333#340C22×C3112yes—
334C2×C31—24no—
335C2×C31—82no—
336C2×C31#241611yescharacteristic
337C2×C31C2×#46111yescharacteristic
338C2×C31#303511yescharacteristic
339C2×C31C2×#46511yescharacteristic
340C2×C31#305611yescharacteristic
341C2×C31#324611yescharacteristic
342C2×C31#324811yescharacteristic
343C22×Q8C23×C3112yes—
344C22×D4—21no—
345C22×D4C23×C3111yescharacteristic
346C23×C4—21no—
347C23×C4D4×C3111yescharacteristic
348C23×C4D4×C3111yescharacteristic
349C23×C4C23×C3111yescharacteristic
350C2×SD16—24no—
351C2×SD16—24no—
352C2×SD16—42no—
353C22×C8—24no—
354C22×C8—24no—
355C22×C8—42no—
356C22×C8C23×C3112yes—
357C22×C8C23×C3112yes—
358C4⋊Q8—22no—
359C4⋊Q8—22no—
360C22⋊Q8—24no—
361C22⋊Q8—24no—
362#121—22no—
363#121D4×C3114yes—
364C4×D4—22no—
365C4×D4—22no—
366C4×D4—44no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—22no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4C23×C3111yescharacteristic
374C2×C4⋊C4C23×C3112yes—
375C2×C4⋊C4C23×C3111yescharacteristic
376C2×C4⋊C4C23×C3111yescharacteristic
377C2×C22⋊C4—21no—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C42—22no—
384C2×C42—22no—
385C2×C42—21no—
386C2×C42D4×C3111yescharacteristic
387C2×C42D4×C3111yescharacteristic
388C2×C42C23×C3111yescharacteristic
389C2×C42C23×C3111yescharacteristic
390#106—24no—
391C4⋊C8—24no—
392C4⋊C8—22no—
393C4⋊C8D4×C3114yescomplemented
394#103—24no—
395#103—24no—
396#103—42no—
397#102—24no—
398#95—21no—
399#95—21no—
400#95—21no—
401#95—22no—
402#95—22no—
403C31#2475111yescharacteristic, direct product factor, Sylow
404C24—21no—
405C24C2×D4×C3111yescharacteristic
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×D4—24no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4C2×D4×C3112yes—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—41no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4SD16×C3114yescomplemented
439C22×C4C2×D4×C3111yescharacteristic
440C22×C4C2×D4×C3111yescharacteristic
441C22×C4C2×D4×C3111yescharacteristic
442C22×C4C2×D4×C3112yes—
443C22×C4C2×D4×C3111yescharacteristic
444C22×C4C2×D4×C3111yescharacteristic
445C22×C4#41×C3112yes—
446C22×C4C24×C3111yescharacteristic
447SD16—48no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—48no—
454C2×C8C2×D4×C3114yes—
455C4⋊C4—21no—
456C4⋊C4—22no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—22no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—42no—
463C4⋊C4—44no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4C2×D4×C3112yes—
469C22⋊C4—24no—
470C22⋊C4—42no—
471C22⋊C4—44no—
472C22⋊C4—42no—
473C22⋊C4—42no—
474C22⋊C4—42no—
475C42—22no—
476C42—22no—
477C42—21no—
478C42—21no—
479C42—42no—
480C42C2×D4×C3112yes—
481C42#41×C3112yes—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—22no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23C31×#12112yes—
494C23C2×SD16×C3112yes—
495C23C31×#13712yes—
496C23C22×D4×C3111yescharacteristic
497Q8—42no—
498Q8—42no—
499Q8—42no—
500Q8—42no—
501D4—22no—
502D4—22no—
503D4—48no—
504D4—41no—
505D4—41no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—28no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—44no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—41no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—84no—
537C2×C4C31×#12112yes—
538C2×C4C31×#12112yes—
539C2×C4C31×#12112yes—
540C2×C4C2×SD16×C3112yes—
541C2×C4C2×SD16×C3112yes—
542C2×C4C31×#13512yes—
543C2×C4C22×D4×C3111yescharacteristic
544C2×C4C22×D4×C3111yescharacteristic
545C2×C4C2×#41×C3111yescharacteristic
546C2×C4C2×#41×C3111yescharacteristic
547C2×C4C31×H(5,𝔽2)11yescharacteristic
548C2×C4C31×2−1+411yescharacteristic
549C8—24no—
550C8—42no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—81no—
561C22—81no—
562C22—81no—
563C22—81no—
564C22C31×#46114yescomplemented
565C22C2×C31×#12111yescharacteristic
566C22C2×C31×#12111yescharacteristic
567C22C31×#53711yescharacteristic
568C22C22×SD16×C3111yescharacteristic
569C22C2×C31×#13511yescharacteristic
570C22C2×C31×#13711yescharacteristic
571C22C31×#57511yescharacteristic
572C4—22no—
573C4—22no—
574C4—22no—
575C4—42no—
576C4—41no—
577C4—82no—
578C4—81no—
579C4—82no—
580C4—81no—
581C4C31×#46514yescomplemented
582C4D42×C3111yescharacteristic
583C4C31×#54711yescharacteristic
584C4C31×#54811yescharacteristic
585C4C31×#55311yescharacteristic
586C2—24no—
587C2—82no—
588C2C31×#241611yescharacteristic
589C2C2×C31×#46111yescharacteristic
590C2C31×#303511yescharacteristic
591C2C2×C31×#46511yescharacteristic
592C2C31×#305611yescharacteristic
593C2C31×#324611yescharacteristic
594C2C31×#324811yescharacteristic
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC2×C31×#121C22
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×#461C2semidirect
3. C22×C31×C4⋊C8C2semidirect
4. C31×#2857C2semidirect
6. C31×#2066C2semidirect
7. C31×#2055C2semidirect
8. C31×#1981C2semidirect
9. C31×#1948C2semidirect
10. C31×#1892C2semidirect
11. C31×#1802C2semidirect
96. C31×C4⋊C8D4semidirect
140. C22×C4×C31SD16semidirect
184. #24751C31direct (N × H)
255. C2×#461C2×C31semidirect
256. C22×C4⋊C8C2×C31semidirect
257. #2857C2×C31semidirect
259. #2066C2×C31semidirect
260. #2055C2×C31semidirect
261. #1981C2×C31semidirect
262. #1948C2×C31semidirect
263. #1892C2×C31semidirect
264. #1802C2×C31semidirect
278. C22×C31#461semidirect
295. C4×C31#465semidirect
393. C4⋊C8D4×C31semidirect
403. C31#24751direct (N × H)
438. C22×C4SD16×C31semidirect
564. C22C31×#461semidirect
581. 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×#4612
C22×C31×C4⋊C82
C31×#28572
C4×C31×#1212
C31×#20662
C31×#20552
C31×#19812
C31×#19482
C31×#18922
C31×#18022
#2475131