Finite Group Databasebeta

← C31×#24350 #21481109

Subgroups — C31×#24350 #21481109

Order 7936.

1462 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3501C211yesmaximal, characteristic, complemented
2C31×#2848C212yesmaximal, complemented
3C4×C31×#120C211yesmaximal, characteristic
4C31×#2062C212yesmaximal, complemented
5C31×#2029C212yesmaximal, complemented
6C31×#1996C212yesmaximal, complemented
7C31×#1889C212yesmaximal
8C31×#1867C212yesmaximal, complemented
9C31×#1799C211yesmaximal, characteristic, complemented
10C24×C4×C31C2211yescharacteristic
11C31×#554—22no—
12C31×#554—24no—
13D4×Q8×C31—22no—
14C31×#544—21no—
15C31×#543—21no—
16C31×#543C2212yes—
17C31×#533—21no—
18C2×C31×C22⋊Q8—21no—
19C2×C31×C22⋊Q8—22no—
20C2×C31×C22⋊Q8C2212yes—
21C2×C31×#120C2211yescharacteristic
22C31×#518C2212yes—
23C2×C4×D4×C31—21no—
24C2×C4×D4×C31C2211yescharacteristic
25C2×C31×#99C2212yes—
26C31×#386—21no—
27C31×#386C2211yescharacteristic
28C31×#385—21no—
29C31×#385—24no—
30C31×#385C2211yescharacteristic
31C31×#385C2214yes—
32C31×#380—22no—
33C31×#380C2212yes—
34C31×#379C2214yes—
35C31×#379C2211yescharacteristic
36C4×C22⋊C4×C31—21no—
37C4×C22⋊C4×C31C2211yescharacteristic
38C31×#342C2212yes—
39C31×#342C2214yes—
40C31×#342C2214yes—
41C25×C31C2311yescharacteristic
42C22×Q8×C31—21no—
43C22×Q8×C31D412yescomplemented
44C22×D4×C31—21no—
45C22×D4×C31C2311yescharacteristic
46C23×C4×C31—21no—
47C23×C4×C31—22no—
48C23×C4×C31—22no—
49C23×C4×C31—21no—
50C23×C4×C31—44no—
51C23×C4×C31D411yescharacteristic
52C23×C4×C31C2311yescharacteristic
53C23×C4×C31C2311yescharacteristic
54C31×C4⋊Q8—21no—
55C31×C4⋊Q8—21no—
56C31×C4⋊Q8—24no—
57C31×#125—22no—
58C31×#125—24no—
59C31×C22⋊Q8—22no—
60C31×C22⋊Q8—22no—
61C31×C22⋊Q8—24no—
62C31×C22⋊Q8—24no—
63C31×C22⋊Q8—22no—
64C31×C22⋊Q8—22no—
65C31×C22⋊Q8—22no—
66C31×C22⋊Q8—22no—
67C31×C22⋊Q8—22no—
68C31×C22⋊Q8—22no—
69C31×C22⋊Q8—24no—
70C31×C22⋊Q8—24no—
71C31×C22⋊Q8—48no—
72C31×#120—22no—
73C31×#120Q814yescomplemented
74C4×Q8×C31—22no—
75C4×Q8×C31—22no—
76C4×D4×C31—22no—
77C4×D4×C31—22no—
78C4×D4×C31—24no—
79C4×D4×C31—24no—
80C4×D4×C31—44no—
81C31×#117—22no—
82C2×C4⋊C4×C31—21no—
83C2×C4⋊C4×C31—22no—
84C2×C4⋊C4×C31—21no—
85C2×C4⋊C4×C31—22no—
86C2×C4⋊C4×C31—24no—
87C2×C4⋊C4×C31D411yescharacteristic
88C2×C4⋊C4×C31D412yescomplemented
89C2×C4⋊C4×C31D414yescomplemented
90C2×C22⋊C4×C31—22no—
91C2×C22⋊C4×C31—21no—
92C2×C22⋊C4×C31—21no—
93C2×C22⋊C4×C31—21no—
94C2×C22⋊C4×C31—24no—
95C2×C22⋊C4×C31—21no—
96C2×C22⋊C4×C31—24no—
97C2×C22⋊C4×C31D411yescharacteristic
98C2×C22⋊C4×C31C2314yes—
99C2×C22⋊C4×C31C2311yescharacteristic
100C2×C22⋊C4×C31C2311yescharacteristic
101C2×C22⋊C4×C31C2314yes—
102C2×C22⋊C4×C31C2311yescharacteristic
103C2×C42×C31—21no—
104C2×C42×C31D411yescharacteristic
105C31×#99—24no—
106C31×#95—21no—
107C31×#95—21no—
108C31×#95—21no—
109C31×#95—24no—
110C31×#95—22no—
