Finite Group Databasebeta

← C31×#24371 #21481130

Subgroups — C31×#24371 #21481130

Order 7936.

1540 conjugacy classes of subgroups.

The subgroup lattice has 596 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×#464C212yesmaximal, complemented
3C2×C31×#452C211yesmaximal, characteristic, complemented
4C31×#2435C211yesmaximal, characteristic, complemented
5C31×#2341C211yesmaximal, characteristic, complemented
6C31×#2054C212yesmaximal, complemented
7C31×#2012C212yesmaximal, complemented
8C31×#1894C211yesmaximal, characteristic, complemented
9C31×#1880C212yesmaximal, 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×Q8×C31C2212yescomplemented
24C31×#464—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×#390—22no—
36C31×#390C2212yes—
37C31×#390C2212yescomplemented
38C31×#380—21no—
39C31×#380C2211yescharacteristic
40C31×#379—22no—
41C4×C22⋊C4×C31—21no—
42C4×C22⋊C4×C31C2211yescharacteristic
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×C31C2311yescharacteristic
62C31×AGL(1,ℤ/8)—232no—
63C2×Q16×C31—22no—
64C2×Q16×C31—24no—
65C2×SD16×C31—24no—
66C2×SD16×C31—24no—
67C2×SD16×C31—22no—
68C2×SD16×C31—42no—
69C2×SD16×C31D418yes—
70C2×D8×C31—24no—
71C2×D8×C31—24no—
72C2×D8×C31—24no—
73C2×M16×C31—22no—
74C2×M16×C31—24no—
75C22×C8×C31C2311yescharacteristic
76C22×C8×C31C2311yescharacteristic
77C22×C8×C31C2312yes—
78C31×#124—24no—
79C31×#124—42no—
80C31×C22⋊Q8—22no—
81C31×C22⋊Q8—22no—
82C4×Q8×C31—24no—
83C4×Q8×C31—42no—
84C2×C4⋊C4×C31—22no—
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×C31—22no—
96C2×C22⋊C4×C31D412yes—
97C2×C42×C31—22no—
98C2×C42×C31C2312yes—
99C31×C4⋊C8—24no—
100C31×#103—22no—
101C31×#103—24no—
102C31×#102—24no—
103C31×#102—22no—
104C31×#102—42no—
105C31×C22⋊C8—22no—
106C31×#95—21no—
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×C31C2×D412yes—
127C2×Q8×C31C2×D414yes—
128C2×D4×C31—28no—
129C2×D4×C31—24no—
130C2×D4×C31—22no—
131C2×D4×C31—22no—
132C2×D4×C31—22no—
133C2×D4×C31—22no—
134C2×D4×C31—21no—
135C2×D4×C31—22no—
136C2×D4×C31—22no—
137C2×D4×C31—22no—
138C2×D4×C31—48no—
139C2×D4×C31—44no—
140C2×D4×C31—44no—
141C2×D4×C31—44no—
142C2×D4×C31—42no—
143C2×D4×C31C2×D414yes—
144C2×D4×C31C2×D412yes—
145C22×C4×C31—21no—
146C22×C4×C31—22no—
147C22×C4×C31—22no—
148C22×C4×C31—24no—
149C22×C4×C31—22no—
150C22×C4×C31—21no—
151C22×C4×C31—21no—
152C22×C4×C31—44no—
153C22×C4×C31C2×D412yes—
154C22×C4×C31C2×D412yes—
155C22×C4×C31C2×D411yescharacteristic
156C22×C4×C31C2×D412yes—
157C22×C4×C31#4112yes—
158C22×C4×C31C2411yescharacteristic
159Q16×C31—44no—
160SD16×C31—216no—
161SD16×C31—44no—
162D8×C31—216no—
163M16×C31—28no—
164C2×C8×C31—22no—
165C2×C8×C31—22no—
166C2×C8×C31—21no—
167C2×C8×C31—21no—
168C2×C8×C31—22no—
169C2×C8×C31—22no—
170C2×C8×C31C2×D414yes—
171C4⋊C4×C31—22no—
172C4⋊C4×C31—21no—
173C4⋊C4×C31—21no—
174C4⋊C4×C31—22no—
175C4⋊C4×C31—44no—
176C4⋊C4×C31—42no—
177C22⋊C4×C31—42no—
178C22⋊C4×C31—44no—
179C22⋊C4×C31—44no—
180C22⋊C4×C31—44no—
181C22⋊C4×C31—42no—
182C22⋊C4×C31—82no—
183C22⋊C4×C31—82no—
