Finite Group Databasebeta

← C31×#24358 #21481117

Subgroups — C31×#24358 #21481117

Order 7936.

1598 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×SD16×C31C211yesmaximal, characteristic
2C2×C31×#461C212yesmaximal, complemented
3C2×C31×#450C211yesmaximal, characteristic
4C31×#2434C211yesmaximal, characteristic, complemented
5D4×C22⋊C4×C31C211yesmaximal, characteristic
6C31×#2055C212yesmaximal, complemented
7C31×#2013C212yesmaximal, complemented
8C31×#1892C211yesmaximal, characteristic, complemented
9C31×#1877C212yesmaximal, complemented
10C31×#1850C211yesmaximal, characteristic, complemented
11C31×#1787C211yesmaximal, characteristic, complemented
12C23×Q8×C31C2211yescharacteristic
13C23×D4×C31C2211yescharacteristic
14C22×SD16×C31—24no—
15C22×SD16×C31—28no—
16C22×SD16×C31C2212yes—
17C22×SD16×C31C2211yescharacteristic
18C22×SD16×C31C2212yes—
19C22×SD16×C31C2211yescharacteristic
20C23×C8×C31C2211yescharacteristic
21C2×C31×C4⋊Q8C2212yescomplemented
22C2×C31×C22⋊Q8C2211yescharacteristic
23C2×C4×D4×C31C2212yes—
24C22×C22⋊C4×C31—21no—
25C31×#461—28no—
26C31×#450—24no—
27C2×C31×C4⋊C8C2212yescomplemented
28C2×C31×#103C2212yescomplemented
29C2×C31×#103C2212yescomplemented
30C2×C31×#103C2212yescomplemented
31C2×C31×#102C2211yescharacteristic
32C2×C31×#102C2211yescharacteristic
33C2×C31×C22⋊C8C2211yescharacteristic
34C31×#395—22no—
35C31×#391C2212yescomplemented
36C31×#390C2212yes—
37C31×#386—22no—
38C31×#385—21no—
39C31×#385—21no—
40C31×#380C2211yescharacteristic
41C31×#379—21no—
42C4×C22⋊C4×C31C2211yescharacteristic
43C31×#340C2212yescomplemented
44C31×#336C2212yescomplemented
45C25×C31—21no—
46C22×Q8×C31—21no—
47C22×Q8×C31—24no—
48C22×Q8×C31—22no—
49C22×Q8×C31—22no—
50C22×Q8×C31—42no—
51C22×Q8×C31C2311yescharacteristic
52C22×Q8×C31C2312yes—
53C22×D4×C31—24no—
54C22×D4×C31—21no—
55C22×D4×C31—22no—
56C22×D4×C31—42no—
57C22×D4×C31D414yes—
58C22×D4×C31C2311yescharacteristic
59C22×D4×C31C2311yescharacteristic
60C23×C4×C31—21no—
61C23×C4×C31—22no—
62C23×C4×C31C2311yescharacteristic
63C2×SD16×C31—232no—
64C2×SD16×C31—24no—
65C2×SD16×C31—24no—
66C2×SD16×C31—24no—
67C2×SD16×C31—24no—
68C2×SD16×C31—22no—
69C2×SD16×C31—22no—
70C2×SD16×C31—24no—
71C2×SD16×C31—24no—
72C2×SD16×C31—42no—
73C2×SD16×C31D418yes—
74C22×C8×C31—22no—
75C22×C8×C31—24no—
76C22×C8×C31C2311yescharacteristic
77C22×C8×C31C2311yescharacteristic
78C22×C8×C31C2312yes—
79C31×C4⋊Q8—24no—
80C31×C4⋊Q8—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×C31—22no—
87C2×C4⋊C4×C31—22no—
88C2×C4⋊C4×C31C2311yescharacteristic
89C2×C4⋊C4×C31C2312yes—
90C2×C22⋊C4×C31—21no—
91C2×C22⋊C4×C31—21no—
92C2×C22⋊C4×C31—22no—
93C2×C22⋊C4×C31—22no—
94C2×C22⋊C4×C31—22no—
95C2×C22⋊C4×C31—44no—
96C2×C22⋊C4×C31D411yescharacteristic
