Finite Group Databasebeta

← #227895

Subgroups — #227895

Order 512.

888 conjugacy classes of subgroups.

The subgroup lattice has 669 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22986C211yesmaximal, characteristic, complemented
2#22530C211yesmaximal, characteristic, complemented
3C4×#449C211yesmaximal, characteristic
4C2×SD16⋊C8C211yesmaximal, characteristic, complemented
5C2×#1489C211yesmaximal, characteristic, complemented
6#9663C211yesmaximal, characteristic, complemented
7#9556C211yesmaximal, characteristic, complemented
8#9545C211yesmaximal, characteristic, complemented
9#9078C211yesmaximal, characteristic, complemented
10#8948C211yesmaximal, characteristic, complemented
11#8947C211yesmaximal, characteristic, complemented
12#8844C211yesmaximal, characteristic, complemented
13#8832C211yesmaximal, characteristic, complemented
14#8203C211yesmaximal, characteristic, complemented
15#8147C211yesmaximal, characteristic, complemented
16C2×C4×#41C2211yescharacteristic
17C2×#449C411yescharacteristic
18C2×C8×Q8C2211yescharacteristic
19C2×C8×Q8C2211yescharacteristic
20C2×C4×SD16C2211yescharacteristic
21C2×C4×D8C2211yescharacteristic
22C2×#435—21no—
23C2×C8×D4C2211yescharacteristic
24C2×#423C2211yescharacteristic
25C2×C4×M16C2211yescharacteristic
26C4×#121C2211yescharacteristic
27#1954—21no—
28#1954—21no—
29#1954C2211yescharacteristic
30#1954C2211yescharacteristic
31#1954C2211yescharacteristic
32#1939C2211yescharacteristic
33#1924—21no—
34#1880C411yescharacteristic
35#1879C411yescharacteristic
36#1875C411yescharacteristic
37#1870—21no—
38#1864C411yescharacteristic
39#1863C411yescharacteristic
40#1854C411yescharacteristic
41#1850—21no—
42#1797—21no—
43#1797C2211yescharacteristic
44#1788C411yescharacteristic
45#1770C2211yescharacteristic
46C4×#103C2211yescharacteristic
47C4×#102C2211yescharacteristic
48C4×C22⋊C8—21no—
49C4×C22⋊C8C2211yescharacteristic
50C4×C22⋊C8C2211yescharacteristic
51SD16⋊C8—24no—
52C2×#335C2211yescharacteristic
53C2×#335C2211yescharacteristic
54#1489—24no—
55C2×Q8⋊C8C2211yescharacteristic
56C2×Q8⋊C8C2211yescharacteristic
57C2×D4⋊C8C2211yescharacteristic
58C2×D4⋊C8C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62C2×#322C2211yescharacteristic
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—
74C23×C8—21no—
75C2×#121C2×C411yescharacteristic
76C4×#41—22no—
77C4×#41—22no—
78C4×#41—41no—
79C4×#41—41no—
80C2×C4×Q8C2×C411yescharacteristic
81C2×C4×Q8C2311yescharacteristic
82C2×C4×D4—21no—
83C2×C4×D4—21no—
84C2×C4×D4—21no—
85C2×C4×D4C2×C411yescharacteristic
86C2×C4×D4C2311yescharacteristic
87C2×C4×D4C2311yescharacteristic
88C2×#117—21no—
89C2×#117C2×C411yescharacteristic
90C22×C42—21no—
91C22×C42C2311yescharacteristic
92#449—24no—
93C8×Q8—22no—
94C8×Q8—22no—
95C8×Q8—41no—
96C8×Q8—41no—
97C4×SD16—22no—
98C4×SD16—22no—
99C4×D8—41no—
100C4×D8—41no—
101#435—44no—
102C8×D4—22no—
103C8×D4—41no—
104C2×#107C2×C411yescharacteristic
105C2×#106C2×C411yescharacteristic
106#423—22no—
107#423—41no—
108C2×C4⋊C8—21no—
109C2×C4⋊C8—21no—
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⋊C8—21no—
131C2×C22⋊C8—21no—
132C2×C22⋊C8C2×C411yescharacteristic
133C2×C22⋊C8C2×C411yescharacteristic
134C2×C22⋊C8D411yescharacteristic
135C2×C22⋊C8D411yescharacteristic
136C4×M16—22no—
137C4×M16—41no—
138C2×#97—21no—
139C2×#97—21no—
140C2×C4×C8—21no—
141C2×C4×C8—21no—
142C2×C4×C8C2311yescharacteristic
143C2×C4×C8C2311yescharacteristic
144C2×C4×C8C2311yescharacteristic
145C2×C4×C8C2311yescharacteristic
146C2×C4×C8C2311yescharacteristic
147C2×C4×C8C2311yescharacteristic
148#390C2×C411yescharacteristic
149#389C2×C411yescharacteristic
150#388—21no—
151#386—21no—
152#380C2×C411yescharacteristic
