Finite Group Databasebeta

← #228130

Subgroups — #228130

Order 512.

873 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23570C211yesmaximal, characteristic
2#22957C211yesmaximal, characteristic, complemented
3#22402C211yesmaximal, characteristic
4C2×#1590C211yesmaximal, characteristic, complemented
5C2×#1492C211yesmaximal, characteristic
6#9655C211yesmaximal, characteristic, complemented
7#9554C211yesmaximal, characteristic, complemented
8#9542C211yesmaximal, characteristic, complemented
9#9071C211yesmaximal, characteristic, complemented
10#8952C211yesmaximal, characteristic, complemented
11#8945C211yesmaximal, characteristic, complemented
12#8849C211yesmaximal, characteristic, complemented
13#8833C211yesmaximal, characteristic, complemented
14C8×#102C211yesmaximal, characteristic, complemented
15C8×C22⋊C8C211yesmaximal, characteristic, complemented
16C2×#532C2211yescharacteristic
17C2×#446C2211yescharacteristic
18C2×C4×SD16C2211yescharacteristic
19C2×C4×SD16C2211yescharacteristic
20C2×#436C2211yescharacteristic
21C2×C8×D4C2211yescharacteristic
22C2×#423C2211yescharacteristic
23C2×#408—21no—
24C22×C4×C8C2211yescharacteristic
25C4×C22⋊Q8C2211yescharacteristic
26#2058C411yescharacteristic
27#2041C411yescharacteristic
28#2034C411yescharacteristic
29#2034C411yescharacteristic
30#2024C411yescharacteristic
31#2014C411yescharacteristic
32#2000C411yescharacteristic
33#2000C411yescharacteristic
34#1984C2211yescharacteristic
35#1954C2211yescharacteristic
36#1954C2211yescharacteristic
37#1939—21no—
38#1939C2211yescharacteristic
39#1917—21no—
40#1870—21no—
41#1869—21no—
42#1851—21no—
43#1797C2211yescharacteristic
44#1780—21no—
45C8×C4⋊C4C2211yescharacteristic
46C4×#102C2211yescharacteristic
47C4×#102C2211yescharacteristic
48#1753—21no—
49C4×C22⋊C8C2211yescharacteristic
50C4×C22⋊C8C2211yescharacteristic
51#1590—24no—
52C2×#334C2211yescharacteristic
53C2×#334C2211yescharacteristic
54#1492—24no—
55C2×Q8⋊C8C2211yescharacteristic
56C2×Q8⋊C8C2211yescharacteristic
57C2×D4⋊C8C2211yescharacteristic
58C2×D4⋊C8C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×C82C2211yescharacteristic
62C2×C82C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67#1273C2211yescharacteristic
68#1273C2211yescharacteristic
69C22×#41—21no—
70C22×SD16C2×C411yescharacteristic
71C22×SD16C2×C411yescharacteristic
72C22×M16—21no—
73C22×M16—21no—
74C23×C8—21no—
75C23×C8—21no—
76#532—22no—
77#532—22no—
78#532—41no—
79#532—41no—
80C2×C4⋊Q8C2×C411yescharacteristic
81C2×#127C2×C411yescharacteristic
82C2×#124—21no—
83C2×C22⋊Q8C2×C411yescharacteristic
84C2×#121—21no—
85C2×#121C2×C411yescharacteristic
86C2×#121C2×C411yescharacteristic
87C2×C4×Q8C2311yescharacteristic
88C2×C4×D4—21no—
89C2×C4×D4C2311yescharacteristic
90C2×C4×D4C2311yescharacteristic
91C22×C42C2311yescharacteristic
92#446—22no—
93#446—41no—
94C4×SD16—22no—
95C4×SD16—22no—
96C4×SD16—41no—
97C4×SD16—41no—
98#436—22no—
99#436—41no—
100C8×D4—22no—
101C8×D4—41no—
102C2×#106C2×C411yescharacteristic
103C2×#106C2×C411yescharacteristic
104#423—22no—
105#423—41no—
106C2×C4⋊C8—21no—
107C2×C4⋊C8—21no—
108C2×C4⋊C8—21no—
