Finite Group Databasebeta

← C31×#23901 #21480660

Subgroups — C31×#23901 #21480660

Order 7936.

2056 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3484C211yesmaximal, characteristic
2C22×C31×#123C211yesmaximal, characteristic, complemented
3C2×C31×#458C212yesmaximal, complemented
4C31×#2849C212yesmaximal, complemented
5C31×#2022C214yesmaximal, complemented
6C31×#1898C212yesmaximal, complemented
7C31×#1890C212yesmaximal, complemented
8C31×#1781C211yesmaximal, characteristic, complemented
9C2×C31×H(5,𝔽2)C2211yescharacteristic
10C23×D4×C31C2211yescharacteristic
11C24×C4×C31C2211yescharacteristic, complemented
12C31×#546—24no—
13D42×C31—24no—
14C31×#535—21no—
15C31×#534—21no—
16C31×#534C2212yes—
17C31×#525—21no—
18C2×C31×#123—21no—
19C2×C31×#123—21no—
20C2×C31×#123—28no—
21C2×C31×#123C2212yes—
22C2×C31×#123C2212yescomplemented
23C2×C31×#123C2214yescomplemented
24C2×C31×#120—21no—
25C2×C31×#120C2212yes—
26C31×#518C2211yescharacteristic
27C22×C4⋊C4×C31—21no—
28C22×C22⋊C4×C31—21no—
29C22×C22⋊C4×C31C2211yescharacteristic, complemented
30C31×#458—28no—
31C2×C31×#99C2212yes—
32C2×C31×#99C2214yescomplemented
33C2×C31×#99C2212yescomplemented
34C31×#386—24no—
35C31×#385—24no—
36C31×#381C2214yescomplemented
37C31×#379C2212yescomplemented
38C31×#342C2214yescomplemented
39C25×C31D411yescharacteristic
40C25×C31C2311yescharacteristic
41C31×H(5,𝔽2)—24no—
42C31×H(5,𝔽2)—42no—
43C2×#41×C31—21no—
44C2×#41×C31—22no—
45C22×D4×C31—21no—
46C22×D4×C31—22no—
47C22×D4×C31—24no—
48C22×D4×C31—22no—
49C22×D4×C31—24no—
50C22×D4×C31—42no—
51C22×D4×C31D411yescharacteristic
52C22×D4×C31D411yescharacteristic
53C22×D4×C31D414yes—
54C22×D4×C31C2312yes—
55C22×D4×C31C2311yescharacteristic
56C22×D4×C31C2311yescharacteristic
57C23×C4×C31—21no—
58C23×C4×C31—21no—
59C23×C4×C31—21no—
60C23×C4×C31—24no—
61C23×C4×C31—24no—
62C23×C4×C31—42no—
63C23×C4×C31D411yescharacteristic
64C23×C4×C31C2312yes—
65C31×#127—22no—
66C31×#124—22no—
67C31×#123—28no—
68C31×#123—28no—
69C31×#123—22no—
70C31×#123—24no—
71C31×#123—22no—
72C31×#123—22no—
73C31×#123—24no—
74C31×#123—24no—
75C31×#123—28no—
76C31×#123—44no—
77C31×#123D418yescomplemented
78C31×C22⋊Q8—22no—
79C31×#121—22no—
80C31×#121—24no—
81C31×#121—24no—
82C31×#121—24no—
83C31×#120—24no—
84C31×#120—22no—
85C31×#120—22no—
86C31×#120—22no—
87C31×#120—22no—
88C31×#120—24no—
89C31×#120—22no—
90C4×D4×C31—24no—
91C4×D4×C31—24no—
92C31×#117—21no—
93C2×C4⋊C4×C31—28no—
94C2×C4⋊C4×C31—22no—
95C2×C4⋊C4×C31—22no—
96C2×C4⋊C4×C31—24no—
97C2×C22⋊C4×C31—28no—
98C2×C22⋊C4×C31—22no—
99C2×C22⋊C4×C31—22no—
100C2×C22⋊C4×C31—22no—
101C2×C22⋊C4×C31—21no—
102C2×C22⋊C4×C31—21no—
103C2×C22⋊C4×C31—24no—
104C2×C22⋊C4×C31—24no—
105C2×C22⋊C4×C31—24no—
106C2×C22⋊C4×C31D414yes—
107C2×C22⋊C4×C31C2314yes—
108C2×C22⋊C4×C31C2312yes—
109C2×C22⋊C4×C31C2312yes—
110C31×#99—24no—
111C31×#99—28no—
112C31×#99—42no—
113C31×#95—28no—
114C24×C31—22no—
115C24×C31—22no—
116C24×C31—21no—
