Finite Group Databasebeta

← C31×#24012 #21480771

Subgroups — C31×#24012 #21480771

Order 7936.

1340 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#537C211yesmaximal, characteristic
2C2×C31×#481C211yesmaximal, characteristic, complemented
3C2×C31×#483C211yesmaximal, characteristic, complemented
4C31×#2311C211yesmaximal, characteristic, complemented
5C31×#2035C212yesmaximal, complemented
6C31×#2034C212yesmaximal, complemented
7C31×#1875C211yesmaximal, characteristic, complemented
8C31×#1873C211yesmaximal, characteristic, complemented
9C31×#1853C212yesmaximal, complemented
10C31×#1788C212yesmaximal, complemented
11C31×#1780C211yesmaximal, characteristic, complemented
12C22×#41×C31C2211yescharacteristic
13C22×#41×C31C2211yescharacteristic
14C31×#537—24no—
15C31×#537—24no—
16C2×C31×C22⋊Q8—21no—
17C2×C31×C22⋊Q8C2211yescharacteristic
18C2×C31×C22⋊Q8C2211yescharacteristic
19C2×C31×#121—21no—
20C2×C31×#121—21no—
21C2×C31×#121—21no—
22C2×C31×#121C2211yescharacteristic
23C2×C31×#121C2211yescharacteristic
24C2×C31×#117C2211yescharacteristic
25C22×C4⋊C4×C31C2211yescharacteristic
26C22×C4⋊C4×C31C2211yescharacteristic
27C31×#483—24no—
28C31×#481—24no—
29C2×C31×#106C2212yes—
30C2×C31×#106C2212yes—
31C2×C31×#103C2212yes—
32C2×C31×#103C2212yes—
33C2×C31×#102C2212yes—
34C2×C31×#102C2212yes—
35C2×C31×C22⋊C8C2211yescharacteristic
36C2×C31×C22⋊C8C2211yescharacteristic
37C2×C31×C22⋊C8C2212yes—
38C2×C31×C22⋊C8C2212yes—
39C31×#391—21no—
40C31×#390—21no—
41C31×#386—21no—
42C31×#386—21no—
43C31×#380C2212yes—
44C31×#380C2212yes—
45C31×#340C2212yes—
46C31×#340C2212yes—
47C2×#41×C31—22no—
48C2×#41×C31—22no—
49C2×#41×C31—22no—
50C2×#41×C31—22no—
51C2×#41×C31—24no—
52C2×#41×C31—24no—
53C2×#41×C31—42no—
54C2×#41×C31—42no—
55C22×Q8×C31D411yescharacteristic
56C22×Q8×C31C2311yescharacteristic
57C22×D4×C31—21no—
58C22×D4×C31—21no—
59C22×D4×C31—21no—
60C22×D4×C31D411yescharacteristic
61C22×D4×C31D411yescharacteristic
62C22×D4×C31D411yescharacteristic
63C22×D4×C31C2311yescharacteristic
64C23×C4×C31—21no—
65C23×C4×C31—21no—
66C23×C4×C31—21no—
67C23×C4×C31C2311yescharacteristic
68C23×C4×C31C2311yescharacteristic
69C23×C4×C31C2311yescharacteristic
70C22×C8×C31C2312yes—
71C22×C8×C31C2312yes—
72C31×C22⋊Q8—28no—
73C31×C22⋊Q8—22no—
74C31×C22⋊Q8—22no—
75C31×C22⋊Q8—22no—
76C31×C22⋊Q8D414yes—
77C31×#121—28no—
78C31×#121—28no—
79C31×#121—28no—
80C31×#121—22no—
81C31×#121—22no—
82C31×#121—22no—
83C31×#121D414yes—
84C31×#117—42no—
85C2×C4⋊C4×C31—21no—
86C2×C4⋊C4×C31—21no—
87C2×C4⋊C4×C31—21no—
88C2×C4⋊C4×C31—22no—
89C2×C4⋊C4×C31—22no—
90C2×C4⋊C4×C31—22no—
91C2×C4⋊C4×C31—22no—
92C2×C4⋊C4×C31—21no—
93C2×C4⋊C4×C31C2312yes—
94C2×C4⋊C4×C31C2312yes—
95C2×C4⋊C4×C31C2311yescharacteristic
96C2×C4⋊C4×C31C2311yescharacteristic
97C2×C22⋊C4×C31—21no—
