Finite Group Databasebeta

← #105688

Subgroups — #105688

Order 512.

1211 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
1D4×#117C211yesmaximal, characteristic
2#20732C211yesmaximal, characteristic
3C2×#1486C211yesmaximal, characteristic
4#10836C211yesmaximal, characteristic, complemented
5#7981C211yesmaximal, characteristic, complemented
6#7957C211yesmaximal, characteristic, complemented
7#7854C211yesmaximal, characteristic, complemented
8C22×C4×D4C2211yescharacteristic
9C22×#117—21no—
10C2×#431C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#423C2211yescharacteristic
13#2354—21no—
14#2352—22no—
15D4×C4⋊C4—21no—
16#2344—21no—
17#2342—22no—
18D4×C22⋊C4—21no—
19#2293C412yescomplemented
20#2290—22no—
21#2285C2211yescharacteristic
22#2285C2211yescharacteristic
23#2280—21no—
24C42×D4C412yescomplemented
25C4×#117C2211yescharacteristic
26#1823—21no—
27#1802—22no—
28#1786—21no—
29C4×#102C414yescomplemented
30C2×H(3,ℤ/4)—21no—
31C2×H(3,ℤ/4)—21no—
32#1486—22no—
33#1486—21no—
34#1486—21no—
35C2×D4⋊C8C414yescomplemented
36#1289—22no—
37#1286—22no—
38#1274—22no—
39C23×D4D411yescharacteristic
40C24×C4—21no—
41C22×M16—21no—
42C2×C4×D4—22no—
43C2×C4×D4—21no—
44C2×C4×D4—21no—
45C2×C4×D4—21no—
46C2×C4×D4—21no—
47C2×C4×D4—21no—
48C2×C4×D4—21no—
49C2×C4×D4C2×C411yescharacteristic
50C2×C4×D4C2×C412yes—
51C2×C4×D4C2×C411yescharacteristic
52C2×C4×D4C2×C412yes—
53C2×C4×D4D412yes—
54C2×C4×D4D411yescharacteristic
55C2×C4×D4D411yescharacteristic
56C2×C4×D4Q812yes—
57C2×#117—22no—
58C2×#117—22no—
59C2×#117—21no—
60C2×#117—21no—
61C2×#117—22no—
62C2×#117—42no—
63C2×#117—42no—
64C2×#117—42no—
65C2×#117—42no—
66C2×#117D411yescharacteristic
67C2×#117D411yescharacteristic
68C22×C4⋊C4—22no—
69C22×C4⋊C4D411yescharacteristic
70C22×C22⋊C4—21no—
71C22×C22⋊C4—22no—
72C22×C42—22no—
73C22×C42C2311yescharacteristic
74#431—21no—
75#431—21no—
76#431—21no—
77#431—21no—
78#431—21no—
79#431—21no—
80#431—21no—
81#431—21no—
82#423—21no—
83#423—21no—
84#423—21no—
85#423—21no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8C2×C412yescomplemented
90C2×#102—24no—
91C2×C22⋊C8—21no—
92C2×C22⋊C8—21no—
93C2×C4×C8C2×C412yescomplemented
94C2×C4×C8C2×C412yescomplemented
95#390—22no—
96#389—22no—
97#388—24no—
98#388—24no—
99#387—24no—
100#386—24no—
101#385—24no—
102#384—21no—
103#384—21no—
104#382—21no—
105#382D412yes—
106#381—22no—
107#380—22no—
108#380—22no—
109#380—21no—
110#380—21no—
111#380—21no—
112#380D412yes—
113#379—22no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4C2×C412yes—
120C4×C4⋊C4C2×C412yescomplemented
121C4×C22⋊C4—21no—
122C4×C22⋊C4—21no—
123C4×C22⋊C4—22no—
124C4×C22⋊C4—22no—
125C4×C22⋊C4—24no—
126#376D412yes—
127C43C2×C412yes—
128#340—24no—
129H(3,ℤ/4)—42no—
130H(3,ℤ/4)—42no—
131D4⋊C8—24no—
132D4⋊C8—24no—
133C25—21no—
134C22×D4—21no—
135C22×D4—21no—
136C22×D4—21no—
137C22×D4—44no—
138C22×D4C22⋊C411yescharacteristic
139C22×D4C22⋊C412yes—
140C22×D4C22⋊C411yescharacteristic
141C22×D4C4⋊C412yes—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—24no—
