Finite Group Databasebeta

← #227669

Subgroups — #227669

Order 512.

920 conjugacy classes of subgroups.

The subgroup lattice has 665 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23083C211yesmaximal, characteristic, complemented
2#22530C211yesmaximal, characteristic, complemented
3C4×#449C211yesmaximal, characteristic
4C2×C8×SD16C211yesmaximal, characteristic, complemented
5C2×#1476C211yesmaximal, characteristic, complemented
6#9657C211yesmaximal, characteristic, complemented
7#9554C211yesmaximal, characteristic, complemented
8#9542C211yesmaximal, characteristic, complemented
9#9072C211yesmaximal, characteristic, complemented
10#8948C211yesmaximal, characteristic, complemented
11#8947C211yesmaximal, characteristic, complemented
12#8844C211yesmaximal, characteristic, complemented
13#8831C211yesmaximal, characteristic, complemented
14C8×#103C211yesmaximal, characteristic, complemented
15C8×C22⋊C8C211yesmaximal, characteristic, complemented
16C2×C4×#41C2211yescharacteristic
17C2×#449C411yescharacteristic
18C2×C8×Q8C2211yescharacteristic
19C2×C8×Q8C2211yescharacteristic
20C2×C4×SD16C2211yescharacteristic
21C2×C4×D8C2211yescharacteristic
22C2×C8×D4C2211yescharacteristic
23C2×#431C2211yescharacteristic
24C22×C4×C8C2211yescharacteristic
25C4×#121C2211yescharacteristic
26#1954—21no—
27#1954—21no—
28#1954—21no—
29#1954C2211yescharacteristic
30#1954C2211yescharacteristic
31#1954C2211yescharacteristic
32#1939C2211yescharacteristic
33#1923—21no—
34#1880C411yescharacteristic
35#1879C411yescharacteristic
36#1875C411yescharacteristic
37#1869—21no—
38#1864C411yescharacteristic
39#1863C411yescharacteristic
40#1854C411yescharacteristic
41#1797—21no—
42#1797C2211yescharacteristic
43#1788C411yescharacteristic
44C8×C4⋊C4C2211yescharacteristic
45C4×#103C2211yescharacteristic
46C4×#102C2211yescharacteristic
47C8×C22⋊C4—21no—
48C4×C22⋊C8—21no—
49C4×C22⋊C8C2211yescharacteristic
50C4×C22⋊C8C2211yescharacteristic
51C8×SD16—24no—
52C2×#335C2211yescharacteristic
53C2×#334C2211yescharacteristic
54#1476—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×#41C2×C411yescharacteristic
70C22×SD16C2×C411yescharacteristic
71C22×D8C2×C411yescharacteristic
72C23×C8—21no—
73C23×C8—21no—
74C2×#121C2×C411yescharacteristic
75C4×#41—22no—
76C4×#41—22no—
77C4×#41—41no—
78C4×#41—41no—
79C2×C4×Q8C2×C411yescharacteristic
80C2×C4×Q8C2311yescharacteristic
81C2×C4×D4—21no—
82C2×C4×D4—21no—
83C2×C4×D4—21no—
84C2×C4×D4C2×C411yescharacteristic
85C2×C4×D4C2311yescharacteristic
86C2×C4×D4C2311yescharacteristic
87C2×#117—21no—
88C2×#117C2×C411yescharacteristic
89C22×C42—21no—
90C22×C42C2311yescharacteristic
91#449C818yescomplemented
92C8×Q8—22no—
93C8×Q8—22no—
94C8×Q8—41no—
95C8×Q8—41no—
96C4×SD16—22no—
97C4×SD16—22no—
98C4×D8—41no—
99C4×D8—41no—
100C8×D4—22no—
101C8×D4—41no—
102#431—22no—
103#431—41no—
104C2×#107C2×C411yescharacteristic
105C2×#106C2×C411yescharacteristic
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×#103C2×C411yescharacteristic
116C2×#102C2×C411yescharacteristic