98C2×C22⋊C4×C31—21no—
99C2×C22⋊C4×C31—21no—
100C2×C22⋊C4×C31—21no—
101C2×C22⋊C4×C31—21no—
102C2×C22⋊C4×C31—22no—
103C2×C22⋊C4×C31—22no—
104C2×C42×C31—21no—
105C31×#106—24no—
106C31×#106—24no—
107C31×#103—24no—
108C31×#103—42no—
109C31×#102—24no—
110C31×#102—42no—
111C31×C22⋊C8—22no—
112C31×C22⋊C8—22no—
113C31×C22⋊C8—24no—
114C31×C22⋊C8—42no—
115C31×#95—22no—
116C31×#95—22no—
117C24×C31—21no—
118C24×C31C2×D411yescharacteristic
119C24×C31C2×D411yescharacteristic
120C24×C31C2×D411yescharacteristic
121#41×C31—48no—
122#41×C31—48no—
123#41×C31—48no—
124#41×C31—48no—
125C2×Q8×C31—22no—
126C2×Q8×C31—22no—
127C2×Q8×C31—21no—
128C2×Q8×C31—22no—
129C2×Q8×C31—22no—
130C2×Q8×C31—41no—
131C2×Q8×C31C2×D412yes—
132C2×D4×C31—24no—
133C2×D4×C31—22no—
134C2×D4×C31—22no—
135C2×D4×C31—22no—
136C2×D4×C31—22no—
137C2×D4×C31—24no—
138C2×D4×C31—24no—
139C2×D4×C31—22no—
140C2×D4×C31—22no—
141C2×D4×C31—21no—
142C2×D4×C31—22no—
143C2×D4×C31—22no—
144C2×D4×C31—41no—
145C2×D4×C31—41no—
146C2×D4×C31—41no—
147C2×D4×C31—42no—
148C2×D4×C31—42no—
149C2×D4×C31—82no—
150C2×D4×C31—82no—
151C2×D4×C31C2×D412yes—
152C22×C4×C31—21no—
153C22×C4×C31—21no—
154C22×C4×C31—22no—
155C22×C4×C31—22no—
156C22×C4×C31—22no—
157C22×C4×C31—24no—
158C22×C4×C31—21no—
159C22×C4×C31—21no—
160C22×C4×C31—21no—
161C22×C4×C31—24no—
162C22×C4×C31—21no—
163C22×C4×C31—21no—
164C22×C4×C31—22no—
165C22×C4×C31—22no—
166C22×C4×C31—22no—
167C22×C4×C31—22no—
168C22×C4×C31—42no—
169C22×C4×C31—42no—
170C22×C4×C31—42no—
171C22×C4×C31—82no—
172C22×C4×C31—82no—
173C22×C4×C31C2×D411yescharacteristic
174C22×C4×C31C2×D411yescharacteristic
175C22×C4×C31C2×D412yes—
176C22×C4×C31C2×D412yes—
177C22×C4×C31C2×D411yescharacteristic
178C22×C4×C31#4114yes—
179C22×C4×C31C2411yescharacteristic
180C2×C8×C31—22no—
181C2×C8×C31—22no—
182C2×C8×C31—22no—
183C2×C8×C31—22no—
184C2×C8×C31—22no—
185C2×C8×C31—22no—
186C4⋊C4×C31—24no—
187C4⋊C4×C31—24no—
188C4⋊C4×C31—24no—
189C4⋊C4×C31—21no—
190C4⋊C4×C31—21no—
191C4⋊C4×C31—22no—
192C4⋊C4×C31—22no—
193C4⋊C4×C31—22no—
194C4⋊C4×C31—22no—
195C4⋊C4×C31—42no—
196C4⋊C4×C31C2×D412yes—
197C4⋊C4×C31C2×D412yes—
198C22⋊C4×C31—24no—
199C22⋊C4×C31—24no—
200C22⋊C4×C31—24no—
201C22⋊C4×C31—24no—
202C22⋊C4×C31—42no—
203C22⋊C4×C31—82no—
204C22⋊C4×C31—82no—
205C42×C31—42no—
206#24012C3111yesmaximal, characteristic, direct product factor, Sylow
207C23×C31—21no—
208C23×C31—21no—
209C23×C31—21no—
210C23×C31—21no—
211C23×C31—22no—
212C23×C31—22no—
213C23×C31—22no—
214C23×C31—22no—
215C23×C31—21no—
216C23×C31—22no—
217C23×C31—21no—
218C23×C31—21no—
219C23×C31—22no—
220C23×C31—22no—
221C23×C31—41no—
222C23×C31—41no—
223C23×C31—42no—
224C23×C31—42no—
225C23×C31#12012yes—
226C23×C31#12312yes—
227C23×C31C22×D411yescharacteristic
228Q8×C31—24no—
229Q8×C31—44no—
230Q8×C31—42no—
