Finite Group Databasebeta

← C31×#24112 #21480871

Subgroups — C31×#24112 #21480871

Order 7936.

1518 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3490C211yesmaximal, characteristic
2C31×#2849C212yesmaximal, complemented
3C2×C31×#385C211yesmaximal, characteristic, complemented
4C31×#2062C212yesmaximal, complemented
5C31×#2044C212yesmaximal, complemented
6C31×#1898C212yesmaximal, complemented
7C31×#1867C214yesmaximal, complemented
8C31×#1819C211yesmaximal, characteristic, complemented
9C24×C4×C31C2211yescharacteristic
10C31×#551—22no—
11C31×#551—22no—
12C31×#546—22no—
13C31×#546—22no—
14C31×#543C2211yescharacteristic
15C31×#538—21no—
16C31×#538—21no—
17C31×#534C2211yescharacteristic
18C31×#525—21no—
19C31×#525—21no—
20C2×C31×#123—22no—
21C2×C31×#120C2212yes—
22C31×#518C2212yes—
23C22×C4⋊C4×C31—21no—
24C22×C22⋊C4×C31C2211yescharacteristic
25C22×C22⋊C4×C31C2211yescharacteristic
26C2×C31×#99C2214yes—
27C31×#386—22no—
28C31×#386C2214yes—
29C31×#385—22no—
30C31×#385—21no—
31C31×#385—21no—
32C31×#385C2214yes—
33C31×#385C2212yes—
34C31×#385C2212yes—
35C31×#381—22no—
36C31×#379C2212yes—
37C2×C31×#95—21no—
38C31×#342C2212yes—
39C31×#342C2214yes—
40C31×#342C2212yes—
41C25×C31C2311yescharacteristic
42C2×#41×C31—21no—
43C2×#41×C31—21no—
44C22×D4×C31D412yes—
45C22×D4×C31D412yes—
46C22×D4×C31C2312yes—
47C23×C4×C31—21no—
48C23×C4×C31—21no—
49C23×C4×C31—22no—
50C23×C4×C31—21no—
51C23×C4×C31—22no—
52C23×C4×C31—44no—
53C23×C4×C31D411yescharacteristic
54C23×C4×C31D411yescharacteristic
55C23×C4×C31C2312yes—
56C31×#126—22no—
57C31×#126—22no—
58C31×#124—22no—
59C31×#124—22no—
60C31×#123—22no—
61C31×#123—22no—
62C31×#123—22no—
63C31×#123—22no—
64C31×#123—22no—
65C31×#123—48no—
66C31×C22⋊Q8—21no—
67C31×C22⋊Q8—21no—
68C31×C22⋊Q8—21no—
69C31×C22⋊Q8—21no—
70C31×C22⋊Q8—22no—
71C31×C22⋊Q8—22no—
72C31×#121—21no—
73C31×#121—21no—
74C31×#121—22no—
75C31×#121—22no—
76C31×#121—21no—
77C31×#121—21no—
78C31×#120—22no—
79C31×#120—22no—
80C31×#120—22no—
81C31×#120—22no—
82C31×#120—22no—
83C4×D4×C31—22no—
84C4×D4×C31—22no—
85C4×D4×C31—22no—
86C4×D4×C31—22no—
87C4×D4×C31—22no—
88C4×D4×C31—22no—
89C31×#117—21no—
90C31×#117—21no—
91C31×#117—22no—
92C2×C4⋊C4×C31—22no—
93C2×C4⋊C4×C31—22no—
94C2×C4⋊C4×C31—24no—
95C2×C4⋊C4×C31—44no—
96C2×C22⋊C4×C31—22no—
97C2×C22⋊C4×C31—22no—
98C2×C22⋊C4×C31—21no—
99C2×C22⋊C4×C31—22no—
100C2×C22⋊C4×C31—24no—
101C2×C22⋊C4×C31—22no—
102C2×C22⋊C4×C31—24no—
103C2×C22⋊C4×C31—22no—
104C2×C22⋊C4×C31D411yescharacteristic
105C2×C22⋊C4×C31D411yescharacteristic
106C2×C22⋊C4×C31D412yes—
107C2×C22⋊C4×C31D412yes—
108C2×C22⋊C4×C31C2314yes—
109C2×C22⋊C4×C31C2311yescharacteristic
110C2×C22⋊C4×C31C2311yescharacteristic
111C2×C22⋊C4×C31C2312yes—
112C2×C22⋊C4×C31C2312yes—
113C31×#99—28no—