117C2×#102C2×C411yescharacteristic
118C2×#102C2×C411yescharacteristic
119C2×#102C2×C411yescharacteristic
120C2×#102C2×C411yescharacteristic
121C2×C22⋊C8—21no—
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⋊C8—21no—
130C2×C22⋊C8C2×C411yescharacteristic
131C2×C22⋊C8C2×C411yescharacteristic
132C2×C22⋊C8D411yescharacteristic
133C2×C22⋊C8D411yescharacteristic
134C2×C4×C8—22no—
135C2×C4×C8—21no—
136C2×C4×C8—21no—
137C2×C4×C8—21no—
138C2×C4×C8—21no—
139C2×C4×C8—21no—
140C2×C4×C8—41no—
141C2×C4×C8C2311yescharacteristic
142C2×C4×C8C2311yescharacteristic
143C2×C4×C8C2311yescharacteristic
144C2×C4×C8C2311yescharacteristic
145C2×C4×C8C2311yescharacteristic
146C2×C4×C8C2311yescharacteristic
147#390C2×C411yescharacteristic
148#389C2×C411yescharacteristic
149#388—21no—
150#386—21no—
151#380C2×C411yescharacteristic
152C4×C4⋊C4C2311yescharacteristic
153C4×C22⋊C4—21no—
154#340C2×C411yescharacteristic
155#340C2×C411yescharacteristic
156#340C2×C411yescharacteristic
157#340C2×C411yescharacteristic
158#336—21no—
159#336—21no—
160#336—21no—
161#336C2×C411yescharacteristic
162#336C2×C411yescharacteristic
163#336D411yescharacteristic
164#336D411yescharacteristic
165#335—41no—
166#334—22no—
167Q8⋊C8—22no—
168Q8⋊C8D414yescomplemented
169D4⋊C8—22no—
170D4⋊C8—22no—
171D4⋊C8—22no—
172D4⋊C8—41no—
173C82—22no—
174C82—41no—
175C2×#41—22no—
176C2×#41—22no—
177C2×#41—22no—
178C2×#41—41no—
179C2×#41—41no—
180C2×#41C2×C814yescomplemented
181C22×Q8C22×C411yescharacteristic
182C22×D4—21no—
183C22×D4—21no—
184C22×D4C22×C411yescharacteristic
185C22×D4C22×C411yescharacteristic
186C23×C4—21no—
187C23×C4—21no—
188C23×C4—21no—
189C23×C4—21no—
190C23×C4C22×C411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4#4111yescharacteristic
193C2×SD16—22no—
194C2×SD16—22no—
195C2×SD16C2×C814yescomplemented
196C2×D8—41no—
197C2×D8—41no—
198C2×D8C2×C814yescomplemented
199C22×C8—21no—
200C22×C8—24no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—42no—
207C22×C8—42no—
208C22×C8—42no—
209C22×C8—42no—
210C22×C8—81no—
211C22×C8C22×C411yescharacteristic
212C22×C8C22×C411yescharacteristic
213C22×C8C22×C411yescharacteristic
214C22×C8C22×C411yescharacteristic
215C22×C8C2×D411yescharacteristic
216C22×C8C2×D411yescharacteristic
217C22×C8C2×D411yescharacteristic
218C22×C8C2×D411yescharacteristic
219#121—22no—
220#121C2×C814yes—
221C4×Q8—21no—
222C4×Q8—21no—
223C4×Q8—22no—
224C4×Q8—41no—
225C4×Q8C2×D412yes—
226C4×Q8#4112yes—
227C4×D4—22no—
228C4×D4—21no—
229C4×D4—21no—
230C4×D4—21no—
231C4×D4—21no—
232C4×D4—21no—
233C4×D4—21no—
234C4×D4—21no—
235C4×D4—21no—
236C4×D4—41no—
237C4×D4—42no—
238C4×D4—42no—
239C4×D4—42no—
240C4×D4—42no—
241C4×D4—81no—
242C4×D4—81no—
243#117—22no—
244#117—42no—
245#117—42no—
246#117—41no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4C22×C411yescharacteristic
251C2×C4⋊C4C22×C411yescharacteristic
252C2×C4⋊C4C22×C411yescharacteristic
253C2×C4⋊C4C22×C411yescharacteristic
254C2×C4⋊C4C22×C411yescharacteristic
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42—42no—
