Finite Group Databasebeta

← C31×#24355 #21481114

Subgroups — C31×#24355 #21481114

Order 7936.

1742 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×D8×C31C211yesmaximal, characteristic
2C2×C31×#463C212yesmaximal, complemented
3C2×C31×#450C211yesmaximal, characteristic, complemented
4C31×#2435C211yesmaximal, characteristic, complemented
5D4×C22⋊C4×C31C211yesmaximal, characteristic, complemented
6C31×#2054C212yesmaximal, complemented
7C31×#2012C212yesmaximal, complemented
8C31×#1892C211yesmaximal, characteristic, complemented
9C31×#1879C212yesmaximal, complemented
10C31×#1850C211yesmaximal, characteristic, complemented
11C31×#1786C211yesmaximal, characteristic, complemented
12C23×D4×C31C2211yescharacteristic
13C23×D4×C31C2211yescharacteristic
14C22×SD16×C31C2211yescharacteristic, complemented
15C22×SD16×C31C2211yescharacteristic, complemented
16C22×D8×C31—24no—
17C22×D8×C31—28no—
18C22×D8×C31C2212yes—
19C22×D8×C31C2212yes—
20C23×C8×C31C2211yescharacteristic
21C2×C31×#124C2212yescomplemented
22C2×C31×C22⋊Q8C2211yescharacteristic
23C2×C4×D4×C31C2212yescomplemented
24C22×C22⋊C4×C31—21no—
25C31×#463—28no—
26C31×#450—24no—
27C2×C31×C4⋊C8C2212yescomplemented
28C2×C31×#103C2212yescomplemented
29C2×C31×#102C2211yescharacteristic, complemented
30C2×C31×#102C2212yescomplemented
31C2×C31×#102C2211yescharacteristic, complemented
32C2×C31×#102C2212yescomplemented
33C2×C31×C22⋊C8C2211yescharacteristic
34C31×#394—22no—
35C31×#390C2212yescomplemented
36C31×#390C2212yescomplemented
37C31×#386—22no—
38C31×#380C2211yescharacteristic
39C31×#379—21no—
40C31×#379—21no—
41C31×#379—21no—
42C4×C22⋊C4×C31C2211yescharacteristic
43C31×#340C2212yescomplemented
44C31×#336C2212yescomplemented
45C25×C31—21no—
46C25×C31—21no—
47C22×Q8×C31C2311yescharacteristic
48C22×D4×C31—24no—
49C22×D4×C31—24no—
50C22×D4×C31—22no—
51C22×D4×C31—21no—
52C22×D4×C31—21no—
53C22×D4×C31—22no—
54C22×D4×C31—22no—
55C22×D4×C31—42no—
56C22×D4×C31—42no—
57C22×D4×C31D414yes—
58C22×D4×C31C2312yes—
59C22×D4×C31C2311yescharacteristic
60C22×D4×C31C2311yescharacteristic
61C23×C4×C31—22no—
62C23×C4×C31C2311yescharacteristic
63C2×SD16×C31—22no—
64C2×SD16×C31—22no—
65C2×SD16×C31—24no—
66C2×SD16×C31—42no—
67C2×D8×C31—232no—
68C2×D8×C31—24no—
69C2×D8×C31—24no—
70C2×D8×C31—24no—
71C2×D8×C31—24no—
72C2×D8×C31—24no—
73C2×D8×C31D418yes—
74C22×C8×C31—22no—
75C22×C8×C31—24no—
76C22×C8×C31C2311yescharacteristic
77C22×C8×C31C2311yescharacteristic
78C22×C8×C31C2312yes—
79C31×#124—24no—
80C31×#124—42no—
81C31×C22⋊Q8—22no—
82C31×C22⋊Q8—22no—
83C4×D4×C31—24no—
84C4×D4×C31—42no—
85C2×C4⋊C4×C31—21no—
86C2×C4⋊C4×C31C2312yes—
87C2×C4⋊C4×C31C2311yescharacteristic
88C2×C22⋊C4×C31—21no—
89C2×C22⋊C4×C31—21no—
90C2×C22⋊C4×C31—22no—
91C2×C22⋊C4×C31—22no—
92C2×C22⋊C4×C31—22no—
93C2×C22⋊C4×C31—22no—
94C2×C22⋊C4×C31—22no—
95C2×C22⋊C4×C31—22no—
96C2×C22⋊C4×C31—22no—
97C2×C22⋊C4×C31—44no—
98C2×C22⋊C4×C31D411yescharacteristic
99C2×C22⋊C4×C31D411yescharacteristic
100C2×C42×C31C2312yes—
101C31×C4⋊C8—24no—
102C31×#103—24no—
103C31×#102—22no—