111C24×C31—22no—
112C24×C31—21no—
113C24×C31—21no—
114C24×C31—22no—
115C24×C31—42no—
116C24×C31—42no—
117C24×C31C2×D411yescharacteristic
118C24×C31C2×Q812yescomplemented
119C24×C31C2411yescharacteristic
120C2×Q8×C31—22no—
121C2×Q8×C31—22no—
122C2×Q8×C31—22no—
123C2×Q8×C31—22no—
124C2×Q8×C31—44no—
125C2×Q8×C31—82no—
126C2×D4×C31—22no—
127C2×D4×C31—22no—
128C2×D4×C31—24no—
129C2×D4×C31—42no—
130C2×D4×C31—42no—
131C2×D4×C31C2×Q812yes—
132C2×D4×C31#4112yes—
133C22×C4×C31—24no—
134C22×C4×C31—28no—
135C22×C4×C31—21no—
136C22×C4×C31—21no—
137C22×C4×C31—22no—
138C22×C4×C31—22no—
139C22×C4×C31—22no—
140C22×C4×C31—21no—
141C22×C4×C31—21no—
142C22×C4×C31—21no—
143C22×C4×C31—24no—
144C22×C4×C31—42no—
145C22×C4×C31—42no—
146C22×C4×C31—42no—
147C22×C4×C31—44no—
148C22×C4×C31—42no—
149C22×C4×C31—44no—
150C22×C4×C31—42no—
151C22×C4×C31—44no—
152C22×C4×C31—42no—
153C22×C4×C31—41no—
154C22×C4×C31—42no—
155C22×C4×C31—41no—
156C22×C4×C31C2×D411yescharacteristic
157C22×C4×C31C2×D412yes—
158C22×C4×C31C2×D411yescharacteristic
159C22×C4×C31C2×D411yescharacteristic
160C22×C4×C31C2×D414yes—
161C22×C4×C31C2×D411yescharacteristic
162C22×C4×C31C2×D414yes—
163C22×C4×C31C2×D411yescharacteristic
164C22×C4×C31C2×D412yes—
165C4⋊C4×C31—22no—
166C4⋊C4×C31—24no—
167C4⋊C4×C31—22no—
168C4⋊C4×C31—22no—
169C4⋊C4×C31—24no—
170C4⋊C4×C31—22no—
171C4⋊C4×C31—24no—
172C4⋊C4×C31—22no—
173C4⋊C4×C31—22no—
174C4⋊C4×C31—22no—
175C4⋊C4×C31—24no—
176C4⋊C4×C31—42no—
177C4⋊C4×C31—44no—
178C4⋊C4×C31—48no—
179C22⋊C4×C31—24no—
180C22⋊C4×C31—24no—
181C22⋊C4×C31—22no—
182C22⋊C4×C31—24no—
183C22⋊C4×C31—21no—
184C22⋊C4×C31—22no—
185C22⋊C4×C31—24no—
186C22⋊C4×C31—24no—
187C22⋊C4×C31—22no—
188C22⋊C4×C31—24no—
189C22⋊C4×C31—42no—
190C22⋊C4×C31—48no—
191C22⋊C4×C31—42no—
192C22⋊C4×C31—42no—
193C22⋊C4×C31—48no—
194C22⋊C4×C31C2×Q812yescomplemented
195C42×C31—24no—
196C42×C31—21no—
197C42×C31—21no—
198C42×C31—42no—
199#24350C3111yesmaximal, characteristic, direct product factor, Sylow
200C23×C31—24no—
201C23×C31—24no—
202C23×C31—22no—
203C23×C31—22no—
204C23×C31—21no—
205C23×C31—22no—
206C23×C31—24no—
207C23×C31—21no—
208C23×C31—41no—
209C23×C31—44no—
210C23×C31—41no—
211C23×C31—42no—
212C23×C31—41no—
213C23×C31—42no—
214C23×C31—41no—
215C23×C31—44no—
216C23×C31—48no—
217C23×C31—41no—
218C23×C31—41no—
219C23×C31—41no—
220C23×C31—41no—
221C23×C31—42no—
222C23×C31C22⋊Q812yescomplemented
223C23×C31C22×D411yescharacteristic
224C23×C31C22×D412yes—
225C23×C31C22×Q811yescharacteristic
226C23×C31C2×#4111yescharacteristic
227C23×C312−1+412yes—
228Q8×C31—44no—
229Q8×C31—82no—
230Q8×C31—82no—
231Q8×C31—82no—
232D4×C31—24no—
233D4×C31—24no—
234D4×C31—48no—
235C2×C4×C31—21no—
236C2×C4×C31—21no—
237C2×C4×C31—22no—
238C2×C4×C31—24no—
239C2×C4×C31—22no—
240C2×C4×C31—24no—
241C2×C4×C31—21no—
242C2×C4×C31—24no—
243C2×C4×C31—24no—
244C2×C4×C31—21no—