272C2×C42—42no—
273C2×C42C22×C411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C2×C42#4111yescharacteristic
276C2×C42C2411yescharacteristic
277#107—41no—
278#106—22no—
279C4⋊C8—21no—
280C4⋊C8—21no—
281C4⋊C8—21no—
282C4⋊C8—21no—
283C4⋊C8—21no—
284C4⋊C8—21no—
285C4⋊C8—21no—
286C4⋊C8—42no—
287C4⋊C8—42no—
288C4⋊C8—81no—
289C4⋊C8C2×D412yes—
290#103—22no—
291#103—22no—
292#103C2×C814yescomplemented
293#102—22no—
294#102—41no—
295#102—41no—
296#102—41no—
297#102C2×C814yescomplemented
298C22⋊C8—22no—
299C22⋊C8—22no—
300C22⋊C8—22no—
301C22⋊C8—42no—
302C22⋊C8—42no—
303C22⋊C8—42no—
304C22⋊C8—42no—
305C22⋊C8—42no—
306C22⋊C8—81no—
307C22⋊C8—81no—
308C22⋊C8—81no—
309C22⋊C8—81no—
310C22⋊C8C2×C814yescomplemented
311C4×C8—21no—
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—42no—
323C4×C8—42no—
324C4×C8—42no—
325C4×C8—42no—
326C4×C8—81no—
327C4×C8C2×D412yes—
328#95—21no—
329#95—21no—
330#95—21no—
331C24—21no—
332C24—21no—
333C24C4×D411yescharacteristic
334#41—28no—
335#41—44no—
336#41—44no—
337#41—44no—
338C2×Q8—21no—
339C2×Q8—21no—
340C2×Q8—21no—
341C2×Q8C4×D412yes—
342C2×Q8C22×C812yes—
343C2×Q8#13112yes—
344C2×D4—24no—
345C2×D4—24no—
346C2×D4—21no—
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—42no—
357C2×D4—42no—
358C2×D4—81no—
359C2×D4—81no—
360C2×D4C22×C812yes—
361C2×D4C22×C812yes—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
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—24no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—41no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—42no—
399C22×C4—81no—
400C22×C4C4×D411yescharacteristic
401C22×C4C4×D411yescharacteristic
402C22×C4C4×D411yescharacteristic
403C22×C4C22×C812yes—
404C22×C4#13112yes—
405C22×C4C23×C411yescharacteristic
406C22×C4C22×D411yescharacteristic
407C22×C4C2×#4111yescharacteristic
408SD16—28no—
409D8—44no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
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—22no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—48no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—41no—
439C2×C8—41no—
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—84no—
450C2×C8C22×C812yes—
451C2×C8C22×C812yes—
452C2×C8#13512yes—
453C2×C8#13512yes—
454C4⋊C4—21no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—42no—
465C4⋊C4—81no—
466C4⋊C4C4×D412yes—
467C4⋊C4C22×C812yes—
468C22⋊C4—24no—
469C22⋊C4—42no—
470C22⋊C4—42no—
471C22⋊C4—41no—
472C22⋊C4—41no—
473C22⋊C4—41no—
474C22⋊C4—41no—
475C22⋊C4—42no—
476C22⋊C4—81no—
477C22⋊C4—81no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—22no—
482C42—21no—
483C42—22no—
484C42—21no—
485C42—21no—
486C42—42no—
487C42—42no—
488C42—41no—
489C42—41no—
490C42C22×D411yescharacteristic
491C42C2×#4111yescharacteristic
492C42H(5,𝔽2)11yescharacteristic
493C42H(5,𝔽2)11yescharacteristic
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—22no—
501C23—22no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23C8×D412yes—
515C23C2×C4×D411yescharacteristic
