Finite Group Databasebeta

← #6585880

Subgroups — #6585880

Order 512.

1698 conjugacy classes of subgroups.

The subgroup lattice has 677 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36180C211yesmaximal, characteristic, complemented
2#36134C211yesmaximal, characteristic, complemented
3#33289C211yesmaximal, characteristic, complemented
4#33197C211yesmaximal, characteristic, complemented
5#33195C211yesmaximal, characteristic, complemented
6#28798C212yesmaximal, complemented
7#27822C212yesmaximal, complemented
8#26447C212yesmaximal, complemented
9#26282C212yesmaximal, complemented
10#26110C212yesmaximal, complemented
11#26025C212yesmaximal, complemented
12#25477C212yesmaximal, complemented
13#24811C212yesmaximal
14#24327C212yesmaximal
15#22968C212yesmaximal, complemented
16#22758C212yesmaximal
17#22522C212yesmaximal, complemented
18#21712C211yesmaximal, characteristic, complemented
19C42×D8C211yesmaximal, characteristic
20C2×#554C2212yes—
21C2×#533—21no—
22C2×#476C2212yes—
23C2×#474C2212yes—
24C2×#446C2212yes—
25C2×C4×D8C2211yescharacteristic
26C2×C4×D8C2211yescharacteristic
27C2×C4×D8C2211yescharacteristic
28#2868C2211yescharacteristic
29#2868C2211yescharacteristic
30#2857—21no—
31#2857—21no—
32#2567C2212yes—
33#2552C2212yes—
34#2552C2212yes—
35#2552C2212yes—
36#2542C2212yes—
37#2538C2212yes—
38#2502—22no—
39#2441—22no—
40#2440—22no—
41#2440—22no—
42#2429—22no—
43#2423—22no—
44#2365C2212yes—
45#2354C2212yes—
46Q8×C4⋊C4C2212yes—
47D4×C4⋊C4C2212yes—
48D4×C4⋊C4C2212yes—
49C4×C4⋊Q8C2211yescharacteristic
50C2×#384—21no—
51C42×D4C2211yescharacteristic
52C42×D4C2211yescharacteristic
53#2100C2214yes—
54#2088C2214yes—
55#2083C2214yes—
56#2067C2214yes—
57#2059C2214yes—
58#2025C2214yes—
59#1992C2214yes—
60#1990C2212yes—
61#1990C2212yes—
62#1990C2212yes—
63#1990C2212yes—
64#1989C2214yes—
65#1988C2212yes—
66#1987C2212yes—
67#1984C2212yes—
68#1948C2212yes—
69#1948C2212yes—
70#1947C2212yes—
71#1947C2212yes—
72#1943C2214yes—
73#1939C2212yes—
74#1939C2212yes—
75#1936C2214yes—
76#1921C2214yes—
77#1921C2214yes—
78#1879C2214yes—
79#1864C2214yes—
80#1842C2211yescharacteristic
81#1834C2212yes—
82#1834C2212yes—
83#1802C2212yes—
84#1802C2212yes—
85#1802C2212yes—
86#1802C2212yes—
87#1802C2214yes—
88C4×#107C2211yescharacteristic
89C4×#107C2211yescharacteristic
90C4×#107C2211yescharacteristic
91#1770C2212yes—
92#1770C2212yes—
93#1762C2214yes—
94C4×#102C2211yescharacteristic
95C4×#102C2211yescharacteristic
96C4×#102C2211yescharacteristic
97C4×#102C2211yescharacteristic
98C4×#102C2211yescharacteristic
99C4×#102C2211yescharacteristic
100C42×C8C2211yescharacteristic
101C22×D8C2311yescharacteristic
102#554—28no—
103#554—44no—
104#533—48no—
105C2×C4⋊Q8—21no—
106C2×C4⋊Q8—21no—
107C2×C4⋊Q8—21no—
108C2×C4⋊Q8—21no—
109C2×C4⋊Q8D411yescharacteristic
110C2×C4⋊Q8C2312yes—
111C2×C4⋊Q8C2312yes—
112C2×#125—22no—
113C2×#125C2314yes—
114C2×C22⋊Q8—22no—
115C2×C22⋊Q8—22no—
116C2×C22⋊Q8—22no—
117C2×C22⋊Q8—24no—
118C2×C22⋊Q8—22no—
119C2×C4×Q8—22no—
120C2×C4×Q8—22no—
