Finite Group Databasebeta

← #204590

Subgroups — #204590

Order 512.

1371 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35085C211yesmaximal, characteristic
2#20525C211yesmaximal, characteristic, complemented
3C2×#1504C211yesmaximal, characteristic
4C2×#1508C211yesmaximal, characteristic, complemented
5C2×#1501C211yesmaximal, characteristic
6C22×#328C211yesmaximal, characteristic, complemented
7#13782C211yesmaximal, characteristic, complemented
8#9683C212yesmaximal, complemented
9#9682C212yesmaximal, complemented
10#9681C212yesmaximal, complemented
11#9679C212yesmaximal, complemented
12C2×#525C2211yescharacteristic
13C22×C22⋊Q8C2211yescharacteristic
14C2×#407C2211yescharacteristic
15C22×C22⋊C8C2211yescharacteristic
16#2857C2212yes—
17#2850C2212yes—
18#2663—22no—
19#2564—22no—
20#2503C411yescharacteristic, complemented
21#2501C411yescharacteristic, complemented
22#2499C412yescomplemented
23#2436C411yescharacteristic, complemented
24#2434C411yescharacteristic, complemented
25#2432C412yescomplemented
26#2412—21no—
27#2412—21no—
28#2322—21no—
29#2322—21no—
30C2×#385—21no—
31C2×C4×C22⋊C4C2211yescharacteristic
32C2×C4×C22⋊C4C2211yescharacteristic
33C22×#95C2211yescharacteristic
34#1753—22no—
35#1752—21no—
36C8×C22⋊C4—21no—
37#1749C2212yes—
38C4×C22⋊C8C2212yes—
39#1508—24no—
40#1504—24no—
41#1501—24no—
42C2×#328—24no—
43C2×#328C2212yes—
44C2×#328C2212yes—
45C2×#328C2211yescharacteristic
46C2×#328C2211yescharacteristic
47C2×#328C2212yes—
48C2×#328C2212yes—
49C2×#328C2212yes—
50C2×#327C2211yescharacteristic
51C2×#327C2211yescharacteristic
52C2×#327C2212yes—
53#1436—22no—
54C4⋊#95C2212yes—
55#1434C2212yes—
56C22⋊#95—21no—
57C22×#41C2×C411yescharacteristic
58C23×Q8C2×C411yescharacteristic
59C24×C4—21no—
60C24×C4C2311yescharacteristic
61C22×M16—21no—
62C23×C8—21no—
63C2×C4⋊Q8C2×C412yescomplemented
64C2×#124C2×C412yescomplemented
65#525—22no—
66#525—22no—
67#525—22no—
68#525—22no—
69C2×#123—21no—
70C2×#123—21no—
71C2×#123—21no—
72C2×C22⋊Q8—21no—
73C2×C22⋊Q8—21no—
74C2×C22⋊Q8—22no—
75C2×C22⋊Q8—22no—
76C2×C22⋊Q8—22no—
77C2×C22⋊Q8—22no—
78C2×C22⋊Q8C2×C412yes—
79C2×C22⋊Q8C2×C411yescharacteristic
80C2×C22⋊Q8C2312yes—
81C2×C22⋊Q8C2311yescharacteristic
82C2×#121C2×C411yescharacteristic
83C2×#121C2311yescharacteristic
84C2×#120—21no—
85C2×C4×D4—21no—
86C2×C4×D4—21no—
87C2×#117C2×C411yescharacteristic
88C22×C4⋊C4—21no—
89C22×C4⋊C4—22no—
90C22×C4⋊C4C2×C411yescharacteristic
91C22×C22⋊C4—21no—
92C22×C22⋊C4—21no—
93C22×C22⋊C4D411yescharacteristic
94C22×C22⋊C4D411yescharacteristic
95C22×C42—21no—
96C22×C42C2312yes—
97#407—22no—
98#407D414yes—
99C2×C22⋊C8—21no—
100C2×C22⋊C8—21no—
101C2×C22⋊C8—22no—
102C2×C22⋊C8D414yes—
103C2×C22⋊C8C2311yescharacteristic
104C2×C22⋊C8C2311yescharacteristic
105C2×C22⋊C8C2312yes—
106C2×#97—22no—
107C2×C4×C8—22no—
108#400—22no—
109#399—22no—
110#397—22no—
111#397—22no—
112#395—22no—
113#394—22no—
114#394—22no—
115#393—22no—
116#391C2×C412yescomplemented
