Finite Group Databasebeta

← #195871

Subgroups — #195871

Order 512.

1878 conjugacy classes of subgroups.

The subgroup lattice has 679 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2920C211yesmaximal, characteristic, complemented
2C2×C4×#121C211yesmaximal, characteristic
3#20626C211yesmaximal, characteristic, complemented
4C2×#1486C212yesmaximal, complemented
5#9078C214yesmaximal, complemented
6#9074C214yesmaximal, complemented
7#7901C212yesmaximal, complemented
8C22×#121C411yescharacteristic
9C22×C4×D4—21no—
10C22×C4×D4C2211yescharacteristic
11C23×C42C2211yescharacteristic
12#2920—28no—
13C2×#432—21no—
14C2×#432—21no—
15C2×#431C2212yes—
16C2×#431C2212yescomplemented
17C2×#423—21no—
18C2×#423C2212yes—
19C2×#407—21no—
20C2×#407—21no—
21C2×C4×M16C2212yescomplemented
22C4×#121—21no—
23C4×#121—21no—
24C4×#121—21no—
25C4×#121—21no—
26C4×#121C412yes—
27C4×#121C412yes—
28C4×#121C2212yes—
29C4×#121C2212yes—
30C2×#390C411yescharacteristic
31C2×#389C411yescharacteristic
32C2×#388—21no—
33C2×#386—21no—
34C2×#380C411yescharacteristic
35C2×C4×C4⋊C4C2211yescharacteristic
36C2×C4×C22⋊C4—21no—
37#1797—21no—
38#1797—21no—
39#1797C2212yescomplemented
40#1797C2212yescomplemented
41#1728—21no—
42#1486—28no—
43C2×D4⋊C8C2218yescomplemented
44#1279C2218yescomplemented
45C23×D4—21no—
46C23×D4C2×C411yescharacteristic
47C24×C4—21no—
48C24×C4C2×C411yescharacteristic
49C24×C4D411yescharacteristic
50C24×C4D411yescharacteristic
51C22×M16—22no—
52C22×M16—22no—
53C2×#121—21no—
54C2×#121—21no—
55C2×#121—28no—
56C2×#121C2×C412yes—
57C2×#121C2×C412yes—
58C2×C4×D4—22no—
59C2×C4×D4—22no—
60C2×C4×D4—22no—
61C2×C4×D4—22no—
62C2×C4×D4—21no—
63C2×C4×D4—21no—
64C2×C4×D4—24no—
65C2×C4×D4—24no—
66C2×C4×D4—48no—
67C2×C4×D4C2×C412yes—
68C2×C4×D4C2312yes—
69C22×C4⋊C4—21no—
70C22×C4⋊C4C2×C411yescharacteristic
71C22×C4⋊C4C2×C411yescharacteristic
72C22×C4⋊C4C2×C411yescharacteristic
73C22×C22⋊C4—21no—
74C22×C22⋊C4—21no—
75C22×C22⋊C4—21no—
76C22×C22⋊C4—21no—
77C22×C22⋊C4—21no—
78C22×C42—24no—
79C22×C42—24no—
80C22×C42—21no—
81C22×C42C2×C411yescharacteristic
82C22×C42C2×C411yescharacteristic
83C22×C42C2×C411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42D411yescharacteristic
88C22×C42D411yescharacteristic
89C22×C42D411yescharacteristic
90C22×C42C2311yescharacteristic
91C22×C42C2311yescharacteristic
92C22×C42C2311yescharacteristic
93#432—28no—
94#432—28no—
95#431—24no—
96#431—28no—
97#431D418yescomplemented
98#423—28no—
99#423—24no—
100#423—24no—
101C2×C4⋊C8—22no—
102C2×C4⋊C8—22no—
103C2×C4⋊C8—22no—
104C2×C4⋊C8C2314yes—
105#407—28no—
106#407—44no—
107C2×C22⋊C8—22no—
108C2×C22⋊C8—22no—
109C2×C22⋊C8—22no—
110C2×C22⋊C8—22no—
111C4×M16—28no—
112C2×#97—22no—
113C2×C4×C8C2314yes—
114#390—21no—
115#390—21no—
116#390C2×C412yes—
117#390C2×C412yes—
118#389—21no—
119#389—21no—
120#389C2×C412yes—
121#389C2×C412yes—
122#388—22no—
123#388—22no—
124#388—22no—
125#388—22no—
126#386—22no—
127#386—22no—
128#386—22no—
129#386—22no—
130#380—21no—
131#380—21no—
132#380C2×C412yes—
133#380C2×C412yes—
134C4×C4⋊C4—21no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4C2×C412yes—
137C4×C4⋊C4C2312yes—
138C4×C22⋊C4—22no—
139C4×C22⋊C4—22no—
140C4×C22⋊C4—22no—
141C4×C22⋊C4—22no—
142C2×#95—21no—
143C2×#95—21no—
144C2×#95—21no—
145#336—22no—
146#336—22no—
147D4⋊C8—216no—
148C25—21no—
149C25C22⋊C411yescharacteristic
150C22×D4—28no—
151C22×D4—22no—
152C22×D4—22no—
153C22×D4—24no—
154C22×D4—21no—
155C22×D4—24no—
156C22×D4—42no—
157C22×D4—44no—
158C22×D4—44no—
159C22×D4C22×C412yes—
160C23×C4—21no—
161C23×C4—24no—
162C23×C4—22no—
163C23×C4—24no—
164C23×C4—22no—
165C23×C4—24no—
166C23×C4—24no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—42no—
174C23×C4—42no—
175C23×C4—42no—
176C23×C4—44no—