104C31×#102—22no—
105C31×#102—24no—
106C31×#102—42no—
107C31×C22⋊C8—22no—
108C31×#95—21no—
109C31×#95D412yes—
110C24×C31—21no—
111C24×C31—21no—
112C24×C31—22no—
113C24×C31—24no—
114C24×C31—44no—
115C24×C31—44no—
116C24×C31—42no—
117C24×C31—42no—
118C24×C31—42no—
119C24×C31—42no—
120C24×C31—42no—
121C24×C31—82no—
122C24×C31—82no—
123C24×C31C2×D411yescharacteristic
124C2×Q8×C31—22no—
125C2×Q8×C31—21no—
126C2×Q8×C31—21no—
127C2×Q8×C31—22no—
128C2×D4×C31—28no—
129C2×D4×C31—28no—
130C2×D4×C31—24no—
131C2×D4×C31—22no—
132C2×D4×C31—22no—
133C2×D4×C31—22no—
134C2×D4×C31—21no—
135C2×D4×C31—22no—
136C2×D4×C31—21no—
137C2×D4×C31—22no—
138C2×D4×C31—22no—
139C2×D4×C31—22no—
140C2×D4×C31—22no—
141C2×D4×C31—44no—
142C2×D4×C31—48no—
143C2×D4×C31—48no—
144C2×D4×C31—44no—
145C2×D4×C31—44no—
146C2×D4×C31—44no—
147C2×D4×C31—42no—
148C2×D4×C31C2×D414yes—
149C2×D4×C31C2×D412yes—
150C2×D4×C31C2×D412yes—
151C2×D4×C31C2×D414yes—
152C22×C4×C31—22no—
153C22×C4×C31—22no—
154C22×C4×C31—22no—
155C22×C4×C31—21no—
156C22×C4×C31—21no—
157C22×C4×C31—44no—
158C22×C4×C31—44no—
159C22×C4×C31—82no—
160C22×C4×C31C2×D412yes—
161C22×C4×C31C2×D412yes—
162C22×C4×C31C2×D411yescharacteristic
163C22×C4×C31C2×D412yes—
164C22×C4×C31#4112yes—
165C22×C4×C31C2411yescharacteristic
166SD16×C31—44no—
167SD16×C31—44no—
168D8×C31—216no—
169D8×C31—216no—
170C2×C8×C31—28no—
171C2×C8×C31—22no—
172C2×C8×C31—22no—
173C2×C8×C31—21no—
174C2×C8×C31—21no—
175C2×C8×C31—22no—
176C2×C8×C31—22no—
177C2×C8×C31C2×D414yes—
178C4⋊C4×C31—21no—
179C4⋊C4×C31—22no—
180C4⋊C4×C31—21no—
181C4⋊C4×C31—22no—
182C4⋊C4×C31—42no—
183C22⋊C4×C31—22no—
184C22⋊C4×C31—22no—
185C22⋊C4×C31—44no—
186C22⋊C4×C31—44no—
187C22⋊C4×C31—44no—
188C22⋊C4×C31—44no—
189C22⋊C4×C31—42no—
190C22⋊C4×C31—42no—
191C22⋊C4×C31—82no—
192C22⋊C4×C31—82no—
193C22⋊C4×C31—82no—
194C42×C31—22no—
195C42×C31—22no—
196#24355C3111yesmaximal, characteristic, direct product factor, Sylow
197C23×C31—22no—
198C23×C31—22no—
199C23×C31—22no—
200C23×C31—24no—
201C23×C31—22no—
202C23×C31—24no—
203C23×C31—48no—
204C23×C31—48no—
205C23×C31—41no—
206C23×C31—41no—
207C23×C31—42no—
208C23×C31—42no—
209C23×C31—42no—
210C23×C31—42no—
211C23×C31—42no—
212C23×C31—41no—
213C23×C31—41no—
214C23×C31—42no—
215C23×C31—42no—
216C23×C31—42no—
217C23×C31—42no—
218C23×C31—48no—
219C23×C31—84no—
220C23×C31—88no—
221C23×C31—88no—
222C23×C31—88no—
223C23×C31—88no—
224C23×C31—84no—
225C23×C31—84no—
226C23×C31#12112yes—
227C23×C31#13512yes—
228C23×C31AGL(1,ℤ/8)12yes—
229C23×C31C22×D411yescharacteristic
230Q8×C31—41no—
231Q8×C31—41no—
232Q8×C31—42no—
233D4×C31—22no—
234D4×C31—24no—
235D4×C31—22no—
236D4×C31—22no—
237D4×C31—24no—
238D4×C31—24no—
239D4×C31—48no—
240D4×C31—48no—
241D4×C31—42no—
242D4×C31—42no—
243D4×C31—42no—
244D4×C31#12014yes—
245C2×C4×C31—22no—
246C2×C4×C31—22no—
247C2×C4×C31—22no—
248C2×C4×C31—24no—
249C2×C4×C31—22no—
250C2×C4×C31—22no—
251C2×C4×C31—22no—
252C2×C4×C31—22no—
253C2×C4×C31—22no—
254C2×C4×C31—22no—
255C2×C4×C31—41no—
256C2×C4×C31—41no—
257C2×C4×C31—42no—
258C2×C4×C31—42no—