117#391C2×C412yescomplemented
118#391C2×C412yescomplemented
119#390C2×C412yescomplemented
120#389—22no—
121#389—22no—
122#388—22no—
123#388—22no—
124#387—22no—
125#387—22no—
126#386—22no—
127#386—22no—
128#386—21no—
129#386—21no—
130#385—28no—
131#385—21no—
132#385—21no—
133#385—22no—
134#385—22no—
135#384C2×C412yescomplemented
136#382C2×C412yescomplemented
137#381—21no—
138#381—21no—
139#380—22no—
140#380—22no—
141C4×C22⋊C4—22no—
142C4×C22⋊C4—22no—
143C4×C22⋊C4C2×C412yes—
144C4×C22⋊C4C2×C411yescharacteristic
145C4×C22⋊C4C2×C411yescharacteristic
146C4×C22⋊C4D411yescharacteristic
147C4×C22⋊C4D411yescharacteristic
148C4×C22⋊C4D412yes—
149C2×#95—22no—
150C2×#95—22no—
151C2×#95—22no—
152C2×#95—21no—
153C2×#95—22no—
154C2×#95—22no—
155C2×#95D412yes—
156C2×#95C2311yescharacteristic
157C2×#95C2312yes—
158C2×#95C2311yescharacteristic
159#336—22no—
160#336—22no—
161#328—24no—
162#328—24no—
163#328—22no—
164#328—22no—
165#328—24no—
166#328—24no—
167#328—24no—
168#327—22no—
169#327—22no—
170#327—24no—
171C25C2×D411yescharacteristic
172C2×#41—22no—
173C2×#41—22no—
174C2×#41—82no—
175C22×Q8—21no—
176C22×Q8—22no—
177C22×Q8—22no—
178C22×Q8—82no—
179C22×Q8C22×C412yes—
180C22×Q8C22×C411yescharacteristic
181C22×D4—21no—
182C22×D4—21no—
183C22×D4—21no—
184C22×D4C22×C411yescharacteristic
185C23×C4—21no—
186C23×C4—21no—
187C23×C4—21no—
188C23×C4—22no—
189C23×C4—24no—
190C23×C4—21no—
191C23×C4—21no—
192C23×C4—42no—
193C23×C4—42no—
194C23×C4—82no—
195C23×C4—82no—
196C23×C4—82no—
197C23×C4C22×C411yescharacteristic
198C23×C4C22×C411yescharacteristic
199C23×C4C2×D412yes—
200C23×C4C2×D412yes—
201C23×C4C2×D411yescharacteristic
202C23×C4C2×D411yescharacteristic
203C23×C4C2×D411yescharacteristic
204C23×C4C2×D412yes—
205C23×C4C2×D412yes—
206C23×C4#4112yes—
207C23×C4#4112yes—
208C23×C4C2411yescharacteristic
209C2×M16—24no—
210C2×M16—82no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—22no—
214C22×C8—24no—
215C22×C8—82no—
216C4⋊Q8—82no—
217#124—82no—
218#123—44no—
219#123—42no—
220#123—42no—
221#123—44no—
222C22⋊Q8—24no—
223C22⋊Q8—24no—
224C22⋊Q8—22no—
225C22⋊Q8—22no—
226C22⋊Q8—22no—
227C22⋊Q8—22no—
228C22⋊Q8—21no—
229C22⋊Q8—21no—
230C22⋊Q8—22no—
231C22⋊Q8—22no—
232C22⋊Q8—21no—
233C22⋊Q8—21no—
234C22⋊Q8—42no—
235C22⋊Q8—42no—
236C22⋊Q8—42no—
237C22⋊Q8—42no—
238#121—22no—
239#121—22no—
240#121—21no—
241#121—21no—
242#121—21no—
243#121—21no—
244#120—42no—
245#120—42no—
246C4×D4—42no—
247C4×D4—42no—
248C4×D4—42no—
249C4×D4—42no—
250#117—22no—
251#117—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—82no—
267C2×C4⋊C4—82no—
268C2×C4⋊C4C22×C412yes—
269C2×C4⋊C4C22×C411yescharacteristic
270C2×C4⋊C4C22×C411yescharacteristic
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—42no—
285C2×C22⋊C4—42no—
286C2×C22⋊C4—42no—
287C2×C22⋊C4—42no—
288C2×C22⋊C4—42no—
289C2×C22⋊C4C22⋊C411yescharacteristic
290C2×C22⋊C4C22⋊C411yescharacteristic
291C2×C22⋊C4C22⋊C411yescharacteristic
