Finite Group Databasebeta

← #182872

Subgroups — #182872

Order 512.

1071 conjugacy classes of subgroups.

The subgroup lattice has 683 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60233C211yesmaximal, characteristic
2#20528C211yesmaximal, characteristic, complemented
3#20527C211yesmaximal, characteristic, complemented
4#17572C211yesmaximal, characteristic
5#17559C211yesmaximal, characteristic
6#8240C212yesmaximal, complemented
7#8240C212yesmaximal, complemented
8#8239C212yesmaximal, complemented
9#8238C212yesmaximal, complemented
10#8057C211yesmaximal, characteristic, complemented
11#8049C211yesmaximal, characteristic, complemented
12C2×#525C2211yescharacteristic
13#2977—22no—
14#2976—21no—
15#2976—21no—
16#2974—22no—
17#2973—21no—
18#2972—21no—
19#2933C412yescomplemented
20#2933C412yescomplemented
21#2929C2212yes—
22#2929C2212yes—
23#2924C412yescomplemented
24#2880C412yescomplemented
25C2×#407C2211yescharacteristic
26C2×#407C2211yescharacteristic
27C22×C22⋊C8C2211yescharacteristic
28C22×C22⋊C8C2211yescharacteristic
29#2280C2211yescharacteristic
30#2280C2211yescharacteristic
31#1753—22no—
32#1752—21no—
33#1752C2211yescharacteristic, complemented
34#1752C2212yescomplemented
35#1752C2211yescharacteristic, complemented
36C8×C22⋊C4—21no—
37C8×C22⋊C4C2211yescharacteristic, complemented
38C8×C22⋊C4C2212yescomplemented
39C8×C22⋊C4C2211yescharacteristic, complemented
40#1750—22no—
41#1750—21no—
42#1750—21no—
43#1520C412yescomplemented
44#1519C412yescomplemented
45#1516C414yescomplemented
46#1513C414yescomplemented
47#1511C412yescomplemented
48#1510C412yescomplemented
49#1501C2212yes—
50#1501C2212yes—
51#1501C2212yes—
52#1501C2212yes—
53#1298C2212yescomplemented
54#1298C2212yescomplemented
55#1297C2212yescomplemented
56#1297C2212yescomplemented
57C22×#41C2×C411yescharacteristic
58C24×C4C2311yescharacteristic
59C2×#131C2×C412yes—
60C2×#131C2×C412yes—
61C22×M16—21no—
62C22×M16C2×C411yescharacteristic
63C22×M16C2×C411yescharacteristic
64C23×C8—21no—
65C23×C8C2311yescharacteristic
66C23×C8C2311yescharacteristic
67#525—21no—
68#525—21no—
69#525—21no—
70#525—21no—
71#525C2×C412yes—
72#525C2×C412yes—
73#525C2×C412yes—
74#525C2312yes—
75C2×#123—21no—
76C2×C22⋊Q8C2×C411yescharacteristic
77C2×C22⋊Q8C2×C411yescharacteristic
78C2×#121C2×C411yescharacteristic
79C2×#121C2×C411yescharacteristic
80C2×#120—21no—
81C2×C4×D4—21no—
82C2×C4×D4—21no—
83C2×#117—21no—
84C2×#117—21no—
85C2×#117C2×C411yescharacteristic
86#436—22no—
87#436—22no—
88#435—22no—
89#435—22no—
90#435—22no—
91#435—22no—
92#435—22no—
93#435—22no—
94#435—22no—
95#435—22no—
96C8×D4—22no—
97C8×D4—22no—
98C8×D4—22no—
99C8×D4—22no—
100C8×D4—22no—
101C8×D4—22no—
102#433—22no—
103#433—22no—
104#432—22no—
105#431—22no—
106#424C2×C412yes—
107#424C2×C412yes—
108#408—22no—
109#408—22no—
110#408C2×C412yes—
111#408C2×C412yes—
112#407—21no—
113#407—21no—
114#407—21no—
115#407—21no—
116#407—22no—
117#407C2×C412yes—
118#407C2×C412yes—
119#407C2×C412yes—
120#407C2312yes—
121C2×C22⋊C8—21no—
122C2×C22⋊C8—21no—
123C2×C22⋊C8—21no—
124C2×C22⋊C8—21no—
125C2×C22⋊C8—21no—
126C2×C22⋊C8—21no—
127C2×C22⋊C8—22no—
128C2×C22⋊C8C2×C412yes—
129C2×C22⋊C8C2×C412yes—
130C2×C22⋊C8C2×C412yes—
131C2×C22⋊C8C2×C412yes—
