Finite Group Databasebeta

← C31×#23932 #21480691

Subgroups — C31×#23932 #21480691

Order 7936.

1556 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#532C211yesmaximal, characteristic, complemented
2C2×C31×#449C212yesmaximal, complemented
3C31×#2297C211yesmaximal, characteristic, complemented
4C31×#2024C211yesmaximal, characteristic, complemented
5C31×#2023C211yesmaximal, characteristic, complemented
6C31×#2014C211yesmaximal, characteristic, complemented
7C31×#2011C211yesmaximal, characteristic, complemented
8C31×#1879C212yesmaximal, complemented
9C31×#1877C212yesmaximal, complemented
10C31×#1875C212yesmaximal, complemented
11C31×#1817C211yesmaximal, characteristic, complemented
12C22×#41×C31—21no—
13C22×#41×C31C2211yescharacteristic
14C22×SD16×C31C2212yes—
15C22×D8×C31C2212yes—
16C31×#532—28no—
17C2×C31×C4⋊Q8C2211yescharacteristic
18C2×C31×#127C2211yescharacteristic
19C2×C31×#124—21no—
20C2×C31×#121—21no—
21C2×C31×#121C2212yes—
22C2×C4×D4×C31—21no—
23C2×C4×D4×C31C2212yes—
24C2×C31×#117C2211yescharacteristic
25C22×C42×C31C2211yescharacteristic
26C31×#449—28no—
27C2×C31×#103C2212yes—
28C2×C31×#102C2212yes—
29C2×C31×#102C2212yes—
30C2×C31×#102C2212yes—
31C2×C31×C22⋊C8C2212yes—
32C31×#391C2211yescharacteristic
33C31×#391C2211yescharacteristic
34C31×#390C2211yescharacteristic
35C31×#390C2211yescharacteristic
36C31×#386—21no—
37C31×#386—21no—
38C4×C22⋊C4×C31—21no—
39C4×C22⋊C4×C31—21no—
40C31×#340C2212yes—
41C31×#340C2212yes—
42C31×#336C2212yes—
43C31×#336C2211yescharacteristic
44C31×#336C2211yescharacteristic
45C2×#41×C31—28no—
46C2×#41×C31—22no—
47C2×#41×C31—22no—
48C2×#41×C31—24no—
49C2×#41×C31—42no—
50C2×#41×C31—44no—
51C2×#41×C31—44no—
52C22×Q8×C31—21no—
53C22×Q8×C31C2311yescharacteristic
54C22×D4×C31—22no—
55C22×D4×C31—21no—
56C22×D4×C31—21no—
57C22×D4×C31—21no—
58C22×D4×C31C2312yes—
59C22×D4×C31C2311yescharacteristic
60C23×C4×C31—22no—
61C23×C4×C31—21no—
62C23×C4×C31D411yescharacteristic
63C23×C4×C31D411yescharacteristic
64C23×C4×C31C2311yescharacteristic
65C2×SD16×C31—24no—
66C2×SD16×C31—24no—
67C2×SD16×C31—42no—
68C2×D8×C31—24no—
69C2×D8×C31—24no—
70C2×D8×C31—42no—
71C22×C8×C31C2312yes—
72C22×C8×C31C2312yes—
73C31×C4⋊Q8—24no—
74C31×#127—24no—
75C31×#124—28no—
76C31×#121—28no—
77C31×#121—24no—
78C31×#121D418yes—
79C4×D4×C31—28no—
80C4×D4×C31—24no—
81C4×D4×C31—24no—
82C31×#117—42no—
83C2×C4⋊C4×C31—21no—
84C2×C4⋊C4×C31C2311yescharacteristic
85C2×C4⋊C4×C31C2311yescharacteristic
86C2×C4⋊C4×C31C2312yes—
87C2×C22⋊C4×C31—22no—
88C2×C22⋊C4×C31—22no—
89C2×C22⋊C4×C31—21no—
90C2×C22⋊C4×C31—21no—
91C2×C22⋊C4×C31—21no—
92C2×C22⋊C4×C31—21no—
93C2×C22⋊C4×C31—21no—
94C2×C42×C31—24no—
95C2×C42×C31—21no—
96C2×C42×C31D411yescharacteristic
97C2×C42×C31D411yescharacteristic
98C2×C42×C31C2311yescharacteristic
99C2×C42×C31C2311yescharacteristic
100C31×#103—24no—
101C31×#102—24no—
102C31×#102—42no—
103C31×#102—42no—
