Finite Group Databasebeta

← C31×#24352 #21481111

Subgroups — C31×#24352 #21481111

Order 7936.

1198 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3501C211yesmaximal, characteristic, complemented
2C31×#2626C212yesmaximal, complemented
3C31×#2321C211yesmaximal, characteristic
4C31×#2062C212yesmaximal, complemented
5C31×#2029C212yesmaximal, complemented
6C31×#1996C212yesmaximal, complemented
7C31×#1889C212yesmaximal
8C31×#1867C212yesmaximal, complemented
9C31×#1799C211yesmaximal, characteristic, complemented
10C31×#554—24no—
11C31×#554—22no—
12D4×Q8×C31—22no—
13C31×#544—21no—
14C31×#543—21no—
15C31×#543C2212yes—
16C31×#533—21no—
17C2×C31×C22⋊Q8—21no—
18C2×C31×C22⋊Q8C2212yes—
19C2×C31×#120C2211yescharacteristic
20C31×#518C2212yes—
21C2×C4×D4×C31C2211yescharacteristic
22C22×C22⋊C4×C31C2211yescharacteristic
23C2×C31×#99C2212yes—
24C31×#399—22no—
25C31×#396—24no—
26C31×#393—22no—
27C31×#390—21no—
28C31×#388—21no—
29C31×#388—21no—
30C31×#386—21no—
31C31×#386C2211yescharacteristic
32C31×#385C2211yescharacteristic
33C31×#385C2214yes—
34C31×#380C2212yes—
35C31×#379C2211yescharacteristic
36C31×#379C2214yes—
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×C31D411yescharacteristic
47C23×C4×C31C2311yescharacteristic
48C23×C4×C31C2311yescharacteristic
49C31×C4⋊Q8—21no—
50C31×C4⋊Q8—21no—
51C31×C4⋊Q8—24no—
52C31×#125—22no—
53C31×#125—24no—
54C31×C22⋊Q8—22no—
55C31×C22⋊Q8—22no—
56C31×C22⋊Q8—24no—
57C31×C22⋊Q8—24no—
58C31×C22⋊Q8—22no—
59C31×C22⋊Q8—22no—
60C31×C22⋊Q8—22no—
61C31×C22⋊Q8—22no—
62C31×C22⋊Q8—22no—
63C31×C22⋊Q8—22no—
64C31×C22⋊Q8—24no—
65C31×C22⋊Q8—24no—
66C31×#120—22no—
67C31×#120Q814yescomplemented
68C4×Q8×C31—22no—
69C4×Q8×C31—22no—
70C4×D4×C31—22no—
71C4×D4×C31—22no—
72C4×D4×C31—24no—
73C4×D4×C31—24no—
74C31×#117—22no—
75C2×C4⋊C4×C31—21no—
76C2×C4⋊C4×C31—22no—
77C2×C4⋊C4×C31—22no—
78C2×C4⋊C4×C31—24no—
79C2×C4⋊C4×C31—21no—
80C2×C4⋊C4×C31D411yescharacteristic
81C2×C4⋊C4×C31D414yescomplemented
82C2×C4⋊C4×C31D412yescomplemented
83C2×C22⋊C4×C31—22no—
84C2×C22⋊C4×C31—21no—
85C2×C22⋊C4×C31—22no—
86C2×C22⋊C4×C31—22no—
87C2×C22⋊C4×C31—21no—
88C2×C22⋊C4×C31—21no—
89C2×C22⋊C4×C31—21no—
90C2×C22⋊C4×C31—24no—
91C2×C22⋊C4×C31—24no—
92C2×C22⋊C4×C31—21no—
93C2×C22⋊C4×C31—21no—
94C2×C22⋊C4×C31D411yescharacteristic
95C2×C22⋊C4×C31C2314yes—
96C2×C22⋊C4×C31C2311yescharacteristic
97C2×C22⋊C4×C31C2311yescharacteristic
98C2×C22⋊C4×C31C2314yes—
99C2×C22⋊C4×C31C2311yescharacteristic
100C2×C42×C31—21no—
101C2×C42×C31D411yescharacteristic
102C31×#99—24no—
103C31×#95—24no—
104C31×#95—22no—
105C31×#95—21no—
106C31×#95—21no—
107C31×#95—21no—
108C24×C31—22no—
109C24×C31—21no—
