Finite Group Databasebeta

← #106946

Subgroups — #106946

Order 512.

1186 conjugacy classes of subgroups.

The subgroup lattice has 693 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34183C211yesmaximal, characteristic
2C2×#1806C211yesmaximal, characteristic
3C2×#1477C211yesmaximal, characteristic
4#10849C211yesmaximal, characteristic, complemented
5#8045C211yesmaximal, characteristic, complemented
6#8043C211yesmaximal, characteristic, complemented
7#8014C211yesmaximal, characteristic, complemented
8C2×#520—21no—
9C2×C4×#41C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#423C2211yescharacteristic
12C22×C4⋊C8C2211yescharacteristic
13C22×#104—21no—
14Q8×C4⋊C4—21no—
15Q8×C4⋊C4C412yescomplemented
16D4×C4⋊C4—22no—
17D4×C4⋊C4—21no—
18D4×C4⋊C4—22no—
19D4×C4⋊C4C412yescomplemented
20#2317—22no—
21#2317—21no—
22#2285—21no—
23#2285C2211yescharacteristic
24C2×#384—21no—
25C2×#384C2211yescharacteristic
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—
42#520—42no—
43#520—42no—
44#520—42no—
45#520—42no—
46C4×#41—21no—
47C4×#41—21no—
48C4×#41—21no—
49C4×#41—21no—
50C4×#41C2×C412yes—
51C4×#41C2×C412yes—
52C4×#41D412yes—
53C4×#41Q812yes—
54C2×C4×Q8—22no—
55C2×C4×Q8C2×C411yescharacteristic
56C2×C4×Q8D411yescharacteristic
57C2×C4×D4—21no—
58C2×C4×D4—21no—
59C2×C4×D4—24no—
60C2×C4×D4—22no—
61C2×C4×D4C2×C411yescharacteristic
62C2×C4×D4D411yescharacteristic
63C2×#117—22no—
64C2×#117—21no—
65C2×#117—21no—
66C2×#117D411yescharacteristic
67C22×C4⋊C4—21no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4D412yes—
74C22×C42—21no—
75C22×C42—21no—
76C22×C42C2311yescharacteristic
77#431—21no—
78#431—21no—
79#431—21no—
80#431C2×C412yescomplemented
81#423—21no—
82#423—21no—
83#423—21no—
84#423C2×C412yescomplemented
85C2×C4⋊C8—21no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8C2×C412yescomplemented
94C2×#104—24no—
95C2×#104—42no—
96C2×#104—42no—
97C2×C22⋊C8—21no—
98C2×C4×C8—21no—
99#389—24no—
100#389—22no—
101#389—24no—
102#389—22no—
103#385—24no—
104#385—24no—
105#385—24no—
106#384—22no—
107#384—22no—
108#384—22no—
109#384—21no—
110#384—21no—
111#384C2×C412yes—
112#384D412yes—
113#384D411yescharacteristic
114#384D411yescharacteristic
115#382—22no—
116#382—21no—
117#380—22no—
118#380—21no—
119#380—22no—
120#380D412yes—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4C2×C412yes—
128C4×C4⋊C4C2×C412yescomplemented
129C2×#95—21no—
130#340—44no—
131#339—24no—
132Q8⋊C8—24no—
133D4⋊C8—24no—
134C2×#41—21no—
135C2×#41—21no—
136C2×#41—21no—
137C2×#41—21no—
138C2×#41—21no—
139C2×#41—21no—
140C2×#41C22⋊C412yes—
141C2×#41C4⋊C412yes—
142C2×#41D812yes—
143C2×#41SD1612yes—
144C2×#41SD1612yes—
145C2×#41Q1612yes—
146C22×Q8C22⋊C411yescharacteristic
147C22×D4—21no—
148C22×D4C22⋊C411yescharacteristic
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—24no—
153C23×C4—22no—
154C23×C4—41no—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C2×M16—22no—
159C2×M16—41no—
160C2×M16—41no—
161C2×M16—41no—
162C2×M16—41no—
163C2×M16—41no—
164C22×C8—22no—
165C22×C8—42no—
166C22×C8—42no—
167C22×C8—41no—
168C22×C8—41no—
169C22×C8—41no—
170C22×C8—41no—
171C4×Q8—21no—
172C4×Q8—21no—
173C4×Q8—21no—
174C4×Q8—21no—
175C4×Q8—44no—
176C4×Q8—44no—
177C4×Q8C4212yes—
178C4×Q8C22⋊C412yes—
179C4×Q8C4⋊C412yes—
180C4×Q8C4⋊C412yes—
181C4×D4—22no—
182C4×D4—22no—
183C4×D4—22no—
184C4×D4—22no—
185C4×D4—22no—
186C4×D4—22no—
187C4×D4—22no—
188C4×D4—22no—
189C4×D4—21no—
190C4×D4—21no—
191C4×D4—21no—
192C4×D4—21no—
193C4×D4—48no—
194C4×D4—48no—
195C4×D4—44no—
196C4×D4—44no—
197C4×D4C4212yes—
198C4×D4C22⋊C412yes—
199C4×D4C4⋊C412yes—
200C4×D4C4⋊C412yes—
201#117—21no—
202#117—21no—
203#117—22no—
204#117—22no—
205#117—22no—
206#117—22no—
207#117—44no—
208#117—44no—
209#117—41no—
210#117—41no—
211#117—41no—
212#117—41no—
