Finite Group Databasebeta

← #188488

Subgroups — #188488

Order 512.

1124 conjugacy classes of subgroups.

The subgroup lattice has 659 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#529C211yesmaximal, characteristic
2#20808C211yesmaximal, characteristic, complemented
3C2×#1521C212yesmaximal
4#14688C211yesmaximal, characteristic, complemented
5#8578C212yesmaximal, complemented
6#8577C212yesmaximal
7#8574C212yesmaximal, complemented
8#8573C212yesmaximal
9#8025C212yesmaximal, complemented
10C2×#529C2211yescharacteristic
11C2×#431C2212yes—
12C4×#125C411yescharacteristic, complemented
13C4×#125C411yescharacteristic, complemented
14C4×#126—21no—
15C4×#126—21no—
16C4×#124C411yescharacteristic, complemented
17C4×#124C411yescharacteristic, complemented
18C4×C22⋊Q8C2211yescharacteristic
19C4×C22⋊Q8C2211yescharacteristic
20C4×#123—21no—
21C4×#123—21no—
22C4×#121C2211yescharacteristic
23C4×#121C2211yescharacteristic
24#2298—21no—
25#2295C411yescharacteristic, complemented
26#2293C411yescharacteristic, complemented
27#2290—21no—
28#2283C2211yescharacteristic
29C2×#383C2212yes—
30C42×Q8C411yescharacteristic, complemented
31C42×D4—21no—
32C42×D4C411yescharacteristic, complemented
33C4×#117—21no—
34C2×C43C2211yescharacteristic
35#1842—21no—
36#1831—21no—
37#1797C2212yes—
38#1797C2212yes—
39C4×C4⋊C8—21no—
40C4×C4⋊C8—21no—
41#1521—28no—
42C2×#328C2214yes—
43C2×#327C2214yes—
44C4⋊#95—21no—
45C4⋊#95—21no—
46C4⋊#95C2212yes—
47#1434—21no—
48#1434—21no—
49#1434C2212yes—
50#1294C2214yes—
51#1294C2214yes—
52#529—22no—
53#529—22no—
54#529D414yes—
55#529Q814yes—
56C2×#126—21no—
57C2×#125C2×C411yescharacteristic
58C2×#124C2×C411yescharacteristic
59C2×#123—21no—
60C2×C22⋊Q8C2311yescharacteristic
61C2×#121C2311yescharacteristic
62C2×C4×Q8C2×C411yescharacteristic
63C2×C4×Q8C2×C411yescharacteristic
64C2×C4×Q8C2×C411yescharacteristic
65C2×C4×D4—21no—
66C2×C4×D4—21no—
67C2×C4×D4—21no—
68C2×C4×D4C2×C411yescharacteristic
69C2×C4×D4C2×C411yescharacteristic
70C2×C4×D4C2×C411yescharacteristic
71C2×#117—21no—
72C2×#117—21no—
73C22×C4⋊C4—22no—
74C22×C42—21no—
75C22×C42D411yescharacteristic
76C22×C42D411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42D411yescharacteristic
79C22×C42C2311yescharacteristic
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82#431—24no—
83#431—24no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—22no—
87C2×C22⋊C8C2314yes—
88C2×C4×C8—22no—
89#391C2×C411yescharacteristic
90#391C2×C411yescharacteristic
91#390C2×C411yescharacteristic
92#390C2×C411yescharacteristic
93#389C2×C411yescharacteristic
94#389C2×C411yescharacteristic
95#389C2×C411yescharacteristic
96#389C2×C411yescharacteristic
97#388—21no—
98#388—21no—
99#388—21no—
100#388—21no—
101#388—21no—
102#388—21no—
103#388—21no—
104#388—21no—
105#387—21no—
106#387—21no—
107#387—21no—
108#387—21no—
109#387—21no—
110#387—21no—
111#387—21no—
112#387—21no—
113#386—21no—
114#386—21no—
115#385—21no—
116#385—21no—
117#383—21no—
118#383—24no—
119#383—24no—
120#382C2×C411yescharacteristic
121#382C2×C411yescharacteristic
122#381—21no—
123#380C2311yescharacteristic
124#380C2311yescharacteristic
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4C2×C411yescharacteristic
136C4×C4⋊C4C2×C411yescharacteristic
137C4×C4⋊C4C2×C411yescharacteristic
138C4×C4⋊C4C2×C411yescharacteristic
139C4×C4⋊C4C2×C411yescharacteristic
140C4×C4⋊C4C2×C411yescharacteristic
141C4×C22⋊C4—21no—
142C4×C22⋊C4—21no—
143C4×C22⋊C4—21no—
144C4×C22⋊C4—21no—
145C4×C22⋊C4—21no—
146C4×C22⋊C4—21no—
147C4×C22⋊C4—21no—
148C4×C22⋊C4—21no—
149C4×C22⋊C4—21no—
150C4×C22⋊C4—21no—
151C4×C22⋊C4—21no—
152C4×C22⋊C4—21no—
153C4×C22⋊C4—21no—
154#376—21no—
155#376—21no—
156C2×#95—22no—
157C2×#95—22no—
158C2×#95—22no—
159C2×#95—22no—
160C2×#95—22no—
