Finite Group Databasebeta

← #227962

Subgroups — #227962

Order 512.

903 conjugacy classes of subgroups.

The subgroup lattice has 671 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22990C211yesmaximal, characteristic, complemented
2#22469C211yesmaximal, characteristic, complemented
3C4×#449C211yesmaximal, characteristic
4C2×#1586C211yesmaximal, characteristic, complemented
5C2×#1489C211yesmaximal, characteristic, complemented
6#9655C211yesmaximal, characteristic, complemented
7#9559C211yesmaximal, characteristic, complemented
8#9543C211yesmaximal, characteristic, complemented
9#9078C211yesmaximal, characteristic, complemented
10#8949C211yesmaximal, characteristic, complemented
11#8947C211yesmaximal, characteristic, complemented
12#8845C211yesmaximal, characteristic, complemented
13#8833C211yesmaximal, characteristic, complemented
14C8×#102C211yesmaximal, characteristic, complemented
15#8146C211yesmaximal, characteristic, complemented
16C2×C4×#41C2211yescharacteristic
17C2×#449C411yescharacteristic
18C2×C4×SD16C2211yescharacteristic
19C2×C4×D8C2211yescharacteristic
20C2×#436—21no—
21C2×#436C2211yescharacteristic
22C2×C8×D4C2211yescharacteristic
23C2×C8×D4C2211yescharacteristic
24C2×#423C2211yescharacteristic
25C2×C4×M16C2211yescharacteristic
26C4×#121C2211yescharacteristic
27#1984C2211yescharacteristic
28#1984C2211yescharacteristic
29#1954C2211yescharacteristic
30#1939C2211yescharacteristic
31#1923—21no—
32#1880C411yescharacteristic
33#1879C411yescharacteristic
34#1875C411yescharacteristic
35#1870—21no—
36#1869—21no—
37#1869—21no—
38#1864C411yescharacteristic
39#1863C411yescharacteristic
40#1854C411yescharacteristic
41#1851—21no—
42#1817—21no—
43#1797C2211yescharacteristic
44#1788C411yescharacteristic
45C8×C4⋊C4C2211yescharacteristic
46C4×#103C2211yescharacteristic
47C4×#102C2211yescharacteristic
48C4×C22⋊C8C2211yescharacteristic
49C4×C22⋊C8C2211yescharacteristic
50C2×#336—21no—
51#1586—24no—
52C2×#334C2211yescharacteristic
53C2×#334C2211yescharacteristic
54#1489—24no—
55C2×Q8⋊C8C2211yescharacteristic
56C2×Q8⋊C8C2211yescharacteristic
57C2×D4⋊C8C2211yescharacteristic
58C2×D4⋊C8C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62C2×C82C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67#1273C2211yescharacteristic
68#1273C2211yescharacteristic
69C22×#41C2×C411yescharacteristic
70C22×SD16C2×C411yescharacteristic
71C22×D8C2×C411yescharacteristic
72C22×M16—21no—
73C22×M16—21no—
74C22×M16—21no—
75C23×C8—21no—
76C23×C8—21no—
77C2×#121C2×C411yescharacteristic
78C4×#41—22no—
79C4×#41—22no—
80C4×#41—41no—
81C4×#41—41no—
82C2×C4×Q8C2×C411yescharacteristic
83C2×C4×Q8C2311yescharacteristic
84C2×C4×D4—21no—
85C2×C4×D4—21no—
86C2×C4×D4—21no—
87C2×C4×D4C2×C411yescharacteristic
88C2×C4×D4C2311yescharacteristic
89C2×C4×D4C2311yescharacteristic
90C2×#117—21no—
91C2×#117C2×C411yescharacteristic
92C22×C42—21no—
93C22×C42C2311yescharacteristic
94#449—24no—
95C4×SD16—41no—
96C4×SD16—41no—
97C4×D8—22no—
98C4×D8—22no—
99#436—22no—
100#436—41no—
101#436—44no—
102C8×D4—22no—
103C8×D4—22no—
104C8×D4—41no—
105C8×D4—41no—