153C4×C4⋊C4C2311yescharacteristic
154C4×C22⋊C4—21no—
155#340C2×C411yescharacteristic
156#340C2×C411yescharacteristic
157#340C2×C411yescharacteristic
158#340C2×C411yescharacteristic
159#336—21no—
160#336—21no—
161#336—21no—
162#336C2×C411yescharacteristic
163#336C2×C411yescharacteristic
164#336D411yescharacteristic
165#336D411yescharacteristic
166#335—22no—
167#335—41no—
168Q8⋊C8—22no—
169Q8⋊C8D414yescomplemented
170D4⋊C8—22no—
171D4⋊C8—22no—
172D4⋊C8—22no—
173D4⋊C8—41no—
174#322—22no—
175#322—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×SD16M1614yescomplemented
197C2×D8—22no—
198C2×D8—22no—
199C2×D8—41no—
200C2×M16—42no—
201C2×M16—42no—
202C2×M16—42no—
203C2×M16—42no—
204C2×M16—42no—
205C2×M16—81no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—42no—
213C22×C8—42no—
214C22×C8—81no—
215C22×C8C22×C411yescharacteristic
216C22×C8C22×C411yescharacteristic
217C22×C8C22×C411yescharacteristic
218C22×C8C22×C411yescharacteristic
219C22×C8C2×D411yescharacteristic
220C22×C8C2×D411yescharacteristic
221C22×C8C2×D411yescharacteristic
222C22×C8C2×D411yescharacteristic
223#121—22no—
224#121—41no—
225C4×Q8—21no—
226C4×Q8—21no—
227C4×Q8—22no—
228C4×Q8—41no—
229C4×Q8C2×D412yes—
230C4×Q8#4112yes—
231C4×D4—22no—
232C4×D4—21no—
233C4×D4—21no—
234C4×D4—21no—
235C4×D4—21no—
236C4×D4—21no—
237C4×D4—21no—
238C4×D4—21no—
239C4×D4—21no—
240C4×D4—41no—
241C4×D4—42no—
242C4×D4—42no—
243C4×D4—42no—
244C4×D4—42no—
245C4×D4—42no—
246C4×D4—42no—
247#117—22no—
248#117—42no—
249#117—42no—
250#117—41no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
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×C42—21no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—42no—
276C2×C42—42no—
277C2×C42C22×C411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42#4111yescharacteristic
280C2×C42C2411yescharacteristic
281#107—41no—
282#106—22no—
283C4⋊C8—21no—
284C4⋊C8—21no—
285C4⋊C8—21no—
286C4⋊C8—21no—
287C4⋊C8—21no—
288C4⋊C8—21no—
289C4⋊C8—21no—
290C4⋊C8—42no—
291C4⋊C8—42no—
292C4⋊C8—42no—
293C4⋊C8—42no—
294C4⋊C8C2×D412yes—
295#103—22no—
296#103—22no—
297#103—22no—
298#102—22no—
299#102—22no—
300#102—22no—
301#102—41no—
302#102—41no—
303C22⋊C8—22no—
304C22⋊C8—22no—
305C22⋊C8—41no—
306C22⋊C8—41no—
307C22⋊C8—42no—
308C22⋊C8—42no—
309C22⋊C8—42no—
310C22⋊C8—42no—
311C22⋊C8—81no—
312C22⋊C8—81no—
313C22⋊C8—81no—
314C22⋊C8—81no—
315#97—42no—
316#97—42no—
317C4×C8—21no—
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—42no—
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×Q8C4×D412yes—
345C2×Q8C2×M1612yes—
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—21no—
358C2×D4—42no—
359C2×D4—42no—
360C2×D4—42no—
361C2×D4—42no—
362C2×D4—81no—
363C2×D4C2×M1612yes—
364C2×D4#13112yes—
365C22×C4—21no—
366C22×C4—24no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
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—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—41no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—41no—
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×C4—42no—
404C22×C4—81no—
405C22×C4C4×D411yescharacteristic
406C22×C4C4×D411yescharacteristic
407C22×C4C4×D411yescharacteristic
408C22×C4C23×C411yescharacteristic
409C22×C4C22×D411yescharacteristic
410C22×C4C2×#4111yescharacteristic
411SD16—28no—
412D8—44no—
413M16—48no—
414M16—84no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—22no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—22no—
426C2×C8—21no—
427C2×C8—22no—
428C2×C8—21no—
429C2×C8—21no—
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—41no—
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—41no—
455C2×C8—84no—
456C2×C8C2×M1612yes—