110C24×C31—21no—
111C24×C31—22no—
112C24×C31—81no—
113C24×C31—81no—
114C24×C31C2×D411yescharacteristic
115C24×C31C2×Q812yescomplemented
116C24×C31C2411yescharacteristic
117C2×Q8×C31—22no—
118C2×Q8×C31—22no—
119C2×Q8×C31—22no—
120C2×Q8×C31—22no—
121C2×Q8×C31—82no—
122C2×Q8×C31—82no—
123C2×D4×C31—22no—
124C2×D4×C31—22no—
125C2×D4×C31—24no—
126C2×D4×C31—81no—
127C2×D4×C31—81no—
128C2×D4×C31C2×Q812yes—
129C2×D4×C31#4112yes—
130C22×C4×C31—21no—
131C22×C4×C31—22no—
132C22×C4×C31—22no—
133C22×C4×C31—21no—
134C22×C4×C31—21no—
135C22×C4×C31—21no—
136C22×C4×C31—21no—
137C22×C4×C31—24no—
138C22×C4×C31—22no—
139C22×C4×C31—81no—
140C22×C4×C31—81no—
141C22×C4×C31—81no—
142C22×C4×C31C2×D411yescharacteristic
143C22×C4×C31C2×D411yescharacteristic
144C22×C4×C31C2×D414yes—
145C22×C4×C31C2×D411yescharacteristic
146C22×C4×C31C2×D414yes—
147C22×C4×C31C2×D411yescharacteristic
148C22×C4×C31C2×D412yes—
149C22×C4×C31C2×D411yescharacteristic
150C22×C4×C31C2×D412yes—
151C4⋊C4×C31—22no—
152C4⋊C4×C31—24no—
153C4⋊C4×C31—22no—
154C4⋊C4×C31—22no—
155C4⋊C4×C31—24no—
156C4⋊C4×C31—22no—
157C4⋊C4×C31—24no—
158C4⋊C4×C31—22no—
159C4⋊C4×C31—22no—
160C4⋊C4×C31—22no—
161C4⋊C4×C31—24no—
162C4⋊C4×C31—82no—
163C4⋊C4×C31—81no—
164C4⋊C4×C31—84no—
165C22⋊C4×C31—24no—
166C22⋊C4×C31—24no—
167C22⋊C4×C31—22no—
168C22⋊C4×C31—24no—
169C22⋊C4×C31—21no—
170C22⋊C4×C31—22no—
171C22⋊C4×C31—24no—
172C22⋊C4×C31—24no—
173C22⋊C4×C31—22no—
174C22⋊C4×C31—24no—
175C22⋊C4×C31—41no—
176C22⋊C4×C31—41no—
177C22⋊C4×C31—42no—
178C22⋊C4×C31—81no—
179C22⋊C4×C31—84no—
180C22⋊C4×C31—81no—
181C22⋊C4×C31—84no—
182C22⋊C4×C31—81no—
183C22⋊C4×C31C2×Q812yescomplemented
184C42×C31—24no—
185C42×C31—21no—
186C42×C31—21no—
187C42×C31—81no—
188#24352C3111yesmaximal, characteristic, direct product factor, Sylow
189C23×C31—21no—
190C23×C31—21no—
191C23×C31—22no—
192C23×C31—24no—
193C23×C31—41no—
194C23×C31—41no—
195C23×C31—41no—
196C23×C31—41no—
197C23×C31—44no—
198C23×C31—41no—
199C23×C31—44no—
200C23×C31—41no—
201C23×C31—41no—
202C23×C31—41no—
203C23×C31—41no—
204C23×C31—41no—
205C23×C31—42no—
206C23×C31—82no—
207C23×C31—82no—
208C23×C31—84no—
209C23×C31C22⋊Q812yes—
210C23×C31C22×D411yescharacteristic
211C23×C31C22×D412yes—
212C23×C31C22×Q811yescharacteristic
213C23×C31C2×#4111yescharacteristic
214C23×C312−1+412yes—
215Q8×C31—44no—
216Q8×C31—82no—
217Q8×C31—82no—
218Q8×C31—82no—
219D4×C31—24no—
220D4×C31—24no—
221D4×C31—84no—
222C2×C4×C31—21no—
223C2×C4×C31—21no—
224C2×C4×C31—21no—
225C2×C4×C31—24no—
226C2×C4×C31—24no—
227C2×C4×C31—21no—
228C2×C4×C31—22no—
229C2×C4×C31—41no—
230C2×C4×C31—42no—
231C2×C4×C31—41no—
232C2×C4×C31—42no—