104C31×C22⋊C8—24no—
105C31×#95—21no—
106C31×#95—21no—
107C31×#95—21no—
108C31×#95—21no—
109C24×C31—21no—
110C24×C31—22no—
111C24×C31C2×D411yescharacteristic
112#41×C31—48no—
113#41×C31—48no—
114#41×C31—432no—
115C2×Q8×C31—24no—
116C2×Q8×C31—21no—
117C2×Q8×C31—22no—
118C2×Q8×C31—22no—
119C2×Q8×C31—42no—
120C2×Q8×C31—42no—
121C2×Q8×C31#4112yescomplemented
122C2×D4×C31—24no—
123C2×D4×C31—24no—
124C2×D4×C31—24no—
125C2×D4×C31—28no—
126C2×D4×C31—22no—
127C2×D4×C31—22no—
128C2×D4×C31—22no—
129C2×D4×C31—22no—
130C2×D4×C31—21no—
131C2×D4×C31—22no—
132C2×D4×C31—22no—
133C2×D4×C31—22no—
134C2×D4×C31—42no—
135C2×D4×C31—42no—
136C2×D4×C31—42no—
137C2×D4×C31—44no—
138C2×D4×C31—44no—
139C2×D4×C31—42no—
140C2×D4×C31—42no—
141C2×D4×C31—42no—
142C2×D4×C31C2×D414yes—
143C2×D4×C31#4112yescomplemented
144C22×C4×C31—22no—
145C22×C4×C31—22no—
146C22×C4×C31—22no—
147C22×C4×C31—21no—
148C22×C4×C31—21no—
149C22×C4×C31—22no—
150C22×C4×C31—28no—
151C22×C4×C31—24no—
152C22×C4×C31—22no—
153C22×C4×C31—21no—
154C22×C4×C31—22no—
155C22×C4×C31—22no—
156C22×C4×C31—21no—
157C22×C4×C31—21no—
158C22×C4×C31—41no—
159C22×C4×C31—42no—
160C22×C4×C31—44no—
161C22×C4×C31—42no—
162C22×C4×C31—44no—
163C22×C4×C31C2×D411yescharacteristic
164C22×C4×C31C2×D411yescharacteristic
165C22×C4×C31C2×D414yes—
166C22×C4×C31C2×D411yescharacteristic
167C22×C4×C31C2×D411yescharacteristic
168C22×C4×C31C2×D411yescharacteristic
169C22×C4×C31C2411yescharacteristic
170SD16×C31—48no—
171D8×C31—48no—
172C2×C8×C31—22no—
173C2×C8×C31—22no—
174C2×C8×C31—22no—
175C2×C8×C31—22no—
176C2×C8×C31—22no—
177C2×C8×C31—22no—
178C4⋊C4×C31—24no—
179C4⋊C4×C31—22no—
180C4⋊C4×C31—22no—
181C4⋊C4×C31—22no—
182C4⋊C4×C31C2×D414yes—
183C22⋊C4×C31—28no—
184C22⋊C4×C31—28no—
185C22⋊C4×C31—42no—
186C22⋊C4×C31—42no—
187C22⋊C4×C31—42no—
188C22⋊C4×C31—42no—
189C22⋊C4×C31—42no—
190C42×C31—22no—
191C42×C31—22no—
192C42×C31—22no—
193C42×C31—22no—
194C42×C31—42no—
195#23932C3111yesmaximal, characteristic, direct product factor, Sylow
196C23×C31—21no—
197C23×C31—21no—
198C23×C31—21no—
199C23×C31—21no—
200C23×C31—21no—
201C23×C31—21no—
202C23×C31—24no—
203C23×C31—22no—
204C23×C31—41no—
205C23×C31—42no—
206C23×C31—41no—
207C23×C31—42no—
208C23×C31—41no—
209C23×C31—42no—
210C23×C31—41no—
211C23×C31—41no—
212C23×C31—42no—
213C23×C31—42no—
214C23×C31—41no—
215C23×C31—42no—
216C23×C31#12012yes—
217C23×C31C22×D411yescharacteristic
218Q8×C31—24no—
219Q8×C31—48no—
220Q8×C31—42no—
221D4×C31—24no—
222D4×C31—24no—
223D4×C31—24no—
224D4×C31—48no—
225D4×C31—48no—
226D4×C31—48no—
227D4×C31—416no—
228D4×C31—42no—
229D4×C31—42no—
230D4×C31—42no—
231C2×C4×C31—28no—
232C2×C4×C31—21no—
233C2×C4×C31—21no—
234C2×C4×C31—21no—
235C2×C4×C31—21no—
236C2×C4×C31—21no—
237C2×C4×C31—21no—
238C2×C4×C31—21no—
239C2×C4×C31—21no—
240C2×C4×C31—24no—
241C2×C4×C31—24no—
242C2×C4×C31—21no—
243C2×C4×C31—24no—
244C2×C4×C31—22no—
245C2×C4×C31—21no—