231D4×C31—24no—
232D4×C31—44no—
233D4×C31—44no—
234D4×C31—44no—
235D4×C31—48no—
236D4×C31—42no—
237D4×C31—84no—
238D4×C31—84no—
239C2×C4×C31—21no—
240C2×C4×C31—21no—
241C2×C4×C31—21no—
242C2×C4×C31—21no—
243C2×C4×C31—24no—
244C2×C4×C31—24no—
245C2×C4×C31—24no—
246C2×C4×C31—24no—
247C2×C4×C31—22no—
248C2×C4×C31—22no—
249C2×C4×C31—22no—
250C2×C4×C31—21no—
251C2×C4×C31—21no—
252C2×C4×C31—21no—
253C2×C4×C31—48no—
254C2×C4×C31—41no—
255C2×C4×C31—41no—
256C2×C4×C31—42no—
257C2×C4×C31—42no—
258C2×C4×C31—42no—
259C2×C4×C31—42no—
260C2×C4×C31—42no—
261C2×C4×C31—42no—
262C2×C4×C31—42no—
263C2×C4×C31—42no—
264C2×C4×C31—42no—
265C2×C4×C31—42no—
266C2×C4×C31—84no—
267C2×C4×C31—84no—
268C2×C4×C31#12012yes—
269C2×C4×C31#12312yes—
270C2×C4×C31C22×D411yescharacteristic
271C2×C4×C31C22×D411yescharacteristic
272C2×C4×C31C2×#4112yes—
273C2×C4×C31H(5,𝔽2)12yes—
274C8×C31—42no—
275C8×C31—42no—
276C2×#537C2×C3111yescharacteristic
277C2×#481C2×C3111yescharacteristic, complemented
278C2×#483C2×C3111yescharacteristic, complemented
279#2311C2×C3111yescharacteristic, complemented
280#2035C2×C3112yescomplemented
281#2034C2×C3112yescomplemented
282#1875C2×C3111yescharacteristic, complemented
283#1873C2×C3111yescharacteristic, complemented
284#1853C2×C3112yescomplemented
285#1788C2×C3112yescomplemented
286#1780C2×C3111yescharacteristic, complemented
287C22×C31—24no—
288C22×C31—22no—
289C22×C31—22no—
290C22×C31—22no—
291C22×C31—22no—
292C22×C31—42no—
293C22×C31—42no—
294C22×C31—42no—
295C22×C31—41no—
296C22×C31—41no—
297C22×C31—42no—
298C22×C31—42no—
299C22×C31—41no—
300C22×C31—42no—
301C22×C31—41no—
302C22×C31—42no—
303C22×C31—41no—
304C22×C31—42no—
305C22×C31—82no—
306C22×C31—82no—
307C22×C31C2×#12011yescharacteristic
308C22×C31C2×#12311yescharacteristic
309C22×C31#53411yescharacteristic
310C22×C31#57712yes—
311C22×C31#57712yes—
312C4×C31—22no—
313C4×C31—22no—
314C4×C31—41no—
315C4×C31—42no—
316C4×C31—42no—
317C4×C31—42no—
318C4×C31—82no—
319C4×C31—82no—
320C4×C31#54612yes—
321C4×C31#54612yes—
322C22×#41C22×C3111yescharacteristic
323C22×#41C22×C3111yescharacteristic
324#537—24no—
325#537—24no—
326C2×C22⋊Q8—21no—
327C2×C22⋊Q8C22×C3111yescharacteristic
328C2×C22⋊Q8C22×C3111yescharacteristic
329C2×#121—21no—
330C2×#121—21no—
331C2×#121—21no—
332C2×#121C22×C3111yescharacteristic
333C2×#121C22×C3111yescharacteristic
334C2×#117C22×C3111yescharacteristic
335C22×C4⋊C4C22×C3111yescharacteristic
336C22×C4⋊C4C22×C3111yescharacteristic
337#483—24no—
338#481—24no—
339C2×#106C22×C3112yes—
340C2×#106C22×C3112yes—
341C2×#103C22×C3112yes—
342C2×#103C22×C3112yes—
343C2×#102C22×C3112yes—
344C2×#102C22×C3112yes—
345C2×C22⋊C8C22×C3111yescharacteristic
346C2×C22⋊C8C22×C3111yescharacteristic
347C2×C22⋊C8C22×C3112yes—
348C2×C22⋊C8C22×C3112yes—
349#391—21no—