132C2×C22⋊C8D411yescharacteristic
133C2×C22⋊C8D411yescharacteristic
134C2×C22⋊C8D412yes—
135C2×C22⋊C8D411yescharacteristic
136C2×C22⋊C8D411yescharacteristic
137C2×C22⋊C8D412yes—
138C2×C22⋊C8C2311yescharacteristic
139C2×C22⋊C8C2311yescharacteristic
140C2×C22⋊C8C2312yes—
141#405C2×C412yes—
142#405C2×C412yes—
143C2×#97—21no—
144C2×#97—21no—
145C2×#97—22no—
146C2×C4×C8—21no—
147C2×C4×C8—21no—
148C2×C4×C8—22no—
149#381C2×C412yes—
150#381C2×C412yes—
151#380—21no—
152#380—21no—
153#380C2×C412yes—
154#380C2×C412yes—
155#379—21no—
156#379—21no—
157C4×C22⋊C4—21no—
158C4×C22⋊C4—21no—
159C4×C22⋊C4C2311yescharacteristic
160C4×C22⋊C4C2312yes—
161C4×C22⋊C4C2311yescharacteristic
162#336—21no—
163#336—21no—
164#336—21no—
165#336—21no—
166#336—22no—
167#336—22no—
168#328C2×C414yes—
169#328C2×C414yes—
170#328C2×C414yes—
171#328C2×C414yes—
172#327C2×C414yes—
173#327C2×C414yes—
174#327C2×C414yes—
175#327C2×C414yes—
176C25#4111yescharacteristic
177C2×#41—21no—
178C2×#41—21no—
179C2×#41—81no—
180C2×#41—81no—
181C2×#41C4212yes—
182C2×#41C22×C412yes—
183C22×Q8C4211yescharacteristic
184C22×D4—21no—
185C22×D4—21no—
186C22×D4—21no—
187C22×D4C4211yescharacteristic
188C23×C4—21no—
189C23×C4—81no—
190C23×C4—81no—
191C23×C4C22×C411yescharacteristic
192C23×C4C22×C411yescharacteristic
193C23×C4C22×C411yescharacteristic
194C23×C4C22×C411yescharacteristic
195C23×C4C22×C411yescharacteristic
196C23×C4C22×C411yescharacteristic
197C23×C4C2×D411yescharacteristic
198C23×C4C2×D411yescharacteristic
199C23×C4C2×D411yescharacteristic
200C23×C4C2×D411yescharacteristic
201C23×C4#4111yescharacteristic
202C23×C4#4111yescharacteristic
203C23×C4#4111yescharacteristic
204C23×C4C2411yescharacteristic
205#131—82no—
206#131—82no—
207C2×M16—21no—
208C2×M16—21no—
209C2×M16—22no—
210C2×M16—22no—
211C2×M16—22no—
212C2×M16—22no—
213C2×M16—81no—
214C2×M16—81no—
215C2×M16—81no—
216C2×M16—81no—
217C2×M16C22×C412yes—
218C2×M16C22×C412yes—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—22no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—22no—
225C22×C8—22no—
226C22×C8—22no—
227C22×C8—22no—
228C22×C8—81no—
229C22×C8—81no—
230C22×C8—81no—
231C22×C8—81no—
232C22×C8C22×C412yes—
233C22×C8C22×C412yes—
234C22×C8C2×D411yescharacteristic
235C22×C8C2×D412yes—
236C22×C8C2×D411yescharacteristic
237C22×C8C2×D411yescharacteristic
238C22×C8C2×D412yes—
239C22×C8C2×D411yescharacteristic
240#123—22no—
241#123—22no—
242#123—22no—
243#123—22no—
244C22⋊Q8—21no—
245C22⋊Q8—21no—
246C22⋊Q8—21no—
247C22⋊Q8—21no—
248C22⋊Q8C4212yes—
249C22⋊Q8C4212yes—
250C22⋊Q8C22×C412yes—
251C22⋊Q8C22×C412yes—
252#121—21no—
253#121—21no—
254#121—21no—
255#121—21no—
256#121C4212yes—
257#121C4212yes—
258#121C22×C412yes—
259#121C22×C412yes—
260#120—22no—
261#120—22no—
262#120—22no—
263#120—22no—
264C4×D4—22no—
265C4×D4—22no—
266C4×D4—22no—
267C4×D4—22no—
268C4×D4—22no—
269C4×D4—22no—
270C4×D4—22no—
271C4×D4—22no—
272#117—21no—
273#117—21no—
274#117—81no—
275#117—81no—
276#117—81no—
277#117—81no—
278#117C4212yes—
279#117C22×C412yes—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4C4211yescharacteristic
285C2×C4⋊C4C4211yescharacteristic
286C2×C22⋊C4—21no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C42—21no—
299C2×C42—21no—
