Finite Group Databasebeta

← #392812

Subgroups — #392812

Order 512.

1998 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C4×C8C211yesmaximal, characteristic
2C2×#1933C212yesmaximal, complemented
3C2×#1892C211yesmaximal, characteristic
4C2×#1892C211yesmaximal, characteristic
5#14702C211yesmaximal, characteristic
6#14697C211yesmaximal, characteristic
7#10926C212yesmaximal, complemented
8#10880C212yesmaximal, complemented
9#10820C212yesmaximal, complemented
10#10721C211yesmaximal, characteristic
11#10721C211yesmaximal, characteristic
12C24×C8C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×C22⋊Q8C2211yescharacteristic
15C23×C42C2211yescharacteristic
16C22×#103C2211yescharacteristic
17C22×#103C2211yescharacteristic
18C22×#103C2212yes—
19C22×C4×C8—28no—
20C22×C4×C8—24no—
21C22×C4×C8C2212yes—
22C22×C4×C8C2212yes—
23C2×#391C2212yes—
24C2×#389C2212yes—
25C2×#385—21no—
26C2×#380C2211yescharacteristic
27C2×#380C2211yescharacteristic
28C2×#380C2211yescharacteristic
29#1933—28no—
30#1892—24no—
31#1892C418yescomplemented
32C2×#340C2211yescharacteristic
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2211yescharacteristic
37C2×#340C2212yes—
38C2×#340C2212yes—
39#1436—22no—
40#1436—22no—
41C4⋊#95C2212yes—
42C4⋊#95C2212yes—
43C22⋊#95—21no—
44C22⋊#95—21no—
45C22⋊#95—21no—
46C23×Q8C2311yescharacteristic
47C24×C4D411yescharacteristic
48C24×C4D411yescharacteristic
49C24×C4C2311yescharacteristic
50C23×C8—22no—
51C23×C8—22no—
52C23×C8—22no—
53C23×C8—22no—
54C23×C8—24no—
55C23×C8—24no—
56C23×C8—24no—
57C23×C8—24no—
58C23×C8C2×C414yes—
59C23×C8D414yes—
60C23×C8C2312yes—
61C23×C8C2311yescharacteristic
62C23×C8C2311yescharacteristic
63C2×C22⋊Q8—22no—
64C2×C22⋊Q8—22no—
65C2×C22⋊Q8—22no—
66C2×C22⋊Q8—22no—
67C2×C22⋊Q8—22no—
68C2×C22⋊Q8C2×C414yes—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C22⋊C4—21no—
77C22×C22⋊C4—21no—
78C22×C22⋊C4—21no—
79C22×C22⋊C4—21no—
80C22×C42—22no—
81C22×C42—22no—
82C22×C42—24no—
83C22×C42—24no—
84C22×C42D412yes—
85C22×C42C2312yes—
86C2×#103—24no—
87C2×#103—24no—
88C2×#103—22no—
89C2×#103—22no—
90C2×#103—24no—
91C2×#103—24no—
92C2×#103C2×C418yescomplemented
93C2×C4×C8—232no—
94C2×C4×C8—24no—
95C2×C4×C8—24no—
96C2×C4×C8—24no—
97C2×C4×C8—24no—
98C2×C4×C8—24no—
99C2×C4×C8D418yes—
100#391—24no—
101#391—24no—
102#389—24no—
103#389—24no—
104#385—44no—
105#380—22no—
106#380—22no—
107#380—22no—
108#380—22no—
109#380—22no—
110#380C2×C414yes—
111C2×#95—21no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—21no—
120C2×#95—21no—
121#340—22no—
122#340—22no—
123#340—24no—
124#340—24no—
125#340—24no—
126#340—24no—
127#340C2×C418yescomplemented
128C25C2×D411yescharacteristic
129C22×Q8—21no—
130C22×Q8—21no—
131C22×Q8—22no—
132C22×Q8—21no—
133C22×Q8—21no—
134C22×Q8—21no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8C22×C412yes—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—24no—
144C23×C4—24no—
145C23×C4—24no—
146C23×C4—24no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—22no—
156C23×C4—21no—
157C23×C4SD1614yes—
158C23×C4Q1614yes—
159C23×C4C22×C412yes—
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D412yes—
162C23×C4C2×D412yes—
163C23×C4C2×D412yes—
164C23×C4#4112yes—
165C23×C4#4112yes—
166C23×C4#4112yes—
167C23×C4#4112yes—
168C23×C4C2411yescharacteristic
169C22×C8—216no—
