Finite Group Databasebeta

← C31×#24373 #21481132

Subgroups — C31×#24373 #21481132

Order 7936.

1568 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C31×AGL(1,ℤ/8)C211yesmaximal, characteristic
2C2×C31×#463C212yesmaximal, complemented
3C2×C31×#452C211yesmaximal, characteristic, complemented
4C31×#2435C211yesmaximal, characteristic, complemented
5C31×#2339C211yesmaximal, characteristic, complemented
6C31×#2055C212yesmaximal, complemented
7C31×#2015C212yesmaximal, complemented
8C31×#1894C211yesmaximal, characteristic, complemented
9C31×#1879C212yesmaximal, complemented
10C31×#1851C211yesmaximal, characteristic, complemented
11C31×#1786C211yesmaximal, characteristic, complemented
12C22×#41×C31C2211yescharacteristic
13C23×D4×C31C2211yescharacteristic
14C2×C31×AGL(1,ℤ/8)—24no—
15C2×C31×AGL(1,ℤ/8)—28no—
16C22×Q16×C31C2211yescharacteristic, complemented
17C22×SD16×C31C2212yes—
18C22×SD16×C31C2211yescharacteristic, complemented
19C22×D8×C31C2212yes—
20C22×M16×C31C2211yescharacteristic
21C2×C31×#124C2212yes—
22C2×C31×C22⋊Q8C2211yescharacteristic
23C2×C4×D4×C31C2212yescomplemented
24C31×#463—28no—
25C31×#452—24no—
26C2×C31×C4⋊C8C2212yescomplemented
27C2×C31×#103C2211yescharacteristic, complemented
28C2×C31×#103C2212yescomplemented
29C2×C31×#102C2212yes—
30C2×C31×#102C2212yes—
31C2×C31×#102C2211yescharacteristic, complemented
32C2×C31×C22⋊C8C2211yescharacteristic
33C31×#394—21no—
34C31×#394—21no—
35C31×#391C2212yescomplemented
36C31×#390C2212yes—
37C31×#386—22no—
38C31×#381—21no—
39C31×#380C2211yescharacteristic
40C31×#379—22no—
41C4×C22⋊C4×C31C2211yescharacteristic
42C2×C31×#95—21no—
43C31×#340C2212yescomplemented
44C31×#336C2212yescomplemented
45C25×C31—21no—
46C2×#41×C31—22no—
47C2×#41×C31—24no—
48C2×#41×C31—42no—
49C2×#41×C31D414yes—
50C22×Q8×C31C2311yescharacteristic
51C22×Q8×C31C2311yescharacteristic
52C22×D4×C31—24no—
53C22×D4×C31—21no—
54C22×D4×C31—22no—
55C22×D4×C31—21no—
56C22×D4×C31—22no—
57C22×D4×C31—42no—
58C22×D4×C31C2312yes—
59C22×D4×C31C2311yescharacteristic
60C23×C4×C31—21no—
61C23×C4×C31—22no—
62C23×C4×C31C2311yescharacteristic
63C31×AGL(1,ℤ/8)—232no—
64C2×Q16×C31—22no—
65C2×Q16×C31—42no—
66C2×SD16×C31—24no—
67C2×SD16×C31—24no—
68C2×SD16×C31—22no—
69C2×SD16×C31—24no—
70C2×SD16×C31—24no—
71C2×D8×C31—24no—
72C2×D8×C31—24no—
73C2×D8×C31D418yes—
74C2×M16×C31—22no—
75C2×M16×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—22no—
90C2×C22⋊C4×C31—22no—
91C2×C22⋊C4×C31—22no—
92C2×C22⋊C4×C31—21no—
93C2×C22⋊C4×C31—22no—
94C2×C22⋊C4×C31—22no—
95C2×C22⋊C4×C31D412yes—
96C2×C42×C31C2312yes—
97C31×C4⋊C8—24no—
98C31×#103—22no—
99C31×#103—24no—
100C31×#102—24no—
101C31×#102—22no—
102C31×#102—42no—
103C31×C22⋊C8—22no—
104C31×#95—21no—
105C31×#95—22no—