259C2×C4×C31—42no—
260C2×C4×C31—42no—
261C2×C4×C31—42no—
262C2×C4×C31—88no—
263C2×C4×C31#12112yes—
264C2×C4×C31C22×D411yescharacteristic
265C2×C4×C31C22×D412yes—
266C2×C4×C31C2×#4111yescharacteristic
267C2×C4×C31H(5,𝔽2)12yes—
268C8×C31—24no—
269C8×C31—41no—
270C8×C31—41no—
271C23×D8C2×C3111yescharacteristic
272C2×#463C2×C3112yescomplemented
273C2×#450C2×C3111yescharacteristic, complemented
274#2435C2×C3111yescharacteristic, complemented
275D4×C22⋊C4C2×C3111yescharacteristic, complemented
276#2054C2×C3112yescomplemented
277#2012C2×C3112yescomplemented
278#1892C2×C3111yescharacteristic, complemented
279#1879C2×C3112yescomplemented
280#1850C2×C3111yescharacteristic, complemented
281#1786C2×C3111yescharacteristic, complemented
282C22×C31—22no—
283C22×C31—24no—
284C22×C31—24no—
285C22×C31—22no—
286C22×C31—24no—
287C22×C31—44no—
288C22×C31—44no—
289C22×C31—42no—
290C22×C31—42no—
291C22×C31—42no—
292C22×C31—44no—
293C22×C31—44no—
294C22×C31—44no—
295C22×C31—42no—
296C22×C31—42no—
297C22×C31—816no—
298C22×C31—816no—
299C22×C31—82no—
300C22×C31—82no—
301C22×C31—82no—
302C22×C31—82no—
303C22×C31—82no—
304C22×C31—82no—
305C22×C31#46314yescomplemented
306C22×C31C2×#12111yescharacteristic
307C22×C31D4212yes—
308C22×C31C2×#13511yescharacteristic
309C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
310C22×C31#57612yes—
311C4×C31—22no—
312C4×C31—42no—
313C4×C31—42no—
314C4×C31—81no—
315C4×C31—82no—
316C4×C31—81no—
317C4×C31C2×#12011yescharacteristic
318C4×C31#53411yescharacteristic
319C4×C31#54612yes—
320C23×D4C22×C3111yescharacteristic
321C23×D4C22×C3111yescharacteristic
322C22×SD16C22×C3111yescharacteristic, complemented
323C22×SD16C22×C3111yescharacteristic, complemented
324C22×D8—24no—
325C22×D8—28no—
326C22×D8C22×C3112yes—
327C22×D8C22×C3112yes—
328C23×C8C22×C3111yescharacteristic
329C2×#124C22×C3112yescomplemented
330C2×C22⋊Q8C22×C3111yescharacteristic
331C2×C4×D4C22×C3112yescomplemented
332C22×C22⋊C4—21no—
333#463—28no—
334#450—24no—
335C2×C4⋊C8C22×C3112yescomplemented
336C2×#103C22×C3112yescomplemented
337C2×#102C22×C3111yescharacteristic, complemented
338C2×#102C22×C3112yescomplemented
339C2×#102C22×C3111yescharacteristic, complemented
340C2×#102C22×C3112yescomplemented
341C2×C22⋊C8C22×C3111yescharacteristic
342#394—22no—
343#390C22×C3112yescomplemented
344#390C22×C3112yescomplemented
345#386—22no—
346#380C22×C3111yescharacteristic
347#379—21no—
348#379—21no—
349#379—21no—
350C4×C22⋊C4C22×C3111yescharacteristic
351#340C22×C3112yescomplemented
352#336C22×C3112yescomplemented
353C2×C31—24no—
354C2×C31—44no—
355C2×C31—84no—
356C2×C31—82no—
357C2×C31#241011yescharacteristic
358C2×C31C2×#46311yescharacteristic
359C2×C31#304111yescharacteristic
360C2×C31#326412yes—
361C2×C31#328012yes—
362C25—21no—
363C25—21no—
364C22×Q8C23×C3111yescharacteristic
365C22×D4—24no—
366C22×D4—24no—
367C22×D4—22no—
368C22×D4—21no—
369C22×D4—21no—
370C22×D4—22no—
371C22×D4—22no—
372C22×D4—42no—
373C22×D4—42no—
374C22×D4D4×C3114yes—
375C22×D4C23×C3112yes—
376C22×D4C23×C3111yescharacteristic
