Finite Group Databasebeta

← #393365

Subgroups — #393365

Order 512.

1922 conjugacy classes of subgroups.

The subgroup lattice has 623 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#478C211yesmaximal, characteristic
2C2×#2275C211yesmaximal, characteristic
3C2×#1989C211yesmaximal, characteristic, complemented
4C2×#1802C211yesmaximal, characteristic
5#14283C211yesmaximal, characteristic, complemented
6#10908C212yesmaximal, complemented
7#10907C212yesmaximal, complemented
8#10849C211yesmaximal, characteristic, complemented
9#10833C211yesmaximal, characteristic, complemented
10#10788C212yesmaximal
11#10788C212yesmaximal
12C22×#125C2211yescharacteristic
13C22×C4×D4C2212yes—
14C22×C4×D4C2211yescharacteristic
15C2×#478—24no—
16C2×#478—24no—
17C2×#478C418yescomplemented
18C22×#107C2211yescharacteristic
19C22×#106C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C22×#102C2212yes—
23C22×#102C2211yescharacteristic
24C22×#102C2211yescharacteristic
25C2×#384C2211yescharacteristic
26#2275—24no—
27#2275—24no—
28C2×C4×C4⋊C4C2212yes—
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C22⋊C4—21no—
31C2×C4×C22⋊C4—22no—
32C2×#376C2211yescharacteristic
33C22×#95—21no—
34#1989—24no—
35#1802—24no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2212yes—
42C2×#336C2212yes—
43#1441—21no—
44#1441—22no—
45#1441—21no—
46C4⋊#95C2212yes—
47C4⋊#95C2212yes—
48C23×D4C2311yescharacteristic
49C24×C4—22no—
50C24×C4—21no—
51C23×C8C2312yes—
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C2×#125—22no—
55C2×#125—22no—
56C2×#125—22no—
57C2×#125—22no—
58C2×#125—22no—
59C2×#125C2×C414yescomplemented
60C2×C4×D4—24no—
61C2×C4×D4—24no—
62C2×C4×D4—24no—
63C2×C4×D4—24no—
64C2×C4×D4—22no—
65C2×C4×D4—24no—
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C2×C4×D4C2×C414yes—
69C2×C4×D4D414yes—
70C2×C4×D4D414yes—
71C2×C4×D4Q818yes—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2312yes—
81C22×C22⋊C4—22no—
82C22×C22⋊C4—21no—
83C22×C42—22no—
84C22×C42—21no—
85C22×C42C2312yes—
86C22×C42C2311yescharacteristic
87#478—232no—
88C2×#107—22no—
89C2×#107—22no—
90C2×#107C2×C414yescomplemented
91C2×#106—22no—
92C2×#106—22no—
93C2×#106C2×C414yescomplemented
94C2×C4⋊C8—22no—
95C2×C4⋊C8—22no—
96C2×C4⋊C8—22no—
97C2×C4⋊C8—22no—
98C2×C4⋊C8—22no—
99C2×C4⋊C8C2×C414yescomplemented
100C2×#102—22no—
101C2×#102—22no—
102C2×#102—22no—
103C2×#102—22no—
104C2×#102—24no—
105C2×#102—24no—
106C2×#102—24no—
107C2×#102C2×C414yes—
108C2×#102C2×C414yes—
109#384—22no—
110#384—22no—
111C4×C4⋊C4—24no—
112C4×C4⋊C4—24no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4D414yes—
115C4×C22⋊C4—216no—
116C4×C22⋊C4—28no—
117#376—22no—
118#376—22no—
119C2×#95—28no—
120C2×#95—22no—
121C2×#95—21no—
122C2×#95—22no—
123C2×#95—22no—
124C2×#95—22no—
125C2×#95—22no—
126C2×#95D411yescharacteristic
127C2×#95D411yescharacteristic
128C2×#95D412yes—
129#340—24no—
130#340—24no—
131#340—24no—
132#340—24no—
133#340—24no—
134#336—24no—
135#336—24no—
136C25—21no—
137C22×D4—21no—
138C22×D4—21no—
139C22×D4—22no—
140C22×D4—22no—
141C22×D4—21no—
142C22×D4C22×C412yes—
143C22×D4C2×D412yes—
144C22×D4C2×D412yes—
145C22×D4C2×Q814yes—
146C22×D4#4114yes—
147C23×C4—21no—
148C23×C4—22no—
149C23×C4—28no—
150C23×C4—28no—
151C23×C4—24no—
152C23×C4—24no—
153C23×C4—24no—
154C23×C4—28no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—22no—