106C2×#107C2×C411yescharacteristic
107C2×#106C2×C411yescharacteristic
108#423—22no—
109#423—41no—
110C2×C4⋊C8—21no—
111C2×C4⋊C8—21no—
112C2×C4⋊C8C2311yescharacteristic
113C2×C4⋊C8C2311yescharacteristic
114C2×C4⋊C8C2311yescharacteristic
115C2×C4⋊C8C2311yescharacteristic
116C2×#103C2×C411yescharacteristic
117C2×#103C2×C411yescharacteristic
118C2×#103C2×C411yescharacteristic
119C2×#102C2×C411yescharacteristic
120C2×#102C2×C411yescharacteristic
121C2×#102C2×C411yescharacteristic
122C2×#102C2×C411yescharacteristic
123C2×#102C2×C411yescharacteristic
124C2×C22⋊C8—21no—
125C2×C22⋊C8—21no—
126C2×C22⋊C8—21no—
127C2×C22⋊C8—21no—
128C2×C22⋊C8—21no—
129C2×C22⋊C8—21no—
130C2×C22⋊C8C2×C411yescharacteristic
131C2×C22⋊C8C2×C411yescharacteristic
132C2×C22⋊C8D411yescharacteristic
133C2×C22⋊C8D411yescharacteristic
134C4×M16—22no—
135C4×M16—41no—
136C2×#97—21no—
137C2×C4×C8—21no—
138C2×C4×C8C2311yescharacteristic
139C2×C4×C8C2311yescharacteristic
140C2×C4×C8C2311yescharacteristic
141C2×C4×C8C2311yescharacteristic
142C2×C4×C8C2311yescharacteristic
143C2×C4×C8C2311yescharacteristic
144#390C2×C411yescharacteristic
145#389C2×C411yescharacteristic
146#388—21no—
147#386—21no—
148#380C2×C411yescharacteristic
149C4×C4⋊C4C2311yescharacteristic
150C4×C22⋊C4—21no—
151#340C2×C411yescharacteristic
152#340C2×C411yescharacteristic
153#340C2×C411yescharacteristic
154#340C2×C411yescharacteristic
155#336—21no—
156#336—21no—
157#336—21no—
158#336—21no—
159#336—21no—
160#336—21no—
161#336—21no—
162#336C2×C411yescharacteristic
163#336C2×C411yescharacteristic
164#336D411yescharacteristic
165#336D411yescharacteristic
166#334—22no—
167#334—41no—
168Q8⋊C8—22no—
169Q8⋊C8—22no—
170D4⋊C8—22no—
171D4⋊C8—22no—
172D4⋊C8—41no—
173D4⋊C8D414yescomplemented
174#322—22no—
175C82—41no—
176C2×#41—22no—
177C2×#41—22no—
178C2×#41—22no—
179C2×#41—22no—
180C2×#41—41no—
181C2×#41—41no—
182C22×Q8C22×C411yescharacteristic
183C22×D4—21no—
184C22×D4—21no—
185C22×D4C22×C411yescharacteristic
186C22×D4C22×C411yescharacteristic
187C23×C4—21no—
188C23×C4—21no—
189C23×C4—21no—
190C23×C4—21no—
191C23×C4C22×C411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4#4111yescharacteristic
194C2×SD16—22no—
195C2×SD16—22no—
196C2×SD16—41no—
197C2×D8—22no—
198C2×D8—22no—
199C2×D8—22no—
200C2×M16—24no—
201C2×M16—42no—
202C2×M16—42no—
203C2×M16—42no—
204C2×M16—42no—
205C2×M16—42no—
206C2×M16—42no—
207C2×M16—81no—
208C2×M16—81no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—42no—
216C22×C8—42no—
217C22×C8—42no—
218C22×C8—42no—
219C22×C8—81no—
220C22×C8—81no—
221C22×C8C22×C411yescharacteristic
222C22×C8C22×C411yescharacteristic
223C22×C8C22×C411yescharacteristic
224C22×C8C22×C411yescharacteristic
225C22×C8C2×D411yescharacteristic
226C22×C8C2×D411yescharacteristic
227C22×C8C2×D411yescharacteristic
228C22×C8C2×D411yescharacteristic
229#121—22no—
230#121—41no—
231C4×Q8—22no—
232C4×Q8—21no—
233C4×Q8—21no—
234C4×Q8—21no—
235C4×Q8—21no—
236C4×Q8—41no—
237C4×D4—22no—
238C4×D4—21no—
239C4×D4—21no—
240C4×D4—21no—
241C4×D4—21no—
242C4×D4—21no—
243C4×D4—21no—
244C4×D4—41no—
245C4×D4—42no—
246C4×D4—42no—
247C4×D4—42no—