213#117C22⋊C412yes—
214#117C4⋊C412yes—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4—41no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4—41no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—84no—
244C2×C4⋊C4—82no—
245C2×C4⋊C4—82no—
246C2×C4⋊C4—82no—
247C2×C4⋊C4C22⋊C411yescharacteristic
248C2×C4⋊C4C22⋊C411yescharacteristic
249C2×C4⋊C4C22⋊C412yes—
250C2×C4⋊C4C22⋊C414yes—
251C2×C4⋊C4C22⋊C412yes—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—22no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—24no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—41no—
268C2×C42—41no—
269C2×C42—41no—
270C2×C42—41no—
271C2×C42—41no—
272C2×C42C22×C411yescharacteristic
273C2×C42C22×C411yescharacteristic
274C2×C42C22×C411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42C2×Q811yescharacteristic
280C2×C42#4111yescharacteristic
281C2×C42#4111yescharacteristic
282C2×C42#4111yescharacteristic
283C2×C42#4111yescharacteristic
284C4⋊C8—22no—
285C4⋊C8—22no—
286C4⋊C8—22no—
287C4⋊C8—22no—
288C4⋊C8—22no—
289C4⋊C8—41no—
290C4⋊C8—41no—
291C4⋊C8—41no—
292C4⋊C8—41no—
293C4⋊C8—41no—
294#104—48no—
295C22⋊C8—22no—
296C22⋊C8—41no—
297C4×C8—22no—
298C4×C8—41no—
299#95—24no—
300#95—24no—
301#95—22no—
302#95—22no—
303#95—22no—
304#95—41no—
305C24—21no—
306C24#12011yescharacteristic
307#41—22no—
308#41—22no—
309#41—22no—
310#41—22no—
311#41—22no—
312#41—22no—
313#41—22no—
314#41—22no—
315#41—22no—
316#41—22no—
317#41—22no—
318#41—22no—
319#41#10214yes—
320#41#10314yes—
321#41#10614yes—
322#41#10714yes—
323C2×Q8—21no—
324C2×Q8—21no—
325C2×Q8—21no—
326C2×Q8#9512yes—
327C2×Q8#10212yes—
328C2×Q8#10312yes—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—21no—
336C2×D4—21no—
337C2×D4—21no—
338C2×D4#9512yes—
339C2×D4#10212yes—
340C2×D4#10312yes—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—48no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—82no—
376C22×C4—82no—
377C22×C4—84no—
378C22×C4—84no—
379C22×C4—81no—
380C22×C4—81no—
381C22×C4—81no—
382C22×C4—81no—
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×C4C2×SD1611yescharacteristic
404C22×C4C2×Q1611yescharacteristic
405C22×C4AGL(1,ℤ/8)11yescharacteristic
406C22×C4AGL(1,ℤ/8)11yescharacteristic
407C22×C4#13711yescharacteristic
408C22×C4#13711yescharacteristic
409M16—42no—
410M16—42no—
411M16—81no—
412M16—81no—
413C2×C8—44no—
414C2×C8—42no—
415C2×C8—42no—
416C2×C8—42no—
417C2×C8—81no—
418C2×C8—81no—
419C2×C8—82no—
420C2×C8—81no—
421C2×C8—81no—
422C2×C8—81no—
423C2×C8—81no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—24no—
431C4⋊C4—44no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—48no—
438C4⋊C4—44no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—41no—
442C4⋊C4—42no—
443C4⋊C4—41no—
444C4⋊C4—41no—
445C4⋊C4—42no—
446C4⋊C4—41no—
447C4⋊C4—42no—
448C4⋊C4—41no—
449C4⋊C4—41no—
450C4⋊C4—42no—
451C4⋊C4—41no—
452C4⋊C4—41no—
453C4⋊C4—82no—
454C4⋊C4—84no—
455C4⋊C4—84no—
456C4⋊C4#9512yes—
457C4⋊C4#9512yes—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—22no—
461C22⋊C4—22no—
462C22⋊C4—22no—
463C22⋊C4—22no—
464C22⋊C4—48no—
465C22⋊C4—42no—
466C22⋊C4—42no—
467C42—22no—
468C42—22no—
469C42—22no—
470C42—22no—
471C42—22no—
472C42—22no—
473C42—48no—
474C42—42no—
475C42—42no—
476C42—42no—
477C42—42no—
478C42—81no—
479C42—81no—
480C42C2×C4211yescharacteristic
481C42C2×C22⋊C411yescharacteristic
482C42C2×C4⋊C411yescharacteristic
483C42C2×C4⋊C411yescharacteristic
484C42#11711yescharacteristic
485C42#11711yescharacteristic
486C42C4×D411yescharacteristic
487C42C4×D411yescharacteristic
488C42C4×D411yescharacteristic
489C42C4×D411yescharacteristic
490C42#12111yescharacteristic
491C42#12111yescharacteristic
492C42C22⋊Q811yescharacteristic
493C42C22⋊Q811yescharacteristic
494C42#12311yescharacteristic