106C31×#95—22no—
107C31×#95D411yescharacteristic
108C31×#95D411yescharacteristic
109C24×C31—21no—
110C24×C31—22no—
111C24×C31—44no—
112C24×C31—44no—
113C24×C31—42no—
114C24×C31—81no—
115C24×C31—81no—
116C24×C31C2×D411yescharacteristic
117#41×C31—28no—
118#41×C31—44no—
119#41×C31—48no—
120C2×Q8×C31—21no—
121C2×Q8×C31—22no—
122C2×Q8×C31—22no—
123C2×Q8×C31—21no—
124C2×Q8×C31—21no—
125C2×Q8×C31—22no—
126C2×Q8×C31—22no—
127C2×Q8×C31C2×D412yes—
128C2×D4×C31—28no—
129C2×D4×C31—24no—
130C2×D4×C31—22no—
131C2×D4×C31—22no—
132C2×D4×C31—21no—
133C2×D4×C31—22no—
134C2×D4×C31—22no—
135C2×D4×C31—22no—
136C2×D4×C31—22no—
137C2×D4×C31—48no—
138C2×D4×C31—44no—
139C2×D4×C31—44no—
140C2×D4×C31—44no—
141C2×D4×C31—42no—
142C2×D4×C31C2×D414yes—
143C2×D4×C31C2×D412yes—
144C2×D4×C31C2×D414yes—
145C22×C4×C31—22no—
146C22×C4×C31—22no—
147C22×C4×C31—21no—
148C22×C4×C31—24no—
149C22×C4×C31—22no—
150C22×C4×C31—21no—
151C22×C4×C31—21no—
152C22×C4×C31—44no—
153C22×C4×C31—44no—
154C22×C4×C31—44no—
155C22×C4×C31—82no—
156C22×C4×C31C2×D412yes—
157C22×C4×C31C2×D412yes—
158C22×C4×C31C2×D411yescharacteristic
159C22×C4×C31C2×D412yes—
160C22×C4×C31#4112yes—
161C22×C4×C31C2411yescharacteristic
162Q16×C31—44no—
163SD16×C31—216no—
164SD16×C31—44no—
165D8×C31—216no—
166M16×C31—28no—
167C2×C8×C31—22no—
168C2×C8×C31—22no—
169C2×C8×C31—21no—
170C2×C8×C31—21no—
171C2×C8×C31—22no—
172C2×C8×C31—22no—
173C2×C8×C31C2×D414yes—
174C4⋊C4×C31—22no—
175C4⋊C4×C31—21no—
176C4⋊C4×C31—21no—
177C4⋊C4×C31—22no—
178C4⋊C4×C31—42no—
179C22⋊C4×C31—42no—
180C22⋊C4×C31—44no—
181C22⋊C4×C31—44no—
182C22⋊C4×C31—44no—
183C22⋊C4×C31—42no—
184C22⋊C4×C31—82no—
185C22⋊C4×C31—82no—
186C22⋊C4×C31—81no—
187C42×C31—22no—
188C42×C31—22no—
189#24373C3111yesmaximal, characteristic, direct product factor, Sylow
190C23×C31—22no—
191C23×C31—24no—
192C23×C31—22no—
193C23×C31—22no—
194C23×C31—21no—
195C23×C31—21no—
196C23×C31—24no—
197C23×C31—48no—
198C23×C31—42no—
199C23×C31—41no—
200C23×C31—41no—
201C23×C31—42no—
202C23×C31—42no—
203C23×C31—42no—
204C23×C31—42no—
205C23×C31—42no—
206C23×C31—42no—
207C23×C31—84no—
208C23×C31—84no—
209C23×C31—88no—
210C23×C31—88no—
211C23×C31#12112yes—
212C23×C31C22×D411yescharacteristic
213Q8×C31—22no—
214Q8×C31—22no—
215Q8×C31—42no—
216Q8×C31—41no—
217Q8×C31—41no—
218Q8×C31—42no—
219D4×C31—24no—
220D4×C31—22no—
221D4×C31—24no—
222D4×C31—24no—
223D4×C31—48no—
224D4×C31—48no—
225D4×C31—42no—
226D4×C31—42no—
227D4×C31#12014yes—
228C2×C4×C31—22no—
229C2×C4×C31—22no—
230C2×C4×C31—22no—
231C2×C4×C31—24no—
232C2×C4×C31—22no—
233C2×C4×C31—22no—
234C2×C4×C31—22no—
235C2×C4×C31—22no—
236C2×C4×C31—22no—
237C2×C4×C31—22no—
238C2×C4×C31—22no—
239C2×C4×C31—48no—
240C2×C4×C31—42no—
241C2×C4×C31—41no—
242C2×C4×C31—41no—