246C2×C4×C31—21no—
247C2×C4×C31—44no—
248C2×C4×C31—48no—
249C2×C4×C31—416no—
250C2×C4×C31—41no—
251C2×C4×C31—41no—
252C2×C4×C31—42no—
253C2×C4×C31—42no—
254C2×C4×C31—41no—
255C2×C4×C31—41no—
256C2×C4×C31—41no—
257C2×C4×C31—41no—
258C2×C4×C31—41no—
259C2×C4×C31—41no—
260C2×C4×C31#12012yes—
261C2×C4×C31#13512yes—
262C2×C4×C31AGL(1,ℤ/8)12yes—
263C2×C4×C31AGL(1,ℤ/8)12yes—
264C2×C4×C31C22×D412yes—
265C2×C4×C31C2×#4111yescharacteristic
266C2×C4×C31C2×#4111yescharacteristic
267C2×C4×C31H(5,𝔽2)12yes—
268C8×C31—42no—
269C8×C31—42no—
270C2×#532C2×C3111yescharacteristic, complemented
271C2×#449C2×C3112yescomplemented
272#2297C2×C3111yescharacteristic, complemented
273#2024C2×C3111yescharacteristic, complemented
274#2023C2×C3111yescharacteristic, complemented
275#2014C2×C3111yescharacteristic, complemented
276#2011C2×C3111yescharacteristic, complemented
277#1879C2×C3112yescomplemented
278#1877C2×C3112yescomplemented
279#1875C2×C3112yescomplemented
280#1817C2×C3111yescharacteristic, complemented
281C22×C31—22no—
282C22×C31—22no—
283C22×C31—22no—
284C22×C31—41no—
285C22×C31—41no—
286C22×C31—41no—
287C22×C31—41no—
288C22×C31—42no—
289C22×C31—42no—
290C22×C31—42no—
291C22×C31—44no—
292C22×C31—44no—
293C22×C31—44no—
294C22×C31—82no—
295C22×C31—82no—
296C22×C31—81no—
297C22×C31—82no—
298C22×C31—81no—
299C22×C31—82no—
300C22×C31—81no—
301C22×C31—81no—
302C22×C31C2×#12011yescharacteristic
303C22×C31#52511yescharacteristic
304C22×C31#53511yescharacteristic
305C22×C31C2×#13511yescharacteristic
306C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
307C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
308C22×C31#57511yescharacteristic
309C4×C31—22no—
310C4×C31—22no—
311C4×C31—22no—
312C4×C31—22no—
313C4×C31—42no—
314C4×C31—41no—
315C4×C31—44no—
316C4×C31—81no—
317C4×C31—81no—
318C4×C31#54612yes—
319C4×C31#54612yes—
320C22×#41—21no—
321C22×#41C22×C3111yescharacteristic
322C22×SD16C22×C3112yes—
323C22×D8C22×C3112yes—
324#532—28no—
325C2×C4⋊Q8C22×C3111yescharacteristic
326C2×#127C22×C3111yescharacteristic
327C2×#124—21no—
328C2×#121—21no—
329C2×#121C22×C3112yes—
330C2×C4×D4—21no—
331C2×C4×D4C22×C3112yes—
332C2×#117C22×C3111yescharacteristic
333C22×C42C22×C3111yescharacteristic
334#449—28no—
335C2×#103C22×C3112yes—
336C2×#102C22×C3112yes—
337C2×#102C22×C3112yes—
338C2×#102C22×C3112yes—
339C2×C22⋊C8C22×C3112yes—
340#391C22×C3111yescharacteristic
341#391C22×C3111yescharacteristic
342#390C22×C3111yescharacteristic
343#390C22×C3111yescharacteristic
344#386—21no—
345#386—21no—
346C4×C22⋊C4—21no—
347C4×C22⋊C4—21no—
348#340C22×C3112yes—
349#340C22×C3112yes—
350#336C22×C3112yes—
351#336C22×C3111yescharacteristic
352#336C22×C3111yescharacteristic
353C2×C31—42no—
354C2×C31—82no—
355C2×C31—84no—
356C2×C31#240411yescharacteristic
357C2×C31#300211yescharacteristic
358C2×C31#300311yescharacteristic
359C2×C31#310611yescharacteristic