248C4×D4—42no—
249C4×D4—42no—
250C4×D4—42no—
251C4×D4C2×D412yes—
252C4×D4#4112yes—
253#117—22no—
254#117—42no—
255#117—42no—
256#117—41no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4C22×C411yescharacteristic
261C2×C4⋊C4C22×C411yescharacteristic
262C2×C4⋊C4C22×C411yescharacteristic
263C2×C4⋊C4C22×C411yescharacteristic
264C2×C4⋊C4C22×C411yescharacteristic
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×C42—21no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—42no—
282C2×C42—42no—
283C2×C42C22×C411yescharacteristic
284C2×C42C2×D411yescharacteristic
285C2×C42#4111yescharacteristic
286C2×C42C2411yescharacteristic
287#107—22no—
288#106—41no—
289C4⋊C8—21no—
290C4⋊C8—21no—
291C4⋊C8—21no—
292C4⋊C8—21no—
293C4⋊C8—21no—
294C4⋊C8—21no—
295C4⋊C8—21no—
296C4⋊C8—42no—
297C4⋊C8—42no—
298C4⋊C8C2×D412yes—
299#103—22no—
300#103—22no—
301#103—41no—
302#102—22no—
303#102—22no—
304#102—22no—
305#102—41no—
306#102M1614yescomplemented
307C22⋊C8—22no—
308C22⋊C8—22no—
309C22⋊C8—41no—
310C22⋊C8—41no—
311C22⋊C8—42no—
312C22⋊C8—42no—
313C22⋊C8—42no—
314C22⋊C8—42no—
315C22⋊C8—81no—
316C22⋊C8—81no—
317#97—42no—
318C4×C8—21no—
319C4×C8—21no—
320C4×C8—21no—
321C4×C8—21no—
322C4×C8—21no—
323C4×C8—21no—
324C4×C8—21no—
325C4×C8—21no—
326C4×C8—21no—
327C4×C8—21no—
328C4×C8—21no—
329C4×C8—42no—
330C4×C8C2×D412yes—
331#95—21no—
332#95—21no—
333#95—21no—
334C24—21no—
335C24—21no—
336C24C4×D411yescharacteristic
337#41—44no—
338#41—44no—
339#41—44no—
340#41—44no—
341C2×Q8—21no—
342C2×Q8—21no—
343C2×Q8—21no—
344C2×Q8—21no—
345C2×Q8—21no—
346C2×Q8#13112yes—
347C2×D4—24no—
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—42no—
358C2×D4—42no—
359C2×D4—42no—
360C2×D4—42no—
361C2×D4—81no—
362C2×D4C4×D412yes—
363C2×D4C2×M1612yes—
364C2×D4#13112yes—
365C22×C4—24no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—41no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
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×C4—42no—
404C22×C4—81no—
405C22×C4C4×D411yescharacteristic
406C22×C4C4×D411yescharacteristic
407C22×C4C4×D411yescharacteristic
408C22×C4C23×C411yescharacteristic
409C22×C4C22×D411yescharacteristic
410C22×C4C2×#4111yescharacteristic
411SD16—44no—
412D8—44no—
413M16—48no—
414M16—84no—
415M16—84no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—22no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—22no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—22no—
430C2×C8—21no—
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—21no—
441C2×C8—21no—
442C2×C8—21no—
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—41no—
455C2×C8—41no—
456C2×C8—84no—
457C2×C8—84no—
458C2×C8C2×M1612yes—
459C2×C8AGL(1,ℤ/8)12yes—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—41no—
471C4⋊C4—41no—
472C4⋊C4C4×D412yes—
473C4⋊C4C2×M1612yes—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—42no—
477C22⋊C4—42no—
478C22⋊C4—42no—
479C22⋊C4—42no—
480C22⋊C4—42no—
481C22⋊C4—41no—
482C22⋊C4—41no—
483C22⋊C4—42no—
484C42—21no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—42no—
491C42—42no—
492C42—41no—
493C42—41no—
494C42—41no—
495C42—41no—