152C23×C4—41no—
153C23×C4—41no—
154C23×C4—41no—
155C23×C4—41no—
156C23×C4—41no—
157C23×C4—41no—
158C23×C4—41no—
159C23×C4—41no—
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C2×M16—41no—
164C2×M16—41no—
165C2×M16—41no—
166C2×M16—41no—
167C2×M16—41no—
168C2×M16—41no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C4×D4—21no—
173C4×D4—21no—
174C4×D4—21no—
175C4×D4—21no—
176C4×D4—22no—
177C4×D4—22no—
178C4×D4—22no—
179C4×D4—22no—
180C4×D4—22no—
181C4×D4—22no—
182C4×D4—22no—
183C4×D4—22no—
184C4×D4—21no—
185C4×D4—21no—
186C4×D4—21no—
187C4×D4—21no—
188C4×D4—48no—
189C4×D4C4212yes—
190C4×D4C4212yes—
191C4×D4C22⋊C412yes—
192C4×D4C22⋊C412yes—
193C4×D4C4⋊C412yes—
194C4×D4C4⋊C412yes—
195C4×D4C4⋊C412yes—
196C4×D4C4⋊C412yes—
197#117—21no—
198#117—21no—
199#117—21no—
200#117—21no—
201#117—21no—
202#117—21no—
203#117—44no—
204#117—44no—
205#117—44no—
206#117—44no—
207#117—44no—
208#117—41no—
209#117—41no—
210#117—41no—
211#117—41no—
212#117—41no—
213#117—41no—
214#117—41no—
215#117—41no—
216#117—44no—
217#117D812yes—
218#117SD1612yes—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4C22⋊C412yes—
234C2×C4⋊C4C22⋊C411yescharacteristic
235C2×C4⋊C4C22⋊C411yescharacteristic
236C2×C4⋊C4C4⋊C412yes—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—41no—
254C2×C42—24no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—44no—
260C2×C42—44no—
261C2×C42C22⋊C412yes—
262C2×C42C22⋊C412yes—
263C2×C42C22×C411yescharacteristic
264C2×C42C22×C411yescharacteristic
265C2×C42C22×C411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C2×C42C2×D411yescharacteristic
268C2×C42C2×D411yescharacteristic
269C2×C42C2×D411yescharacteristic
270C2×C42C2×Q811yescharacteristic
271C2×C42#4111yescharacteristic
272C2×C42#4111yescharacteristic
273C2×C42#4111yescharacteristic
274C2×C42#4111yescharacteristic
275C4⋊C8—22no—
276C4⋊C8—22no—
277C4⋊C8—41no—
278C4⋊C8—41no—
279C4⋊C8—41no—
280C4⋊C8—41no—
281C4⋊C8—41no—
282C4⋊C8—41no—
283#102—48no—
284C22⋊C8—41no—
285C22⋊C8—41no—
286C22⋊C8—41no—
287C22⋊C8—41no—
288C4×C8—22no—
289C4×C8—22no—
290C4×C8—22no—
291C4×C8—22no—
292#95—24no—
293#95—24no—
294#95—22no—
295#95—22no—
296#95—22no—
297#95—22no—
298#95—41no—
299#95C22⋊C414yes—
300C24—21no—
301C24—21no—
302C24—22no—
303C24—22no—
304C24—44no—
305C24—41no—
306C24—41no—
307C24—41no—
308C24—41no—
309C24—82no—
310C24#12011yescharacteristic
311C2×D4—24no—
312C2×D4—24no—
313C2×D4—24no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—42no—
319C2×D4—42no—
320C2×D4—42no—
321C2×D4—44no—
322C2×D4—44no—
323C2×D4—44no—
324C2×D4#9512yes—
325C2×D4#9512yes—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—42no—
342C22×C4—48no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
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—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—48no—
380C22×C4—42no—
381C22×C4—81no—
382C22×C4—84no—
383C22×C4C2×C22⋊C411yescharacteristic
384C22×C4C2×C22⋊C411yescharacteristic
385C22×C4C2×C22⋊C411yescharacteristic
386C22×C4C2×C22⋊C411yescharacteristic