117C24×C31—22no—
118C24×C31—22no—
119C24×C31—44no—
120C24×C31—41no—
121C24×C31—44no—
122C24×C31C2×D411yescharacteristic
123C24×C31C2×D414yes—
124C24×C31C2×D411yescharacteristic
125C24×C31C2×D412yes—
126C24×C31C2×D411yescharacteristic
127C24×C31C2×D411yescharacteristic
128C24×C31C2411yescharacteristic
129#41×C31—48no—
130#41×C31—42no—
131#41×C31—42no—
132C2×Q8×C31—21no—
133C2×D4×C31—22no—
134C2×D4×C31—22no—
135C2×D4×C31—22no—
136C2×D4×C31—21no—
137C2×D4×C31—24no—
138C2×D4×C31—22no—
139C2×D4×C31—22no—
140C2×D4×C31—22no—
141C2×D4×C31—24no—
142C2×D4×C31—22no—
143C2×D4×C31—22no—
144C2×D4×C31—22no—
145C2×D4×C31—22no—
146C2×D4×C31—22no—
147C2×D4×C31—24no—
148C2×D4×C31—24no—
149C2×D4×C31—24no—
150C2×D4×C31—22no—
151C2×D4×C31—41no—
152C2×D4×C31—41no—
153C2×D4×C31—416no—
154C2×D4×C31—48no—
155C2×D4×C31—42no—
156C2×D4×C31—42no—
157C2×D4×C31—48no—
158C2×D4×C31—84no—
159C2×D4×C31C2×D414yes—
160C22×C4×C31—24no—
161C22×C4×C31—24no—
162C22×C4×C31—21no—
163C22×C4×C31—22no—
164C22×C4×C31—24no—
165C22×C4×C31—22no—
166C22×C4×C31—24no—
167C22×C4×C31—22no—
168C22×C4×C31—22no—
169C22×C4×C31—22no—
170C22×C4×C31—24no—
171C22×C4×C31—22no—
172C22×C4×C31—22no—
173C22×C4×C31—24no—
174C22×C4×C31—41no—
175C22×C4×C31—416no—
176C22×C4×C31—44no—
177C22×C4×C31—42no—
178C22×C4×C31—42no—
179C22×C4×C31—42no—
180C22×C4×C31—42no—
181C22×C4×C31C2×D412yes—
182C22×C4×C31C2×D411yescharacteristic
183C22×C4×C31C2×D411yescharacteristic
184C22×C4×C31C2×D414yes—
185C22×C4×C31C2×D414yescomplemented
186C4⋊C4×C31—28no—
187C4⋊C4×C31—22no—
188C4⋊C4×C31—24no—
189C4⋊C4×C31—28no—
190C4⋊C4×C31—48no—
191C22⋊C4×C31—24no—
192C22⋊C4×C31—28no—
193C22⋊C4×C31—22no—
194C22⋊C4×C31—24no—
195C22⋊C4×C31—24no—
196C22⋊C4×C31—28no—
197C22⋊C4×C31—24no—
198C22⋊C4×C31—24no—
199C22⋊C4×C31—22no—
200C22⋊C4×C31—22no—
201C22⋊C4×C31—22no—
202C22⋊C4×C31—24no—
203C22⋊C4×C31—24no—
204C22⋊C4×C31—48no—
205C22⋊C4×C31—48no—
206C22⋊C4×C31C2×D414yescomplemented
207C42×C31—22no—
208#23901C3111yesmaximal, characteristic, direct product factor, Sylow
209C23×C31—21no—
210C23×C31—21no—
211C23×C31—24no—
212C23×C31—21no—
213C23×C31—21no—
214C23×C31—21no—
215C23×C31—28no—
216C23×C31—28no—
217C23×C31—21no—
218C23×C31—21no—
219C23×C31—21no—
220C23×C31—22no—
221C23×C31—21no—
222C23×C31—24no—
223C23×C31—44no—
224C23×C31—42no—
225C23×C31—42no—
226C23×C31—44no—
227C23×C31—48no—
228C23×C31—48no—
229C23×C31—42no—
230C23×C31—42no—
231C23×C31—48no—
232C23×C31—42no—
233C23×C31—44no—
234C23×C31—44no—
235C23×C31—82no—
236C23×C31—84no—
237C23×C31—82no—
238C23×C31#12012yes—
239C23×C31#12012yes—
240C23×C31#12012yes—
241C23×C31C22×D411yescharacteristic
242C23×C31C22×D412yes—
243C23×C31C22×D412yes—
244C23×C31H(5,𝔽2)12yes—
245Q8×C31—42no—
246D4×C31—28no—
247D4×C31—44no—
248D4×C31—42no—
249D4×C31—44no—
250D4×C31—44no—
251D4×C31—42no—
252D4×C31—42no—
253D4×C31—44no—
254D4×C31—44no—
255D4×C31—48no—
256D4×C31—84no—
257D4×C31—84no—
258C2×C4×C31—21no—
259C2×C4×C31—21no—