177C23×C4—44no—
178C23×C4—42no—
179C23×C4—42no—
180C23×C4C22⋊C411yescharacteristic
181C23×C4C22⋊C411yescharacteristic
182C23×C4C22⋊C411yescharacteristic
183C23×C4C22⋊C411yescharacteristic
184C23×C4C22⋊C411yescharacteristic
185C23×C4C22⋊C411yescharacteristic
186C23×C4C22⋊C411yescharacteristic
187C23×C4C22⋊C411yescharacteristic
188C23×C4C22⋊C411yescharacteristic
189C23×C4C22×C411yescharacteristic
190C23×C4C22×C411yescharacteristic
191C23×C4C22×C411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C23×C4C2×D411yescharacteristic
196C23×C4C2×D411yescharacteristic
197C23×C4C2×D411yescharacteristic
198C2×M16—28no—
199C2×M16—44no—
200C2×M16—44no—
201C2×M16—44no—
202C2×M16—44no—
203C2×M16—44no—
204C22×C8—24no—
205C22×C8—24no—
206#121—28no—
207#121—28no—
208#121—24no—
209#121—24no—
210#121—24no—
211#121M1618yescomplemented
212C4×D4—28no—
213C4×D4—24no—
214C4×D4—24no—
215C4×D4—48no—
216C4×D4—48no—
217C4×D4—48no—
218C4×D4—48no—
219C4×D4—42no—
220C4×D4—44no—
221C4×D4—42no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4C22×C412yes—
232C2×C4⋊C4C22×C412yes—
233C2×C4⋊C4C22×C412yes—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—42no—
250C2×C42—24no—
251C2×C42—22no—
252C2×C42—24no—
253C2×C42—22no—
254C2×C42—24no—
255C2×C42—24no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—24no—
259C2×C42—24no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—44no—
269C2×C42C22⋊C411yescharacteristic
270C2×C42C22⋊C411yescharacteristic
271C2×C42C22⋊C411yescharacteristic
272C2×C42C22⋊C411yescharacteristic
273C2×C42C22⋊C411yescharacteristic
274C2×C42C22⋊C411yescharacteristic
275C2×C42C22×C411yescharacteristic
276C2×C42C22×C411yescharacteristic
277C2×C42C22×C411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42C2×D411yescharacteristic
280C2×C42C2×D411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C2×C42C2×D414yes—
283C2×C42C2×D411yescharacteristic
284C2×C42C2×D411yescharacteristic
285C2×C42#4114yes—
286C2×C42C2411yescharacteristic
287C4⋊C8—28no—
288C4⋊C8—28no—
289C4⋊C8—24no—
290C4⋊C8—24no—
291C4⋊C8—44no—
292C22⋊C8—28no—
293C22⋊C8—28no—
294C22⋊C8—44no—
295C22⋊C8—44no—
296#97—28no—
297C4×C8—24no—
298C4×C8C2×D418yescomplemented
299#95—22no—
300#95—22no—
301#95—22no—
302#95—22no—
303#95—22no—
304#95—22no—
305C24—22no—
306C24—22no—
307C24—22no—
308C24—24no—
309C24—22no—
310C24—22no—
311C24—22no—
312C24—41no—
313C24—41no—
314C24—42no—
315C24—42no—
316C24—42no—
317C24—42no—
318C24—42no—
319C24—42no—
320C24C2×C22⋊C411yescharacteristic
321C24C2×C22⋊C411yescharacteristic
322C24C2×C22⋊C411yescharacteristic
323C2×D4—24no—
324C2×D4—28no—
325C2×D4—28no—
326C2×D4—24no—
327C2×D4—22no—
328C2×D4—24no—
329C2×D4—416no—
330C2×D4—42no—
331C2×D4—44no—
332C2×D4—44no—
333C2×D4—432no—
334C2×D4—44no—
335C2×D4—44no—
336C2×D4—42no—
337C2×D4C2×M1614yes—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—416no—
395C22×C4—48no—
396C22×C4—48no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—48no—
400C22×C4—48no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—44no—
408C22×C4—44no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—88no—
415C22×C4—88no—
416C22×C4C2×C22⋊C411yescharacteristic
417C22×C4C2×C22⋊C411yescharacteristic
418C22×C4C2×C22⋊C411yescharacteristic
419C22×C4C2×C22⋊C411yescharacteristic
420C22×C4C2×C22⋊C411yescharacteristic
421C22×C4C2×C22⋊C411yescharacteristic
422C22×C4C2×C22⋊C411yescharacteristic
423C22×C4C2×C22⋊C411yescharacteristic
424C22×C4C2×C22⋊C411yescharacteristic
425C22×C4C4×D414yes—
426C22×C4#12012yes—
427C22×C4#12114yes—
428C22×C4#12312yes—
429C22×C4C2×M1614yes—
430C22×C4AGL(1,ℤ/8)12yes—
431C22×C4AGL(1,ℤ/8)14yes—
432C22×C4#13712yes—
433C22×C4C23×C411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436M16—416no—
437M16—88no—