97C2×C22⋊C4×C31D411yescharacteristic
98C2×C42×C31C2312yes—
99C31×C4⋊C8—24no—
100C31×#103—24no—
101C31×#103—24no—
102C31×#103—42no—
103C31×#102—22no—
104C31×#102—22no—
105C31×C22⋊C8—22no—
106C31×#95—22no—
107C31×#95—22no—
108C31×#95—21no—
109C31×#95D412yes—
110C24×C31—21no—
111C24×C31—21no—
112C24×C31—24no—
113C24×C31—44no—
114C24×C31—42no—
115C24×C31—42no—
116C24×C31—82no—
117C24×C31C2×D411yescharacteristic
118C2×Q8×C31—22no—
119C2×Q8×C31—28no—
120C2×Q8×C31—22no—
121C2×Q8×C31—22no—
122C2×Q8×C31—22no—
123C2×Q8×C31—21no—
124C2×Q8×C31—22no—
125C2×Q8×C31—21no—
126C2×Q8×C31—22no—
127C2×Q8×C31—22no—
128C2×Q8×C31—44no—
129C2×Q8×C31—48no—
130C2×Q8×C31—44no—
131C2×Q8×C31—42no—
132C2×Q8×C31C2×D414yes—
133C2×D4×C31—28no—
134C2×D4×C31—24no—
135C2×D4×C31—22no—
136C2×D4×C31—21no—
137C2×D4×C31—21no—
138C2×D4×C31—22no—
139C2×D4×C31—22no—
140C2×D4×C31—44no—
141C2×D4×C31—48no—
142C2×D4×C31—44no—
143C2×D4×C31C2×D412yes—
144C2×D4×C31C2×D412yes—
145C2×D4×C31C2×D414yes—
146C22×C4×C31—22no—
147C22×C4×C31—22no—
148C22×C4×C31—22no—
149C22×C4×C31—21no—
150C22×C4×C31—22no—
151C22×C4×C31—21no—
152C22×C4×C31—44no—
153C22×C4×C31—44no—
154C22×C4×C31—42no—
155C22×C4×C31—44no—
156C22×C4×C31—82no—
157C22×C4×C31C2×D412yes—
158C22×C4×C31C2×D411yescharacteristic
159C22×C4×C31C2×D412yes—
160C22×C4×C31C2×D412yes—
161C22×C4×C31#4112yes—
162C22×C4×C31C2411yescharacteristic
163SD16×C31—216no—
164SD16×C31—216no—
165SD16×C31—44no—
166SD16×C31—44no—
167C2×C8×C31—28no—
168C2×C8×C31—22no—
169C2×C8×C31—22no—
170C2×C8×C31—21no—
171C2×C8×C31—21no—
172C2×C8×C31—22no—
173C2×C8×C31—22no—
174C2×C8×C31C2×D414yes—
175C4⋊C4×C31—21no—
176C4⋊C4×C31—22no—
177C4⋊C4×C31—21no—
178C4⋊C4×C31—22no—
179C4⋊C4×C31—44no—
180C4⋊C4×C31—44no—
181C4⋊C4×C31—42no—
182C22⋊C4×C31—22no—
183C22⋊C4×C31—44no—
184C22⋊C4×C31—42no—
185C22⋊C4×C31—42no—
186C22⋊C4×C31—44no—
187C22⋊C4×C31—82no—
188C22⋊C4×C31SD1614yescomplemented
189C42×C31—22no—
190C42×C31—22no—
191#24358C3111yesmaximal, characteristic, direct product factor, Sylow
192C23×C31—22no—
193C23×C31—22no—
194C23×C31—22no—
195C23×C31—24no—
196C23×C31—22no—
197C23×C31—48no—
198C23×C31—41no—
199C23×C31—41no—
200C23×C31—42no—
201C23×C31—42no—
202C23×C31—41no—
203C23×C31—41no—
204C23×C31—42no—
205C23×C31—48no—
206C23×C31—84no—
207C23×C31—88no—
208C23×C31—88no—
209C23×C31#12112yes—
210C23×C31C2×SD1612yes—
211C23×C31#13712yes—
212C23×C31C22×D411yescharacteristic
213Q8×C31—24no—
214Q8×C31—24no—
215Q8×C31—48no—
216Q8×C31—42no—
217Q8×C31—42no—
218Q8×C31—41no—
219Q8×C31—41no—
220Q8×C31—42no—
221D4×C31—22no—
222D4×C31—22no—
223D4×C31—22no—
224D4×C31—24no—
225D4×C31—48no—
226D4×C31—42no—
227D4×C31#12014yes—
228C2×C4×C31—22no—
229C2×C4×C31—22no—
230C2×C4×C31—22no—
231C2×C4×C31—22no—
232C2×C4×C31—22no—