245C2×C4×C31—22no—
246C2×C4×C31—42no—
247C2×C4×C31—41no—
248C2×C4×C31—44no—
249C2×C4×C31—48no—
250C2×C4×C31—41no—
251C2×C4×C31—42no—
252C2×C4×C31—41no—
253C2×C4×C31—41no—
254C2×C4×C31—41no—
255C2×C4×C31—44no—
256C2×C4×C31—42no—
257C2×C4×C31—41no—
258C2×C4×C31—44no—
259C2×C4×C31—44no—
260C2×C4×C31—41no—
261C2×C4×C31—41no—
262C2×C4×C31—44no—
263C2×C4×C31—44no—
264C2×C4×C31—41no—
265C2×C4×C31—41no—
266C2×C4×C31—44no—
267C2×C4×C31—81no—
268C2×C4×C31—82no—
269C2×C4×C31—82no—
270C2×C4×C31#12011yescharacteristic
271C2×C4×C31#12012yes—
272C2×C4×C31#12011yescharacteristic
273C2×C4×C31C22⋊Q812yescomplemented
274#3501C2×C3111yescharacteristic, complemented
275#2848C2×C3112yescomplemented
276C4×#120C2×C3111yescharacteristic
277#2062C2×C3112yescomplemented
278#2029C2×C3112yescomplemented
279#1996C2×C3112yescomplemented
280#1889C2×C3112yes—
281#1867C2×C3112yescomplemented
282#1799C2×C3111yescharacteristic, complemented
283C22×C31—22no—
284C22×C31—21no—
285C22×C31—22no—
286C22×C31—21no—
287C22×C31—24no—
288C22×C31—28no—
289C22×C31—22no—
290C22×C31—41no—
291C22×C31—41no—
292C22×C31—42no—
293C22×C31—42no—
294C22×C31—42no—
295C22×C31—44no—
296C22×C31—44no—
297C22×C31—42no—
298C22×C31—42no—
299C22×C31—44no—
300C22×C31—42no—
301C22×C31—42no—
302C22×C31—41no—
303C22×C31—41no—
304C22×C31—42no—
305C22×C31—81no—
306C22×C31C2×#12011yescharacteristic
307C22×C31C2×C22⋊Q811yescharacteristic
308C22×C31#53611yescharacteristic
309C22×C31#54812yes—
310C22×C31D4×Q812yes—
311C4×C31—22no—
312C4×C31—21no—
313C4×C31—22no—
314C4×C31—42no—
315C4×C31—42no—
316C4×C31—41no—
317C4×C31—44no—
318C4×C31—44no—
319C4×C31—41no—
320C4×C31—84no—
321C4×C31—81no—
322C4×C31B(4,𝔽2)12yes—
323C24×C4C22×C3111yescharacteristic
324#554—22no—
325#554—24no—
326D4×Q8—22no—
327#544—21no—
328#543—21no—
329#543C22×C3112yes—
330#533—21no—
331C2×C22⋊Q8—21no—
332C2×C22⋊Q8—22no—
333C2×C22⋊Q8C22×C3112yes—
334C2×#120C22×C3111yescharacteristic
335#518C22×C3112yes—
336C2×C4×D4—21no—
337C2×C4×D4C22×C3111yescharacteristic
338C2×#99C22×C3112yes—
339#386—21no—
340#386C22×C3111yescharacteristic
341#385—21no—
342#385—24no—
343#385C22×C3111yescharacteristic
344#385C22×C3114yes—
345#380—22no—
346#380C22×C3112yes—
347#379C22×C3114yes—
348#379C22×C3111yescharacteristic
349C4×C22⋊C4—21no—
350C4×C22⋊C4C22×C3111yescharacteristic
351#342C22×C3112yes—
352#342C22×C3114yes—
353#342C22×C3114yes—
354C2×C31—21no—
355C2×C31—22no—
356C2×C31—21no—
357C2×C31—22no—
358C2×C31—44no—
359C2×C31—42no—
360C2×C31#240911yescharacteristic
361C2×C31C2×B(4,𝔽2)11yescharacteristic
362C2×C31#302511yescharacteristic
363C25C23×C3111yescharacteristic
364C22×Q8—21no—
365C22×Q8D4×C3112yescomplemented
366C22×D4—21no—
367C22×D4C23×C3111yescharacteristic
368C23×C4—21no—
369C23×C4—22no—
370C23×C4—22no—
371C23×C4—21no—
372C23×C4—44no—
373C23×C4D4×C3111yescharacteristic
374C23×C4C23×C3111yescharacteristic