360C2×C31#311511yescharacteristic
361C2×C31#320711yescharacteristic
362C2×C31#321411yescharacteristic
363C2×#41—28no—
364C2×#41—22no—
365C2×#41—22no—
366C2×#41—24no—
367C2×#41—42no—
368C2×#41—44no—
369C2×#41—44no—
370C22×Q8—21no—
371C22×Q8C23×C3111yescharacteristic
372C22×D4—22no—
373C22×D4—21no—
374C22×D4—21no—
375C22×D4—21no—
376C22×D4C23×C3112yes—
377C22×D4C23×C3111yescharacteristic
378C23×C4—22no—
379C23×C4—21no—
380C23×C4D4×C3111yescharacteristic
381C23×C4D4×C3111yescharacteristic
382C23×C4C23×C3111yescharacteristic
383C2×SD16—24no—
384C2×SD16—24no—
385C2×SD16—42no—
386C2×D8—24no—
387C2×D8—24no—
388C2×D8—42no—
389C22×C8C23×C3112yes—
390C22×C8C23×C3112yes—
391C4⋊Q8—24no—
392#127—24no—
393#124—28no—
394#121—28no—
395#121—24no—
396#121D4×C3118yes—
397C4×D4—28no—
398C4×D4—24no—
399C4×D4—24no—
400#117—42no—
401C2×C4⋊C4—21no—
402C2×C4⋊C4C23×C3111yescharacteristic
403C2×C4⋊C4C23×C3111yescharacteristic
404C2×C4⋊C4C23×C3112yes—
405C2×C22⋊C4—22no—
406C2×C22⋊C4—22no—
407C2×C22⋊C4—21no—
408C2×C22⋊C4—21no—
409C2×C22⋊C4—21no—
410C2×C22⋊C4—21no—
411C2×C22⋊C4—21no—
412C2×C42—24no—
413C2×C42—21no—
414C2×C42D4×C3111yescharacteristic
415C2×C42D4×C3111yescharacteristic
416C2×C42C23×C3111yescharacteristic
417C2×C42C23×C3111yescharacteristic
418#103—24no—
419#102—24no—
420#102—42no—
421#102—42no—
422C22⋊C8—24no—
423#95—21no—
424#95—21no—
425#95—21no—
426#95—21no—
427C31#2393211yescharacteristic, direct product factor, Sylow
428C24—21no—
429C24—22no—
430C24C2×D4×C3111yescharacteristic
431#41—48no—
432#41—48no—
433#41—432no—
434C2×Q8—24no—
435C2×Q8—21no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—42no—
439C2×Q8—42no—
440C2×Q8#41×C3112yescomplemented
441C2×D4—24no—
442C2×D4—24no—
443C2×D4—24no—
444C2×D4—28no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—21no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—42no—
454C2×D4—42no—
455C2×D4—42no—
456C2×D4—44no—
457C2×D4—44no—
458C2×D4—42no—
459C2×D4—42no—
460C2×D4—42no—
461C2×D4C2×D4×C3114yes—
462C2×D4#41×C3112yescomplemented
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—28no—
470C22×C4—24no—
471C22×C4—22no—
472C22×C4—21no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—41no—
478C22×C4—42no—
479C22×C4—44no—
480C22×C4—42no—
481C22×C4—44no—
482C22×C4C2×D4×C3111yescharacteristic
483C22×C4C2×D4×C3111yescharacteristic
484C22×C4C2×D4×C3114yes—
485C22×C4C2×D4×C3111yescharacteristic
486C22×C4C2×D4×C3111yescharacteristic
487C22×C4C2×D4×C3111yescharacteristic
488C22×C4C24×C3111yescharacteristic
489SD16—48no—
490D8—48no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C4⋊C4—24no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4C2×D4×C3114yes—
502C22⋊C4—28no—
503C22⋊C4—28no—
504C22⋊C4—42no—
505C22⋊C4—42no—
506C22⋊C4—42no—
507C22⋊C4—42no—
508C22⋊C4—42no—
509C42—22no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—42no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—24no—