184C22⋊C4×C31—82no—
185C22⋊C4×C31—81no—
186C42×C31—22no—
187C42×C31—22no—
188C42×C31—44no—
189#24371C3111yesmaximal, characteristic, direct product factor, Sylow
190C23×C31—22no—
191C23×C31—22no—
192C23×C31—21no—
193C23×C31—21no—
194C23×C31—22no—
195C23×C31—24no—
196C23×C31—48no—
197C23×C31—42no—
198C23×C31—42no—
199C23×C31—42no—
200C23×C31—42no—
201C23×C31—42no—
202C23×C31—42no—
203C23×C31—41no—
204C23×C31—42no—
205C23×C31—42no—
206C23×C31—41no—
207C23×C31—84no—
208C23×C31—84no—
209C23×C31—88no—
210C23×C31—88no—
211C23×C31#12112yes—
212C23×C31C22×D411yescharacteristic
213Q8×C31—22no—
214Q8×C31—24no—
215Q8×C31—41no—
216Q8×C31—41no—
217Q8×C31—42no—
218Q8×C31#12014yes—
219D4×C31—22no—
220D4×C31—24no—
221D4×C31—22no—
222D4×C31—24no—
223D4×C31—48no—
224D4×C31—48no—
225D4×C31—42no—
226D4×C31—42no—
227D4×C31—42no—
228C2×C4×C31—22no—
229C2×C4×C31—24no—
230C2×C4×C31—22no—
231C2×C4×C31—22no—
232C2×C4×C31—22no—
233C2×C4×C31—24no—
234C2×C4×C31—22no—
235C2×C4×C31—22no—
236C2×C4×C31—22no—
237C2×C4×C31—22no—
238C2×C4×C31—22no—
239C2×C4×C31—22no—
240C2×C4×C31—48no—
241C2×C4×C31—41no—
242C2×C4×C31—41no—
243C2×C4×C31—42no—
244C2×C4×C31—42no—
245C2×C4×C31—42no—
246C2×C4×C31—42no—
247C2×C4×C31—41no—
248C2×C4×C31—41no—
249C2×C4×C31#12112yes—
250C2×C4×C31C22×D411yescharacteristic
251C2×C4×C31C22×D412yes—
252C2×C4×C31C2×#4111yescharacteristic
253C2×C4×C31H(5,𝔽2)12yes—
254C8×C31—24no—
255C8×C31—41no—
256C8×C31—41no—
257C22×AGL(1,ℤ/8)C2×C3111yescharacteristic
258C2×#464C2×C3112yescomplemented
259C2×#452C2×C3111yescharacteristic, complemented
260#2435C2×C3111yescharacteristic, complemented
261#2341C2×C3111yescharacteristic, complemented
262#2054C2×C3112yescomplemented
263#2012C2×C3112yescomplemented
264#1894C2×C3111yescharacteristic, complemented
265#1880C2×C3112yescomplemented
266#1851C2×C3111yescharacteristic, complemented
267#1786C2×C3111yescharacteristic, complemented
268C22×C31—22no—
269C22×C31—22no—
270C22×C31—22no—
271C22×C31—42no—
272C22×C31—42no—
273C22×C31—42no—
274C22×C31—44no—
275C22×C31—44no—
276C22×C31—44no—
277C22×C31—42no—
278C22×C31—42no—
279C22×C31—816no—
280C22×C31—82no—
281C22×C31—82no—
282C22×C31—82no—
283C22×C31—82no—
284C22×C31—82no—
285C22×C31—82no—
286C22×C31C2×#12111yescharacteristic
287C22×C31D4212yes—
288C22×C31#57512yes—
289C22×C31#57611yescharacteristic
290C22×C31#57711yescharacteristic
291C4×C31—22no—
292C4×C31—24no—
293C4×C31—42no—
294C4×C31—42no—
295C4×C31—81no—
296C4×C31—82no—
297C4×C31—81no—
298C4×C31C2×#12011yescharacteristic
299C4×C31#53411yescharacteristic
300C4×C31#54612yes—
301C22×#41C22×C3111yescharacteristic
302C23×D4C22×C3111yescharacteristic
303C2×AGL(1,ℤ/8)—24no—
304C2×AGL(1,ℤ/8)—28no—
305C22×Q16C22×C3111yescharacteristic, complemented
306C22×SD16C22×C3112yes—
307C22×SD16C22×C3111yescharacteristic, complemented
308C22×D8C22×C3112yes—
309C22×M16C22×C3111yescharacteristic
310C2×#124C22×C3112yes—
311C2×C22⋊Q8C22×C3111yescharacteristic
312C2×C4×Q8C22×C3112yescomplemented
313#464—28no—
314#452—24no—
315C2×C4⋊C8C22×C3112yescomplemented