496C42C22×D411yescharacteristic
497C42C2×#4111yescharacteristic
498C42H(5,𝔽2)11yescharacteristic
499C42H(5,𝔽2)11yescharacteristic
500C23—22no—
501C23—22no—
502C23—22no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23C2×C4×D411yescharacteristic
522Q8—22no—
523Q8—22no—
524Q8—41no—
525Q8—41no—
526D4—22no—
527D4—22no—
528D4—22no—
529D4—48no—
530D4—41no—
531D4—41no—
532D4—41no—
533D4—41no—
534D4—84no—
535D4#43614yescomplemented
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—48no—
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—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—84no—
580C2×C4—81no—
581C2×C4—81no—
582C2×C4—81no—
583C2×C4—81no—
584C2×C4#44312yes—
585C2×C4C2×C4×D411yescharacteristic
586C2×C4#51811yescharacteristic
587C2×C4D4211yescharacteristic
588C2×C4#54611yescharacteristic
589C2×C4#54611yescharacteristic
590C2×C4#55111yescharacteristic
591C2×C4C22×M1611yescharacteristic
592C2×C4C2×#13111yescharacteristic
593C2×C4C2×#13111yescharacteristic
594C2×C4#56811yescharacteristic
595C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
596C2×C4#57511yescharacteristic
597C2×C4#57711yescharacteristic
598C2×C4#57711yescharacteristic
599C8—24no—
600C8—22no—
601C8—22no—
602C8—42no—
603C8—41no—
604C8—41no—
605C8—41no—
606C8—41no—
607C8—42no—
608C8—41no—
609C8—41no—
610C8—81no—
611C8—81no—
612C8—81no—
613C22—24no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22—41no—
625C22—41no—
626C22—41no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22—81no—
631C22—81no—
632C22—81no—
633C22—81no—
634C22#235811yescharacteristic
635C22#292911yescharacteristic
636C22D4×M1611yescharacteristic
637C22C4×AGL(1,ℤ/8)11yescharacteristic
638C22#295211yescharacteristic
639C22C2×#44311yescharacteristic
640C22#295611yescharacteristic
641C4—21no—
642C4—22no—
643C4—21no—
644C4—41no—
645C4—42no—
646C4—41no—
647C4—42no—
648C4—42no—
649C4—81no—
650C4—81no—
651C4—81no—
652C4C2×#43611yescharacteristic
653C4#293311yescharacteristic
654C4#297411yescharacteristic
655C4#297611yescharacteristic
656C4#327211yescharacteristic
657C4#327511yescharacteristic
658C4#330611yescharacteristic
659C4#330711yescharacteristic
660C2—44no—
661C2—81no—
662C2—161no—
663C2#1791211yescharacteristic
664C2#1792711yescharacteristic
665C2#1804611yescharacteristic
666C2#1805711yescharacteristic
667C2#2284811yescharacteristic
668C2#2286611yescharacteristic
669C2#2288811yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC2×#449C4
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
1. #22990C2semidirect
2. #22469C2semidirect
4. C2×#1586C2semidirect
5. C2×#1489C2semidirect
6. #9655C2semidirect
7. #9559C2semidirect
8. #9543C2semidirect
9. #9078C2semidirect
10. #8949C2semidirect
11. #8947C2semidirect
12. #8845C2semidirect
13. #8833C2semidirect
14. C8×#102C2semidirect
15. #8146C2semidirect
173. D4⋊C8D4semidirect
306. #102M16semidirect
535. D4#436semidirect

Maximal subgroups

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

subgroupindex
#229902
#224692
C4×#4492
C2×#15862
C2×#14892
#96552
#95592
#95432
#90782
#89492
#89472
#88452
#88332
C8×#1022
#81462