161C2×#95C2314yes—
162C43—21no—
163C43C2×C411yescharacteristic
164C43C2×C411yescharacteristic
165C43C2×C411yescharacteristic
166C43C2×C411yescharacteristic
167C43D411yescharacteristic
168C43D411yescharacteristic
169C43D411yescharacteristic
170C43D411yescharacteristic
171#336—22no—
172#336—22no—
173#328—28no—
174#327—28no—
175C22×Q8C22×C411yescharacteristic
176C22×D4—21no—
177C22×D4C22×C411yescharacteristic
178C23×C4—21no—
179C23×C4—21no—
180C23×C4—21no—
181C23×C4—24no—
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2411yescharacteristic
189C22×C8—24no—
190#126—24no—
191#126—24no—
192#125—22no—
193#125C4⋊C414yescomplemented
194#124—22no—
195#124C4⋊C414yescomplemented
196#123—24no—
197#123—24no—
198C22⋊Q8—21no—
199C22⋊Q8—21no—
200C22⋊Q8C2×D412yes—
201C22⋊Q8C2×Q812yes—
202#121—21no—
203#121—21no—
204#121C2×D412yes—
205#121C2×Q812yes—
206C4×Q8—22no—
207C4×Q8—42no—
208C4×Q8—42no—
209C4×Q8C4⋊C414yescomplemented
210C4×D4—24no—
211C4×D4—24no—
212C4×D4—22no—
213C4×D4—42no—
214C4×D4—42no—
215C4×D4—44no—
216C4×D4—44no—
217C4×D4C4⋊C414yescomplemented
218#117—24no—
219#117—24no—
220#117—44no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4C22×C411yescharacteristic
235C2×C4⋊C4C22×C411yescharacteristic
236C2×C4⋊C4C22×C411yescharacteristic
237C2×C4⋊C4C22×C411yescharacteristic
238C2×C4⋊C4C22×C411yescharacteristic
239C2×C4⋊C4C22×C411yescharacteristic
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C42—21no—
254C2×C42—24no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—42no—
264C2×C42—42no—
265C2×C42—42no—
266C2×C42—42no—
267C2×C42—44no—
268C2×C42—42no—
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×C42C22⋊C411yescharacteristic
279C2×C42C22⋊C411yescharacteristic
280C2×C42C22⋊C411yescharacteristic
281C2×C42C22⋊C411yescharacteristic
282C2×C42C22⋊C411yescharacteristic
283C2×C42C22⋊C411yescharacteristic
284C2×C42C22⋊C411yescharacteristic
285C2×C42C22×C411yescharacteristic
286C2×C42C22×C411yescharacteristic
287C2×C42C22×C411yescharacteristic
288C2×C42C22×C411yescharacteristic
289C2×C42C22×C411yescharacteristic
290C2×C42C22×C411yescharacteristic
291C2×C42C2×D411yescharacteristic
292C2×C42C2×D411yescharacteristic
293C2×C42C2×D411yescharacteristic
294C2×C42C2×D411yescharacteristic
295C2×C42C2×D412yes—
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D411yescharacteristic
298C2×C42C2×Q812yes—
299C2×C42#4112yes—
300C2×C42#4112yes—
301C4⋊C8—44no—
302C4⋊C8—44no—
303C4⋊C8—44no—
304C22⋊C8—24no—
305C22⋊C8—24no—
306C4×C8—44no—
307#95—21no—
308#95—21no—
309#95—21no—
310#95—21no—
311#95—21no—
312#95—21no—
313#95—21no—
314#95—21no—
315#95—21no—
316#95—21no—
317#95—21no—
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—24no—
328#95—24no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
334C24—21no—
335C24C22×D411yescharacteristic
336C2×Q8—21no—
337C2×Q8—41no—
338C2×Q8—41no—
339C2×Q8C2×C4⋊C412yes—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—21no—
343C2×D4—41no—
344C2×D4—41no—
345C2×D4—42no—
346C2×D4—42no—
347C2×D4C2×C4⋊C412yes—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—44no—
382C22×C4—44no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4C2×C22⋊C411yescharacteristic
389C22×C4C2×C22⋊C411yescharacteristic
390C22×C4C2×C22⋊C411yescharacteristic
391C22×C4C2×C22⋊C411yescharacteristic
392C22×C4C2×C22⋊C411yescharacteristic
393C22×C4C2×C22⋊C411yescharacteristic
394C22×C4C2×C22⋊C411yescharacteristic
395C22×C4C2×C22⋊C411yescharacteristic
396C22×C4C2×C22⋊C411yescharacteristic
397C22×C4C2×C22⋊C411yescharacteristic
398C22×C4C2×C22⋊C411yescharacteristic
399C22×C4C2×C22⋊C411yescharacteristic
400C22×C4#12112yes—
401C22×C4#12112yes—
402C22×C4C22⋊Q812yes—
403C22×C4C22⋊Q812yes—
404C22×C4#13514yes—
405C22×C4C23×C411yescharacteristic