243C2×C4×C31—42no—
244C2×C4×C31—42no—
245C2×C4×C31—42no—
246C2×C4×C31—41no—
247C2×C4×C31—42no—
248C2×C4×C31—41no—
249C2×C4×C31—88no—
250C2×C4×C31#12112yes—
251C2×C4×C31C22×D411yescharacteristic
252C2×C4×C31C22×D412yes—
253C2×C4×C31C2×#4111yescharacteristic
254C2×C4×C31H(5,𝔽2)12yes—
255C8×C31—24no—
256C8×C31—41no—
257C8×C31—41no—
258C22×AGL(1,ℤ/8)C2×C3111yescharacteristic
259C2×#463C2×C3112yescomplemented
260C2×#452C2×C3111yescharacteristic, complemented
261#2435C2×C3111yescharacteristic, complemented
262#2339C2×C3111yescharacteristic, complemented
263#2055C2×C3112yescomplemented
264#2015C2×C3112yescomplemented
265#1894C2×C3111yescharacteristic, complemented
266#1879C2×C3112yescomplemented
267#1851C2×C3111yescharacteristic, complemented
268#1786C2×C3111yescharacteristic, complemented
269C22×C31—22no—
270C22×C31—24no—
271C22×C31—22no—
272C22×C31—22no—
273C22×C31—42no—
274C22×C31—44no—
275C22×C31—44no—
276C22×C31—42no—
277C22×C31—42no—
278C22×C31—42no—
279C22×C31—44no—
280C22×C31—44no—
281C22×C31—44no—
282C22×C31—816no—
283C22×C31—82no—
284C22×C31—82no—
285C22×C31—82no—
286C22×C31—82no—
287C22×C31C2×#12111yescharacteristic
288C22×C31D4212yes—
289C22×C31#57512yes—
290C22×C31#57611yescharacteristic
291C22×C31#57711yescharacteristic
292C4×C31—22no—
293C4×C31—42no—
294C4×C31—42no—
295C4×C31—42no—
296C4×C31—81no—
297C4×C31—82no—
298C4×C31—81no—
299C4×C31C2×#12011yescharacteristic
300C4×C31#53411yescharacteristic
301C4×C31#54612yes—
302C22×#41C22×C3111yescharacteristic
303C23×D4C22×C3111yescharacteristic
304C2×AGL(1,ℤ/8)—24no—
305C2×AGL(1,ℤ/8)—28no—
306C22×Q16C22×C3111yescharacteristic, complemented
307C22×SD16C22×C3112yes—
308C22×SD16C22×C3111yescharacteristic, complemented
309C22×D8C22×C3112yes—
310C22×M16C22×C3111yescharacteristic
311C2×#124C22×C3112yes—
312C2×C22⋊Q8C22×C3111yescharacteristic
313C2×C4×D4C22×C3112yescomplemented
314#463—28no—
315#452—24no—
316C2×C4⋊C8C22×C3112yescomplemented
317C2×#103C22×C3111yescharacteristic, complemented
318C2×#103C22×C3112yescomplemented
319C2×#102C22×C3112yes—
320C2×#102C22×C3112yes—
321C2×#102C22×C3111yescharacteristic, complemented
322C2×C22⋊C8C22×C3111yescharacteristic
323#394—21no—
324#394—21no—
325#391C22×C3112yescomplemented
326#390C22×C3112yes—
327#386—22no—
328#381—21no—
329#380C22×C3111yescharacteristic
330#379—22no—
331C4×C22⋊C4C22×C3111yescharacteristic
332C2×#95—21no—
333#340C22×C3112yescomplemented
334#336C22×C3112yescomplemented
335C2×C31—42no—
336C2×C31—44no—
337C2×C31—84no—
338C2×C31#241011yescharacteristic
339C2×C31#304011yescharacteristic
340C2×C31#304511yescharacteristic
341C2×C31#326712yes—
342C2×C31#327312yes—
343C25—21no—
344C2×#41—22no—
345C2×#41—24no—
346C2×#41—42no—
347C2×#41D4×C3114yes—