114C31×#95—22no—
115C31×#95—22no—
116C31×#95—24no—
117C31×#95—24no—
118C24×C31—22no—
119C24×C31—22no—
120C24×C31—42no—
121C24×C31—42no—
122C24×C31C2×D411yescharacteristic
123C24×C31C2×D412yes—
124C24×C31C2×D412yes—
125C24×C31C2×D412yes—
126C24×C31C2×D411yescharacteristic
127C24×C31C2411yescharacteristic
128#41×C31—82no—
129#41×C31—82no—
130C2×Q8×C31—21no—
131C2×Q8×C31—21no—
132C2×D4×C31—22no—
133C2×D4×C31—22no—
134C2×D4×C31—22no—
135C2×D4×C31—22no—
136C2×D4×C31—24no—
137C2×D4×C31—24no—
138C2×D4×C31—21no—
139C2×D4×C31—21no—
140C2×D4×C31—22no—
141C2×D4×C31—22no—
142C2×D4×C31—44no—
143C2×D4×C31—44no—
144C22×C4×C31—24no—
145C22×C4×C31—24no—
146C22×C4×C31—28no—
147C22×C4×C31—21no—
148C22×C4×C31—21no—
149C22×C4×C31—22no—
150C22×C4×C31—24no—
151C22×C4×C31—22no—
152C22×C4×C31—22no—
153C22×C4×C31—22no—
154C22×C4×C31—21no—
155C22×C4×C31—21no—
156C22×C4×C31—24no—
157C22×C4×C31—44no—
158C22×C4×C31—48no—
159C22×C4×C31—44no—
160C22×C4×C31—42no—
161C22×C4×C31—42no—
162C22×C4×C31—42no—
163C22×C4×C31—44no—
164C22×C4×C31—42no—
165C22×C4×C31—41no—
166C22×C4×C31—41no—
167C22×C4×C31—42no—
168C22×C4×C31—42no—
169C22×C4×C31C2×D412yes—
170C22×C4×C31C2×D411yescharacteristic
171C22×C4×C31C2×D412yes—
172C22×C4×C31C2×D411yescharacteristic
173C22×C4×C31C2×D411yescharacteristic
174C22×C4×C31C2×D412yes—
175C22×C4×C31C2×D411yescharacteristic
176C4⋊C4×C31—22no—
177C4⋊C4×C31—22no—
178C4⋊C4×C31—21no—
179C4⋊C4×C31—21no—
180C4⋊C4×C31—21no—
181C4⋊C4×C31—21no—
182C4⋊C4×C31—22no—
183C4⋊C4×C31—22no—
184C4⋊C4×C31—44no—
185C4⋊C4×C31—44no—
186C4⋊C4×C31—48no—
187C22⋊C4×C31—24no—
188C22⋊C4×C31—22no—
189C22⋊C4×C31—22no—
190C22⋊C4×C31—22no—
191C22⋊C4×C31—22no—
192C22⋊C4×C31—22no—
193C22⋊C4×C31—22no—
194C22⋊C4×C31—22no—
195C22⋊C4×C31—22no—
196C22⋊C4×C31—22no—
197C22⋊C4×C31—22no—
198C22⋊C4×C31—22no—
199C22⋊C4×C31—22no—
200C22⋊C4×C31—22no—
201C22⋊C4×C31—22no—
202C22⋊C4×C31—22no—
203C22⋊C4×C31—24no—
204C22⋊C4×C31—24no—
205C22⋊C4×C31—48no—
206C22⋊C4×C31—44no—
207C22⋊C4×C31—44no—
208C22⋊C4×C31#4114yescomplemented
209C42×C31—22no—
210C42×C31—22no—
211#24112C3111yesmaximal, characteristic, direct product factor, Sylow
212C23×C31—21no—
213C23×C31—21no—
214C23×C31—22no—
215C23×C31—22no—
216C23×C31—21no—
217C23×C31—21no—
218C23×C31—24no—
219C23×C31—24no—
220C23×C31—24no—
221C23×C31—24no—
222C23×C31—22no—
223C23×C31—24no—
224C23×C31—41no—
225C23×C31—42no—
226C23×C31—41no—
227C23×C31—41no—
228C23×C31—42no—
229C23×C31—41no—
230C23×C31—48no—
231C23×C31—48no—
232C23×C31—42no—
233C23×C31—42no—
234C23×C31#12012yes—
235C23×C31#12011yescharacteristic
236C23×C31#12011yescharacteristic
237C23×C31C22×D411yescharacteristic
238C23×C31C22×D412yes—
239C23×C31C2×#4112yes—
240C23×C312−1+412yes—
241Q8×C31—81no—
242Q8×C31—81no—