233C2×C4×C31—42no—
234C2×C4×C31—41no—
235C2×C4×C31—41no—
236C2×C4×C31—44no—
237C2×C4×C31—41no—
238C2×C4×C31—41no—
239C2×C4×C31—44no—
240C2×C4×C31—41no—
241C2×C4×C31—41no—
242C2×C4×C31—44no—
243C2×C4×C31—44no—
244C2×C4×C31—41no—
245C2×C4×C31—44no—
246C2×C4×C31—41no—
247C2×C4×C31—41no—
248C2×C4×C31—41no—
249C2×C4×C31—44no—
250C2×C4×C31—81no—
251C2×C4×C31—82no—
252C2×C4×C31—82no—
253C2×C4×C31#12011yescharacteristic
254C2×C4×C31#12011yescharacteristic
255C2×C4×C31#12012yes—
256C2×C4×C31C22⋊Q812yes—
257#3501C2×C3111yescharacteristic, complemented
258#2626C2×C3112yescomplemented
259#2321C2×C3111yescharacteristic
260#2062C2×C3112yescomplemented
261#2029C2×C3112yescomplemented
262#1996C2×C3112yescomplemented
263#1889C2×C3112yes—
264#1867C2×C3112yescomplemented
265#1799C2×C3111yescharacteristic, complemented
266C22×C31—22no—
267C22×C31—21no—
268C22×C31—22no—
269C22×C31—21no—
270C22×C31—22no—
271C22×C31—41no—
272C22×C31—41no—
273C22×C31—42no—
274C22×C31—42no—
275C22×C31—42no—
276C22×C31—42no—
277C22×C31—48no—
278C22×C31—41no—
279C22×C31—41no—
280C22×C31—42no—
281C22×C31—81no—
282C22×C31—84no—
283C22×C31—81no—
284C22×C31—81no—
285C22×C31—81no—
286C22×C31—81no—
287C22×C31C2×#12011yescharacteristic
288C22×C31C2×C22⋊Q811yescharacteristic
289C22×C31#53611yescharacteristic
290C22×C31#54812yes—
291C22×C31D4×Q812yes—
292C4×C31—22no—
293C4×C31—21no—
294C4×C31—21no—
295C4×C31—22no—
296C4×C31—42no—
297C4×C31—41no—
298C4×C31—44no—
299C4×C31—44no—
300C4×C31—41no—
301C4×C31—84no—
302C4×C31—81no—
303C4×C31—81no—
304#554—24no—
305#554—22no—
306D4×Q8—22no—
307#544—21no—
308#543—21no—
309#543C22×C3112yes—
310#533—21no—
311C2×C22⋊Q8—21no—
312C2×C22⋊Q8C22×C3112yes—
313C2×#120C22×C3111yescharacteristic
314#518C22×C3112yes—
315C2×C4×D4C22×C3111yescharacteristic
316C22×C22⋊C4C22×C3111yescharacteristic
317C2×#99C22×C3112yes—
318#399—22no—
319#396—24no—
320#393—22no—
321#390—21no—
322#388—21no—
323#388—21no—
324#386—21no—
325#386C22×C3111yescharacteristic
326#385C22×C3111yescharacteristic
327#385C22×C3114yes—
328#380C22×C3112yes—
329#379C22×C3111yescharacteristic
330#379C22×C3114yes—
331C4×C22⋊C4C22×C3111yescharacteristic
332#342C22×C3112yes—
333#342C22×C3114yes—
334#342C22×C3114yes—
335C2×C31—21no—
336C2×C31—22no—
337C2×C31—21no—
338C2×C31—22no—
339C2×C31—42no—
340C2×C31—82no—
341C2×C31#240911yescharacteristic
342C2×C31#302611yescharacteristic
343C2×C31#302711yescharacteristic
344C25C23×C3111yescharacteristic
345C22×Q8—21no—
346C22×Q8D4×C3112yescomplemented
347C22×D4—21no—
348C22×D4C23×C3111yescharacteristic
349C23×C4D4×C3111yescharacteristic
350C23×C4C23×C3111yescharacteristic