495C42#12311yescharacteristic
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23#37911yescharacteristic
508C23#38511yescharacteristic
509C23#38611yescharacteristic
510C23#44711yescharacteristic
511C23#44811yescharacteristic
512C23#45011yescharacteristic
513C23#45111yescharacteristic
514Q8—22no—
515Q8—22no—
516Q8—22no—
517Q8#34014yescomplemented
518D4—24no—
519D4—24no—
520D4—24no—
521D4—24no—
522D4—22no—
523D4—22no—
524D4—22no—
525D4#34014yescomplemented
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4—82no—
576C2×C4—82no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4—162no—
580C2×C4—162no—
581C2×C4—168no—
582C2×C4C2×#9511yescharacteristic
583C2×C4C4×C22⋊C411yescharacteristic
584C2×C4C4×C22⋊C411yescharacteristic
585C2×C4#38011yescharacteristic
586C2×C4#38011yescharacteristic
587C2×C4#38111yescharacteristic
588C2×C4#38511yescharacteristic
589C2×C4#38511yescharacteristic
590C2×C4#38511yescharacteristic
591C2×C4#38611yescharacteristic
592C2×C4#38611yescharacteristic
593C2×C4#38611yescharacteristic
594C2×C4C2×#10211yescharacteristic
595C2×C4C2×#10211yescharacteristic
596C2×C4C2×#10311yescharacteristic
597C2×C4C2×#10311yescharacteristic
598C2×C4#41711yescharacteristic
599C2×C4#41711yescharacteristic
600C2×C4#41811yescharacteristic
601C2×C4#41911yescharacteristic
602C2×C4C2×#10611yescharacteristic
603C2×C4C2×#10711yescharacteristic
604C2×C4#42811yescharacteristic
605C2×C4#42811yescharacteristic
606C2×C4#44711yescharacteristic
607C2×C4#44811yescharacteristic
608C2×C4#45011yescharacteristic
609C2×C4#45111yescharacteristic
610C2×C4#45311yescharacteristic
611C2×C4#45411yescharacteristic
612C2×C4#45511yescharacteristic
613C2×C4#45611yescharacteristic
614C2×C4#48011yescharacteristic
615C2×C4#48011yescharacteristic
616C2×C4#48111yescharacteristic
617C2×C4#48111yescharacteristic
618C2×C4#48311yescharacteristic
619C2×C4#48311yescharacteristic
620C2×C4#48411yescharacteristic
621C2×C4#48411yescharacteristic
622C8—82no—
623C8—82no—
624C8—82no—
625C22—21no—
626C22—21no—
627C22—21no—
628C22—21no—
629C22—21no—
630C22—21no—
631C22—21no—
632C22—21no—
633C22—21no—
634C22—21no—
635C22—21no—
636C22—21no—
637C22—24no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22C22⋊#9511yescharacteristic
645C22#178611yescharacteristic
646C22#178711yescharacteristic
647C22#179211yescharacteristic
648C22#179311yescharacteristic
649C22#179511yescharacteristic
650C22#179611yescharacteristic
651C22#185211yescharacteristic
652C22#185311yescharacteristic
653C22#187211yescharacteristic
654C22#187411yescharacteristic
655C4—21no—
656C4—21no—
657C4—21no—
658C4—21no—
659C4—21no—
660C4—21no—
661C4—24no—
662C4—42no—
663C4—42no—
664C4—84no—
665C4—84no—
666C4—82no—
667C4—82no—
668C4C2×#34011yescharacteristic
669C4#173611yescharacteristic
670C4#182311yescharacteristic
671C4#182411yescharacteristic
672C4#185211yescharacteristic
673C4#185311yescharacteristic
674C4#185911yescharacteristic
675C4#186011yescharacteristic
676C4#187211yescharacteristic
677C4#187411yescharacteristic
678C4#188211yescharacteristic
679C4#188311yescharacteristic
680C2—21no—
681C2—21no—
682C2—21no—
683C2—21no—
684C2—44no—
685C2#842011yescharacteristic
686C2#842111yescharacteristic
687C2#842411yescharacteristic
688C2#842511yescharacteristic
689C2#1073011yescharacteristic
690C2#1077511yescharacteristic
691C2#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
15. Q8×C4⋊C4C4semidirect
19. D4×C4⋊C4C4semidirect
28. #1806C4semidirect
32. #1477C4semidirect
80. #431C2×C4semidirect
84. #423C2×C4semidirect
93. C2×C4⋊C8C2×C4semidirect
128. C4×C4⋊C4C2×C4semidirect
517. Q8#340semidirect
525. D4#340semidirect

Maximal subgroups

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

subgroupindex
#341832
C2×#18062
C2×#14772
#108492
#80452
#80432
#80142