348C22×Q8C23×C3111yescharacteristic
349C22×Q8C23×C3111yescharacteristic
350C22×D4—24no—
351C22×D4—21no—
352C22×D4—22no—
353C22×D4—21no—
354C22×D4—22no—
355C22×D4—42no—
356C22×D4C23×C3112yes—
357C22×D4C23×C3111yescharacteristic
358C23×C4—21no—
359C23×C4—22no—
360C23×C4C23×C3111yescharacteristic
361AGL(1,ℤ/8)—232no—
362C2×Q16—22no—
363C2×Q16—42no—
364C2×SD16—24no—
365C2×SD16—24no—
366C2×SD16—22no—
367C2×SD16—24no—
368C2×SD16—24no—
369C2×D8—24no—
370C2×D8—24no—
371C2×D8D4×C3118yes—
372C2×M16—22no—
373C2×M16—24no—
374C22×C8C23×C3111yescharacteristic
375C22×C8C23×C3111yescharacteristic
376C22×C8C23×C3112yes—
377#124—24no—
378#124—42no—
379C22⋊Q8—22no—
380C22⋊Q8—22no—
381C4×D4—24no—
382C4×D4—42no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4C23×C3112yes—
385C2×C4⋊C4C23×C3111yescharacteristic
386C2×C22⋊C4—21no—
387C2×C22⋊C4—22no—
388C2×C22⋊C4—22no—
389C2×C22⋊C4—22no—
390C2×C22⋊C4—21no—
391C2×C22⋊C4—22no—
392C2×C22⋊C4—22no—
393C2×C22⋊C4D4×C3112yes—
394C2×C42C23×C3112yes—
395C4⋊C8—24no—
396#103—22no—
397#103—24no—
398#102—24no—
399#102—22no—
400#102—42no—
401C22⋊C8—22no—
402#95—21no—
403#95—22no—
404#95—22no—
405#95D4×C3111yescharacteristic
406#95D4×C3111yescharacteristic
407C31#2437311yescharacteristic, direct product factor, Sylow
408C24—21no—
409C24—22no—
410C24—44no—
411C24—44no—
412C24—42no—
413C24—81no—
414C24—81no—
415C24C2×D4×C3111yescharacteristic
416#41—28no—
417#41—44no—
418#41—48no—
419C2×Q8—21no—
420C2×Q8—22no—
421C2×Q8—22no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8C2×D4×C3112yes—
427C2×D4—28no—
428C2×D4—24no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—21no—
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—48no—
437C2×D4—44no—
438C2×D4—44no—
439C2×D4—44no—
440C2×D4—42no—
441C2×D4C2×D4×C3114yes—
442C2×D4C2×D4×C3112yes—
443C2×D4C2×D4×C3114yes—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—82no—
455C22×C4C2×D4×C3112yes—
456C22×C4C2×D4×C3112yes—
457C22×C4C2×D4×C3111yescharacteristic
458C22×C4C2×D4×C3112yes—
459C22×C4#41×C3112yes—
460C22×C4C24×C3111yescharacteristic
461Q16—44no—
462SD16—216no—
463SD16—44no—
464D8—216no—
465M16—28no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8C2×D4×C3114yes—
473C4⋊C4—22no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—22no—
477C4⋊C4—42no—
478C22⋊C4—42no—
479C22⋊C4—44no—
480C22⋊C4—44no—
481C22⋊C4—44no—
482C22⋊C4—42no—
483C22⋊C4—82no—
484C22⋊C4—82no—
485C22⋊C4—81no—
486C42—22no—
487C42—22no—
488C23—22no—
489C23—24no—
490C23—22no—
491C23—22no—
492C23—21no—
493C23—21no—
494C23—24no—
495C23—48no—
496C23—42no—
497C23—41no—
498C23—41no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—84no—
506C23—84no—
507C23—88no—
508C23—88no—
509C23C31×#12112yes—
510C23C22×D4×C3111yescharacteristic