292C2×C22⋊C4C22⋊C412yes—
293C2×C22⋊C4C22⋊C412yes—
294C2×C22⋊C4C22⋊C411yescharacteristic
295C2×C42—21no—
296C2×C42—42no—
297C2×C42—82no—
298C2×C42—82no—
299C2×C42C22×C412yes—
300C2×C42C22×C412yes—
301C2×C42C2×D412yes—
302C2×C42C2×D412yes—
303C22⋊C8—24no—
304C22⋊C8—24no—
305C22⋊C8—21no—
306C22⋊C8—21no—
307C22⋊C8—22no—
308C22⋊C8C2×D412yes—
309C22⋊C8C2×D412yes—
310C22⋊C8C2×D414yes—
311#97—82no—
312C4×C8—82no—
313#95—28no—
314#95—22no—
315#95—22no—
316#95—22no—
317#95—22no—
318#95—22no—
319#95—22no—
320#95—22no—
321#95—22no—
322#95—22no—
323#95—21no—
324#95—21no—
325#95—22no—
326#95—21no—
327#95—21no—
328#95—42no—
329#95—82no—
330#95—82no—
331#95C22⋊C412yes—
332#95C22⋊C412yes—
333#95C22⋊C412yes—
334#95C22⋊C412yes—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—21no—
340C24—42no—
341C24C2×C22⋊C411yescharacteristic
342C24C2×C22⋊C411yescharacteristic
343C24#12412yes—
344C24#12712yes—
345C24C22×D411yescharacteristic
346#41—84no—
347#41—84no—
348C2×Q8—22no—
349C2×Q8—22no—
350C2×Q8—21no—
351C2×Q8—21no—
352C2×Q8—42no—
353C2×Q8—42no—
354C2×Q8—42no—
355C2×Q8—42no—
356C2×Q8—82no—
357C2×Q8—84no—
358C2×Q8—84no—
359C2×D4—21no—
360C2×D4—21no—
361C2×D4—42no—
362C2×D4—41no—
363C2×D4—41no—
364C2×D4—42no—
365C2×D4—42no—
366C2×D4—42no—
367C2×D4—42no—
368C2×D4—42no—
369C2×D4—42no—
370C2×D4—82no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—48no—
401C22×C4—42no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—82no—
414C22×C4—82no—
415C22×C4—82no—
416C22×C4—82no—
417C22×C4—82no—
418C22×C4—82no—
419C22×C4—88no—
420C22×C4—84no—
421C22×C4—84no—
422C22×C4—88no—
423C22×C4C2×C22⋊C412yes—
424C22×C4C2×C22⋊C412yes—
425C22×C4C2×C22⋊C412yes—
426C22×C4C2×C22⋊C411yescharacteristic
427C22×C4C2×C22⋊C411yescharacteristic
428C22×C4C2×C22⋊C412yes—
429C22×C4C4×D412yes—
430C22×C4C4×D412yes—
431C22×C4C4×D412yes—
432C22×C4C4×D412yes—
433C22×C4#12112yes—
434C22×C4#12112yes—
435C22×C4#12112yes—
436C22×C4#12112yes—
437C22×C4#12412yes—
438C22×C4#12712yes—
439C22×C4C23×C411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C2×#4111yescharacteristic
444C22×C4C2×#4111yescharacteristic
445M16—84no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—24no—
449C2×C8—42no—
450C2×C8—42no—
451C2×C8—42no—
452C2×C8—42no—
453C2×C8—84no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—44no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—41no—
464C4⋊C4—41no—
465C4⋊C4—42no—
466C4⋊C4—44no—
467C4⋊C4—42no—
468C4⋊C4—44no—
469C4⋊C4—82no—
470C4⋊C4—82no—
471C4⋊C4—88no—
472C22⋊C4—22no—
473C22⋊C4—22no—
474C22⋊C4—22no—
475C22⋊C4—24no—
476C22⋊C4—42no—
477C22⋊C4—44no—
478C22⋊C4—48no—
479C22⋊C4—42no—
480C22⋊C4—41no—
481C22⋊C4—42no—
482C22⋊C4—41no—
483C22⋊C4—44no—
484C22⋊C4—42no—
485C22⋊C4—42no—
486C22⋊C4—41no—
487C22⋊C4—41no—
488C22⋊C4—42no—
489C22⋊C4—44no—
490C22⋊C4—82no—
491C22⋊C4—82no—
492C22⋊C4#10414yes—
493C42—42no—
494C42—42no—
495C42—42no—
496C42—42no—
497C42—42no—