387C22×C4C2×C4⋊C411yescharacteristic
388C22×C4C4×D411yescharacteristic
389C22×C4C4×D411yescharacteristic
390C22×C4C4×D411yescharacteristic
391C22×C4C4×D411yescharacteristic
392C22×C4#12011yescharacteristic
393C22×C4#12011yescharacteristic
394C22×C4#12011yescharacteristic
395C22×C4#12111yescharacteristic
396C22×C4#12111yescharacteristic
397C22×C4C22⋊Q811yescharacteristic
398C22×C4C22⋊Q811yescharacteristic
399C22×C4#12311yescharacteristic
400C22×C4#12311yescharacteristic
401C22×C4C2×D811yescharacteristic
402C22×C4C2×SD1611yescharacteristic
403C22×C4#13511yescharacteristic
404C22×C4#13511yescharacteristic
405C22×C4AGL(1,ℤ/8)11yescharacteristic
406C22×C4AGL(1,ℤ/8)11yescharacteristic
407C22×C4AGL(1,ℤ/8)11yescharacteristic
408C22×C4#13711yescharacteristic
409M16—81no—
410M16—81no—
411M16—81no—
412M16—81no—
413C2×C8—44no—
414C2×C8—42no—
415C2×C8—42no—
416C2×C8—42no—
417C2×C8—42no—
418C2×C8—42no—
419C2×C8—82no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—48no—
427C4⋊C4—44no—
428C4⋊C4—44no—
429C4⋊C4—42no—
430C4⋊C4—41no—
431C4⋊C4—42no—
432C4⋊C4—41no—
433C4⋊C4—82no—
434C4⋊C4—82no—
435C4⋊C4—81no—
436C4⋊C4—81no—
437C4⋊C4#9512yes—
438C4⋊C4#9512yes—
439C22⋊C4—22no—
440C22⋊C4—22no—
441C22⋊C4—22no—
442C22⋊C4—22no—
443C22⋊C4—22no—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C22⋊C4—22no—
447C22⋊C4—22no—
448C22⋊C4—22no—
449C22⋊C4—48no—
450C22⋊C4—48no—
451C22⋊C4—42no—
452C22⋊C4—41no—
453C22⋊C4—42no—
454C22⋊C4—41no—
455C22⋊C4—84no—
456C22⋊C4—81no—
457C22⋊C4—81no—
458C42—22no—
459C42—22no—
460C42—22no—
461C42—48no—
462C42—44no—
463C42—42no—
464C42—42no—
465C42—42no—
466C42—42no—
467C42—82no—
468C42#10214yescomplemented
469C42C2×C4211yescharacteristic
470C42C2×C22⋊C411yescharacteristic
471C42C2×C4⋊C411yescharacteristic
472C42C2×C4⋊C411yescharacteristic
473C42#11711yescharacteristic
474C42#11711yescharacteristic
475C42C4×D411yescharacteristic
476C42C4×D411yescharacteristic
477C42C4×D411yescharacteristic
478C42C4×D411yescharacteristic
479C42#12111yescharacteristic
480C42#12111yescharacteristic
481C42C22⋊Q811yescharacteristic
482C42C22⋊Q811yescharacteristic
483C42#12311yescharacteristic
484C42#12311yescharacteristic
485C23—24no—
486C23—22no—
487C23—22no—
488C23—41no—
489C23—41no—
490C23—44no—
491C23—44no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—41no—
496C23—41no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—41no—
502C23—41no—
503C23—82no—
504C23—82no—
505C23—84no—
506C23—84no—
507C23—84no—
508C23—84no—
509C23#37911yescharacteristic
510C23#38511yescharacteristic
511C23#38611yescharacteristic
512C23#44711yescharacteristic
513C23#44911yescharacteristic
514C23#45011yescharacteristic
515C23#45211yescharacteristic
516D4—24no—
517D4—22no—
518D4—22no—
519D4—22no—
520D4—44no—
521D4—44no—
522D4—42no—
523D4H(3,ℤ/4)14yescomplemented
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—816no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4C2×#9511yescharacteristic
576C2×C4C4×C22⋊C411yescharacteristic