300C2×C42—21no—
301C2×C42—22no—
302C4⋊C8—22no—
303C4⋊C8—22no—
304C4⋊C8—22no—
305C4⋊C8—22no—
306C22⋊C8—22no—
307C22⋊C8—22no—
308C22⋊C8—22no—
309C22⋊C8—22no—
310C22⋊C8—22no—
311C22⋊C8—22no—
312C22⋊C8—22no—
313C22⋊C8—22no—
314C22⋊C8—21no—
315C22⋊C8—21no—
316C22⋊C8—22no—
317C22⋊C8—22no—
318C22⋊C8—22no—
319C22⋊C8—22no—
320C22⋊C8—22no—
321C22⋊C8C22⋊C412yes—
322C22⋊C8C22⋊C412yes—
323C22⋊C8C22⋊C412yes—
324C22⋊C8C22⋊C412yes—
325C22⋊C8C22⋊C412yes—
326C22⋊C8C22⋊C412yes—
327C22⋊C8C22⋊C412yes—
328C22⋊C8C22⋊C412yes—
329C22⋊C8C22×C412yes—
330C22⋊C8C22×C412yes—
331C22⋊C8C22×C414yes—
332#97—22no—
333#97—22no—
334#97—81no—
335#97—81no—
336#97—82no—
337C4×C8—22no—
338C4×C8—22no—
339C4×C8—81no—
340C4×C8—81no—
341C4×C8—82no—
342#95—22no—
343#95—22no—
344#95C22×C412yes—
345#95C22×C412yes—
346#95C22×C414yes—
347C24—21no—
348C24—42no—
349C24—42no—
350C24—81no—
351C24#11711yescharacteristic
352C24#11711yescharacteristic
353C24C4×D411yescharacteristic
354C24C4×D411yescharacteristic
355C24C4×D411yescharacteristic
356C24C4×D411yescharacteristic
357C24C2×#4111yescharacteristic
358#41—42no—
359#41—42no—
360#41—81no—
361#41—81no—
362#41—82no—
363#41—82no—
364C2×Q8—21no—
365C2×Q8—81no—
366C2×Q8C2×C4212yes—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—21no—
374C2×D4—44no—
375C2×D4—81no—
376C2×D4—82no—
377C2×D4—82no—
378C2×D4C2×C4212yes—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—81no—
402C22×C4—81no—
403C22×C4—81no—
404C22×C4—81no—
405C22×C4—81no—
406C22×C4—81no—
407C22×C4—81no—
408C22×C4—81no—
409C22×C4—81no—
410C22×C4C2×C4211yescharacteristic
411C22×C4C2×C4211yescharacteristic
412C22×C4C2×C4211yescharacteristic
413C22×C4C2×C4211yescharacteristic
414C22×C4C2×C22⋊C411yescharacteristic
415C22×C4C2×C22⋊C411yescharacteristic
416C22×C4C2×C22⋊C411yescharacteristic
417C22×C4C2×C22⋊C411yescharacteristic
418C22×C4#11711yescharacteristic
419C22×C4#11711yescharacteristic
420C22×C4C4×D411yescharacteristic
421C22×C4C4×D411yescharacteristic
422C22×C4C4×D411yescharacteristic
423C22×C4C4×D411yescharacteristic
424C22×C4C4×D411yescharacteristic
425C22×C4C4×D411yescharacteristic
426C22×C4C4×D411yescharacteristic
427C22×C4C4×D411yescharacteristic
428C22×C4C4×D411yescharacteristic
429C22×C4C4×D411yescharacteristic
430C22×C4C4×D411yescharacteristic
431C22×C4C4×D411yescharacteristic
432C22×C4C23×C411yescharacteristic
433C22×C4C23×C411yescharacteristic
434C22×C4C23×C411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C2×#4111yescharacteristic
438M16—42no—
439M16—42no—
440M16—82no—
441M16—82no—
442M16—81no—
443M16—81no—
444M16—82no—
445M16—82no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—24no—
451C2×C8—41no—
452C2×C8—42no—
453C2×C8—41no—
454C2×C8—41no—
455C2×C8—42no—
456C2×C8—41no—
457C2×C8—42no—
458C2×C8—42no—
459C2×C8—41no—
460C2×C8—41no—
461C2×C8—42no—
462C2×C8—41no—
463C2×C8—41no—
464C2×C8—42no—
465C2×C8—82no—
466C2×C8—82no—
467C2×C8—81no—
468C2×C8—81no—
469C2×C8—82no—
470C2×C8—82no—
471C2×C8C2×C22⋊C412yes—
472C2×C8C2×C22⋊C412yes—
473C2×C8C2×C22⋊C412yes—
474C2×C8C2×C22⋊C412yes—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—82no—
482C4⋊C4—82no—
483C4⋊C4C2×C4212yes—
484C4⋊C4C2×C4212yes—
485C22⋊C4—22no—
486C22⋊C4—22no—
487C22⋊C4—22no—
488C22⋊C4—22no—