457C2×C8AGL(1,ℤ/8)12yes—
458C4⋊C4—21no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4C4×D412yes—
472C22⋊C4—42no—
473C22⋊C4—42no—
474C22⋊C4—42no—
475C22⋊C4—42no—
476C22⋊C4—42no—
477C22⋊C4—41no—
478C22⋊C4—41no—
479C22⋊C4—41no—
480C22⋊C4—41no—
481C22⋊C4—42no—
482C42—21no—
483C42—21no—
484C42—22no—
485C42—21no—
486C42—22no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—42no—
491C42—42no—
492C42—41no—
493C42—41no—
494C42C22×D411yescharacteristic
495C42C2×#4111yescharacteristic
496C42H(5,𝔽2)11yescharacteristic
497C42H(5,𝔽2)11yescharacteristic
498C23—21no—
499C23—21no—
500C23—22no—
501C23—22no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23C2×C4×D411yescharacteristic
520Q8—22no—
521Q8—22no—
522Q8—22no—
523Q8#43514yescomplemented
524D4—22no—
525D4—22no—
526D4—22no—
527D4—22no—
528D4—48no—
529D4—41no—
530D4—41no—
531D4—41no—
532D4—41no—
533D4—84no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—48no—
551C2×C4—41no—
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—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—84no—
574C2×C4—81no—
575C2×C4—81no—
576C2×C4—81no—
577C2×C4—81no—
578C2×C4—81no—
579C2×C4—81no—
580C2×C4—81no—
581C2×C4—81no—
582C2×C4#44412yes—
583C2×C4C2×C4×D411yescharacteristic
584C2×C4#51811yescharacteristic
585C2×C4D4211yescharacteristic
586C2×C4#54611yescharacteristic
587C2×C4#54611yescharacteristic
588C2×C4#55111yescharacteristic
589C2×C4C22×M1611yescharacteristic
590C2×C4C2×#13111yescharacteristic
591C2×C4C2×#13111yescharacteristic
592C2×C4#56811yescharacteristic
593C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
594C2×C4#57511yescharacteristic
595C2×C4#57611yescharacteristic
596C2×C4#57811yescharacteristic
597C8—22no—
598C8—22no—
599C8—42no—
600C8—42no—
601C8—41no—
602C8—41no—
603C8—41no—
604C8—41no—
605C8—42no—
606C8—41no—
607C8—41no—
608C8—81no—
609C8—81no—
610C8—81no—
611C22—41no—
612C22—42no—
613C22—42no—
614C22—41no—
615C22—42no—
616C22—41no—
617C22—41no—
618C22—41no—
619C22—42no—
620C22—41no—
621C22—42no—
622C22—41no—
623C22—42no—
624C22—81no—
625C22—81no—
626C22—81no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22—81no—
631C22—81no—
632C22#235811yescharacteristic
633C22#292911yescharacteristic
634C22D4×M1611yescharacteristic
635C22C4×AGL(1,ℤ/8)11yescharacteristic
636C22#295111yescharacteristic
637C22C2×#44411yescharacteristic
638C22#295611yescharacteristic
639C4—24no—
640C4—21no—
641C4—22no—
642C4—21no—
643C4—41no—
644C4—41no—
645C4—42no—
646C4—81no—
647C4—81no—
648C4—81no—
649C4—81no—
650C4C2×#43511yescharacteristic
651C4#293311yescharacteristic
652C4#297411yescharacteristic
653C4#297611yescharacteristic
654C4#327411yescharacteristic
655C4#327711yescharacteristic
656C4#330811yescharacteristic
657C4#330911yescharacteristic
658C2—81no—
659C2—82no—
660C2—82no—
661C2#1791711yescharacteristic
662C2#1792611yescharacteristic
663C2#1804811yescharacteristic
664C2#1806111yescharacteristic
665C2#2284611yescharacteristic
666C2#2286311yescharacteristic
667C2#2288611yescharacteristic
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. #22986C2semidirect
2. #22530C2semidirect
4. C2×SD16⋊C8C2semidirect
5. C2×#1489C2semidirect
6. #9663C2semidirect
7. #9556C2semidirect
8. #9545C2semidirect
9. #9078C2semidirect
10. #8948C2semidirect
11. #8947C2semidirect
12. #8844C2semidirect
13. #8832C2semidirect
14. #8203C2semidirect
15. #8147C2semidirect
169. Q8⋊C8D4semidirect
196. C2×SD16M16semidirect
523. Q8#435semidirect

Maximal subgroups

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

subgroupindex
#229862
#225302
C4×#4492
C2×SD16⋊C82
C2×#14892
#96632
#95562
#95452
#90782
#89482
#89472
#88442
#88322
#82032
#81472