577C2×C4C4×C22⋊C411yescharacteristic
578C2×C4#38011yescharacteristic
579C2×C4#38011yescharacteristic
580C2×C4#38111yescharacteristic
581C2×C4#38511yescharacteristic
582C2×C4#38511yescharacteristic
583C2×C4#38511yescharacteristic
584C2×C4#38611yescharacteristic
585C2×C4#38611yescharacteristic
586C2×C4#38611yescharacteristic
587C2×C4C2×#10211yescharacteristic
588C2×C4#41611yescharacteristic
589C2×C4#41711yescharacteristic
590C2×C4#41811yescharacteristic
591C2×C4C4×D811yescharacteristic
592C2×C4C4×D811yescharacteristic
593C2×C4C4×SD1611yescharacteristic
594C2×C4C4×SD1611yescharacteristic
595C2×C4#44011yescharacteristic
596C2×C4#44011yescharacteristic
597C2×C4#44211yescharacteristic
598C2×C4#44211yescharacteristic
599C2×C4#44711yescharacteristic
600C2×C4#44911yescharacteristic
601C2×C4#45011yescharacteristic
602C2×C4#45211yescharacteristic
603C2×C4#45911yescharacteristic
604C2×C4#46111yescharacteristic
605C2×C4#46311yescharacteristic
606C2×C4#46311yescharacteristic
607C2×C4#47411yescharacteristic
608C2×C4#47611yescharacteristic
609C2×C4#47811yescharacteristic
610C2×C4#47811yescharacteristic
611C8—82no—
612C8—82no—
613C8—82no—
614C22—21no—
615C22—21no—
616C22—21no—
617C22—21no—
618C22—21no—
619C22—21no—
620C22—24no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22—41no—
625C22—44no—
626C22—44no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—84no—
633C22—84no—
634C22—84no—
635C22—84no—
636C22—82no—
637C22—82no—
638C22C22⋊#9511yescharacteristic
639C22#178611yescharacteristic
640C22#178811yescharacteristic
641C22#186211yescharacteristic
642C22#186411yescharacteristic
643C22#187711yescharacteristic
644C22#187911yescharacteristic
645C4—21no—
646C4—21no—
647C4—21no—
648C4—21no—
649C4—21no—
650C4—21no—
651C4—21no—
652C4—44no—
653C4—42no—
654C4—42no—
655C4—82no—
656C4—84no—
657C4—162no—
658C4C2×H(3,ℤ/4)12yes—
659C4C4×#10211yescharacteristic
660C4#176211yescharacteristic
661C4#180211yescharacteristic
662C4#180411yescharacteristic
663C4#186211yescharacteristic
664C4#186411yescharacteristic
665C4#187711yescharacteristic
666C4#187911yescharacteristic
667C2—21no—
668C2—21no—
669C2—21no—
670C2—21no—
671C2—21no—
672C2—21no—
673C2—44no—
674C2—82no—
675C2#1071611yescharacteristic
676C2#1078811yescharacteristic
677C2#1079211yescharacteristic
subgroupG/H
centerC2×C4C4×C22⋊C4
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
socleC22C22⋊#95
prime radicalC23×D4D4
LCSG ▷ C42 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C4×#117 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #10836C2semidirect
5. #7981C2semidirect
6. #7957C2semidirect
7. #7854C2semidirect
19. #2293C4semidirect
24. C42×D4C4semidirect
29. C4×#102C4semidirect
35. C2×D4⋊C8C4semidirect
89. C2×C4⋊C8C2×C4semidirect
93. C2×C4×C8C2×C4semidirect
94. C2×C4×C8C2×C4semidirect
120. C4×C4⋊C4C2×C4semidirect
468. C42#102semidirect
523. D4H(3,ℤ/4)semidirect

Maximal subgroups

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

subgroupindex
D4×#1172
#207322
C2×#14862
#108362
#79812
#79572
#78542