377C22×D4C23×C3111yescharacteristic
378C23×C4—22no—
379C23×C4C23×C3111yescharacteristic
380C2×SD16—22no—
381C2×SD16—22no—
382C2×SD16—24no—
383C2×SD16—42no—
384C2×D8—232no—
385C2×D8—24no—
386C2×D8—24no—
387C2×D8—24no—
388C2×D8—24no—
389C2×D8—24no—
390C2×D8D4×C3118yes—
391C22×C8—22no—
392C22×C8—24no—
393C22×C8C23×C3111yescharacteristic
394C22×C8C23×C3111yescharacteristic
395C22×C8C23×C3112yes—
396#124—24no—
397#124—42no—
398C22⋊Q8—22no—
399C22⋊Q8—22no—
400C4×D4—24no—
401C4×D4—42no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4C23×C3112yes—
404C2×C4⋊C4C23×C3111yescharacteristic
405C2×C22⋊C4—21no—
406C2×C22⋊C4—21no—
407C2×C22⋊C4—22no—
408C2×C22⋊C4—22no—
409C2×C22⋊C4—22no—
410C2×C22⋊C4—22no—
411C2×C22⋊C4—22no—
412C2×C22⋊C4—22no—
413C2×C22⋊C4—22no—
414C2×C22⋊C4—44no—
415C2×C22⋊C4D4×C3111yescharacteristic
416C2×C22⋊C4D4×C3111yescharacteristic
417C2×C42C23×C3112yes—
418C4⋊C8—24no—
419#103—24no—
420#102—22no—
421#102—22no—
422#102—24no—
423#102—42no—
424C22⋊C8—22no—
425#95—21no—
426#95D4×C3112yes—
427C31#2435511yescharacteristic, direct product factor, Sylow
428C24—21no—
429C24—21no—
430C24—22no—
431C24—24no—
432C24—44no—
433C24—44no—
434C24—42no—
435C24—42no—
436C24—42no—
437C24—42no—
438C24—42no—
439C24—82no—
440C24—82no—
441C24C2×D4×C3111yescharacteristic
442C2×Q8—22no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—22no—
446C2×D4—28no—
447C2×D4—28no—
448C2×D4—24no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—21no—
453C2×D4—22no—
454C2×D4—21no—
455C2×D4—22no—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—44no—
460C2×D4—48no—
461C2×D4—48no—
462C2×D4—44no—
463C2×D4—44no—
464C2×D4—44no—
465C2×D4—42no—
466C2×D4C2×D4×C3114yes—
467C2×D4C2×D4×C3112yes—
468C2×D4C2×D4×C3112yes—
469C2×D4C2×D4×C3114yes—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—82no—
478C22×C4C2×D4×C3112yes—
479C22×C4C2×D4×C3112yes—
480C22×C4C2×D4×C3111yescharacteristic
481C22×C4C2×D4×C3112yes—
482C22×C4#41×C3112yes—
483C22×C4C24×C3111yescharacteristic
484SD16—44no—
485SD16—44no—
486D8—216no—
487D8—216no—
488C2×C8—28no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8C2×D4×C3114yes—
496C4⋊C4—21no—
497C4⋊C4—22no—
498C4⋊C4—21no—
499C4⋊C4—22no—
500C4⋊C4—42no—
501C22⋊C4—22no—
502C22⋊C4—22no—
503C22⋊C4—44no—
504C22⋊C4—44no—
505C22⋊C4—44no—
506C22⋊C4—44no—
507C22⋊C4—42no—
508C22⋊C4—42no—
509C22⋊C4—82no—
510C22⋊C4—82no—
511C22⋊C4—82no—
512C42—22no—
513C42—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—24no—
518C23—22no—
519C23—24no—
520C23—48no—
521C23—48no—
522C23—41no—
523C23—41no—
524C23—42no—
525C23—48no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—41no—
531C23—41no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—84no—
537C23—88no—
538C23—88no—
539C23—88no—
540C23—88no—
541C23—84no—
542C23—84no—
543C23C31×#12112yes—
544C23C31×#13512yes—
545C23C31×AGL(1,ℤ/8)12yes—
546C23C22×D4×C3111yescharacteristic
547Q8—41no—
548Q8—41no—
549Q8—42no—
550D4—22no—
551D4—24no—
552D4—22no—
553D4—22no—
554D4—24no—
555D4—24no—
556D4—48no—
557D4—48no—
558D4—42no—