233C2×C4×C31—24no—
234C2×C4×C31—22no—
235C2×C4×C31—24no—
236C2×C4×C31—22no—
237C2×C4×C31—22no—
238C2×C4×C31—48no—
239C2×C4×C31—42no—
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—42no—
247C2×C4×C31—42no—
248C2×C4×C31—88no—
249C2×C4×C31#12112yes—
250C2×C4×C31C2×SD1614yescomplemented
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—
258C23×SD16C2×C3111yescharacteristic
259C2×#461C2×C3112yescomplemented
260C2×#450C2×C3111yescharacteristic
261#2434C2×C3111yescharacteristic, complemented
262D4×C22⋊C4C2×C3111yescharacteristic
263#2055C2×C3112yescomplemented
264#2013C2×C3112yescomplemented
265#1892C2×C3111yescharacteristic, complemented
266#1877C2×C3112yescomplemented
267#1850C2×C3111yescharacteristic, complemented
268#1787C2×C3111yescharacteristic, complemented
269C22×C31—22no—
270C22×C31—24no—
271C22×C31—24no—
272C22×C31—22no—
273C22×C31—24no—
274C22×C31—44no—
275C22×C31—44no—
276C22×C31—42no—
277C22×C31—42no—
278C22×C31—42no—
279C22×C31—42no—
280C22×C31—42no—
281C22×C31—816no—
282C22×C31—82no—
283C22×C31—82no—
284C22×C31#46114yescomplemented
285C22×C31C2×#12111yescharacteristic
286C22×C31D4212yes—
287C22×C31C22×SD1611yescharacteristic
288C22×C31C2×#13711yescharacteristic
289C22×C31#57712yes—
290C4×C31—22no—
291C4×C31—24no—
292C4×C31—44no—
293C4×C31—42no—
294C4×C31—81no—
295C4×C31—82no—
296C4×C31—81no—
297C4×C31C2×#12011yescharacteristic
298C4×C31#53411yescharacteristic
299C4×C31#54612yes—
300C23×Q8C22×C3111yescharacteristic
301C23×D4C22×C3111yescharacteristic
302C22×SD16—24no—
303C22×SD16—28no—
304C22×SD16C22×C3112yes—
305C22×SD16C22×C3111yescharacteristic
306C22×SD16C22×C3112yes—
307C22×SD16C22×C3111yescharacteristic
308C23×C8C22×C3111yescharacteristic
309C2×C4⋊Q8C22×C3112yescomplemented
310C2×C22⋊Q8C22×C3111yescharacteristic
311C2×C4×D4C22×C3112yes—
312C22×C22⋊C4—21no—
313#461—28no—
314#450—24no—
315C2×C4⋊C8C22×C3112yescomplemented
316C2×#103C22×C3112yescomplemented
317C2×#103C22×C3112yescomplemented
318C2×#103C22×C3112yescomplemented
319C2×#102C22×C3111yescharacteristic
320C2×#102C22×C3111yescharacteristic
321C2×C22⋊C8C22×C3111yescharacteristic
322#395—22no—
323#391C22×C3112yescomplemented
324#390C22×C3112yes—
325#386—22no—
326#385—21no—
327#385—21no—
328#380C22×C3111yescharacteristic
329#379—21no—
330C4×C22⋊C4C22×C3111yescharacteristic
331#340C22×C3112yescomplemented
332#336C22×C3112yescomplemented
333C2×C31—24no—
334C2×C31—44no—
335C2×C31—82no—
336C2×C31#241011yescharacteristic
337C2×C31C2×#46111yescharacteristic
338C2×C31#304211yescharacteristic
339C2×C31#326612yes—
340C2×C31D4×SD1612yes—
341C25—21no—
342C22×Q8—21no—
343C22×Q8—24no—
344C22×Q8—22no—
345C22×Q8—22no—
346C22×Q8—42no—
347C22×Q8C23×C3111yescharacteristic
348C22×Q8C23×C3112yes—
