Finite Group Databasebeta

← #216800

Subgroups — #216800

Order 512.

1313 conjugacy classes of subgroups.

The subgroup lattice has 762 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36621C211yesmaximal, characteristic
2#20715C211yesmaximal, characteristic, complemented
3#14686C211yesmaximal, characteristic, complemented
4#9887C212yesmaximal, complemented
5#9682C212yesmaximal, complemented
6#9155C212yesmaximal
7#9151C212yesmaximal, complemented
8#8572C212yesmaximal, complemented
9#8571C212yesmaximal, complemented
10C2×#529C411yescharacteristic, complemented
11C23×C42C2211yescharacteristic
12C2×#432—21no—
13#2855C411yescharacteristic, complemented
14#2854C411yescharacteristic, complemented
15#2853—21no—
16#2852—21no—
17#2852—21no—
18#2851C411yescharacteristic, complemented
19#2851C2211yescharacteristic
20#2850C411yescharacteristic, complemented
21#2850C2211yescharacteristic
22#2849—21no—
23#2848C2211yescharacteristic
24#2847C2211yescharacteristic
25C4×C22⋊Q8C411yescharacteristic, complemented
26C4×C22⋊Q8C2211yescharacteristic
27C4×#123—21no—
28C4×#121C411yescharacteristic, complemented
29C4×#121C2211yescharacteristic
30C4×#120—21no—
31#2290—21no—
32#2290—21no—
33C2×#384—21no—
34#2283C411yescharacteristic, complemented
35C2×#380—21no—
36C2×#380—21no—
37C2×#380C2212yes—
38#1817—21no—
39#1797—21no—
40#1797C2212yes—
41#1779C2212yes—
42#1749—21no—
43#1749C2212yes—
44C2×#328C2214yes—
45C2×#327C2214yes—
46C4⋊#95—21no—
47C4⋊#95C2212yes—
48C4⋊#95C2212yes—
49#1294C2214yes—
50#1294C2214yes—
51C24×C4D411yescharacteristic
52C24×C4D411yescharacteristic
53C24×C4C2311yescharacteristic
54#529—44no—
55C2×#126—21no—
56C2×#125C2×C411yescharacteristic
57C2×#124C2×C411yescharacteristic
58C2×#123—21no—
59C2×#123—21no—
60C2×C22⋊Q8C2×C411yescharacteristic
61C2×C22⋊Q8C2×C411yescharacteristic
62C2×C22⋊Q8C2311yescharacteristic
63C2×#121C2×C411yescharacteristic
64C2×#121C2×C411yescharacteristic
65C2×#121C2311yescharacteristic
66C2×#120—21no—
67C2×C4×Q8C2×C411yescharacteristic
68C2×C4×D4—21no—
69C2×C4×D4—21no—
70C2×C4×D4—21no—
71C2×C4×D4C2×C411yescharacteristic
72C2×#117—21no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4—22no—
76C22×C22⋊C4—22no—
77C22×C42—44no—
78C22×C42C2×C411yescharacteristic
79C22×C42C2×C411yescharacteristic
80C22×C42C2×C411yescharacteristic
81C22×C42C2×C411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42D411yescharacteristic
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90#432—44no—
91C2×C4⋊C8—22no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8C2314yes—
95C2×#97—22no—
96#391C2×C411yescharacteristic
97#391C2×C411yescharacteristic
98#390C2×C411yescharacteristic
99#390C2×C411yescharacteristic
100#389C2×C411yescharacteristic
101#389C2×C411yescharacteristic
102#389C2×C411yescharacteristic
103#389C2×C411yescharacteristic
104#388—21no—
105#388—21no—
106#388—21no—
107#388—21no—
108#388—21no—
109#388—21no—
110#388—21no—