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C22×Q811yescharacteristic
409C22×C4C2×#4111yescharacteristic
410C22×C4C2×#4111yescharacteristic
411C2×C8—44no—
412C2×C8—44no—
413C2×C8—44no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—88no—
432C4⋊C4C2×C4⋊C412yes—
433C4⋊C4C2×C4⋊C412yes—
434C22⋊C4—22no—
435C22⋊C4—22no—
436C22⋊C4—22no—
437C22⋊C4—22no—
438C22⋊C4—24no—
439C22⋊C4—24no—
440C22⋊C4—24no—
441C22⋊C4—42no—
442C22⋊C4—42no—
443C22⋊C4—42no—
444C22⋊C4—42no—
445C22⋊C4—42no—
446C22⋊C4—42no—
447C22⋊C4—42no—
448C22⋊C4—42no—
449C42—24no—
450C42—22no—
451C42—22no—
452C42—22no—
453C42—22no—
454C42—22no—
455C42—22no—
456C42—22no—
457C42—22no—
458C42—22no—
459C42—22no—
460C42—22no—
461C42—22no—
462C42—41no—
463C42—41no—
464C42—41no—
465C42—41no—
466C42—41no—
467C42—41no—
468C42—41no—
469C42—41no—
470C42—41no—
471C42—41no—
472C42—41no—
473C42—41no—
474C42—42no—
475C42—42no—
476C42—42no—
477C42—42no—
478C42—42no—
479C42—42no—
480C42—42no—
481C42—42no—
482C42C2×C4⋊C412yes—
483C42C2×C4⋊C412yes—
484C42#11712yes—
485C42#11712yes—
486C42#12112yes—
487C42#12112yes—
488C42C22⋊Q812yes—
489C42C22⋊Q812yes—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—22no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23C22×C22⋊C411yescharacteristic
502C23C2×#12111yescharacteristic
503C23C2×C22⋊Q811yescharacteristic
504C23C2×#13512yes—
505C23#57512yes—
506Q8—42no—
507Q8—42no—
508D4—42no—
509D4—42no—
510D4—44no—
511D4—44no—
512C2×C4—28no—
513C2×C4—24no—
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—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—24no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—48no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—84no—
577C2×C4—84no—
578C2×C4—84no—
579C2×C4—84no—
580C2×C4—84no—
581C2×C4—84no—
582C2×C4—84no—
583C2×C4—84no—
584C2×C4—84no—
585C2×C4#38012yes—
586C2×C4#38012yes—
587C2×C4#38012yes—
588C2×C4#38012yes—
589C2×C4#41012yes—
590C2×C4#41012yes—
591C2×C4#41612yes—
592C2×C4#41612yes—
593C2×C4#42112yes—
594C2×C4#42112yes—
595C2×C4C22×C4⋊C411yescharacteristic
596C2×C4C2×#11711yescharacteristic
597C2×C4C2×#12111yescharacteristic
598C2×C4C2×C22⋊Q811yescharacteristic
599C8—84no—
600C22—21no—
601C22—21no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—21no—
606C22—21no—
607C22—22no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—81no—
612C22—81no—
613C22—81no—
614C22—81no—
615C22C2×#38011yescharacteristic
616C22C2×#41011yescharacteristic
617C22C2×#41011yescharacteristic
618C22C2×#41611yescharacteristic
619C22#289611yescharacteristic
620C22C2×#42111yescharacteristic
621C22C2×#42111yescharacteristic
622C22#303512yes—
623C22#307612yes—
624C4—21no—
625C4—21no—
626C4—21no—
627C4—21no—
628C4—22no—
629C4—22no—
630C4—22no—
631C4—22no—
632C4—22no—
633C4—22no—
634C4—22no—
635C4—22no—
636C4—22no—
637C4—22no—
638C4—42no—
639C4—42no—
640C4—41no—
641C4—41no—
642C4—42no—
643C4—81no—
644C4—81no—
645C4—81no—
646C4—81no—
647C4—164no—
648C2—22no—
649C2—22no—
650C2—82no—
651C2#1756711yescharacteristic
652C2#1757611yescharacteristic
653C2#1757911yescharacteristic
654C2#1759311yescharacteristic
655C2#2063211yescharacteristic
656C2#2064511yescharacteristic
657C2#2065411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20808C2semidirect
4. #14688C2semidirect
5. #8578C2semidirect
7. #8574C2semidirect
9. #8025C2semidirect
12. C4×#125C4semidirect
13. C4×#125C4semidirect
16. C4×#124C4semidirect
17. C4×#124C4semidirect
25. #2295C4semidirect
26. #2293C4semidirect
30. C42×Q8C4semidirect
32. C42×D4C4semidirect
193. #125C4⋊C4semidirect
195. #124C4⋊C4semidirect
209. C4×Q8C4⋊C4semidirect
217. C4×D4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C4×#5292
#208082
C2×#15212
#146882
#85782
#85772
#85742
#85732
#80252