109C2×C4⋊C8C2311yescharacteristic
110C2×C4⋊C8C2311yescharacteristic
111C2×C4⋊C8C2311yescharacteristic
112C2×C4⋊C8C2311yescharacteristic
113C2×#103C2×C411yescharacteristic
114C2×#103C2×C411yescharacteristic
115C2×#102C2×C411yescharacteristic
116C2×#102C2×C411yescharacteristic
117C2×#102C2×C411yescharacteristic
118C2×#102C2×C411yescharacteristic
119C2×#102C2×C411yescharacteristic
120C2×#102C2×C411yescharacteristic
121#408—44no—
122C2×C22⋊C8—21no—
123C2×C22⋊C8—21no—
124C2×C22⋊C8—21no—
125C2×C22⋊C8—21no—
126C2×C22⋊C8—21no—
127C2×C22⋊C8—21no—
128C2×C22⋊C8—21no—
129C2×C22⋊C8C2×C411yescharacteristic
130C2×C22⋊C8C2×C411yescharacteristic
131C2×C22⋊C8D411yescharacteristic
132C2×C22⋊C8D411yescharacteristic
133C2×#97—21no—
134C2×C4×C8—22no—
135C2×C4×C8—21no—
136C2×C4×C8—21no—
137C2×C4×C8—41no—
138C2×C4×C8C2311yescharacteristic
139C2×C4×C8C2311yescharacteristic
140C2×C4×C8C2311yescharacteristic
141C2×C4×C8C2311yescharacteristic
142C2×C4×C8C2311yescharacteristic
143C2×C4×C8C2311yescharacteristic
144#391C2×C411yescharacteristic
145#389C2×C411yescharacteristic
146#387—21no—
147#385—21no—
148#380C2×C411yescharacteristic
149C4×C4⋊C4—21no—
150C4×C4⋊C4C2311yescharacteristic
151C4×C22⋊C4—21no—
152#340C2×C411yescharacteristic
153#340C2×C411yescharacteristic
154#340C2×C411yescharacteristic
155#340C2×C411yescharacteristic
156#336—21no—
157#336—21no—
158#336—21no—
159#336—21no—
160#336—21no—
161#336C2×C411yescharacteristic
162#336C2×C411yescharacteristic
163#336D411yescharacteristic
164#336D411yescharacteristic
165#334—22no—
166#334—41no—
167Q8⋊C8—22no—
168Q8⋊C8—41no—
169D4⋊C8—22no—
170D4⋊C8—22no—
171D4⋊C8—22no—
172D4⋊C8D414yes—
173C82—22no—
174C82—41no—
175C2×#41—42no—
176C2×#41—42no—
177C2×#41—42no—
178C2×#41—42no—
179C2×#41—81no—
180C2×#41—81no—
181C22×Q8—21no—
182C22×Q8C22×C411yescharacteristic
183C22×D4—21no—
184C22×D4—21no—
185C22×D4—21no—
186C22×D4—21no—
187C22×D4C22×C411yescharacteristic
188C22×D4C22×C411yescharacteristic
189C23×C4—21no—
190C23×C4—21no—
191C23×C4C22×C411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4#4111yescharacteristic
194C2×SD16—22no—
195C2×SD16—22no—
196C2×SD16—22no—
197C2×SD16—41no—
198C2×SD16—41no—
199C2×SD16—41no—
200C2×M16—42no—
201C2×M16—42no—
202C2×M16—42no—
203C2×M16—42no—
204C2×M16—81no—
205C2×M16—81no—
206C22×C8—21no—
207C22×C8—24no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—42no—
214C22×C8—42no—
215C22×C8—42no—
216C22×C8—42no—
217C22×C8—81no—
218C22×C8C22×C411yescharacteristic
219C22×C8C22×C411yescharacteristic
220C22×C8C22×C411yescharacteristic
221C22×C8C22×C411yescharacteristic
222C22×C8C2×D411yescharacteristic
223C22×C8C2×D411yescharacteristic
224C22×C8C2×D411yescharacteristic
225C22×C8C2×D411yescharacteristic
226C4⋊Q8—22no—
227C4⋊Q8—41no—
228#127—22no—
229#127—41no—
230#124—42no—
231#124—42no—
232C22⋊Q8—22no—
233C22⋊Q8—41no—
234#121—22no—
235#121—22no—
236#121—42no—
237#121—42no—
238#121—41no—
239#121—41no—
240C4×Q8—21no—
241C4×Q8—21no—
242C4×Q8—21no—
243C4×Q8—21no—
244C4×D4—21no—
245C4×D4—21no—
246C4×D4—21no—
247C4×D4—21no—
248C4×D4—21no—
249C4×D4—21no—
250C4×D4—42no—
251C4×D4—42no—
252C4×D4C2×D412yes—