349C22×D4—24no—
350C22×D4—21no—
351C22×D4—22no—
352C22×D4—42no—
353C22×D4D4×C3114yes—
354C22×D4C23×C3111yescharacteristic
355C22×D4C23×C3111yescharacteristic
356C23×C4—21no—
357C23×C4—22no—
358C23×C4C23×C3111yescharacteristic
359C2×SD16—232no—
360C2×SD16—24no—
361C2×SD16—24no—
362C2×SD16—24no—
363C2×SD16—24no—
364C2×SD16—22no—
365C2×SD16—22no—
366C2×SD16—24no—
367C2×SD16—24no—
368C2×SD16—42no—
369C2×SD16D4×C3118yes—
370C22×C8—22no—
371C22×C8—24no—
372C22×C8C23×C3111yescharacteristic
373C22×C8C23×C3111yescharacteristic
374C22×C8C23×C3112yes—
375C4⋊Q8—24no—
376C4⋊Q8—42no—
377C22⋊Q8—22no—
378C22⋊Q8—22no—
379C4×D4—24no—
380C4×D4—42no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—22no—
384C2×C4⋊C4C23×C3112yes—
385C2×C4⋊C4C23×C3111yescharacteristic
386C2×C22⋊C4—21no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—22no—
389C2×C22⋊C4—22no—
390C2×C22⋊C4—22no—
391C2×C22⋊C4—44no—
392C2×C22⋊C4D4×C3111yescharacteristic
393C2×C22⋊C4D4×C3111yescharacteristic
394C2×C42C23×C3112yes—
395C4⋊C8—24no—
396#103—24no—
397#103—24no—
398#103—42no—
399#102—22no—
400#102—22no—
401C22⋊C8—22no—
402#95—21no—
403#95—22no—
404#95—22no—
405#95D4×C3112yes—
406C31#2435811yescharacteristic, direct product factor, Sylow
407C24—21no—
408C24—21no—
409C24—24no—
410C24—44no—
411C24—42no—
412C24—42no—
413C24—82no—
414C24C2×D4×C3111yescharacteristic
415C2×Q8—22no—
416C2×Q8—28no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—21no—
421C2×Q8—22no—
422C2×Q8—21no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—44no—
426C2×Q8—48no—
427C2×Q8—44no—
428C2×Q8—42no—
429C2×Q8C2×D4×C3114yes—
430C2×D4—28no—
431C2×D4—24no—
432C2×D4—22no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—22no—
436C2×D4—22no—
437C2×D4—44no—
438C2×D4—48no—
439C2×D4—44no—
440C2×D4C2×D4×C3112yes—
441C2×D4C2×D4×C3112yes—
442C2×D4C2×D4×C3114yes—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—42no—
452C22×C4—44no—
453C22×C4—82no—
454C22×C4C2×D4×C3112yes—
455C22×C4C2×D4×C3111yescharacteristic
456C22×C4C2×D4×C3112yes—
457C22×C4C2×D4×C3112yes—
458C22×C4#41×C3112yes—
459C22×C4C24×C3111yescharacteristic
460SD16—216no—
461SD16—216no—
462SD16—44no—
463SD16—44no—
464C2×C8—28no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8C2×D4×C3114yes—
472C4⋊C4—21no—
473C4⋊C4—22no—
474C4⋊C4—21no—
475C4⋊C4—22no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—42no—
479C22⋊C4—22no—
480C22⋊C4—44no—
481C22⋊C4—42no—
482C22⋊C4—42no—
483C22⋊C4—44no—
484C22⋊C4—82no—
485C22⋊C4SD16×C3114yescomplemented
486C42—22no—
487C42—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—24no—
492C23—22no—
493C23—48no—
494C23—41no—
495C23—41no—
496C23—42no—
497C23—48no—
498C23—42no—
499C23—41no—
500C23—41no—
501C23—42no—
502C23—84no—
503C23—88no—