260C2×C4×C31—24no—
261C2×C4×C31—24no—
262C2×C4×C31—21no—
263C2×C4×C31—24no—
264C2×C4×C31—22no—
265C2×C4×C31—21no—
266C2×C4×C31—21no—
267C2×C4×C31—28no—
268C2×C4×C31—24no—
269C2×C4×C31—44no—
270C2×C4×C31—44no—
271C2×C4×C31—48no—
272C2×C4×C31—44no—
273C2×C4×C31—48no—
274C2×C4×C31—48no—
275C2×C4×C31—42no—
276C2×C4×C31—42no—
277C2×C4×C31—42no—
278C2×C4×C31—44no—
279C2×C4×C31—44no—
280C2×C4×C31—48no—
281C2×C4×C31—44no—
282C2×C4×C31—82no—
283C2×C4×C31#12012yes—
284#3484C2×C3111yescharacteristic
285C22×#123C2×C3111yescharacteristic, complemented
286C2×#458C2×C3112yescomplemented
287#2849C2×C3112yescomplemented
288#2022C2×C3114yescomplemented
289#1898C2×C3112yescomplemented
290#1890C2×C3112yescomplemented
291#1781C2×C3111yescharacteristic, complemented
292C22×C31—24no—
293C22×C31—24no—
294C22×C31—22no—
295C22×C31—24no—
296C22×C31—22no—
297C22×C31—22no—
298C22×C31—22no—
299C22×C31—22no—
300C22×C31—24no—
301C22×C31—42no—
302C22×C31—42no—
303C22×C31—416no—
304C22×C31—42no—
305C22×C31—42no—
306C22×C31—42no—
307C22×C31—42no—
308C22×C31—42no—
309C22×C31—42no—
310C22×C31—41no—
311C22×C31—44no—
312C22×C31—42no—
313C22×C31—81no—
314C22×C31—88no—
315C22×C31—84no—
316C22×C31—81no—
317C22×C31—162no—
318C22×C31#45814yescomplemented
319C22×C31C2×#12011yescharacteristic
320C22×C31C2×#12011yescharacteristic
321C22×C31#53411yescharacteristic
322C22×C31D4214yes—
323C4×C31—22no—
324C4×C31—44no—
325C4×C31—44no—
326C4×C31—41no—
327C4×C31—42no—
328C4×C31—44no—
329C4×C31—84no—
330C2×H(5,𝔽2)C22×C3111yescharacteristic
331C23×D4C22×C3111yescharacteristic
332C24×C4C22×C3111yescharacteristic, complemented
333#546—24no—
334D42—24no—
335#535—21no—
336#534—21no—
337#534C22×C3112yes—
338#525—21no—
339C2×#123—21no—
340C2×#123—21no—
341C2×#123—28no—
342C2×#123C22×C3112yes—
343C2×#123C22×C3112yescomplemented
344C2×#123C22×C3114yescomplemented
345C2×#120—21no—
346C2×#120C22×C3112yes—
347#518C22×C3111yescharacteristic
348C22×C4⋊C4—21no—
349C22×C22⋊C4—21no—
350C22×C22⋊C4C22×C3111yescharacteristic, complemented
351#458—28no—
352C2×#99C22×C3112yes—
353C2×#99C22×C3114yescomplemented
354C2×#99C22×C3112yescomplemented
355#386—24no—
356#385—24no—
357#381C22×C3114yescomplemented
358#379C22×C3112yescomplemented
359#342C22×C3114yescomplemented
360C2×C31—21no—
361C2×C31—24no—
362C2×C31—21no—
363C2×C31—24no—
364C2×C31—42no—
365C2×C31—42no—
366C2×C31—81no—
367C2×C31—82no—
368C2×C31#240311yescharacteristic
369C2×C31C2×#45812yes—
370C25D4×C3111yescharacteristic
371C25C23×C3111yescharacteristic
372H(5,𝔽2)—24no—
373H(5,𝔽2)—42no—
374C2×#41—21no—
375C2×#41—22no—
376C22×D4—21no—
377C22×D4—22no—
378C22×D4—24no—
379C22×D4—22no—
380C22×D4—24no—
381C22×D4—42no—
382C22×D4D4×C3111yescharacteristic
383C22×D4D4×C3111yescharacteristic
384C22×D4D4×C3114yes—
385C22×D4C23×C3112yes—
386C22×D4C23×C3111yescharacteristic
387C22×D4C23×C3111yescharacteristic
388C23×C4—21no—
389C23×C4—21no—
390C23×C4—21no—
391C23×C4—24no—
392C23×C4—24no—