111#388—21no—
112#387—21no—
113#387—21no—
114#387—21no—
115#387—21no—
116#387—21no—
117#387—21no—
118#387—21no—
119#387—21no—
120#386—21no—
121#386—21no—
122#386—21no—
123#386—21no—
124#386—21no—
125#386—21no—
126#386—21no—
127#386—21no—
128#386—21no—
129#386—21no—
130#385—21no—
131#385—21no—
132#385—21no—
133#385—21no—
134#385—21no—
135#385—21no—
136#385—21no—
137#385—21no—
138#385—21no—
139#385—21no—
140#384—44no—
141#383—21no—
142#382C2×C411yescharacteristic
143#382C2×C411yescharacteristic
144#381—21no—
145#381—21no—
146#380—44no—
147#380—44no—
148#380—44no—
149#380C2×C411yescharacteristic
150#380C2×C411yescharacteristic
151#380C2×C411yescharacteristic
152#380C2×C411yescharacteristic
153#380C2311yescharacteristic
154#380C2311yescharacteristic
155#379—21no—
156C4×C4⋊C4—21no—
157C4×C4⋊C4—21no—
158C4×C4⋊C4C2×C411yescharacteristic
159C4×C4⋊C4C2×C411yescharacteristic
160C4×C22⋊C4—21no—
161C4×C22⋊C4—21no—
162C4×C22⋊C4—21no—
163C4×C22⋊C4—21no—
164C4×C22⋊C4—21no—
165#376—21no—
166C2×#95—22no—
167C2×#95—22no—
168C2×#95C2314yes—
169#336—22no—
170#336—22no—
171#328—44no—
172#327—44no—
173C25C2×D411yescharacteristic
174C22×Q8C22×C411yescharacteristic
175C22×D4—21no—
176C22×D4—21no—
177C22×D4—21no—
178C22×D4C22×C411yescharacteristic
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—21no—
186C23×C4—24no—
187C23×C4—42no—
188C23×C4—42no—
189C23×C4—42no—
190C23×C4—42no—
191C23×C4—42no—
192C23×C4—42no—
193C23×C4C22⋊C411yescharacteristic
194C23×C4C22⋊C411yescharacteristic
195C23×C4C22⋊C411yescharacteristic
196C23×C4C22⋊C411yescharacteristic
197C23×C4C22⋊C411yescharacteristic
198C23×C4C22⋊C411yescharacteristic
199C23×C4C22⋊C411yescharacteristic
200C23×C4C22⋊C411yescharacteristic
201C23×C4C22×C411yescharacteristic
202C23×C4C22×C411yescharacteristic
203C23×C4C2×D411yescharacteristic
204C23×C4C2×D411yescharacteristic
205C23×C4C2×D411yescharacteristic
206C23×C4C2×D411yescharacteristic
207C23×C4C2×D411yescharacteristic
208C23×C4C2×D411yescharacteristic
209C23×C4C2×D411yescharacteristic
210C23×C4C2411yescharacteristic
211C22×C8—24no—
212#126—44no—
213#125—42no—
214#124—42no—
215#123—44no—
216#123—44no—
217C22⋊Q8—22no—
218C22⋊Q8—22no—
219C22⋊Q8—42no—
220C22⋊Q8—42no—
221#121—22no—
222#121—22no—
223#121—42no—
224#121—42no—
225#120—42no—
226#120—42no—
227C4×Q8—42no—
228C4×D4—42no—
229C4×D4—44no—
230C4×D4—44no—
231C4×D4—42no—
232C4×D4—42no—
233#117—44no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4C22×C411yescharacteristic
254C2×C4⋊C4C22×C411yescharacteristic
255C2×C4⋊C4C22×C411yescharacteristic
256C2×C4⋊C4C22×C411yescharacteristic
257C2×C4⋊C4C22×C411yescharacteristic
258C2×C4⋊C4C22×C411yescharacteristic
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—42no—
288C2×C42—42no—
289C2×C42—42no—
290C2×C42—42no—
291C2×C42—42no—
292C2×C42—42no—
293C2×C42—42no—
294C2×C42—42no—
295C2×C42—42no—
296C2×C42C22⋊C411yescharacteristic
297C2×C42C22⋊C411yescharacteristic