243D4×C31—42no—
244D4×C31—42no—
245D4×C31—44no—
246D4×C31—44no—
247D4×C31—44no—
248D4×C31—82no—
249D4×C31—82no—
250D4×C31—81no—
251D4×C31—81no—
252C2×C4×C31—28no—
253C2×C4×C31—24no—
254C2×C4×C31—22no—
255C2×C4×C31—21no—
256C2×C4×C31—21no—
257C2×C4×C31—22no—
258C2×C4×C31—22no—
259C2×C4×C31—21no—
260C2×C4×C31—21no—
261C2×C4×C31—24no—
262C2×C4×C31—24no—
263C2×C4×C31—42no—
264C2×C4×C31—48no—
265C2×C4×C31—42no—
266C2×C4×C31—42no—
267C2×C4×C31—44no—
268C2×C4×C31—44no—
269C2×C4×C31—48no—
270C2×C4×C31—42no—
271C2×C4×C31—44no—
272C2×C4×C31—42no—
273C2×C4×C31—42no—
274C2×C4×C31—41no—
275C2×C4×C31—41no—
276C2×C4×C31—42no—
277C2×C4×C31—41no—
278C2×C4×C31—41no—
279C2×C4×C31—42no—
280C2×C4×C31—44no—
281C2×C4×C31—44no—
282C2×C4×C31—44no—
283C2×C4×C31—81no—
284C2×C4×C31—81no—
285C2×C4×C31—82no—
286C2×C4×C31—82no—
287#3490C2×C3111yescharacteristic
288#2849C2×C3112yescomplemented
289C2×#385C2×C3111yescharacteristic, complemented
290#2062C2×C3112yescomplemented
291#2044C2×C3112yescomplemented
292#1898C2×C3112yescomplemented
293#1867C2×C3114yescomplemented
294#1819C2×C3111yescharacteristic, complemented
295C22×C31—24no—
296C22×C31—24no—
297C22×C31—21no—
298C22×C31—21no—
299C22×C31—21no—
300C22×C31—22no—
301C22×C31—21no—
302C22×C31—21no—
303C22×C31—22no—
304C22×C31—21no—
305C22×C31—22no—
306C22×C31—22no—
307C22×C31—22no—
308C22×C31—42no—
309C22×C31—42no—
310C22×C31—48no—
311C22×C31—48no—
312C22×C31—42no—
313C22×C31—42no—
314C22×C31—42no—
315C22×C31—42no—
316C22×C31—42no—
317C22×C31—84no—
318C22×C31B(4,𝔽2)12yes—
319C22×C31#45812yes—
320C22×C31C2×#12011yescharacteristic
321C22×C31#52511yescharacteristic
322C22×C31#53611yescharacteristic
323C22×C31#54714yes—
324C4×C31—44no—
325C4×C31—42no—
326C4×C31—44no—
327C4×C31—41no—
328C4×C31—41no—
329C4×C31—42no—
330C4×C31—41no—
331C4×C31—41no—
332C4×C31—42no—
333C4×C31—84no—
334C24×C4C22×C3111yescharacteristic
335#551—22no—
336#551—22no—
337#546—22no—
338#546—22no—
339#543C22×C3111yescharacteristic
340#538—21no—
341#538—21no—
342#534C22×C3111yescharacteristic
343#525—21no—
344#525—21no—
345C2×#123—22no—
346C2×#120C22×C3112yes—
347#518C22×C3112yes—
348C22×C4⋊C4—21no—
349C22×C22⋊C4C22×C3111yescharacteristic
350C22×C22⋊C4C22×C3111yescharacteristic
351C2×#99C22×C3114yes—
352#386—22no—
353#386C22×C3114yes—
354#385—22no—
355#385—21no—
356#385—21no—
357#385C22×C3114yes—
358#385C22×C3112yes—
359#385C22×C3112yes—
360#381—22no—
361#379C22×C3112yes—
362C2×#95—21no—
363#342C22×C3112yes—
364#342C22×C3114yes—
365#342C22×C3112yes—
366C2×C31—21no—
367C2×C31—24no—
368C2×C31—21no—
369C2×C31—22no—
370C2×C31—22no—
371C2×C31—42no—
372C2×C31—82no—
373C2×C31#240611yescharacteristic
374C2×C31C2×B(4,𝔽2)11yescharacteristic
375C2×C31C2×#45811yescharacteristic
376C25C23×C3111yescharacteristic