521C23—22no—
522C23—41no—
523C23—42no—
524C23—41no—
525C23—42no—
526C23—41no—
527C23—42no—
528C23—41no—
529C23—41no—
530C23—42no—
531C23—42no—
532C23—41no—
533C23—42no—
534C23C31×#12012yes—
535C23C22×D4×C3111yescharacteristic
536Q8—24no—
537Q8—48no—
538Q8—42no—
539D4—24no—
540D4—24no—
541D4—24no—
542D4—48no—
543D4—48no—
544D4—48no—
545D4—416no—
546D4—42no—
547D4—42no—
548D4—42no—
549C2×C4—28no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—21no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—44no—
566C2×C4—48no—
567C2×C4—416no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4C31×#12012yes—
579C2×C4C31×#13512yes—
580C2×C4C31×AGL(1,ℤ/8)12yes—
581C2×C4C31×AGL(1,ℤ/8)12yes—
582C2×C4C22×D4×C3112yes—
583C2×C4C2×#41×C3111yescharacteristic
584C2×C4C2×#41×C3111yescharacteristic
585C2×C4C31×H(5,𝔽2)12yes—
586C8—42no—
587C8—42no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—44no—
599C22—44no—
600C22—44no—
601C22—82no—
602C22—82no—
603C22—81no—
604C22—82no—
605C22—81no—
606C22—82no—
607C22—81no—
608C22—81no—
609C22C2×C31×#12011yescharacteristic
610C22C31×#52511yescharacteristic
611C22C31×#53511yescharacteristic
612C22C2×C31×#13511yescharacteristic
613C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
614C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
615C22C31×#57511yescharacteristic
616C4—22no—
617C4—22no—
618C4—22no—
619C4—22no—
620C4—42no—
621C4—41no—
622C4—44no—
623C4—81no—
624C4—81no—
625C4C31×#54612yes—
626C4C31×#54612yes—
627C2—42no—
628C2—82no—
629C2—84no—
630C2C31×#240411yescharacteristic
631C2C31×#300211yescharacteristic
632C2C31×#300311yescharacteristic
633C2C31×#310611yescharacteristic
634C2C31×#311511yescharacteristic
635C2C31×#320711yescharacteristic
636C2C31×#321411yescharacteristic
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
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×#532C2semidirect
2. C2×C31×#449C2semidirect
3. C31×#2297C2semidirect
4. C31×#2024C2semidirect
5. C31×#2023C2semidirect
6. C31×#2014C2semidirect
7. C31×#2011C2semidirect
8. C31×#1879C2semidirect
9. C31×#1877C2semidirect
10. C31×#1875C2semidirect
11. C31×#1817C2semidirect
121. C2×Q8×C31#41semidirect
143. C2×D4×C31#41semidirect
195. #23932C31direct (N × H)
270. C2×#532C2×C31semidirect
271. C2×#449C2×C31semidirect
272. #2297C2×C31semidirect
273. #2024C2×C31semidirect
274. #2023C2×C31semidirect
275. #2014C2×C31semidirect
276. #2011C2×C31semidirect
277. #1879C2×C31semidirect
278. #1877C2×C31semidirect
279. #1875C2×C31semidirect
280. #1817C2×C31semidirect
427. C31#23932direct (N × H)
440. C2×Q8#41×C31semidirect
462. C2×D4#41×C31semidirect

Maximal subgroups

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

subgroupindex
C2×C31×#5322
C2×C31×#4492
C31×#22972
C31×#20242
C31×#20232
C31×#20142
C31×#20112
C31×#18792
C31×#18772
C31×#18752
C31×#18172
#2393231