298C2×C42C22⋊C411yescharacteristic
299C2×C42C22⋊C411yescharacteristic
300C2×C42C22⋊C411yescharacteristic
301C2×C42C22⋊C411yescharacteristic
302C2×C42C22⋊C411yescharacteristic
303C2×C42C22⋊C411yescharacteristic
304C2×C42C22×C411yescharacteristic
305C2×C42C22×C411yescharacteristic
306C2×C42C22×C411yescharacteristic
307C2×C42C22×C411yescharacteristic
308C2×C42C2×D411yescharacteristic
309C2×C42C2×D411yescharacteristic
310C2×C42C2×D411yescharacteristic
311C2×C42C2×D411yescharacteristic
312C4⋊C8—44no—
313C22⋊C8—24no—
314C22⋊C8—24no—
315C22⋊C8—44no—
316C22⋊C8—44no—
317#97—44no—
318#95—21no—
319#95—21no—
320#95—21no—
321#95—21no—
322#95—21no—
323#95—21no—
324#95—21no—
325#95—21no—
326#95—21no—
327#95—21no—
328#95—21no—
329#95—21no—
330#95—21no—
331#95—21no—
332#95—21no—
333#95—21no—
334#95—21no—
335#95—21no—
336#95—21no—
337#95—21no—
338#95—24no—
339#95—24no—
340#95—44no—
341#95—44no—
342C24—21no—
343C24—21no—
344C24—21no—
345C24—21no—
346C24—21no—
347C24—41no—
348C24—41no—
349C24—41no—
350C24—41no—
351C24C2×C22⋊C411yescharacteristic
352C24C2×C22⋊C411yescharacteristic
353C24AGL(1,ℤ/8)12yes—
354C24#13712yes—
355C24C22×D411yescharacteristic
356C2×Q8—22no—
357C2×Q8—41no—
358C2×Q8—41no—
359C2×D4—22no—
360C2×D4—41no—
361C2×D4—41no—
362C2×D4—42no—
363C2×D4—42no—
364C2×D4—42no—
365C2×D4—42no—
366C2×D4—41no—
367C2×D4—41no—
368C2×D4—41no—
369C2×D4—41no—
370C2×D4—41no—
371C2×D4—41no—
372C2×D4—42no—
373C2×D4—42no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4C2×C22⋊C411yescharacteristic
454C22×C4C2×C22⋊C411yescharacteristic
455C22×C4C2×C22⋊C411yescharacteristic
456C22×C4C2×C22⋊C411yescharacteristic
457C22×C4C2×C22⋊C411yescharacteristic
458C22×C4C2×C22⋊C411yescharacteristic
459C22×C4C2×C22⋊C411yescharacteristic
460C22×C4C2×C22⋊C411yescharacteristic
461C22×C4C2×C22⋊C411yescharacteristic
462C22×C4C2×C22⋊C411yescharacteristic
463C22×C4AGL(1,ℤ/8)12yes—
464C22×C4#13712yes—
465C22×C4C23×C411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4H(5,𝔽2)12yes—
468C22×C42−1+412yes—
469C2×C8—44no—
470C2×C8—44no—
471C2×C8—44no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—41no—
475C4⋊C4—41no—
476C4⋊C4—41no—
477C4⋊C4—41no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—88no—
489C4⋊C4—88no—
490C4⋊C4—88no—
491C22⋊C4—42no—
492C22⋊C4—42no—
493C22⋊C4—42no—
494C22⋊C4—42no—
495C22⋊C4—42no—
496C22⋊C4—42no—
497C22⋊C4—42no—
498C22⋊C4—42no—
499C22⋊C4—41no—
500C22⋊C4—41no—
501C22⋊C4—41no—
502C22⋊C4—41no—
503C22⋊C4—41no—
504C22⋊C4—41no—
505C22⋊C4—42no—
506C22⋊C4—42no—
507C22⋊C4—42no—
508C22⋊C4—42no—
509C22⋊C4—42no—
510C22⋊C4—42no—
511C22⋊C4—42no—
512C22⋊C4—42no—
513C22⋊C4—88no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—22no—
521C42—22no—
522C42—22no—
523C42—41no—
524C42—41no—
525C42—41no—
526C42—41no—
527C42—41no—
528C42—41no—
529C42—42no—
530C42—42no—
531C42—42no—
532C42#9914yes—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23#41712yes—
573C23#41712yes—