351C23×C4C23×C3111yescharacteristic
352C4⋊Q8—21no—
353C4⋊Q8—21no—
354C4⋊Q8—24no—
355#125—22no—
356#125—24no—
357C22⋊Q8—22no—
358C22⋊Q8—22no—
359C22⋊Q8—24no—
360C22⋊Q8—24no—
361C22⋊Q8—22no—
362C22⋊Q8—22no—
363C22⋊Q8—22no—
364C22⋊Q8—22no—
365C22⋊Q8—22no—
366C22⋊Q8—22no—
367C22⋊Q8—24no—
368C22⋊Q8—24no—
369#120—22no—
370#120Q8×C3114yescomplemented
371C4×Q8—22no—
372C4×Q8—22no—
373C4×D4—22no—
374C4×D4—22no—
375C4×D4—24no—
376C4×D4—24no—
377#117—22no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—22no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—24no—
383C2×C4⋊C4D4×C3111yescharacteristic
384C2×C4⋊C4D4×C3114yescomplemented
385C2×C4⋊C4D4×C3112yescomplemented
386C2×C22⋊C4—22no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—22no—
391C2×C22⋊C4—22no—
392C2×C22⋊C4—21no—
393C2×C22⋊C4—21no—
394C2×C22⋊C4—21no—
395C2×C22⋊C4—24no—
396C2×C22⋊C4—24no—
397C2×C22⋊C4D4×C3111yescharacteristic
398C2×C22⋊C4C23×C3111yescharacteristic
399C2×C22⋊C4C23×C3114yes—
400C2×C22⋊C4C23×C3111yescharacteristic
401C2×C22⋊C4C23×C3114yes—
402C2×C22⋊C4C23×C3111yescharacteristic
403C2×C42—21no—
404C2×C42D4×C3111yescharacteristic
405#99—24no—
406#95—21no—
407#95—22no—
408#95—21no—
409#95—24no—
410#95—21no—
411C31#2435211yescharacteristic, direct product factor, Sylow
412C24—22no—
413C24—21no—
414C24—21no—
415C24—22no—
416C24—81no—
417C24—81no—
418C24C2×D4×C3111yescharacteristic
419C24C2×Q8×C3112yescomplemented
420C24C24×C3111yescharacteristic
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—82no—
426C2×Q8—82no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—24no—
430C2×D4—81no—
431C2×D4—81no—
432C2×D4C2×Q8×C3112yes—
433C2×D4#41×C3112yes—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—81no—
444C22×C4—81no—
445C22×C4—81no—
446C22×C4C2×D4×C3111yescharacteristic
447C22×C4C2×D4×C3111yescharacteristic
448C22×C4C2×D4×C3114yes—
449C22×C4C2×D4×C3111yescharacteristic
450C22×C4C2×D4×C3114yes—
451C22×C4C2×D4×C3111yescharacteristic
452C22×C4C2×D4×C3112yes—
453C22×C4C2×D4×C3111yescharacteristic
454C22×C4C2×D4×C3112yes—
455C4⋊C4—22no—
456C4⋊C4—24no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—24no—
460C4⋊C4—22no—
461C4⋊C4—24no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—24no—
466C4⋊C4—82no—
467C4⋊C4—81no—
468C4⋊C4—84no—
469C22⋊C4—24no—
470C22⋊C4—24no—
471C22⋊C4—22no—
472C22⋊C4—24no—
473C22⋊C4—21no—
474C22⋊C4—22no—
475C22⋊C4—24no—
476C22⋊C4—24no—
477C22⋊C4—22no—
478C22⋊C4—24no—
479C22⋊C4—41no—
480C22⋊C4—41no—
481C22⋊C4—42no—
482C22⋊C4—81no—
483C22⋊C4—84no—
484C22⋊C4—81no—
485C22⋊C4—84no—
486C22⋊C4—81no—
487C22⋊C4C2×Q8×C3112yescomplemented
488C42—24no—
489C42—21no—
490C42—21no—