377C2×#41—21no—
378C2×#41—21no—
379C22×D4D4×C3112yes—
380C22×D4D4×C3112yes—
381C22×D4C23×C3112yes—
382C23×C4—21no—
383C23×C4—21no—
384C23×C4—22no—
385C23×C4—21no—
386C23×C4—22no—
387C23×C4—44no—
388C23×C4D4×C3111yescharacteristic
389C23×C4D4×C3111yescharacteristic
390C23×C4C23×C3112yes—
391#126—22no—
392#126—22no—
393#124—22no—
394#124—22no—
395#123—22no—
396#123—22no—
397#123—22no—
398#123—22no—
399#123—22no—
400#123—48no—
401C22⋊Q8—21no—
402C22⋊Q8—21no—
403C22⋊Q8—21no—
404C22⋊Q8—21no—
405C22⋊Q8—22no—
406C22⋊Q8—22no—
407#121—21no—
408#121—21no—
409#121—22no—
410#121—22no—
411#121—21no—
412#121—21no—
413#120—22no—
414#120—22no—
415#120—22no—
416#120—22no—
417#120—22no—
418C4×D4—22no—
419C4×D4—22no—
420C4×D4—22no—
421C4×D4—22no—
422C4×D4—22no—
423C4×D4—22no—
424#117—21no—
425#117—21no—
426#117—22no—
427C2×C4⋊C4—22no—
428C2×C4⋊C4—22no—
429C2×C4⋊C4—24no—
430C2×C4⋊C4—44no—
431C2×C22⋊C4—22no—
432C2×C22⋊C4—22no—
433C2×C22⋊C4—21no—
434C2×C22⋊C4—22no—
435C2×C22⋊C4—24no—
436C2×C22⋊C4—22no—
437C2×C22⋊C4—24no—
438C2×C22⋊C4—22no—
439C2×C22⋊C4D4×C3111yescharacteristic
440C2×C22⋊C4D4×C3111yescharacteristic
441C2×C22⋊C4D4×C3112yes—
442C2×C22⋊C4D4×C3112yes—
443C2×C22⋊C4C23×C3114yes—
444C2×C22⋊C4C23×C3111yescharacteristic
445C2×C22⋊C4C23×C3111yescharacteristic
446C2×C22⋊C4C23×C3112yes—
447C2×C22⋊C4C23×C3112yes—
448#99—28no—
449#95—22no—
450#95—22no—
451#95—24no—
452#95—24no—
453C31#2411211yescharacteristic, direct product factor, Sylow
454C24—22no—
455C24—22no—
456C24—42no—
457C24—42no—
458C24C2×D4×C3111yescharacteristic
459C24C2×D4×C3112yes—
460C24C2×D4×C3112yes—
461C24C2×D4×C3112yes—
462C24C2×D4×C3111yescharacteristic
463C24C24×C3111yescharacteristic
464#41—82no—
465#41—82no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×D4—22no—
469C2×D4—22no—
470C2×D4—22no—
471C2×D4—22no—
472C2×D4—24no—
473C2×D4—24no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—22no—
477C2×D4—22no—
478C2×D4—44no—
479C2×D4—44no—
480C22×C4—24no—
481C22×C4—24no—
482C22×C4—28no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—22no—
486C22×C4—24no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—24no—
493C22×C4—44no—
494C22×C4—48no—
495C22×C4—44no—
496C22×C4—42no—
497C22×C4—42no—
498C22×C4—42no—
499C22×C4—44no—
500C22×C4—42no—
501C22×C4—41no—
502C22×C4—41no—
503C22×C4—42no—
504C22×C4—42no—
505C22×C4C2×D4×C3112yes—
506C22×C4C2×D4×C3111yescharacteristic
507C22×C4C2×D4×C3112yes—
508C22×C4C2×D4×C3111yescharacteristic
509C22×C4C2×D4×C3111yescharacteristic
510C22×C4C2×D4×C3112yes—
511C22×C4C2×D4×C3111yescharacteristic
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—48no—
523C22⋊C4—24no—
524C22⋊C4—22no—
525C22⋊C4—22no—
526C22⋊C4—22no—
527C22⋊C4—22no—
528C22⋊C4—22no—
529C22⋊C4—22no—