393C23×C4—42no—
394C23×C4D4×C3111yescharacteristic
395C23×C4C23×C3112yes—
396#127—22no—
397#124—22no—
398#123—28no—
399#123—28no—
400#123—22no—
401#123—24no—
402#123—22no—
403#123—22no—
404#123—24no—
405#123—24no—
406#123—28no—
407#123—44no—
408#123D4×C3118yescomplemented
409C22⋊Q8—22no—
410#121—22no—
411#121—24no—
412#121—24no—
413#121—24no—
414#120—24no—
415#120—22no—
416#120—22no—
417#120—22no—
418#120—22no—
419#120—24no—
420#120—22no—
421C4×D4—24no—
422C4×D4—24no—
423#117—21no—
424C2×C4⋊C4—28no—
425C2×C4⋊C4—22no—
426C2×C4⋊C4—22no—
427C2×C4⋊C4—24no—
428C2×C22⋊C4—28no—
429C2×C22⋊C4—22no—
430C2×C22⋊C4—22no—
431C2×C22⋊C4—22no—
432C2×C22⋊C4—21no—
433C2×C22⋊C4—21no—
434C2×C22⋊C4—24no—
435C2×C22⋊C4—24no—
436C2×C22⋊C4—24no—
437C2×C22⋊C4D4×C3114yes—
438C2×C22⋊C4C23×C3114yes—
439C2×C22⋊C4C23×C3112yes—
440C2×C22⋊C4C23×C3112yes—
441#99—24no—
442#99—28no—
443#99—42no—
444#95—28no—
445C31#2390111yescharacteristic, direct product factor, Sylow
446C24—22no—
447C24—22no—
448C24—21no—
449C24—22no—
450C24—22no—
451C24—44no—
452C24—41no—
453C24—44no—
454C24C2×D4×C3111yescharacteristic
455C24C2×D4×C3114yes—
456C24C2×D4×C3111yescharacteristic
457C24C2×D4×C3112yes—
458C24C2×D4×C3111yescharacteristic
459C24C2×D4×C3111yescharacteristic
460C24C24×C3111yescharacteristic
461#41—48no—
462#41—42no—
463#41—42no—
464C2×Q8—21no—
465C2×D4—22no—
466C2×D4—22no—
467C2×D4—22no—
468C2×D4—21no—
469C2×D4—24no—
470C2×D4—22no—
471C2×D4—22no—
472C2×D4—22no—
473C2×D4—24no—
474C2×D4—22no—
475C2×D4—22no—
476C2×D4—22no—
477C2×D4—22no—
478C2×D4—22no—
479C2×D4—24no—
480C2×D4—24no—
481C2×D4—24no—
482C2×D4—22no—
483C2×D4—41no—
484C2×D4—41no—
485C2×D4—416no—
486C2×D4—48no—
487C2×D4—42no—
488C2×D4—42no—
489C2×D4—48no—
490C2×D4—84no—
491C2×D4C2×D4×C3114yes—
492C22×C4—24no—
493C22×C4—24no—
494C22×C4—21no—
495C22×C4—22no—
496C22×C4—24no—
497C22×C4—22no—
498C22×C4—24no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—24no—
503C22×C4—22no—
504C22×C4—22no—
505C22×C4—24no—
506C22×C4—41no—
507C22×C4—416no—
508C22×C4—44no—
509C22×C4—42no—
510C22×C4—42no—
511C22×C4—42no—
512C22×C4—42no—
513C22×C4C2×D4×C3112yes—
514C22×C4C2×D4×C3111yescharacteristic
515C22×C4C2×D4×C3111yescharacteristic
516C22×C4C2×D4×C3114yes—
517C22×C4C2×D4×C3114yescomplemented
518C4⋊C4—28no—
519C4⋊C4—22no—
520C4⋊C4—24no—
521C4⋊C4—28no—
522C4⋊C4—48no—
523C22⋊C4—24no—
524C22⋊C4—28no—
525C22⋊C4—22no—
526C22⋊C4—24no—
527C22⋊C4—24no—
528C22⋊C4—28no—
529C22⋊C4—24no—
530C22⋊C4—24no—
531C22⋊C4—22no—
532C22⋊C4—22no—
533C22⋊C4—22no—
534C22⋊C4—24no—
535C22⋊C4—24no—
536C22⋊C4—48no—
537C22⋊C4—48no—
538C22⋊C4C2×D4×C3114yescomplemented
539C42—22no—
540C23—24no—
541C23—21no—
542C23—21no—
543C23—28no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—28no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—22no—
552C23—21no—
553C23—24no—
554C23—44no—
555C23—42no—
556C23—42no—
557C23—44no—
558C23—48no—
559C23—48no—
560C23—42no—
561C23—42no—