170C22×C8—216no—
171C22×C8—216no—
172C22×C8—28no—
173C22×C8—28no—
174C22×C8—28no—
175C22×C8—28no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—21no—
182C22×C8—22no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—216no—
187C22×C8—28no—
188C22×C8—28no—
189C22×C8C22×C414yes—
190C22×C8C2×D414yes—
191C22×C8C2×D414yes—
192C22×C8C2×D412yes—
193C22×C8C2×D412yes—
194C22×C8#4114yes—
195C22×C8#4114yes—
196C22⋊Q8—28no—
197C22⋊Q8—28no—
198C22⋊Q8—44no—
199C22⋊Q8—44no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4C22×C412yes—
231C2×C4⋊C4C22×C414yes—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—42no—
244C2×C42—28no—
245C2×C42—216no—
246C2×C42—28no—
247C2×C42—24no—
248C2×C42—24no—
249C2×C42—24no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42C2×D414yes—
255C2×C42#4114yes—
256#103—216no—
257#103—44no—
258#103—44no—
259C4×C8—216no—
260C4×C8—216no—
261#95—24no—
262#95—42no—
263#95—44no—
264#95—44no—
265#95—44no—
266#95—42no—
267#95—44no—
268#95—44no—
269#95—44no—
270#95—44no—
271C24—22no—
272C24—22no—
273C24—22no—
274C24—22no—
275C24C4×D412yes—
276C24#12112yes—
277C24#12412yes—
278C24C2×SD1612yes—
279C24C2×Q1612yes—
280C24#13512yes—
281C24#13512yes—
282C24C22×D411yescharacteristic
283C2×Q8—22no—
284C2×Q8—24no—
285C2×Q8—22no—
286C2×Q8—24no—
287C2×Q8—41no—
288C2×Q8—42no—
289C2×Q8—41no—
290C2×Q8—42no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8—42no—
294C2×Q8—41no—
295C2×Q8—42no—
296C2×Q8—42no—
297C2×Q8—41no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×Q8—42no—
301C22×C4—28no—
302C22×C4—28no—
303C22×C4—216no—
304C22×C4—216no—
305C22×C4—28no—
306C22×C4—28no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—28no—
318C22×C4—28no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—216no—
334C22×C4—216no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—42no—
369C22×C4#11714yes—
370C22×C4C4×D414yes—
371C22×C4C4×D412yes—
372C22×C4#12112yes—
373C22×C4#12114yes—
374C22×C4#12114yes—
375C22×C4#12314yes—
376C22×C4#12412yes—
377C22×C4#12614yes—
378C22×C4C2×SD1614yes—
379C22×C4C2×Q1614yes—
380C22×C4#13514yes—
381C22×C4#13514yes—
382C22×C4C23×C411yescharacteristic
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D412yes—
385C22×C4C2×#4111yescharacteristic
386C22×C4C2×#4111yescharacteristic
387C22×C4C2×#4111yescharacteristic
388C22×C4C2×#4112yes—
389C22×C4C2×#4112yes—
390C22×C4C2×#4112yes—
391C22×C4C2×#4111yescharacteristic
392C2×C8—232no—
393C2×C8—232no—
394C2×C8—24no—
395C2×C8—24no—
396C2×C8—22no—
397C2×C8—24no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—24no—
402C2×C8—24no—
403C2×C8—24no—
404C2×C8—24no—
405C2×C8—24no—
406C2×C8—24no—
407C2×C8—24no—
408C2×C8C4×D418yes—
409C2×C8#12014yes—
410C2×C8#12314yes—
411C4⋊C4—22no—
412C4⋊C4—24no—
413C4⋊C4—22no—
414C4⋊C4—24no—
415C4⋊C4—41no—
416C4⋊C4—42no—
417C4⋊C4—41no—
418C4⋊C4—42no—
419C4⋊C4—48no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—42no—
423C4⋊C4—41no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—41no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—88no—
431C22⋊C4—48no—
432C22⋊C4—48no—
433C22⋊C4—84no—
434C22⋊C4—84no—
435C42—216no—
436C42—24no—
437C42—24no—
438C42—24no—
439C42—24no—
440C23—22no—
441C23—24no—
442C23—22no—
443C23—24no—
444C23—24no—
445C23—24no—
446C23—24no—
447C23—24no—
448C23—24no—
449C23—24no—
