Finite Group Databasebeta

← #106947

Subgroups — #106947

Order 512.

1020 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34188C211yesmaximal, characteristic
2C2×#1806C211yesmaximal, characteristic
3C2×#1477C211yesmaximal, characteristic
4#10849C211yesmaximal, characteristic, complemented
5#8045C211yesmaximal, characteristic, complemented
6#8043C211yesmaximal, characteristic, complemented
7#8014C211yesmaximal, characteristic, complemented
8C2×C4×#41C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#423C2211yescharacteristic
11C22×C4⋊C8C2211yescharacteristic
12C22×#104—21no—
13#2355—21no—
14#2355C412yescomplemented
15#2354—21no—
16#2354C412yescomplemented
17#2352—22no—
18#2352—22no—
19#2314—22no—
20#2311—21no—
21#2285—21no—
22#2285—21no—
23#2285C2211yescharacteristic
24C2×#384C2211yescharacteristic
25#2274—21no—
26C2×C4×C4⋊C4C2211yescharacteristic
27#1806—22no—
28#1806C414yescomplemented
29C2×#340—22no—
30C2×#339—21no—
31#1477—22no—
32#1477C414yescomplemented
33C2×Q8⋊C8—21no—
34C2×D4⋊C8—21no—
35#1296—22no—
36#1296—21no—
37#1296—21no—
38#1294—22no—
39C22×#41D411yescharacteristic
40C22×M16—21no—
41C23×C8—21no—
42C4×#41—21no—
43C4×#41—21no—
44C4×#41—21no—
45C4×#41—21no—
46C4×#41C2×C412yes—
47C4×#41C2×C412yes—
48C4×#41D412yes—
49C4×#41Q812yes—
50C2×C4×Q8C2×C411yescharacteristic
51C2×C4×Q8D411yescharacteristic
52C2×C4×D4—21no—
53C2×C4×D4—21no—
54C2×C4×D4C2×C411yescharacteristic
55C2×C4×D4D411yescharacteristic
56C2×#117—21no—
57C2×#117—22no—
58C2×#117—22no—
59C2×#117—21no—
60C2×#117D411yescharacteristic
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4D412yes—
65C22×C42—21no—
66C22×C42—21no—
67C22×C42C2311yescharacteristic
68#431—21no—
69#431—21no—
70#431—21no—
71#431C2×C412yes—
72#423—21no—
73#423—21no—
74#423—21no—
75#423C2×C412yes—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8C2×C412yes—
85C2×#104—24no—
86C2×#104—42no—
87C2×#104—42no—
88C2×C22⋊C8—21no—
89C2×C4×C8—21no—
90#391—22no—
91#390—22no—
92#388—24no—
93#388—24no—
94#387—24no—
95#387—24no—
96#386—24no—
97#384—21no—
98#384—21no—
99#384C2×C412yes—
100#384D412yes—
101#384D411yescharacteristic
102#384D411yescharacteristic
103#382—22no—
104#382—22no—
105#382—21no—
106#380—22no—
107#380—22no—
108#380—22no—
109#380—21no—
110#380—22no—
111#380D412yes—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4C2×C412yes—
123C4×C4⋊C4C2×C412yes—
124C4×C22⋊C4—24no—
125C2×#95—21no—
126#340—44no—
127#339—24no—
128Q8⋊C8—24no—
129D4⋊C8—24no—
130C2×#41—21no—
131C2×#41—21no—
132C2×#41—21no—
133C2×#41—21no—
134C2×#41—21no—
135C2×#41—21no—
136C2×#41—21no—
137C2×#41—21no—
138C2×#41—21no—
139C2×#41—21no—
140C2×#41C22⋊C412yes—
141C2×#41C4⋊C412yes—
142C22×Q8C22⋊C411yescharacteristic
143C22×D4—21no—
144C22×D4C22⋊C411yescharacteristic
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—22no—
149C23×C4—41no—
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C2×M16—22no—
154C2×M16—41no—
155C2×M16—41no—
156C2×M16—41no—
157C2×M16—41no—
158C2×M16—41no—
159C22×C8—22no—
160C22×C8—42no—
161C22×C8—42no—
162C22×C8—41no—
163C22×C8—41no—
164C22×C8—41no—
165C22×C8—41no—
166C4×Q8—21no—
167C4×Q8—21no—
168C4×Q8—21no—
169C4×Q8—21no—
170C4×Q8C4212yes—
171C4×Q8C22⋊C412yes—
172C4×Q8C4⋊C412yes—
173C4×Q8C4⋊C412yes—
174C4×D4—22no—
175C4×D4—22no—
176C4×D4—22no—
177C4×D4—22no—
178C4×D4—22no—
179C4×D4—22no—
180C4×D4—22no—
181C4×D4—22no—
182C4×D4—21no—
183C4×D4—21no—
184C4×D4—21no—
185C4×D4—21no—
186C4×D4C4212yes—
187C4×D4C22⋊C412yes—
188C4×D4C4⋊C412yes—
189C4×D4C4⋊C412yes—
190#117—21no—
191#117—21no—
192#117—22no—
193#117—22no—
194#117—22no—
195#117—22no—
196#117—41no—
197#117—41no—
198#117—41no—
199#117—41no—
200#117—84no—
201#117—82no—
202#117—82no—
203#117C22⋊C412yes—
204#117C4⋊C412yes—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—82no—
229C2×C4⋊C4C22⋊C411yescharacteristic
230C2×C4⋊C4C22⋊C411yescharacteristic