375C23×C4C23×C3111yescharacteristic
376C4⋊Q8—21no—
377C4⋊Q8—21no—
378C4⋊Q8—24no—
379#125—22no—
380#125—24no—
381C22⋊Q8—22no—
382C22⋊Q8—22no—
383C22⋊Q8—24no—
384C22⋊Q8—24no—
385C22⋊Q8—22no—
386C22⋊Q8—22no—
387C22⋊Q8—22no—
388C22⋊Q8—22no—
389C22⋊Q8—22no—
390C22⋊Q8—22no—
391C22⋊Q8—24no—
392C22⋊Q8—24no—
393C22⋊Q8—48no—
394#120—22no—
395#120Q8×C3114yescomplemented
396C4×Q8—22no—
397C4×Q8—22no—
398C4×D4—22no—
399C4×D4—22no—
400C4×D4—24no—
401C4×D4—24no—
402C4×D4—44no—
403#117—22no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—22no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—22no—
408C2×C4⋊C4—24no—
409C2×C4⋊C4D4×C3111yescharacteristic
410C2×C4⋊C4D4×C3112yescomplemented
411C2×C4⋊C4D4×C3114yescomplemented
412C2×C22⋊C4—22no—
413C2×C22⋊C4—21no—
414C2×C22⋊C4—21no—
415C2×C22⋊C4—21no—
416C2×C22⋊C4—24no—
417C2×C22⋊C4—21no—
418C2×C22⋊C4—24no—
419C2×C22⋊C4D4×C3111yescharacteristic
420C2×C22⋊C4C23×C3111yescharacteristic
421C2×C22⋊C4C23×C3114yes—
422C2×C22⋊C4C23×C3111yescharacteristic
423C2×C22⋊C4C23×C3114yes—
424C2×C22⋊C4C23×C3111yescharacteristic
425C2×C42—21no—
426C2×C42D4×C3111yescharacteristic
427#99—24no—
428#95—21no—
429#95—21no—
430#95—21no—
431#95—24no—
432#95—22no—
433C31#2435011yescharacteristic, direct product factor, Sylow
434C24—22no—
435C24—21no—
436C24—21no—
437C24—22no—
438C24—42no—
439C24—42no—
440C24C2×D4×C3111yescharacteristic
441C24C2×Q8×C3112yescomplemented
442C24C24×C3111yescharacteristic
443C2×Q8—22no—
444C2×Q8—22no—
445C2×Q8—22no—
446C2×Q8—22no—
447C2×Q8—44no—
448C2×Q8—82no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—24no—
452C2×D4—42no—
453C2×D4—42no—
454C2×D4C2×Q8×C3112yes—
455C2×D4#41×C3112yes—
456C22×C4—24no—
457C22×C4—28no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—24no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—44no—
471C22×C4—42no—
472C22×C4—44no—
473C22×C4—42no—
474C22×C4—44no—
475C22×C4—42no—
476C22×C4—41no—
477C22×C4—42no—
478C22×C4—41no—
479C22×C4C2×D4×C3111yescharacteristic
480C22×C4C2×D4×C3112yes—
481C22×C4C2×D4×C3111yescharacteristic
482C22×C4C2×D4×C3111yescharacteristic
483C22×C4C2×D4×C3114yes—
484C22×C4C2×D4×C3111yescharacteristic
485C22×C4C2×D4×C3114yes—
486C22×C4C2×D4×C3111yescharacteristic
487C22×C4C2×D4×C3112yes—
488C4⋊C4—22no—
489C4⋊C4—24no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—24no—
493C4⋊C4—22no—
494C4⋊C4—24no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—24no—
499C4⋊C4—42no—
500C4⋊C4—44no—
501C4⋊C4—48no—
502C22⋊C4—24no—
503C22⋊C4—24no—
504C22⋊C4—22no—
505C22⋊C4—24no—
506C22⋊C4—21no—
507C22⋊C4—22no—
508C22⋊C4—24no—
509C22⋊C4—24no—
510C22⋊C4—22no—
511C22⋊C4—24no—
512C22⋊C4—42no—
513C22⋊C4—48no—
514C22⋊C4—42no—
515C22⋊C4—42no—
516C22⋊C4—48no—
517C22⋊C4C2×Q8×C3112yescomplemented
518C42—24no—
519C42—21no—
520C42—21no—
521C42—42no—
522C23—24no—
523C23—24no—
524C23—22no—