450C23—24no—
451C23—22no—
452C23—22no—
453C23—24no—
454C23—24no—
455C23—24no—
456C23#38814yes—
457C23C4×SD1614yes—
458C23C4×Q1614yes—
459C23#46514yes—
460C23#46714yes—
461C23#48714yes—
462C23#48814yes—
463C23C2×C4×D411yescharacteristic
464C23C2×#12112yes—
465C23C2×#12111yescharacteristic
466C23C2×#12411yescharacteristic
467C23#53212yes—
468C23C22×SD1611yescharacteristic
469C23C22×Q1611yescharacteristic
470C23C2×#13511yescharacteristic
471C23C2×#13511yescharacteristic
472C23C2×#13512yes—
473C23C2×#13512yes—
474Q8—44no—
475Q8—42no—
476Q8—42no—
477Q8—81no—
478Q8—82no—
479Q8—81no—
480C2×C4—232no—
481C2×C4—232no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—28no—
500C2×C4—24no—
501C2×C4—28no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—81no—
512C2×C4—82no—
513C2×C4—81no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—81no—
520C2×C4—81no—
521C2×C4—81no—
522C2×C4—82no—
523C2×C4—81no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—81no—
536C2×C4—82no—
537C2×C4—81no—
538C2×C4#38814yes—
539C2×C4#46518yes—
540C2×C4#46718yes—
541C2×C4C2×#11711yescharacteristic
542C2×C4C2×C4×D412yes—
543C2×C4C2×C4×D411yescharacteristic
544C2×C4C4×#4112yes—
545C2×C4C2×#12011yescharacteristic
546C2×C4C2×#12311yescharacteristic
547C2×C4C2×#12311yescharacteristic
548C2×C4#52512yes—
549C2×C4#52512yes—
550C2×C4#52512yes—
551C2×C4#52512yes—
552C2×C4#52511yescharacteristic
553C2×C4#52511yescharacteristic
554C2×C4C2×#12611yescharacteristic
555C2×C4#52912yes—
556C2×C4#52912yes—
557C2×C4#52912yes—
558C2×C4#52912yes—
559C8—24no—
560C8—24no—
561C8—28no—
562C22—28no—
563C22—28no—
564C22—28no—
565C22—28no—
566C22—24no—
567C22—24no—
568C22—24no—
569C22—24no—
570C22—24no—
571C22—24no—
572C22#193318yescomplemented
573C22C2×#38811yescharacteristic
574C22C4×#12112yes—
575C22#284711yescharacteristic
576C22#285012yes—
577C22#285411yescharacteristic
578C22C2×C4×SD1611yescharacteristic
579C22C2×C4×Q1611yescharacteristic
580C22C4×#13512yes—
581C22C2×#46512yes—
582C22C2×#46511yescharacteristic
583C22C2×#46712yes—
584C22C2×#46711yescharacteristic
585C22#305012yes—
586C22#305012yes—
587C22#305011yescharacteristic
588C22#305011yescharacteristic
589C22C2×#48711yescharacteristic
590C22C2×#48811yescharacteristic
591C22#313112yes—
592C22#316712yes—
593C22#316812yes—
594C22#316812yes—
595C22#317012yes—
596C4—24no—
597C4—28no—
598C4—28no—
599C4—84no—
600C4—82no—
601C4—82no—
602C4—162no—
603C4—161no—
604C4—161no—
605C4#229012yes—
606C4C4×#12011yescharacteristic
607C4C4×#12311yescharacteristic
608C4#284911yescharacteristic
609C4#285212yes—
610C4#285311yescharacteristic
611C2—28no—
612C2#1469611yescharacteristic
613C2C2×#193311yescharacteristic
614C2#2108311yescharacteristic
615C2C4×#46512yes—
616C2C4×#46712yes—
617C2#3292911yescharacteristic
618C2#3293111yescharacteristic
619C2#3304312yes—
620C2#3304412yes—
621C2#3312811yescharacteristic
622C2#3313011yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1933C2semidirect
7. #10926C2semidirect
8. #10880C2semidirect
9. #10820C2semidirect
31. #1892C4semidirect
92. C2×#103C2×C4semidirect
127. #340C2×C4semidirect
572. C22#1933semidirect

Maximal subgroups

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

subgroupindex
C23×C4×C82
C2×#19332
C2×#18922
#147022
#146972
#109262
#108802
#108202
#107212