121C2×C4×Q8C2312yes—
122C2×C4×D4C2311yescharacteristic
123C2×C4×D4C2311yescharacteristic
124C2×C4×D4C2314yes—
125C2×C4×D4C2311yescharacteristic
126C2×C4×D4C2311yescharacteristic
127C2×C4×D4C2311yescharacteristic
128C2×C4×D4C2311yescharacteristic
129C2×#117—22no—
130C22×C4⋊C4—22no—
131C22×C4⋊C4—22no—
132C22×C4⋊C4—22no—
133C22×C42—21no—
134C22×C42—21no—
135#476—44no—
136#474—28no—
137#446—44no—
138C4×D8—24no—
139C4×D8—24no—
140C4×D8Q818yescomplemented
141C2×#107C2311yescharacteristic
142C2×#107C2314yes—
143C2×#107C2311yescharacteristic
144C2×#107C2311yescharacteristic
145C2×#106C2314yes—
146C2×C4⋊C8C2312yes—
147C2×C4⋊C8C2312yes—
148C2×C4⋊C8C2312yes—
149C2×C4⋊C8C2314yes—
150C2×#102C2311yescharacteristic
151C2×#102C2314yes—
152C2×#102C2311yescharacteristic
153C2×#102C2311yescharacteristic
154C2×#102C2311yescharacteristic
155C2×#102C2314yes—
156C2×#102C2311yescharacteristic
157C2×#102C2311yescharacteristic
158C2×#97C2314yes—
159C2×C4×C8C2311yescharacteristic
160C2×C4×C8C2311yescharacteristic
161C2×C4×C8C2311yescharacteristic
162C2×C4×C8C2311yescharacteristic
163C2×C4×C8C2311yescharacteristic
164C2×C4×C8C2311yescharacteristic
165#399—24no—
166#398—24no—
167#398—24no—
168#398—24no—
169#396—24no—
170#395—24no—
171#391—22no—
172#391—22no—
173#391D412yes—
174#390C2314yes—
175#389—22no—
176#389—22no—
177#389—22no—
178#389—22no—
179#389—24no—
180#389D412yes—
181#389D412yes—
182#389C2314yes—
183#389C2314yes—
184#389C2314yes—
185#388—24no—
186#387—24no—
187#385—24no—
188#385—24no—
189#384—22no—
190#384—22no—
191#384—22no—
192#384—21no—
193#384—21no—
194#384—21no—
195#384—21no—
196#384D411yescharacteristic
197#384C2314yes—
198#384C2312yes—
199#384C2312yes—
200#384C2312yes—
201#384C2312yes—
202#380—22no—
203#380—22no—
204#380—22no—
205#380—22no—
206C4×C4⋊C4C2311yescharacteristic
207C4×C4⋊C4C2311yescharacteristic
208C4×C4⋊C4C2314yes—
209C4×C4⋊C4C2311yescharacteristic
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2312yes—
215C4×C22⋊C4—21no—
216C4×C22⋊C4—21no—
217C4×C22⋊C4—21no—
218C4×C22⋊C4—21no—
219C4×C22⋊C4—21no—
220C4×C22⋊C4—21no—
221C43C2311yescharacteristic
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2311yescharacteristic
229#340C2314yes—
230#340C2314yes—
231#340C2311yescharacteristic
232#340C2311yescharacteristic
233#340C2311yescharacteristic
234#340C2311yescharacteristic
235#340C2311yescharacteristic
236#336C2314yes—
237#336C2314yes—
238C22×Q8—22no—
239C22×Q8—22no—
240C22×Q8C2×D412yes—
241C22×D4C2411yescharacteristic
242C22×D4C2411yescharacteristic
243C23×C4—21no—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—21no—
247C23×C4—21no—
248C23×C4—21no—
249C23×C4—24no—
250C2×D8—22no—
251C2×D8—22no—
252C2×D8C2×Q814yes—
253C22×C8C2411yescharacteristic
254C22×C8C2414yes—
255C22×C8C2411yescharacteristic
256C22×C8C2411yescharacteristic
257C22×C8C2411yescharacteristic
258C4⋊Q8—24no—
259C4⋊Q8—24no—
260C4⋊Q8—24no—
261C4⋊Q8—24no—
262C4⋊Q8—24no—
263C4⋊Q8—44no—
264C4⋊Q8—82no—
265C4⋊Q8—82no—
266C4⋊Q8—82no—
267#125—28no—
268#125—44no—
269#125—48no—