491C42—81no—
492C23—21no—
493C23—21no—
494C23—22no—
495C23—24no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—44no—
501C23—41no—
502C23—44no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—42no—
509C23—82no—
510C23—82no—
511C23—84no—
512C23C31×C22⋊Q812yes—
513C23C22×D4×C3111yescharacteristic
514C23C22×D4×C3112yes—
515C23C22×Q8×C3111yescharacteristic
516C23C2×#41×C3111yescharacteristic
517C23C31×2−1+412yes—
518Q8—44no—
519Q8—82no—
520Q8—82no—
521Q8—82no—
522D4—24no—
523D4—24no—
524D4—84no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—44no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—44no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—41no—
548C2×C4—44no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—44no—
553C2×C4—81no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4C31×#12011yescharacteristic
557C2×C4C31×#12011yescharacteristic
558C2×C4C31×#12012yes—
559C2×C4C31×C22⋊Q812yes—
560C22—22no—
561C22—21no—
562C22—22no—
563C22—21no—
564C22—22no—
565C22—41no—
566C22—41no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—48no—
572C22—41no—
573C22—41no—
574C22—42no—
575C22—81no—
576C22—84no—
577C22—81no—
578C22—81no—
579C22—81no—
580C22—81no—
581C22C2×C31×#12011yescharacteristic
582C22C2×C31×C22⋊Q811yescharacteristic
583C22C31×#53611yescharacteristic
584C22C31×#54812yes—
585C22D4×Q8×C3112yes—
586C4—22no—
587C4—21no—
588C4—21no—
589C4—22no—
590C4—42no—
591C4—41no—
592C4—44no—
593C4—44no—
594C4—41no—
595C4—84no—
596C4—81no—
597C4—81no—
598C2—21no—
599C2—22no—
600C2—21no—
601C2—22no—
602C2—42no—
603C2—82no—
604C2C31×#240911yescharacteristic
605C2C31×#302611yescharacteristic
606C2C31×#302711yescharacteristic
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×#2626C2semidirect
4. C31×#2062C2semidirect
5. C31×#2029C2semidirect
6. C31×#1996C2semidirect
8. C31×#1867C2semidirect
9. C31×#1799C2semidirect
43. C22×Q8×C31D4semidirect
67. C31×#120Q8semidirect
81. C2×C4⋊C4×C31D4semidirect
82. C2×C4⋊C4×C31D4semidirect
115. C24×C31C2×Q8semidirect
183. C22⋊C4×C31C2×Q8semidirect
188. #24352C31direct (N × H)
257. #3501C2×C31semidirect
258. #2626C2×C31semidirect
260. #2062C2×C31semidirect
261. #2029C2×C31semidirect
262. #1996C2×C31semidirect
264. #1867C2×C31semidirect
265. #1799C2×C31semidirect
346. C22×Q8D4×C31semidirect
370. #120Q8×C31semidirect
384. C2×C4⋊C4D4×C31semidirect
385. C2×C4⋊C4D4×C31semidirect
411. C31#24352direct (N × H)
419. C24C2×Q8×C31semidirect
487. C22⋊C4C2×Q8×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#35012
C31×#26262
C31×#23212
C31×#20622
C31×#20292
C31×#19962
C31×#18892
C31×#18672
C31×#17992
#2435231