158C23×C4—82no—
159C23×C4—82no—
160C23×C4—82no—
161C23×C4—82no—
162C23×C4—82no—
163C23×C4—82no—
164C23×C4C2×D412yes—
165C23×C4C2×D412yes—
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2411yescharacteristic
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8C22×C412yes—
185C22×C8C22×C412yes—
186#125—28no—
187#125—28no—
188#125—44no—
189#125—44no—
190C4×D4—216no—
191C4×D4—216no—
192C4×D4—216no—
193C4×D4—216no—
194C4×D4—28no—
195C4×D4—28no—
196C4×D4—28no—
197C4×D4C22⋊C4116yes—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—81no—
229C2×C4⋊C4—82no—
230C2×C4⋊C4—81no—
231C2×C4⋊C4C22×C412yes—
232C2×C4⋊C4C22×C412yes—
233C2×C4⋊C4C22×C412yes—
234C2×C4⋊C4C2×D412yes—
235C2×C4⋊C4C2×D412yes—
236C2×C4⋊C4C2×D414yes—
237C2×C4⋊C4C2×Q814yes—
238C2×C4⋊C4#4114yes—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—28no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—28no—
250C2×C42—24no—
251C2×C42—81no—
252C2×C42—81no—
253C2×C42—82no—
254C2×C42—82no—
255C2×C42C22×C412yes—
256C2×C42C2×D412yes—
257C2×C42C2×D412yes—
258C2×C42C2×D412yes—
259C2×C42C2×Q814yes—
260C2×C42#4114yes—
261C2×C42#4112yes—
262C2×C42#4112yes—
263#107—28no—
264#106—28no—
265C4⋊C8—28no—
266C4⋊C8—44no—
267#102—28no—
268#102—28no—
269#102—48no—
270#95—28no—
271#95—22no—
272#95—24no—
273#95—24no—
274#95—22no—
275#95—82no—
276#95—82no—
277#95—82no—
278#95—82no—
279C24—22no—
280C24—22no—
281C24—22no—
282C24—24no—
283C24—24no—
284C24—41no—
285C24—42no—
286C24—41no—
287C24—42no—
288C24—41no—
289C24—41no—
290C24C22×D411yescharacteristic
291C2×D4—22no—
292C2×D4—24no—
293C2×D4—22no—
294C2×D4—24no—
295C2×D4—22no—
296C2×D4—24no—
297C2×D4—24no—
298C2×D4—24no—
299C2×D4—24no—
300C2×D4—22no—
301C2×D4—22no—
302C2×D4—24no—
303C2×D4—24no—
304C2×D4C2×C22⋊C414yes—
305C2×D4C4×Q818yes—
306C2×D4C22⋊Q818yes—
307C2×D4#12414yes—
308C2×D4C4⋊Q814yes—
309C22×C4—21no—
310C22×C4—22no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—232no—
334C22×C4—216no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—42no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—42no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—88no—
361C22×C4—88no—
362C22×C4—84no—
363C22×C4—84no—
364C22×C4—88no—
365C22×C4—88no—
366C22×C4—88no—
367C22×C4—88no—
368C22×C4—88no—
369C22×C4—88no—
370C22×C4C4×D412yes—
371C22×C4C4×D412yes—
372C22×C4#12012yes—
373C22×C4#12012yes—
374C22×C4#12112yes—
375C22×C4#12112yes—
376C22×C4C22⋊Q814yes—
377C22×C4C22⋊Q814yes—
378C22×C4C23×C411yescharacteristic
379C22×C4C22×D411yescharacteristic
380C22×C4C22×D411yescharacteristic
381C22×C4C22×D411yescharacteristic
382C22×C4C22×Q812yes—
383C22×C4C2×#4111yescharacteristic
384C22×C4C2×#4112yes—
385C22×C4C2×#4112yes—
386C22×C4C2×#4112yes—
387C22×C4C2×#4111yescharacteristic
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—24no—
391C2×C8—24no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—42no—
403C2×C8—42no—
404C2×C8—42no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—24no—
408C4⋊C4—22no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—48no—
420C4⋊C4—48no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—84no—
426C4⋊C4—84no—
427C4⋊C4C2×C22⋊C414yes—
428C4⋊C4C22⋊Q818yes—
429C22⋊C4—232no—
430C22⋊C4—216no—
431C42—24no—
432C42—24no—
433C42—24no—
434C42—24no—
435C42—84no—
436C42—88no—