253C4×D4#4112yes—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4C22×C411yescharacteristic
259C2×C4⋊C4C22×C411yescharacteristic
260C2×C4⋊C4C22×C411yescharacteristic
261C2×C4⋊C4C22×C411yescharacteristic
262C2×C4⋊C4C22×C411yescharacteristic
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×C42—21no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42C22×C411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C2×C42#4111yescharacteristic
278C2×C42C2411yescharacteristic
279#106—22no—
280#106—41no—
281C4⋊C8—21no—
282C4⋊C8—21no—
283C4⋊C8—21no—
284C4⋊C8—21no—
285C4⋊C8—21no—
286C4⋊C8—21no—
287C4⋊C8—21no—
288C4⋊C8—42no—
289C4⋊C8—42no—
290C4⋊C8—81no—
291C4⋊C8C2×D412yes—
292#103—22no—
293#103—41no—
294#102—22no—
295#102—22no—
296#102—22no—
297#102—22no—
298#102—22no—
299#102—41no—
300C22⋊C8—22no—
301C22⋊C8—22no—
302C22⋊C8—22no—
303C22⋊C8—42no—
304C22⋊C8—42no—
305C22⋊C8—42no—
306C22⋊C8—42no—
307C22⋊C8—42no—
308C22⋊C8—81no—
309C22⋊C8—81no—
310C22⋊C8SD1614yescomplemented
311#97—42no—
312C4×C8—21no—
313C4×C8—21no—
314C4×C8—21no—
315C4×C8—21no—
316C4×C8—21no—
317C4×C8—21no—
318C4×C8—21no—
319C4×C8—21no—
320C4×C8—21no—
321C4×C8—21no—
322C4×C8—21no—
323C4×C8—42no—
324C4×C8—81no—
325C4×C8C2×D412yes—
326#95—21no—
327#95—21no—
328#95—21no—
329#95—21no—
330#95—21no—
331C24—21no—
332C24—21no—
333C24C4×D411yescharacteristic
334#41—84no—
335#41—84no—
336#41—84no—
337#41—84no—
338C2×Q8—21no—
339C2×Q8—21no—
340C2×Q8—21no—
341C2×Q8—21no—
342C2×Q8—21no—
343C2×Q8—21no—
344C2×Q8—42no—
345C2×Q8—42no—
346C2×Q8—81no—
347C2×D4—21no—
348C2×D4—21no—
349C2×D4—21no—
350C2×D4—21no—
351C2×D4—21no—
352C2×D4—21no—
353C2×D4—21no—
354C2×D4—21no—
355C2×D4—21no—
356C2×D4—21no—
357C2×D4—21no—
358C2×D4—42no—
359C2×D4—42no—
360C2×D4—42no—
361C2×D4—42no—
362C2×D4—42no—
363C2×D4—42no—
364C2×D4—42no—
365C2×D4—42no—
366C2×D4—81no—
367C2×D4—81no—
368C2×D4—81no—
369C2×D4—81no—
370C2×D4C4×D412yes—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4C4×D411yescharacteristic
404C22×C4C4×D411yescharacteristic
405C22×C4C4×D411yescharacteristic
406C22×C4C2×SD1612yes—
407C22×C4#13512yes—
408C22×C4C23×C411yescharacteristic
409C22×C4C22×D411yescharacteristic
410C22×C4C2×#4111yescharacteristic
411SD16—44no—
412SD16—44no—
413M16—84no—
414M16—84no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—22no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—22no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—22no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—48no—
441C2×C8—41no—
442C2×C8—41no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8—41no—
446C2×C8—41no—
447C2×C8—41no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—41no—
451C2×C8—41no—
452C2×C8—41no—
453C2×C8—41no—
454C2×C8—84no—
455C2×C8#13112yes—
456C2×C8#13112yes—
457C2×C8C2×SD1612yes—
458C2×C8C2×SD1612yes—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—41no—
470C4⋊C4—41no—
471C4⋊C4—42no—
472C4⋊C4C4×D412yes—
473C4⋊C4#13112yes—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—42no—
477C22⋊C4—42no—
478C22⋊C4—42no—
479C22⋊C4—41no—
480C22⋊C4—41no—
481C22⋊C4—81no—
482C42—21no—
483C42—21no—
484C42—21no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42—41no—