350#390—21no—
351#386—21no—
352#386—21no—
353#380C22×C3112yes—
354#380C22×C3112yes—
355#340C22×C3112yes—
356#340C22×C3112yes—
357C2×C31—42no—
358C2×C31—42no—
359C2×C31—81no—
360C2×C31—82no—
361C2×C31#240511yescharacteristic
362C2×C31#300911yescharacteristic
363C2×C31#301011yescharacteristic
364C2×C31#309912yes—
365C2×C31#320412yes—
366C2×#41—22no—
367C2×#41—22no—
368C2×#41—22no—
369C2×#41—22no—
370C2×#41—24no—
371C2×#41—24no—
372C2×#41—42no—
373C2×#41—42no—
374C22×Q8D4×C3111yescharacteristic
375C22×Q8C23×C3111yescharacteristic
376C22×D4—21no—
377C22×D4—21no—
378C22×D4—21no—
379C22×D4D4×C3111yescharacteristic
380C22×D4D4×C3111yescharacteristic
381C22×D4D4×C3111yescharacteristic
382C22×D4C23×C3111yescharacteristic
383C23×C4—21no—
384C23×C4—21no—
385C23×C4—21no—
386C23×C4C23×C3111yescharacteristic
387C23×C4C23×C3111yescharacteristic
388C23×C4C23×C3111yescharacteristic
389C22×C8C23×C3112yes—
390C22×C8C23×C3112yes—
391C22⋊Q8—28no—
392C22⋊Q8—22no—
393C22⋊Q8—22no—
394C22⋊Q8—22no—
395C22⋊Q8D4×C3114yes—
396#121—28no—
397#121—28no—
398#121—28no—
399#121—22no—
400#121—22no—
401#121—22no—
402#121D4×C3114yes—
403#117—42no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—22no—
408C2×C4⋊C4—22no—
409C2×C4⋊C4—22no—
410C2×C4⋊C4—22no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4C23×C3112yes—
413C2×C4⋊C4C23×C3112yes—
414C2×C4⋊C4C23×C3111yescharacteristic
415C2×C4⋊C4C23×C3111yescharacteristic
416C2×C22⋊C4—21no—
417C2×C22⋊C4—21no—
418C2×C22⋊C4—21no—
419C2×C22⋊C4—21no—
420C2×C22⋊C4—21no—
421C2×C22⋊C4—22no—
422C2×C22⋊C4—22no—
423C2×C42—21no—
424#106—24no—
425#106—24no—
426#103—24no—
427#103—42no—
428#102—24no—
429#102—42no—
430C22⋊C8—22no—
431C22⋊C8—22no—
432C22⋊C8—24no—
433C22⋊C8—42no—
434#95—22no—
435#95—22no—
436C31#2401211yescharacteristic, direct product factor, Sylow
437C24—21no—
438C24C2×D4×C3111yescharacteristic
439C24C2×D4×C3111yescharacteristic
440C24C2×D4×C3111yescharacteristic
441#41—48no—
442#41—48no—
443#41—48no—
444#41—48no—
445C2×Q8—22no—
446C2×Q8—22no—
447C2×Q8—21no—
448C2×Q8—22no—
449C2×Q8—22no—
450C2×Q8—41no—
451C2×Q8C2×D4×C3112yes—
452C2×D4—24no—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—22no—
456C2×D4—22no—
457C2×D4—24no—
458C2×D4—24no—
459C2×D4—22no—
460C2×D4—22no—
461C2×D4—21no—
462C2×D4—22no—
463C2×D4—22no—
464C2×D4—41no—
465C2×D4—41no—
466C2×D4—41no—
467C2×D4—42no—
468C2×D4—42no—
469C2×D4—82no—
470C2×D4—82no—
471C2×D4C2×D4×C3112yes—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—24no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—24no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—42no—
491C22×C4—82no—
492C22×C4—82no—
493C22×C4C2×D4×C3111yescharacteristic
494C22×C4C2×D4×C3111yescharacteristic
495C22×C4C2×D4×C3112yes—
496C22×C4C2×D4×C3112yes—