525C23—22no—
526C23—21no—
527C23—22no—
528C23—24no—
529C23—21no—
530C23—41no—
531C23—44no—
532C23—41no—
533C23—42no—
534C23—41no—
535C23—42no—
536C23—41no—
537C23—44no—
538C23—48no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—42no—
544C23C31×C22⋊Q812yescomplemented
545C23C22×D4×C3111yescharacteristic
546C23C22×D4×C3112yes—
547C23C22×Q8×C3111yescharacteristic
548C23C2×#41×C3111yescharacteristic
549C23C31×2−1+412yes—
550Q8—44no—
551Q8—82no—
552Q8—82no—
553Q8—82no—
554D4—24no—
555D4—24no—
556D4—48no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—21no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—44no—
571C2×C4—48no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—44no—
589C2×C4—81no—
590C2×C4—82no—
591C2×C4—82no—
592C2×C4C31×#12011yescharacteristic
593C2×C4C31×#12012yes—
594C2×C4C31×#12011yescharacteristic
595C2×C4C31×C22⋊Q812yescomplemented
596C22—22no—
597C22—21no—
598C22—22no—
599C22—21no—
600C22—24no—
601C22—28no—
602C22—22no—
603C22—41no—
604C22—41no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—44no—
609C22—44no—
610C22—42no—
611C22—42no—
612C22—44no—
613C22—42no—
614C22—42no—
615C22—41no—
616C22—41no—
617C22—42no—
618C22—81no—
619C22C2×C31×#12011yescharacteristic
620C22C2×C31×C22⋊Q811yescharacteristic
621C22C31×#53611yescharacteristic
622C22C31×#54812yes—
623C22D4×Q8×C3112yes—
624C4—22no—
625C4—21no—
626C4—22no—
627C4—42no—
628C4—42no—
629C4—41no—
630C4—44no—
631C4—44no—
632C4—41no—
633C4—84no—
634C4—81no—
635C4C31×B(4,𝔽2)12yes—
636C2—21no—
637C2—22no—
638C2—21no—
639C2—22no—
640C2—44no—
641C2—42no—
642C2C31×#240911yescharacteristic
643C2C2×C31×B(4,𝔽2)11yescharacteristic
644C2C31×#302511yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC24C24×C31
prime radicalC2×C31×#120C22
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#3501C2semidirect
2. C31×#2848C2semidirect
4. C31×#2062C2semidirect
5. C31×#2029C2semidirect
6. C31×#1996C2semidirect
8. C31×#1867C2semidirect
9. C31×#1799C2semidirect
43. C22×Q8×C31D4semidirect
73. C31×#120Q8semidirect
88. C2×C4⋊C4×C31D4semidirect
89. C2×C4⋊C4×C31D4semidirect
118. C24×C31C2×Q8semidirect
194. C22⋊C4×C31C2×Q8semidirect
199. #24350C31direct (N × H)
222. C23×C31C22⋊Q8semidirect
273. C2×C4×C31C22⋊Q8semidirect
274. #3501C2×C31semidirect
275. #2848C2×C31semidirect
277. #2062C2×C31semidirect
278. #2029C2×C31semidirect
279. #1996C2×C31semidirect
281. #1867C2×C31semidirect
282. #1799C2×C31semidirect
365. C22×Q8D4×C31semidirect
395. #120Q8×C31semidirect
410. C2×C4⋊C4D4×C31semidirect
411. C2×C4⋊C4D4×C31semidirect
433. C31#24350direct (N × H)
441. C24C2×Q8×C31semidirect
517. C22⋊C4C2×Q8×C31semidirect
544. C23C31×C22⋊Q8semidirect
595. C2×C4C31×C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C31×#35012
C31×#28482
C4×C31×#1202
C31×#20622
C31×#20292
C31×#19962
C31×#18892
C31×#18672
C31×#17992
#2435031