559D4—42no—
560D4—42no—
561D4C31×#12014yes—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—88no—
580C2×C4C31×#12112yes—
581C2×C4C22×D4×C3111yescharacteristic
582C2×C4C22×D4×C3112yes—
583C2×C4C2×#41×C3111yescharacteristic
584C2×C4C31×H(5,𝔽2)12yes—
585C8—24no—
586C8—41no—
587C8—41no—
588C22—22no—
589C22—24no—
590C22—24no—
591C22—22no—
592C22—24no—
593C22—44no—
594C22—44no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—44no—
599C22—44no—
600C22—44no—
601C22—42no—
602C22—42no—
603C22—816no—
604C22—816no—
605C22—82no—
606C22—82no—
607C22—82no—
608C22—82no—
609C22—82no—
610C22—82no—
611C22C31×#46314yescomplemented
612C22C2×C31×#12111yescharacteristic
613C22D42×C3112yes—
614C22C2×C31×#13511yescharacteristic
615C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
616C22C31×#57612yes—
617C4—22no—
618C4—42no—
619C4—42no—
620C4—81no—
621C4—82no—
622C4—81no—
623C4C2×C31×#12011yescharacteristic
624C4C31×#53411yescharacteristic
625C4C31×#54612yes—
626C2—24no—
627C2—44no—
628C2—84no—
629C2—82no—
630C2C31×#241011yescharacteristic
631C2C2×C31×#46311yescharacteristic
632C2C31×#304111yescharacteristic
633C2C31×#326412yes—
634C2C31×#328012yes—
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC23×D8×C31C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C4×C22⋊C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×#463C2semidirect
3. C2×C31×#450C2semidirect
4. C31×#2435C2semidirect
5. D4×C22⋊C4×C31C2semidirect
6. C31×#2054C2semidirect
7. C31×#2012C2semidirect
8. C31×#1892C2semidirect
9. C31×#1879C2semidirect
10. C31×#1850C2semidirect
11. C31×#1786C2semidirect
14. C22×SD16×C31C22semidirect
15. C22×SD16×C31C22semidirect
21. C2×C31×#124C22semidirect
23. C2×C4×D4×C31C22semidirect
27. C2×C31×C4⋊C8C22semidirect
28. C2×C31×#103C22semidirect
29. C2×C31×#102C22semidirect
30. C2×C31×#102C22semidirect
31. C2×C31×#102C22semidirect
32. C2×C31×#102C22semidirect
35. C31×#390C22semidirect
36. C31×#390C22semidirect
43. C31×#340C22semidirect
44. C31×#336C22semidirect
196. #24355C31direct (N × H)
272. C2×#463C2×C31semidirect
273. C2×#450C2×C31semidirect
274. #2435C2×C31semidirect
275. D4×C22⋊C4C2×C31semidirect
276. #2054C2×C31semidirect
277. #2012C2×C31semidirect
278. #1892C2×C31semidirect
279. #1879C2×C31semidirect
280. #1850C2×C31semidirect
281. #1786C2×C31semidirect
305. C22×C31#463semidirect
322. C22×SD16C22×C31semidirect
323. C22×SD16C22×C31semidirect
329. C2×#124C22×C31semidirect
331. C2×C4×D4C22×C31semidirect
335. C2×C4⋊C8C22×C31semidirect
336. C2×#103C22×C31semidirect
337. C2×#102C22×C31semidirect
338. C2×#102C22×C31semidirect
339. C2×#102C22×C31semidirect
340. C2×#102C22×C31semidirect
343. #390C22×C31semidirect
344. #390C22×C31semidirect
351. #340C22×C31semidirect
352. #336C22×C31semidirect
427. C31#24355direct (N × H)
611. C22C31×#463semidirect

Maximal subgroups

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

subgroupindex
C23×D8×C312
C2×C31×#4632
C2×C31×#4502
C31×#24352
D4×C22⋊C4×C312
C31×#20542
C31×#20122
C31×#18922
C31×#18792
C31×#18502
C31×#17862
#2435531