437C42C2×C22⋊C414yes—
438C42C4×D414yes—
439C42#12114yes—
440C42#12314yes—
441C23—28no—
442C23—24no—
443C23—24no—
444C23—24no—
445C23—28no—
446C23—44no—
447C23—44no—
448C23—44no—
449C23—44no—
450C23—44no—
451C23—44no—
452C23—44no—
453C23—44no—
454C23—44no—
455C23—42no—
456C23—42no—
457C23—44no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23C2×C4×D411yescharacteristic
465C23C2×#12011yescharacteristic
466C23C2×#12111yescharacteristic
467C23C2×C22⋊Q812yes—
468C23#52512yes—
469C23#57611yescharacteristic
470C23#57612yes—
471C23#57711yescharacteristic
472C23#57712yes—
473C23#57711yescharacteristic
474C23#57811yescharacteristic
475D4—24no—
476D4—28no—
477D4—24no—
478D4—24no—
479D4—28no—
480D4#39118yes—
481C2×C4—24no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—28no—
498C2×C4—24no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—816no—
523C2×C4—832no—
524C2×C4—82no—
525C2×C4—81no—
526C2×C4—81no—
527C2×C4—81no—
528C2×C4—81no—
529C2×C4—81no—
530C2×C4—81no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4#38614yes—
539C2×C4#38614yes—
540C2×C4C22×C22⋊C411yescharacteristic
541C2×C4C2×C4×D411yescharacteristic
542C2×C4C2×C4×Q812yes—
543C2×C4C4×#4112yes—
544C2×C4C2×#12111yescharacteristic
545C2×C4C2×C22⋊Q812yes—
546C2×C4C2×C22⋊Q812yes—
547C2×C4C2×#12311yescharacteristic
548C2×C4#52512yes—
549C2×C4#52512yes—
550C2×C4C2×#12411yescharacteristic
551C2×C4#52912yes—
552C2×C4#52912yes—
553C2×C4C2×C4⋊Q811yescharacteristic
554C2×C4#53211yescharacteristic
555C2×C4#53211yescharacteristic
556C2×C4#53312yes—
557C2×C4#53312yes—
558C8—42no—
559C8—42no—
560C8—82no—
561C22—28no—
562C22—44no—
563C22—44no—
564C22—48no—
565C22—48no—
566C22—44no—
567C22—44no—
568C22—44no—
569C22—48no—
570C22—48no—
571C22—44no—
572C22C2×#38611yescharacteristic
573C22C4×C22⋊Q812yes—
574C22#284812yes—
575C22#285011yescharacteristic
576C22#285711yescharacteristic
577C22#294612yes—
578C22#295111yescharacteristic
579C22#295211yescharacteristic
580C22#301212yes—
581C22#301211yescharacteristic
582C22#301311yescharacteristic
583C22#304111yescharacteristic
584C22#304211yescharacteristic
585C22#304511yescharacteristic
586C22#304611yescharacteristic
587C22#308112yes—
588C22#308112yes—
589C22#308312yes—
590C22#308412yes—
591C22#311912yes—
592C22#312012yes—
593C22#312212yes—
594C22#312312yes—
595C4—24no—
596C4—24no—
597C4—44no—
598C4—44no—
599C4—48no—
600C4—44no—
601C4—82no—
602C4—82no—
603C4—162no—
604C4C2×#39111yescharacteristic
605C4#229711yescharacteristic
606C4C4×C22⋊Q812yes—
607C4#285011yescharacteristic
608C4#285512yes—
609C4#285711yescharacteristic
610C2—48no—
611C2#1470211yescharacteristic
612C2#2092911yescharacteristic
613C2#2093711yescharacteristic
614C2#2161912yes—
615C2#2162012yes—
616C2#3296612yes—
617C2#3296812yes—
618C2#3303811yescharacteristic
619C2#3304011yescharacteristic
620C2#3320311yescharacteristic
621C2#3320511yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1989C2semidirect
5. #14283C2semidirect
6. #10908C2semidirect
7. #10907C2semidirect
8. #10849C2semidirect
9. #10833C2semidirect
17. C2×#478C4semidirect
59. C2×#125C2×C4semidirect
90. C2×#107C2×C4semidirect
93. C2×#106C2×C4semidirect
99. C2×C4⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4782
C2×#22752
C2×#19892
C2×#18022
#142832
#109082
#109072
#108492
#108332
#107882