489C42—41no—
490C42—41no—
491C42—41no—
492C42C22×D411yescharacteristic
493C42C2×#4111yescharacteristic
494C42H(5,𝔽2)11yescharacteristic
495C42H(5,𝔽2)11yescharacteristic
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—22no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23C4×SD1612yes—
517C23C2×C4×D411yescharacteristic
518Q8—41no—
519Q8—41no—
520Q8—41no—
521Q8—41no—
522Q8—84no—
523D4—22no—
524D4—22no—
525D4—41no—
526D4—41no—
527D4—41no—
528D4—41no—
529D4—41no—
530D4—41no—
531D4—84no—
532D4—84no—
533D4—84no—
534D4—84no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
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—81no—
577C2×C4—81no—
578C2×C4—81no—
579C2×C4—81no—
580C2×C4C4×SD1612yes—
581C2×C4#44312yes—
582C2×C4C2×C4×D411yescharacteristic
583C2×C4#51811yescharacteristic
584C2×C4D4211yescharacteristic
585C2×C4#54611yescharacteristic
586C2×C4#54611yescharacteristic
587C2×C4#55111yescharacteristic
588C2×C4C2×#13111yescharacteristic
589C2×C4C2×#13111yescharacteristic
590C2×C4C2×#13111yescharacteristic
591C2×C4#56811yescharacteristic
592C2×C4C22×SD1611yescharacteristic
593C2×C4C2×#13511yescharacteristic
594C2×C4#57611yescharacteristic
595C2×C4#57711yescharacteristic
596C8—24no—
597C8—22no—
598C8—22no—
599C8—22no—
600C8—22no—
601C8—42no—
602C8—41no—
603C8—41no—
604C8—41no—
605C8—42no—
606C8—41no—
607C8—81no—
608C8—81no—
609C8—81no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—41no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—81no—
621C22—81no—
622C22—81no—
623C22—81no—
624C22—81no—
625C22—81no—
626C22—81no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22—81no—
631C22#235811yescharacteristic
632C22#293111yescharacteristic
633C22#293311yescharacteristic
634C22C2×C4×SD1611yescharacteristic
635C22#294611yescharacteristic
636C22C2×#44311yescharacteristic
637C22#295611yescharacteristic
638C4—22no—
639C4—21no—
640C4—21no—
641C4—22no—
642C4—41no—
643C4—42no—
644C4—42no—
645C4—81no—
646C4—81no—
647C4—81no—
648C4—81no—
649C4#293111yescharacteristic
650C4#293311yescharacteristic
651C4#297611yescharacteristic
652C4#297911yescharacteristic
653C4#328011yescharacteristic
654C4D4×SD1611yescharacteristic
655C4#329511yescharacteristic
656C4#329611yescharacteristic
657C2—42no—
658C2—82no—
659C2—161no—
660C2#1795711yescharacteristic
661C2#1796811yescharacteristic
662C2#1799711yescharacteristic
663C2#1801511yescharacteristic
664C2#2285311yescharacteristic
665C2#2287411yescharacteristic
666C2#2288811yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×C22⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22957C2semidirect
4. C2×#1590C2semidirect
6. #9655C2semidirect
7. #9554C2semidirect
8. #9542C2semidirect
9. #9071C2semidirect
10. #8952C2semidirect
11. #8945C2semidirect
12. #8849C2semidirect
13. #8833C2semidirect
14. C8×#102C2semidirect
15. C8×C22⋊C8C2semidirect
310. C22⋊C8SD16semidirect

Maximal subgroups

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

subgroupindex
#235702
#229572
#224022
C2×#15902
C2×#14922
#96552
#95542
#95422
#90712
#89522
#89452
#88492
#88332
C8×#1022
C8×C22⋊C82