231C2×C4⋊C4C22⋊C414yes—
232C2×C4⋊C4C22⋊C412yes—
233C2×C4⋊C4C22⋊C412yes—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—24no—
250C2×C42—24no—
251C2×C42—41no—
252C2×C42—41no—
253C2×C42—41no—
254C2×C42—41no—
255C2×C42—41no—
256C2×C42C22×C411yescharacteristic
257C2×C42C22×C411yescharacteristic
258C2×C42C22×C411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D411yescharacteristic
263C2×C42C2×Q811yescharacteristic
264C2×C42#4111yescharacteristic
265C2×C42#4111yescharacteristic
266C2×C42#4111yescharacteristic
267C2×C42#4111yescharacteristic
268C4⋊C8—22no—
269C4⋊C8—22no—
270C4⋊C8—22no—
271C4⋊C8—22no—
272C4⋊C8—22no—
273C4⋊C8—41no—
274C4⋊C8—41no—
275C4⋊C8—41no—
276C4⋊C8—41no—
277C4⋊C8—41no—
278#104—48no—
279C22⋊C8—22no—
280C22⋊C8—41no—
281C4×C8—22no—
282C4×C8—41no—
283#95—24no—
284#95—24no—
285#95—24no—
286#95—24no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—41no—
291C24—21no—
292C24#12011yescharacteristic
293#41—22no—
294#41—22no—
295#41—22no—
296#41—22no—
297#41—22no—
298#41—22no—
299#41—22no—
300#41—22no—
301#41—41no—
302#41—41no—
303#41—41no—
304#41—41no—
305#41—41no—
306#41—41no—
307#41—41no—
308#41—41no—
309C2×Q8—21no—
310C2×Q8—21no—
311C2×Q8—21no—
312C2×Q8—21no—
313C2×Q8—21no—
314C2×Q8#9512yes—
315C2×D4—22no—
316C2×D4—22no—
317C2×D4—21no—
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—21no—
321C2×D4—21no—
322C2×D4—41no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4—41no—
326C2×D4#9512yes—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—82no—
361C22×C4—82no—
362C22×C4—81no—
363C22×C4—81no—
364C22×C4—81no—
365C22×C4—81no—
366C22×C4C2×C22⋊C411yescharacteristic
367C22×C4C2×C22⋊C411yescharacteristic
368C22×C4C2×C22⋊C411yescharacteristic
369C22×C4C2×C22⋊C411yescharacteristic
370C22×C4C2×C4⋊C411yescharacteristic
371C22×C4C4×D411yescharacteristic
372C22×C4C4×D411yescharacteristic
373C22×C4C4×D411yescharacteristic
374C22×C4C4×D411yescharacteristic
375C22×C4#12011yescharacteristic
376C22×C4#12011yescharacteristic
377C22×C4#12011yescharacteristic
378C22×C4#12111yescharacteristic
379C22×C4#12111yescharacteristic
380C22×C4C22⋊Q811yescharacteristic
381C22×C4C22⋊Q811yescharacteristic
382C22×C4#12311yescharacteristic
383C22×C4#12311yescharacteristic
384C22×C4#13511yescharacteristic
385C22×C4#13511yescharacteristic
386C22×C4#13511yescharacteristic
387C22×C4#13511yescharacteristic
388C22×C4AGL(1,ℤ/8)11yescharacteristic
389C22×C4AGL(1,ℤ/8)11yescharacteristic
390C22×C4#13711yescharacteristic
391C22×C4#13711yescharacteristic
392M16—42no—
393M16—42no—
394M16—81no—
395M16—81no—
396C2×C8—44no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—81no—
401C2×C8—81no—
402C2×C8—82no—
403C2×C8—81no—
404C2×C8—81no—
405C2×C8—81no—
406C2×C8—81no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—24no—
414C4⋊C4—44no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—44no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—42no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—42no—
430C4⋊C4—41no—
431C4⋊C4—41no—
432C4⋊C4—42no—
433C4⋊C4—41no—
434C4⋊C4—41no—
435C4⋊C4—82no—
436C4⋊C4#9512yes—
437C4⋊C4#9512yes—
438C22⋊C4—22no—
439C22⋊C4—22no—
440C22⋊C4—22no—
441C22⋊C4—22no—
442C22⋊C4—22no—
443C22⋊C4—22no—
444C22⋊C4—42no—
445C22⋊C4—42no—
446C22⋊C4—84no—
447C22⋊C4—84no—
448C42—22no—
449C42—22no—
450C42—22no—
451C42—22no—
452C42—22no—
453C42—22no—
454C42—42no—
455C42—42no—
456C42—42no—
457C42—42no—
458C42—84no—
459C42—84no—
460C42—81no—
461C42—81no—
462C42C2×C4211yescharacteristic
463C42C2×C22⋊C411yescharacteristic
464C42C2×C4⋊C411yescharacteristic
465C42C2×C4⋊C411yescharacteristic
466C42#11711yescharacteristic
467C42#11711yescharacteristic
468C42C4×D411yescharacteristic
469C42C4×D411yescharacteristic
470C42C4×D411yescharacteristic
471C42C4×D411yescharacteristic
472C42#12111yescharacteristic
473C42#12111yescharacteristic