438C2×C8—28no—
439C2×C8—44no—
440C2×C8—44no—
441C2×C8—44no—
442C2×C8—44no—
443C2×C8—44no—
444C4⋊C4—24no—
445C4⋊C4—22no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—42no—
449C4⋊C4—44no—
450C4⋊C4—44no—
451C4⋊C4—42no—
452C4⋊C4C2×M1614yes—
453C22⋊C4—28no—
454C22⋊C4—28no—
455C22⋊C4—44no—
456C22⋊C4—44no—
457C22⋊C4—44no—
458C22⋊C4—44no—
459C22⋊C4—44no—
460C22⋊C4—42no—
461C22⋊C4—42no—
462C22⋊C4—44no—
463C22⋊C4—44no—
464C42—22no—
465C42—22no—
466C42—21no—
467C42—21no—
468C42—22no—
469C42—22no—
470C42—21no—
471C42—21no—
472C42—22no—
473C42—22no—
474C42—22no—
475C42—22no—
476C42—22no—
477C42—22no—
478C42—44no—
479C42—44no—
480C42C4×D414yes—
481C42#12012yes—
482C42#12114yes—
483C42#12312yes—
484C42C22×D412yes—
485C42C2×#4112yes—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—28no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—24no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—48no—
510C23—48no—
511C23—48no—
512C23—44no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—84no—
520C23—84no—
521C23—84no—
522C23—84no—
523C23#38612yes—
524C23#40712yes—
525C23#41712yes—
526C23#41812yes—
527C23#42112yes—
528C23C22×C22⋊C411yescharacteristic
529D4—28no—
530D4—48no—
531D4—416no—
532D4—416no—
533D4—44no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—28no—
547C2×C4—28no—
548C2×C4—21no—
549C2×C4—24no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
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—42no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—48no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—84no—
589C2×C4—84no—
590C2×C4—88no—
591C2×C4—84no—
592C2×C4—84no—
593C2×C4#38612yes—
594C2×C4#38612yes—
595C2×C4#38612yes—
596C2×C4#40712yes—
597C2×C4#41712yes—
598C2×C4#41812yes—
599C2×C4#42112yes—
600C2×C4#46818yes—
601C2×C4#46918yes—
602C2×C4C2×C4×D412yes—
603C2×C4C2×#12011yescharacteristic
604C2×C4C2×#12112yes—
605C2×C4C2×#12311yescharacteristic
606C2×C4C22×M1612yes—
607C2×C4#56812yes—
608C2×C4C2×AGL(1,ℤ/8)12yes—
609C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
610C2×C4C2×#13711yescharacteristic
611C8—48no—
612C8—84no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—416no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22—41no—
631C22—41no—
632C22—41no—
633C22—41no—
634C22—44no—
635C22—44no—
636C22—44no—
637C22—816no—
638C22—82no—
639C22—82no—
640C22—82no—
641C22—82no—
642C22#189514yes—
643C22C2×#38611yescharacteristic
644C22C2×#40711yescharacteristic
645C22#287911yescharacteristic
646C22C2×#41811yescharacteristic
647C22C2×#41711yescharacteristic
648C22C2×#42111yescharacteristic
649C22C2×#42111yescharacteristic
650C4—22no—
651C4—22no—
652C4—21no—
653C4—21no—
654C4—22no—
655C4—22no—
656C4—44no—
657C4—41no—
658C4—41no—
659C4—41no—
660C4—41no—
661C4—82no—
662C4—82no—
663C4#189514yes—
664C4D4×M1614yes—
665C4C2×#46812yes—
666C4C2×#46912yes—
667C2—24no—
668C2—42no—
669C2—42no—
670C2—84no—
671C2#1744811yescharacteristic
672C2#1745111yescharacteristic
673C2#1749011yescharacteristic
674C2#1749211yescharacteristic
675C2#2089811yescharacteristic
676C2C2×#189511yescharacteristic
677C2#2097911yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC22×#121C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2920C2semidirect
3. #20626C2semidirect
4. C2×#1486C2semidirect
5. #9078C2semidirect
6. #9074C2semidirect
7. #7901C2semidirect
16. C2×#431C22semidirect
21. C2×C4×M16C22semidirect
39. #1797C22semidirect
40. #1797C22semidirect
43. C2×D4⋊C8C22semidirect
44. #1279C22semidirect
97. #431D4semidirect
211. #121M16semidirect
298. C4×C8C2×D4semidirect

Maximal subgroups

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

subgroupindex
C2×#29202
C2×C4×#1212
#206262
C2×#14862
#90782
#90742
#79012