270C22⋊Q8—416no—
271C22⋊Q8—48no—
272C22⋊Q8—48no—
273C22⋊Q8—48no—
274C22⋊Q8—84no—
275C4×Q8—24no—
276C4×Q8—24no—
277C4×Q8—48no—
278C4×Q8—48no—
279C4×D4—28no—
280C4×D4—22no—
281C4×D4—22no—
282C4×D4—22no—
283C4×D4—22no—
284C4×D4—22no—
285C4×D4—22no—
286C4×D4—22no—
287C4×D4—22no—
288C4×D4—44no—
289C4×D4C2×Q814yes—
290C4×D4C2×Q814yes—
291C4×D4#4114yes—
292C4×D4#4114yes—
293#117—48no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—48no—
312C2×C4⋊C4—84no—
313C2×C4⋊C4—84no—
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2×D412yes—
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2×D414yes—
320C2×C4⋊C4C2414yes—
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2414yes—
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2411yescharacteristic
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—24no—
337C2×C42—24no—
338C2×C42—21no—
339C2×C42—21no—
340C2×C42—21no—
341C2×C42—21no—
342C2×C42—21no—
343C2×C42—21no—
344C2×C42—44no—
345C2×C42—44no—
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2414yes—
351C2×C42C2411yescharacteristic
352C2×C42C2411yescharacteristic
353C2×C42C2411yescharacteristic
354#107—22no—
355#107—22no—
356#107—28no—
357#107C2×Q814yes—
358#106—44no—
359C4⋊C8—24no—
360C4⋊C8—24no—
361C4⋊C8—24no—
362C4⋊C8—44no—
363#102—22no—
364#102—22no—
365#102—28no—
366#102—22no—
367#102—22no—
368#102—44no—
369#102C2×Q814yes—
370#102C2×Q814yes—
371#97—44no—
372C4×C8—22no—
373C4×C8—22no—
374C4×C8—22no—
375C4×C8—22no—
376C4×C8—22no—
377C4×C8C2×Q814yes—
378#95—21no—
379#95—21no—
380#95—21no—
381#95—21no—
382#95—21no—
383#95—21no—
384#95—24no—
385#95—24no—
386#95—24no—
387#95—24no—
388#95C2×D414yes—
389#95C2×D414yes—
390C24—21no—
391C24—21no—
392C2×Q8—24no—
393C2×Q8—24no—
394C2×Q8—24no—
395C2×Q8—44no—
396C2×Q8—44no—
397C2×Q8—44no—
398C2×Q8—44no—
399C2×Q8—82no—
400C2×Q8—82no—
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4C22×Q812yes—
410C2×D4C22×Q812yes—
411C2×D4C2×#4112yes—
412C2×D4C2×#4112yes—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—48no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—84no—
441C22×C4—84no—
442C22×C4C22×D414yes—
443C22×C4C22×D414yes—
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C2511yescharacteristic
451D8—28no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—24no—
459C2×C8—21no—
460C2×C8—24no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—24no—
465C2×C8C22×Q812yes—
466C2×C8C22×Q812yes—
467C4⋊C4—24no—
468C4⋊C4—28no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—24no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—24no—
485C4⋊C4—48no—
486C4⋊C4—48no—
487C4⋊C4—48no—
488C4⋊C4—48no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—84no—
496C4⋊C4—84no—
497C4⋊C4—84no—
498C4⋊C4—82no—
499C4⋊C4—82no—
500C4⋊C4—82no—
501C4⋊C4—82no—
502C4⋊C4C22×Q812yes—
503C4⋊C4C22×Q812yes—
504C4⋊C4C2×#4112yes—
505C4⋊C4C2×#4112yes—
506C22⋊C4—24no—
507C22⋊C4—24no—
508C22⋊C4—24no—
509C22⋊C4—24no—
510C22⋊C4—48no—
511C22⋊C4—48no—
512C22⋊C4—42no—
513C22⋊C4—42no—
514C22⋊C4—84no—
515C42—22no—