574C23C22×C22⋊C411yescharacteristic
575C23#53511yescharacteristic
576C23#53611yescharacteristic
577C23C2×AGL(1,ℤ/8)11yescharacteristic
578C23C2×AGL(1,ℤ/8)11yescharacteristic
579C23C2×#13711yescharacteristic
580C23C2×#13711yescharacteristic
581Q8—82no—
582D4—82no—
583D4—84no—
584D4—84no—
585D4—81no—
586D4—81no—
587D4—81no—
588D4—81no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—41no—
646C2×C4—41no—
647C2×C4—41no—
648C2×C4—41no—
649C2×C4—41no—
650C2×C4—41no—
651C2×C4—41no—
652C2×C4—41no—
653C2×C4—41no—
654C2×C4—41no—
655C2×C4—41no—
656C2×C4—41no—
657C2×C4—41no—
658C2×C4—41no—
659C2×C4—41no—
660C2×C4—84no—
661C2×C4—84no—
662C2×C4—84no—
663C2×C4—84no—
664C2×C4—84no—
665C2×C4—84no—
666C2×C4—84no—
667C2×C4—84no—
668C2×C4C2×#9912yes—
669C2×C4C2×#9912yes—
670C2×C4C2×#9912yes—
671C2×C4#41012yes—
672C2×C4#41712yes—
673C2×C4#41712yes—
674C2×C4#42112yes—
675C2×C4#42112yes—
676C2×C4#52012yes—
677C2×C4#53712yes—
678C8—84no—
679C22—21no—
680C22—21no—
681C22—21no—
682C22—21no—
683C22—21no—
684C22—21no—
685C22—21no—
686C22—21no—
687C22—21no—
688C22—21no—
689C22—21no—
690C22—21no—
691C22—22no—
692C22—22no—
693C22—22no—
694C22—22no—
695C22—22no—
696C22—22no—
697C22—44no—
698C22—44no—
699C22—44no—
700C22—44no—
701C22—41no—
702C22—41no—
703C22—41no—
704C22—41no—
705C22—41no—
706C22—41no—
707C22—41no—
708C22—41no—
709C22—81no—
710C22—81no—
711C22—81no—
712C22—81no—
713C22—81no—
714C22—81no—
715C22—81no—
716C22#231111yescharacteristic
717C22C22×#9911yescharacteristic
718C22C2×#41011yescharacteristic
719C22C2×#41711yescharacteristic
720C22C2×#41711yescharacteristic
721C22C2×#42111yescharacteristic
722C22C2×#42111yescharacteristic
723C22#321711yescharacteristic
724C22#321811yescharacteristic
725C22#323811yescharacteristic
726C22#323911yescharacteristic
727C4—22no—
728C4—22no—
729C4—22no—
730C4—22no—
731C4—22no—
732C4—22no—
733C4—22no—
734C4—41no—
735C4—41no—
736C4—41no—
737C4—41no—
738C4—81no—
739C4—81no—
740C4—81no—
741C4—81no—
742C4—81no—
743C4—81no—
744C4—81no—
745C4—164no—
746C4#151314yes—
747C2—21no—
748C2—21no—
749C2—21no—
750C2—21no—
751C2—42no—
752C2—42no—
753C2—161no—
754C2C2×#151311yescharacteristic
755C2#1758111yescharacteristic
756C2#1758511yescharacteristic
757C2#1759211yescharacteristic
758C2#2186711yescharacteristic
759C2#2187411yescharacteristic
760C2#2190011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radical#2847C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20715C2semidirect
3. #14686C2semidirect
4. #9887C2semidirect
5. #9682C2semidirect
7. #9151C2semidirect
8. #8572C2semidirect
9. #8571C2semidirect
10. C2×#529C4semidirect
13. #2855C4semidirect
14. #2854C4semidirect
18. #2851C4semidirect
20. #2850C4semidirect
25. C4×C22⋊Q8C4semidirect
28. C4×#121C4semidirect
34. #2283C4semidirect

Maximal subgroups

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

subgroupindex
#366212
#207152
#146862
#98872
#96822
#91552
#91512
#85722
#85712