516Q8—22no—
517Q8—22no—
518Q8—22no—
519Q8C8×D414yes—
520D4—22no—
521D4—22no—
522D4—22no—
523D4—22no—
524D4—48no—
525D4—41no—
526D4—41no—
527D4—41no—
528D4—41no—
529D4—84no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—22no—
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—84no—
569C2×C4—81no—
570C2×C4—81no—
571C2×C4—81no—
572C2×C4—81no—
573C2×C4—81no—
574C2×C4—81no—
575C2×C4—81no—
576C2×C4—81no—
577C2×C4C8×D412yes—
578C2×C4#44312yes—
579C2×C4C2×C4×D411yescharacteristic
580C2×C4#51811yescharacteristic
581C2×C4D4211yescharacteristic
582C2×C4#54611yescharacteristic
583C2×C4#54611yescharacteristic
584C2×C4#55111yescharacteristic
585C2×C4C23×C811yescharacteristic
586C2×C4C2×#13111yescharacteristic
587C2×C4C2×#13111yescharacteristic
588C2×C4#56811yescharacteristic
589C2×C4C2×#13511yescharacteristic
590C2×C4C2×#13511yescharacteristic
591C2×C4#57611yescharacteristic
592C2×C4#57711yescharacteristic
593C8—24no—
594C8—22no—
595C8—22no—
596C8—22no—
597C8—22no—
598C8—42no—
599C8—41no—
600C8—41no—
601C8—42no—
602C8—41no—
603C8—41no—
604C8—81no—
605C8—81no—
606C8—81no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—41no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—81no—
621C22—81no—
622C22—81no—
623C22—81no—
624C22—81no—
625C22—81no—
626C22—81no—
627C22—81no—
628C22#235811yescharacteristic
629C22C2×C8×D411yescharacteristic
630C22#293311yescharacteristic
631C22C4×#13511yescharacteristic
632C22#294611yescharacteristic
633C22C2×#44311yescharacteristic
634C22#295611yescharacteristic
635C4—22no—
636C4—24no—
637C4—21no—
638C4—22no—
639C4—21no—
640C4—41no—
641C4—42no—
642C4—42no—
643C4—81no—
644C4—81no—
645C4—81no—
646C4C2×C8×D411yescharacteristic
647C4#293311yescharacteristic
648C4#297211yescharacteristic
649C4#297711yescharacteristic
650C4#328211yescharacteristic
651C4#328711yescharacteristic
652C4#330111yescharacteristic
653C4#330511yescharacteristic
654C2—42no—
655C2—82no—
656C2—82no—
657C2C8×#13511yescharacteristic
658C2#1785211yescharacteristic
659C2#1799911yescharacteristic
660C2#1801811yescharacteristic
661C2#2284211yescharacteristic
662C2#2287711yescharacteristic
663C2#2288811yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC2×#449C4
LCSG ▷ C2×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. #23083C2semidirect
2. #22530C2semidirect
4. C2×C8×SD16C2semidirect
5. C2×#1476C2semidirect
6. #9657C2semidirect
7. #9554C2semidirect
8. #9542C2semidirect
9. #9072C2semidirect
10. #8948C2semidirect
11. #8947C2semidirect
12. #8844C2semidirect
13. #8831C2semidirect
14. C8×#103C2semidirect
15. C8×C22⋊C8C2semidirect
91. #449C8semidirect
168. Q8⋊C8D4semidirect
180. C2×#41C2×C8semidirect
195. C2×SD16C2×C8semidirect
198. C2×D8C2×C8semidirect
292. #103C2×C8semidirect
297. #102C2×C8semidirect
310. C22⋊C8C2×C8semidirect

Maximal subgroups

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

subgroupindex
#230832
#225302
C4×#4492
C2×C8×SD162
C2×#14762
#96572
#95542
#95422
#90722
#89482
#89472
#88442
#88312
C8×#1032
C8×C22⋊C82