516C42—22no—
517C42—22no—
518C42—24no—
519C42—24no—
520C42—48no—
521C42—42no—
522C42—42no—
523C42—42no—
524C42—42no—
525C42—42no—
526C42—42no—
527C42AGL(1,ℤ/8)14yes—
528C42C22×Q812yes—
529C42C2×#4112yes—
530C42C2×#4112yes—
531C42H(5,𝔽2)12yes—
532C422−1+412yes—
533C422−1+412yes—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23C23×D411yescharacteristic
549Q8—48no—
550Q8—88no—
551Q8—164no—
552D4—22no—
553D4—22no—
554D4—22no—
555D4—22no—
556D4—22no—
557D4—41no—
558D4—41no—
559D4#53314yescomplemented
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—48no—
585C2×C4—48no—
586C2×C4—48no—
587C2×C4—48no—
588C2×C4—44no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—44no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—41no—
598C2×C4—44no—
599C2×C4—41no—
600C2×C4—84no—
601C2×C4—84no—
602C2×C4—168no—
603C2×C4#54612yes—
604C2×C4#54612yes—
605C2×C4#54812yes—
606C2×C4#54812yes—
607C2×C4D4×Q812yes—
608C2×C4D4×Q812yes—
609C2×C4C2×AGL(1,ℤ/8)12yes—
610C2×C4C2×AGL(1,ℤ/8)12yes—
611C2×C4C2×AGL(1,ℤ/8)12yes—
612C2×C4#57612yes—
613C2×C4#57712yes—
614C2×C4#57812yes—
615C2×C4C23×Q811yescharacteristic
616C2×C4C22×#4111yescharacteristic
617C2×C4C22×#4111yescharacteristic
618C2×C4C2×H(5,𝔽2)11yescharacteristic
619C2×C4C2×2−1+411yescharacteristic
620C2×C4C2×2−1+411yescharacteristic
621C8—22no—
622C8—22no—
623C8—41no—
624C8—41no—
625C8—44no—
626C22—24no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—81no—
637C22—81no—
638C22—81no—
639C22—81no—
640C22C2×#54611yescharacteristic
641C22C2×#54811yescharacteristic
642C22C2×D4×Q811yescharacteristic
643C22C22×AGL(1,ℤ/8)11yescharacteristic
644C22C2×#57611yescharacteristic
645C22C2×#57711yescharacteristic
646C22C2×#57811yescharacteristic
647C4—22no—
648C4—22no—
649C4—22no—
650C4—44no—
651C4—42no—
652C4—42no—
653C4—42no—
654C4—44no—
655C4—42no—
656C4—84no—
657C4—81no—
658C4—84no—
659C4—81no—
660C4#333014yes—
661C4#336614yes—
662C4#338414yes—
663C4C2×#53311yescharacteristic
664C4#350311yescharacteristic
665C4#350911yescharacteristic
666C4#351111yescharacteristic
667C2—44no—
668C2—82no—
669C2#3612911yescharacteristic
670C2C2×#333011yescharacteristic
671C2#6121511yescharacteristic
672C2C2×#336611yescharacteristic
673C2#6133011yescharacteristic
674C2C2×#338411yescharacteristic
675C2#6137311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D8C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36180C2semidirect
2. #36134C2semidirect
3. #33289C2semidirect
4. #33197C2semidirect
5. #33195C2semidirect
6. #28798C2semidirect
7. #27822C2semidirect
8. #26447C2semidirect
9. #26282C2semidirect
10. #26110C2semidirect
11. #26025C2semidirect
12. #25477C2semidirect
15. #22968C2semidirect
17. #22522C2semidirect
18. #21712C2semidirect
140. C4×D8Q8semidirect
559. D4#533semidirect

Maximal subgroups

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

subgroupindex
#361802
#361342
#332892
#331972
#331952
#287982
#278222
#264472
#262822
#261102
#260252
#254772
#248112
#243272
#229682
#227582
#225222
#217122
C42×D82