489C22⋊C4—22no—
490C22⋊C4—22no—
491C22⋊C4—22no—
492C22⋊C4—22no—
493C22⋊C4—22no—
494C22⋊C4—22no—
495C22⋊C4—44no—
496C22⋊C4—42no—
497C22⋊C4—42no—
498C22⋊C4—44no—
499C22⋊C4—82no—
500C22⋊C4—82no—
501C22⋊C4—82no—
502C42—22no—
503C42—22no—
504C42—42no—
505C42—41no—
506C42—41no—
507C42—41no—
508C42—41no—
509C42—42no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—42no—
533C23—81no—
534C23—81no—
535C23—88no—
536C23C4×C22⋊C411yescharacteristic
537C23C4×C22⋊C411yescharacteristic
538C23C4×C22⋊C411yescharacteristic
539C23C4×C22⋊C411yescharacteristic
540C23C2×#11711yescharacteristic
541C23C2×C4×D411yescharacteristic
542C23C2×C4×D411yescharacteristic
543Q8—42no—
544Q8—81no—
545D4—42no—
546D4—44no—
547D4—44no—
548D4—44no—
549D4—44no—
550D4—81no—
551D4—82no—
552D4—82no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—82no—
604C2×C4—81no—
605C2×C4—81no—
606C2×C4C4×C22⋊C411yescharacteristic
607C2×C4C4×C22⋊C411yescharacteristic
608C2×C4C4×C22⋊C411yescharacteristic
609C2×C4C4×C22⋊C411yescharacteristic
610C2×C4C22×C4211yescharacteristic
611C2×C4C22×C22⋊C411yescharacteristic
612C2×C4C2×C4×D411yescharacteristic
613C2×C4C2×C4×D411yescharacteristic
614C8—22no—
615C8—22no—
616C8—22no—
617C8—22no—
618C8—42no—
619C8—42no—
620C8—44no—
621C8—42no—
622C8—42no—
623C22—21no—
624C22—21no—
625C22—21no—
626C22—21no—
627C22—21no—
628C22—21no—
629C22—21no—
630C22—21no—
631C22—21no—
632C22—21no—
633C22—21no—
634C22—21no—
635C22—21no—
636C22—21no—
637C22—21no—
638C22—21no—
639C22—21no—
640C22—21no—
641C22—21no—
642C22—44no—
643C22—44no—
644C22—42no—
645C22—42no—
646C22—42no—
647C22—82no—
648C22—82no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22—162no—
653C22C2×C4×C22⋊C411yescharacteristic
654C22#295512yes—
655C22#295712yes—
656C4—21no—
657C4—21no—
658C4—21no—
659C4—21no—
660C4—21no—
661C4—21no—
662C4—21no—
663C4—42no—
664C4—42no—
665C4—42no—
666C4—82no—
667C4—82no—
668C4—84no—
669C4#288311yescharacteristic
670C4#288311yescharacteristic
671C2—22no—
672C2—22no—
673C2—41no—
674C2—41no—
675C2—41no—
676C2—41no—
677C2—41no—
678C2—82no—
679C2#2039211yescharacteristic
680C2#2042011yescharacteristic
681C2#2042211yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊C4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
socleC22C2×C4×C22⋊C4
prime radicalC2×#525C22
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C22⋊C8 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20528C2semidirect
3. #20527C2semidirect
6. #8240C2semidirect
7. #8240C2semidirect
8. #8239C2semidirect
9. #8238C2semidirect
10. #8057C2semidirect
11. #8049C2semidirect
19. #2933C4semidirect
20. #2933C4semidirect
23. #2924C4semidirect
24. #2880C4semidirect
33. #1752C22semidirect
34. #1752C22semidirect
35. #1752C22semidirect
37. C8×C22⋊C4C22semidirect
38. C8×C22⋊C4C22semidirect
39. C8×C22⋊C4C22semidirect
43. #1520C4semidirect
44. #1519C4semidirect
45. #1516C4semidirect
46. #1513C4semidirect
47. #1511C4semidirect
48. #1510C4semidirect
53. #1298C22semidirect
54. #1298C22semidirect
55. #1297C22semidirect
56. #1297C22semidirect

Maximal subgroups

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

subgroupindex
#602332
#205282
#205272
#175722
#175592
#82402
#82392
#82382
#80572
#80492