474C42C22⋊Q811yescharacteristic
475C42C22⋊Q811yescharacteristic
476C42#12311yescharacteristic
477C42#12311yescharacteristic
478C23—21no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—22no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23#37911yescharacteristic
490C23#38511yescharacteristic
491C23#38611yescharacteristic
492C23#44911yescharacteristic
493C23#44911yescharacteristic
494C23#45211yescharacteristic
495C23#45211yescharacteristic
496Q8—22no—
497Q8—22no—
498Q8—41no—
499Q8—41no—
500D4—22no—
501D4—22no—
502D4—42no—
503D4—42no—
504D4—42no—
505D4—42no—
506D4—41no—
507D4—41no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—162no—
562C2×C4—162no—
563C2×C4C2×#9511yescharacteristic
564C2×C4C4×C22⋊C411yescharacteristic
565C2×C4C4×C22⋊C411yescharacteristic
566C2×C4#38011yescharacteristic
567C2×C4#38011yescharacteristic
568C2×C4#38111yescharacteristic
569C2×C4#38511yescharacteristic
570C2×C4#38511yescharacteristic
571C2×C4#38511yescharacteristic
572C2×C4#38611yescharacteristic
573C2×C4#38611yescharacteristic
574C2×C4#38611yescharacteristic
575C2×C4#41611yescharacteristic
576C2×C4#41611yescharacteristic
577C2×C4#41611yescharacteristic
578C2×C4#41611yescharacteristic
579C2×C4#41711yescharacteristic
580C2×C4#41711yescharacteristic
581C2×C4#41811yescharacteristic
582C2×C4#41911yescharacteristic
583C2×C4#42711yescharacteristic
584C2×C4#42711yescharacteristic
585C2×C4#42811yescharacteristic
586C2×C4#42811yescharacteristic
587C2×C4#44911yescharacteristic
588C2×C4#44911yescharacteristic
589C2×C4#45211yescharacteristic
590C2×C4#45211yescharacteristic
591C2×C4#45311yescharacteristic
592C2×C4#45411yescharacteristic
593C2×C4#45511yescharacteristic
594C2×C4#45611yescharacteristic
595C2×C4#48211yescharacteristic
596C2×C4#48211yescharacteristic
597C2×C4#48211yescharacteristic
598C2×C4#48211yescharacteristic
599C2×C4#48511yescharacteristic
600C2×C4#48511yescharacteristic
601C2×C4#48511yescharacteristic
602C2×C4#48511yescharacteristic
603C8—82no—
604C8—82no—
605C8—82no—
606C22—21no—
607C22—21no—
608C22—21no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—21no—
614C22—21no—
615C22—21no—
616C22—21no—
617C22—21no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—81no—
622C22—81no—
623C22—81no—
624C22—81no—
625C22C22⋊#9511yescharacteristic
626C22#178811yescharacteristic
627C22#178811yescharacteristic
628C22#179211yescharacteristic
629C22#179311yescharacteristic
630C22#179511yescharacteristic
631C22#179611yescharacteristic
632C22#185411yescharacteristic
633C22#185411yescharacteristic
634C22#187311yescharacteristic
635C22#187511yescharacteristic
636C4—21no—
637C4—21no—
638C4—21no—
639C4—21no—
640C4—21no—
641C4—21no—
642C4—42no—
643C4—42no—
644C4—42no—
645C4—84no—
646C4—82no—
647C4—82no—
648C4—84no—
649C4#173511yescharacteristic
650C4#173611yescharacteristic
651C4#182511yescharacteristic
652C4#182511yescharacteristic
653C4#185411yescharacteristic
654C4#185411yescharacteristic
655C4#185911yescharacteristic
656C4#186011yescharacteristic
657C4#187311yescharacteristic
658C4#187511yescharacteristic
659C4#188211yescharacteristic
660C4#188311yescharacteristic
661C2—21no—
662C2—21no—
663C2—21no—
664C2—21no—
665C2—82no—
666C2#842211yescharacteristic
667C2#842311yescharacteristic
668C2#842611yescharacteristic
669C2#842711yescharacteristic
670C2#1073111yescharacteristic
671C2#1077511yescharacteristic
672C2#1078111yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
prime radicalC22×#41D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #10849C2semidirect
5. #8045C2semidirect
6. #8043C2semidirect
7. #8014C2semidirect
14. #2355C4semidirect
16. #2354C4semidirect
28. #1806C4semidirect
32. #1477C4semidirect

Maximal subgroups

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

subgroupindex
#341882
C2×#18062
C2×#14772
#108492
#80452
#80432
#80142