511Q8—22no—
512Q8—22no—
513Q8—42no—
514Q8—41no—
515Q8—41no—
516Q8—42no—
517D4—24no—
518D4—22no—
519D4—24no—
520D4—24no—
521D4—48no—
522D4—48no—
523D4—42no—
524D4—42no—
525D4C31×#12014yes—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—48no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—42no—
546C2×C4—41no—
547C2×C4—88no—
548C2×C4C31×#12112yes—
549C2×C4C22×D4×C3111yescharacteristic
550C2×C4C22×D4×C3112yes—
551C2×C4C2×#41×C3111yescharacteristic
552C2×C4C31×H(5,𝔽2)12yes—
553C8—24no—
554C8—41no—
555C8—41no—
556C22—22no—
557C22—24no—
558C22—22no—
559C22—22no—
560C22—42no—
561C22—44no—
562C22—44no—
563C22—42no—
564C22—42no—
565C22—42no—
566C22—44no—
567C22—44no—
568C22—44no—
569C22—816no—
570C22—82no—
571C22—82no—
572C22—82no—
573C22—82no—
574C22C2×C31×#12111yescharacteristic
575C22D42×C3112yes—
576C22C31×#57512yes—
577C22C31×#57611yescharacteristic
578C22C31×#57711yescharacteristic
579C4—22no—
580C4—42no—
581C4—42no—
582C4—42no—
583C4—81no—
584C4—82no—
585C4—81no—
586C4C2×C31×#12011yescharacteristic
587C4C31×#53411yescharacteristic
588C4C31×#54612yes—
589C2—42no—
590C2—44no—
591C2—84no—
592C2C31×#241011yescharacteristic
593C2C31×#304011yescharacteristic
594C2C31×#304511yescharacteristic
595C2C31×#326712yes—
596C2C31×#327312yes—
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC22×C31×AGL(1,ℤ/8)C2
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×#452C2semidirect
4. C31×#2435C2semidirect
5. C31×#2339C2semidirect
6. C31×#2055C2semidirect
7. C31×#2015C2semidirect
8. C31×#1894C2semidirect
9. C31×#1879C2semidirect
10. C31×#1851C2semidirect
11. C31×#1786C2semidirect
16. C22×Q16×C31C22semidirect
18. C22×SD16×C31C22semidirect
23. C2×C4×D4×C31C22semidirect
26. C2×C31×C4⋊C8C22semidirect
27. C2×C31×#103C22semidirect
28. C2×C31×#103C22semidirect
31. C2×C31×#102C22semidirect
35. C31×#391C22semidirect
43. C31×#340C22semidirect
44. C31×#336C22semidirect
189. #24373C31direct (N × H)
259. C2×#463C2×C31semidirect
260. C2×#452C2×C31semidirect
261. #2435C2×C31semidirect
262. #2339C2×C31semidirect
263. #2055C2×C31semidirect
264. #2015C2×C31semidirect
265. #1894C2×C31semidirect
266. #1879C2×C31semidirect
267. #1851C2×C31semidirect
268. #1786C2×C31semidirect
306. C22×Q16C22×C31semidirect
308. C22×SD16C22×C31semidirect
313. C2×C4×D4C22×C31semidirect
316. C2×C4⋊C8C22×C31semidirect
317. C2×#103C22×C31semidirect
318. C2×#103C22×C31semidirect
321. C2×#102C22×C31semidirect
325. #391C22×C31semidirect
333. #340C22×C31semidirect
334. #336C22×C31semidirect
407. C31#24373direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C31×AGL(1,ℤ/8)2
C2×C31×#4632
C2×C31×#4522
C31×#24352
C31×#23392
C31×#20552
C31×#20152
C31×#18942
C31×#18792
C31×#18512
C31×#17862
#2437331