316C2×#103C22×C3111yescharacteristic, complemented
317C2×#103C22×C3112yescomplemented
318C2×#102C22×C3112yes—
319C2×#102C22×C3112yes—
320C2×#102C22×C3111yescharacteristic, complemented
321C2×C22⋊C8C22×C3111yescharacteristic
322#394—21no—
323#394—21no—
324#390—22no—
325#390C22×C3112yes—
326#390C22×C3112yescomplemented
327#380—21no—
328#380C22×C3111yescharacteristic
329#379—22no—
330C4×C22⋊C4—21no—
331C4×C22⋊C4C22×C3111yescharacteristic
332#340C22×C3112yescomplemented
333#336C22×C3112yescomplemented
334C2×C31—42no—
335C2×C31—82no—
336C2×C31—84no—
337C2×C31#241011yescharacteristic
338C2×C31#304011yescharacteristic
339C2×C31#304511yescharacteristic
340C2×C31#326812yes—
341C2×C31#327512yes—
342C25—21no—
343C2×#41—22no—
344C2×#41—24no—
345C2×#41—42no—
346C2×#41D4×C3114yes—
347C22×Q8C23×C3111yescharacteristic
348C22×Q8C23×C3111yescharacteristic
349C22×D4—24no—
350C22×D4—21no—
351C22×D4—22no—
352C22×D4—21no—
353C22×D4—22no—
354C22×D4—42no—
355C22×D4C23×C3112yes—
356C22×D4C23×C3111yescharacteristic
357C23×C4—21no—
358C23×C4C23×C3111yescharacteristic
359AGL(1,ℤ/8)—232no—
360C2×Q16—22no—
361C2×Q16—24no—
362C2×SD16—24no—
363C2×SD16—24no—
364C2×SD16—22no—
365C2×SD16—42no—
366C2×SD16D4×C3118yes—
367C2×D8—24no—
368C2×D8—24no—
369C2×D8—24no—
370C2×M16—22no—
371C2×M16—24no—
372C22×C8C23×C3111yescharacteristic
373C22×C8C23×C3111yescharacteristic
374C22×C8C23×C3112yes—
375#124—24no—
376#124—42no—
377C22⋊Q8—22no—
378C22⋊Q8—22no—
379C4×Q8—24no—
380C4×Q8—42no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4C23×C3112yes—
384C2×C4⋊C4C23×C3111yescharacteristic
385C2×C22⋊C4—21no—
386C2×C22⋊C4—22no—
387C2×C22⋊C4—22no—
388C2×C22⋊C4—22no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—22no—
391C2×C22⋊C4—22no—
392C2×C22⋊C4—22no—
393C2×C22⋊C4D4×C3112yes—
394C2×C42—22no—
395C2×C42C23×C3112yes—
396C4⋊C8—24no—
397#103—22no—
398#103—24no—
399#102—24no—
400#102—22no—
401#102—42no—
402C22⋊C8—22no—
403#95—21no—
404#95D4×C3111yescharacteristic
405#95D4×C3111yescharacteristic
406C31#2437111yescharacteristic, direct product factor, Sylow
407C24—21no—
408C24—22no—
409C24—44no—
410C24—44no—
411C24—42no—
412C24—81no—
413C24—81no—
414C24C2×D4×C3111yescharacteristic
415#41—28no—
416#41—44no—
417#41—48no—
418C2×Q8—21no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—22no—
424C2×Q8C2×D4×C3112yes—
425C2×Q8C2×D4×C3114yes—
426C2×D4—28no—
427C2×D4—24no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—21no—
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—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—44no—
451C22×C4C2×D4×C3112yes—
452C22×C4C2×D4×C3112yes—
453C22×C4C2×D4×C3111yescharacteristic
454C22×C4C2×D4×C3112yes—
455C22×C4#41×C3112yes—
456C22×C4C24×C3111yescharacteristic
457Q16—44no—
458SD16—216no—
459SD16—44no—
460D8—216no—
461M16—28no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8C2×D4×C3114yes—
469C4⋊C4—22no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—22no—
473C4⋊C4—44no—
474C4⋊C4—42no—
475C22⋊C4—42no—
476C22⋊C4—44no—