497C22×C4C2×D4×C3111yescharacteristic
498C22×C4#41×C3114yes—
499C22×C4C24×C3111yescharacteristic
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C4⋊C4—24no—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—42no—
516C4⋊C4C2×D4×C3112yes—
517C4⋊C4C2×D4×C3112yes—
518C22⋊C4—24no—
519C22⋊C4—24no—
520C22⋊C4—24no—
521C22⋊C4—24no—
522C22⋊C4—42no—
523C22⋊C4—82no—
524C22⋊C4—82no—
525C42—42no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—21no—
535C23—22no—
536C23—21no—
537C23—21no—
538C23—22no—
539C23—22no—
540C23—41no—
541C23—41no—
542C23—42no—
543C23—42no—
544C23C31×#12012yes—
545C23C31×#12312yes—
546C23C22×D4×C3111yescharacteristic
547Q8—24no—
548Q8—44no—
549Q8—42no—
550D4—24no—
551D4—44no—
552D4—44no—
553D4—44no—
554D4—48no—
555D4—42no—
556D4—84no—
557D4—84no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—48no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—84no—
586C2×C4—84no—
587C2×C4C31×#12012yes—
588C2×C4C31×#12312yes—
589C2×C4C22×D4×C3111yescharacteristic
590C2×C4C22×D4×C3111yescharacteristic
591C2×C4C2×#41×C3112yes—
592C2×C4C31×H(5,𝔽2)12yes—
593C8—42no—
594C8—42no—
595C22—24no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—42no—
601C22—42no—
602C22—42no—
603C22—41no—
604C22—41no—
605C22—42no—
606C22—42no—
607C22—41no—
608C22—42no—
609C22—41no—
610C22—42no—
611C22—41no—
612C22—42no—
613C22—82no—
614C22—82no—
615C22C2×C31×#12011yescharacteristic
616C22C2×C31×#12311yescharacteristic
617C22C31×#53411yescharacteristic
618C22C31×#57712yes—
619C22C31×#57712yes—
620C4—22no—
621C4—22no—
622C4—41no—
623C4—42no—
624C4—42no—
625C4—42no—
626C4—82no—
627C4—82no—
628C4C31×#54612yes—
629C4C31×#54612yes—
630C2—42no—
631C2—42no—
632C2—81no—
633C2—82no—
634C2C31×#240511yescharacteristic
635C2C31×#300911yescharacteristic
636C2C31×#301011yescharacteristic
637C2C31×#309912yes—
638C2C31×#320412yes—
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC2×C31×#537C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C22×#41×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×#481C2semidirect
3. C2×C31×#483C2semidirect
4. C31×#2311C2semidirect
5. C31×#2035C2semidirect
6. C31×#2034C2semidirect
7. C31×#1875C2semidirect
8. C31×#1873C2semidirect
9. C31×#1853C2semidirect
10. C31×#1788C2semidirect
11. C31×#1780C2semidirect
206. #24012C31direct (N × H)
277. C2×#481C2×C31semidirect
278. C2×#483C2×C31semidirect
279. #2311C2×C31semidirect
280. #2035C2×C31semidirect
281. #2034C2×C31semidirect
282. #1875C2×C31semidirect
283. #1873C2×C31semidirect
284. #1853C2×C31semidirect
285. #1788C2×C31semidirect
286. #1780C2×C31semidirect
436. C31#24012direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5372
C2×C31×#4812
C2×C31×#4832
C31×#23112
C31×#20352
C31×#20342
C31×#18752
C31×#18732
C31×#18532
C31×#17882
C31×#17802
#2401231