562C23—48no—
563C23—42no—
564C23—44no—
565C23—44no—
566C23—82no—
567C23—84no—
568C23—82no—
569C23C31×#12012yes—
570C23C31×#12012yes—
571C23C31×#12012yes—
572C23C22×D4×C3111yescharacteristic
573C23C22×D4×C3112yes—
574C23C22×D4×C3112yes—
575C23C31×H(5,𝔽2)12yes—
576Q8—42no—
577D4—28no—
578D4—44no—
579D4—42no—
580D4—44no—
581D4—44no—
582D4—48no—
583D4—42no—
584D4—42no—
585D4—44no—
586D4—44no—
587D4—84no—
588D4—84no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—28no—
599C2×C4—24no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—48no—
603C2×C4—44no—
604C2×C4—48no—
605C2×C4—48no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—48no—
612C2×C4—44no—
613C2×C4—82no—
614C2×C4C31×#12012yes—
615C22—24no—
616C22—24no—
617C22—22no—
618C22—24no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—24no—
624C22—42no—
625C22—42no—
626C22—416no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—41no—
634C22—44no—
635C22—42no—
636C22—81no—
637C22—88no—
638C22—81no—
639C22—84no—
640C22—162no—
641C22C31×#45814yescomplemented
642C22C2×C31×#12011yescharacteristic
643C22C2×C31×#12011yescharacteristic
644C22C31×#53411yescharacteristic
645C22D42×C3114yes—
646C4—22no—
647C4—44no—
648C4—44no—
649C4—41no—
650C4—42no—
651C4—44no—
652C4—84no—
653C2—21no—
654C2—24no—
655C2—24no—
656C2—21no—
657C2—42no—
658C2—42no—
659C2—81no—
660C2—82no—
661C2C31×#240311yescharacteristic
662C2C2×C31×#45812yes—
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC24C24×C31
prime radicalC31×#3484C2
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. C22×C31×#123C2semidirect
3. C2×C31×#458C2semidirect
4. C31×#2849C2semidirect
5. C31×#2022C2semidirect
6. C31×#1898C2semidirect
7. C31×#1890C2semidirect
8. C31×#1781C2semidirect
11. C24×C4×C31C22semidirect
22. C2×C31×#123C22semidirect
23. C2×C31×#123C22semidirect
29. C22×C22⋊C4×C31C22semidirect
32. C2×C31×#99C22semidirect
33. C2×C31×#99C22semidirect
36. C31×#381C22semidirect
37. C31×#379C22semidirect
38. C31×#342C22semidirect
77. C31×#123D4semidirect
185. C22×C4×C31C2×D4semidirect
206. C22⋊C4×C31C2×D4semidirect
208. #23901C31direct (N × H)
285. C22×#123C2×C31semidirect
286. C2×#458C2×C31semidirect
287. #2849C2×C31semidirect
288. #2022C2×C31semidirect
289. #1898C2×C31semidirect
290. #1890C2×C31semidirect
291. #1781C2×C31semidirect
318. C22×C31#458semidirect
332. C24×C4C22×C31semidirect
343. C2×#123C22×C31semidirect
344. C2×#123C22×C31semidirect
350. C22×C22⋊C4C22×C31semidirect
353. C2×#99C22×C31semidirect
354. C2×#99C22×C31semidirect
357. #381C22×C31semidirect
358. #379C22×C31semidirect
359. #342C22×C31semidirect
408. #123D4×C31semidirect
445. C31#23901direct (N × H)
517. C22×C4C2×D4×C31semidirect
538. C22⋊C4C2×D4×C31semidirect
641. C22C31×#458semidirect

Maximal subgroups

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

subgroupindex
C31×#34842
C22×C31×#1232
C2×C31×#4582
C31×#28492
C31×#20222
C31×#18982
C31×#18902
C31×#17812
#2390131