498C42—84no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—21no—
503C23—21no—
504C23—24no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—21no—
509C23—21no—
510C23—22no—
511C23—24no—
512C23—24no—
513C23—42no—
514C23—42no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—42no—
520C23—42no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23#39012yes—
525C23#39012yes—
526C23C2×#10412yes—
527C23#42112yes—
528C23C22×C22⋊C411yescharacteristic
529C23C2×#12411yescharacteristic
530C23C2×#12711yescharacteristic
531C23#57712yes—
532C23#57812yes—
533Q8—82no—
534Q8—82no—
535Q8—81no—
536Q8—81no—
537Q8—84no—
538D4—81no—
539D4—81no—
540D4—84no—
541D4—82no—
542D4—81no—
543D4—81no—
544D4—84no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—84no—
597C2×C4—82no—
598C2×C4—82no—
599C2×C4—81no—
600C2×C4—81no—
601C2×C4—82no—
602C2×C4—82no—
603C2×C4—82no—
604C2×C4—82no—
605C2×C4—82no—
606C2×C4—84no—
607C2×C4—816no—
608C2×C4—81no—
609C2×C4—82no—
610C2×C4—81no—
611C2×C4—84no—
612C2×C4#39012yes—
613C2×C4#39012yes—
614C2×C4C2×#10414yes—
615C2×C4C2×C4×D411yescharacteristic
616C2×C4C2×C4×D411yescharacteristic
617C2×C4C2×#12111yescharacteristic
618C2×C4C2×#12111yescharacteristic
619C8—42no—
620C8—42no—
621C8—44no—
622C22—21no—
623C22—21no—
624C22—22no—
625C22—22no—
626C22—24no—
627C22—22no—
628C22—22no—
629C22—24no—
630C22—22no—
631C22—21no—
632C22—22no—
633C22—21no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—44no—
638C22—44no—
639C22—48no—
640C22—42no—
641C22—42no—
642C22—81no—
643C22—82no—
644C22—82no—
645C22—81no—
646C22—81no—
647C22#198014yes—
648C22C2×#39011yescharacteristic
649C22#288312yes—
650C22#289812yes—
651C22C22×#10411yescharacteristic
652C22C2×#42111yescharacteristic
653C22#313712yes—
654C22#315411yescharacteristic
655C22#315611yescharacteristic
656C4—24no—
657C4—21no—
658C4—21no—
659C4—22no—
660C4—21no—
661C4—21no—
662C4—22no—
663C4—42no—
664C4—42no—
665C4—42no—
666C4—82no—
667C4—82no—
668C4—84no—
669C4—82no—
670C4—81no—
671C4—81no—
672C4—81no—
673C4—82no—
674C2—21no—
675C2—21no—
676C2—22no—
677C2—24no—
678C2—42no—
679C2—161no—
680C2#1754812yes—
681C2#1759812yes—
682C2#2120511yescharacteristic
683C2#2121311yescharacteristic
684C2C2×#198011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20525C2semidirect
4. C2×#1508C2semidirect
6. C22×#328C2semidirect
7. #13782C2semidirect
8. #9683C2semidirect
9. #9682C2semidirect
10. #9681C2semidirect
11. #9679C2semidirect
20. #2503C4semidirect
21. #2501C4semidirect
22. #2499C4semidirect
23. #2436C4semidirect
24. #2434C4semidirect
25. #2432C4semidirect
63. C2×C4⋊Q8C2×C4semidirect
64. C2×#124C2×C4semidirect
116. #391C2×C4semidirect
117. #391C2×C4semidirect
118. #391C2×C4semidirect
119. #390C2×C4semidirect
135. #384C2×C4semidirect
136. #382C2×C4semidirect

Maximal subgroups

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

subgroupindex
#350852
#205252
C2×#15042
C2×#15082
C2×#15012
C22×#3282
#137822
#96832
#96822
#96812
#96792