477C22⋊C4—44no—
478C22⋊C4—44no—
479C22⋊C4—42no—
480C22⋊C4—82no—
481C22⋊C4—82no—
482C22⋊C4—82no—
483C22⋊C4—81no—
484C42—22no—
485C42—22no—
486C42—44no—
487C23—22no—
488C23—22no—
489C23—21no—
490C23—21no—
491C23—22no—
492C23—24no—
493C23—48no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—41no—
501C23—42no—
502C23—42no—
503C23—41no—
504C23—84no—
505C23—84no—
506C23—88no—
507C23—88no—
508C23C31×#12112yes—
509C23C22×D4×C3111yescharacteristic
510Q8—22no—
511Q8—24no—
512Q8—41no—
513Q8—41no—
514Q8—42no—
515Q8C31×#12014yes—
516D4—22no—
517D4—24no—
518D4—22no—
519D4—24no—
520D4—48no—
521D4—48no—
522D4—42no—
523D4—42no—
524D4—42no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—48no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4C31×#12112yes—
547C2×C4C22×D4×C3111yescharacteristic
548C2×C4C22×D4×C3112yes—
549C2×C4C2×#41×C3111yescharacteristic
550C2×C4C31×H(5,𝔽2)12yes—
551C8—24no—
552C8—41no—
553C8—41no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—44no—
561C22—44no—
562C22—44no—
563C22—42no—
564C22—42no—
565C22—816no—
566C22—82no—
567C22—82no—
568C22—82no—
569C22—82no—
570C22—82no—
571C22—82no—
572C22C2×C31×#12111yescharacteristic
573C22D42×C3112yes—
574C22C31×#57512yes—
575C22C31×#57611yescharacteristic
576C22C31×#57711yescharacteristic
577C4—22no—
578C4—24no—
579C4—42no—
580C4—42no—
581C4—81no—
582C4—82no—
583C4—81no—
584C4C2×C31×#12011yescharacteristic
585C4C31×#53411yescharacteristic
586C4C31×#54612yes—
587C2—42no—
588C2—84no—
589C2—82no—
590C2C31×#241011yescharacteristic
591C2C31×#304011yescharacteristic
592C2C31×#304511yescharacteristic
593C2C31×#326812yes—
594C2C31×#327512yes—
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×#464C2semidirect
3. C2×C31×#452C2semidirect
4. C31×#2435C2semidirect
5. C31×#2341C2semidirect
6. C31×#2054C2semidirect
7. C31×#2012C2semidirect
8. C31×#1894C2semidirect
9. C31×#1880C2semidirect
10. C31×#1851C2semidirect
11. C31×#1786C2semidirect
16. C22×Q16×C31C22semidirect
18. C22×SD16×C31C22semidirect
23. C2×C4×Q8×C31C22semidirect
26. C2×C31×C4⋊C8C22semidirect
27. C2×C31×#103C22semidirect
28. C2×C31×#103C22semidirect
31. C2×C31×#102C22semidirect
37. C31×#390C22semidirect
43. C31×#340C22semidirect
44. C31×#336C22semidirect
189. #24371C31direct (N × H)
258. C2×#464C2×C31semidirect
259. C2×#452C2×C31semidirect
260. #2435C2×C31semidirect
261. #2341C2×C31semidirect
262. #2054C2×C31semidirect
263. #2012C2×C31semidirect
264. #1894C2×C31semidirect
265. #1880C2×C31semidirect
266. #1851C2×C31semidirect
267. #1786C2×C31semidirect
305. C22×Q16C22×C31semidirect
307. C22×SD16C22×C31semidirect
312. C2×C4×Q8C22×C31semidirect
315. C2×C4⋊C8C22×C31semidirect
316. C2×#103C22×C31semidirect
317. C2×#103C22×C31semidirect
320. C2×#102C22×C31semidirect
326. #390C22×C31semidirect
332. #340C22×C31semidirect
333. #336C22×C31semidirect
406. C31#24371direct (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×#4642
C2×C31×#4522
C31×#24352
C31×#23412
C31×#20542
C31×#20122
C31×#18942
C31×#18802
C31×#18512
C31×#17862
#2437131