530C22⋊C4—22no—
531C22⋊C4—22no—
532C22⋊C4—22no—
533C22⋊C4—22no—
534C22⋊C4—22no—
535C22⋊C4—22no—
536C22⋊C4—22no—
537C22⋊C4—22no—
538C22⋊C4—22no—
539C22⋊C4—24no—
540C22⋊C4—24no—
541C22⋊C4—48no—
542C22⋊C4—44no—
543C22⋊C4—44no—
544C22⋊C4#41×C3114yescomplemented
545C42—22no—
546C42—22no—
547C23—22no—
548C23—22no—
549C23—21no—
550C23—21no—
551C23—24no—
552C23—24no—
553C23—21no—
554C23—21no—
555C23—24no—
556C23—24no—
557C23—22no—
558C23—24no—
559C23—41no—
560C23—42no—
561C23—41no—
562C23—41no—
563C23—42no—
564C23—41no—
565C23—48no—
566C23—48no—
567C23—42no—
568C23—42no—
569C23C31×#12012yes—
570C23C31×#12011yescharacteristic
571C23C31×#12011yescharacteristic
572C23C22×D4×C3111yescharacteristic
573C23C22×D4×C3112yes—
574C23C2×#41×C3112yes—
575C23C31×2−1+412yes—
576Q8—81no—
577Q8—81no—
578D4—44no—
579D4—44no—
580D4—42no—
581D4—42no—
582D4—44no—
583D4—82no—
584D4—82no—
585D4—81no—
586D4—81no—
587C2×C4—28no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—42no—
599C2×C4—48no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—48no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—81no—
619C2×C4—81no—
620C2×C4—82no—
621C2×C4—82no—
622C22—24no—
623C22—24no—
624C22—21no—
625C22—21no—
626C22—21no—
627C22—22no—
628C22—21no—
629C22—21no—
630C22—22no—
631C22—21no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—42no—
636C22—42no—
637C22—48no—
638C22—48no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—84no—
645C22C31×B(4,𝔽2)12yes—
646C22C31×#45812yes—
647C22C2×C31×#12011yescharacteristic
648C22C31×#52511yescharacteristic
649C22C31×#53611yescharacteristic
650C22C31×#54714yes—
651C4—44no—
652C4—42no—
653C4—44no—
654C4—41no—
655C4—41no—
656C4—42no—
657C4—41no—
658C4—41no—
659C4—42no—
660C4—84no—
661C2—21no—
662C2—24no—
663C2—22no—
664C2—22no—
665C2—21no—
666C2—42no—
667C2—82no—
668C2C31×#240611yescharacteristic
669C2C2×C31×B(4,𝔽2)11yescharacteristic
670C2C2×C31×#45811yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC24C24×C31
prime radicalC31×#3490C2
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C25×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#2849C2semidirect
3. C2×C31×#385C2semidirect
4. C31×#2062C2semidirect
5. C31×#2044C2semidirect
6. C31×#1898C2semidirect
7. C31×#1867C2semidirect
8. C31×#1819C2semidirect
208. C22⋊C4×C31#41semidirect
211. #24112C31direct (N × H)
288. #2849C2×C31semidirect
289. C2×#385C2×C31semidirect
290. #2062C2×C31semidirect
291. #2044C2×C31semidirect
292. #1898C2×C31semidirect
293. #1867C2×C31semidirect
294. #1819C2×C31semidirect
453. C31#24112direct (N × H)
544. C22⋊C4#41×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#34902
C31×#28492
C2×C31×#3852
C31×#20622
C31×#20442
C31×#18982
C31×#18672
C31×#18192
#2411231