504C23—88no—
505C23C31×#12112yes—
506C23C2×SD16×C3112yes—
507C23C31×#13712yes—
508C23C22×D4×C3111yescharacteristic
509Q8—24no—
510Q8—24no—
511Q8—48no—
512Q8—42no—
513Q8—42no—
514Q8—41no—
515Q8—41no—
516Q8—42no—
517D4—22no—
518D4—22no—
519D4—22no—
520D4—24no—
521D4—48no—
522D4—42no—
523D4C31×#12014yes—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—48no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—88no—
545C2×C4C31×#12112yes—
546C2×C4C2×SD16×C3114yescomplemented
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—24no—
556C22—24no—
557C22—22no—
558C22—24no—
559C22—44no—
560C22—44no—
561C22—42no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—42no—
566C22—816no—
567C22—82no—
568C22—82no—
569C22C31×#46114yescomplemented
570C22C2×C31×#12111yescharacteristic
571C22D42×C3112yes—
572C22C22×SD16×C3111yescharacteristic
573C22C2×C31×#13711yescharacteristic
574C22C31×#57712yes—
575C4—22no—
576C4—24no—
577C4—44no—
578C4—42no—
579C4—81no—
580C4—82no—
581C4—81no—
582C4C2×C31×#12011yescharacteristic
583C4C31×#53411yescharacteristic
584C4C31×#54612yes—
585C2—24no—
586C2—44no—
587C2—82no—
588C2C31×#241011yescharacteristic
589C2C2×C31×#46111yescharacteristic
590C2C31×#304211yescharacteristic
591C2C31×#326612yes—
592C2D4×SD16×C3112yes—
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC23×D4×C31C22
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×#461C2semidirect
4. C31×#2434C2semidirect
6. C31×#2055C2semidirect
7. C31×#2013C2semidirect
8. C31×#1892C2semidirect
9. C31×#1877C2semidirect
10. C31×#1850C2semidirect
11. C31×#1787C2semidirect
21. C2×C31×C4⋊Q8C22semidirect
27. C2×C31×C4⋊C8C22semidirect
28. C2×C31×#103C22semidirect
29. C2×C31×#103C22semidirect
30. C2×C31×#103C22semidirect
35. C31×#391C22semidirect
43. C31×#340C22semidirect
44. C31×#336C22semidirect
188. C22⋊C4×C31SD16semidirect
191. #24358C31direct (N × H)
250. C2×C4×C31C2×SD16semidirect
259. C2×#461C2×C31semidirect
261. #2434C2×C31semidirect
263. #2055C2×C31semidirect
264. #2013C2×C31semidirect
265. #1892C2×C31semidirect
266. #1877C2×C31semidirect
267. #1850C2×C31semidirect
268. #1787C2×C31semidirect
284. C22×C31#461semidirect
309. C2×C4⋊Q8C22×C31semidirect
315. C2×C4⋊C8C22×C31semidirect
316. C2×#103C22×C31semidirect
317. C2×#103C22×C31semidirect
318. C2×#103C22×C31semidirect
323. #391C22×C31semidirect
331. #340C22×C31semidirect
332. #336C22×C31semidirect
406. C31#24358direct (N × H)
485. C22⋊C4SD16×C31semidirect
546. C2×C4C2×SD16×C31semidirect
569. C22C31×#461semidirect

Maximal subgroups

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

subgroupindex
C23×SD16×C312
C2×C31×#4612
C2×C31×#4502
C31×#24342
D4×C22⋊C4×C312
C31×#20552
C31×#20132
C31×#18922
C31×#18772
C31×#18502
C31×#17872
#2435831