Finite Group Databasebeta

← #6585879

Subgroups — #6585879

Order 512.

1754 conjugacy classes of subgroups.

The subgroup lattice has 679 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36194C211yesmaximal, characteristic, complemented
2D4×C4⋊Q8C211yesmaximal, characteristic, complemented
3#33289C211yesmaximal, characteristic, complemented
4#33197C211yesmaximal, characteristic, complemented
5#33195C211yesmaximal, characteristic, complemented
6#28797C212yesmaximal, complemented
7#27821C212yesmaximal, complemented
8#26446C212yesmaximal, complemented
9#26283C212yesmaximal, complemented
10#26111C212yesmaximal, complemented
11#26024C212yesmaximal, complemented
12#25478C212yesmaximal, complemented
13#24812C212yesmaximal
14#24326C212yesmaximal
15#22969C212yesmaximal, complemented
16#22758C212yesmaximal
17#22523C212yesmaximal, complemented
18#21712C211yesmaximal, characteristic, complemented
19C42×D8C211yesmaximal, characteristic
20C2×D4×Q8C2212yes—
21C22×C4⋊Q8—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#2571C2212yes—
33#2563C2212yes—
34#2551C2212yes—
35#2551C2212yes—
36#2546C2212yes—
37#2538C2212yes—
38#2497—22no—
39#2492—22no—
40#2440—22no—
41#2440—22no—
42#2439—22no—
43#2434—22no—
44#2369C2212yes—
45#2355C2212yes—
46#2354C2212yes—
47D4×C4⋊C4C2212yes—
48D4×C4⋊C4C2212yes—
49C4×C4⋊Q8C2211yescharacteristic
50#2285—21no—
51C42×D4C2211yescharacteristic
52C42×D4C2211yescharacteristic
53#2101C2214yes—
54#2087C2214yes—
55#2085C2214yes—
56#2065C2214yes—
57#2057C2214yes—
58#2023C2214yes—
59#1993C2214yes—
60#1990C2212yes—
61#1990C2212yes—
62#1990C2212yes—
63#1990C2212yes—
64#1988C2212yes—
65#1987C2214yes—
66#1987C2212yes—
67#1984C2212yes—
68#1948C2212yes—
69#1948C2212yes—
70#1947C2212yes—
71#1947C2212yes—
72#1944C2214yes—
73#1939C2212yes—
74#1939C2212yes—
75#1936C2214yes—
76#1921C2214yes—
77#1920C2214yes—
78#1879C2214yes—
79#1862C2214yes—
80#1842C2211yescharacteristic
81#1834C2212yes—
82#1834C2212yes—
83#1804C2214yes—
84#1802C2212yes—
85#1802C2212yes—
86#1802C2212yes—
87#1802C2212yes—
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
101C23×Q8—22no—
102C22×D8C2311yescharacteristic
103D4×Q8—28no—
104D4×Q8—44no—
105C2×C4⋊Q8—22no—
106C2×C4⋊Q8—22no—
107C2×C4⋊Q8—22no—
108C2×C4⋊Q8—21no—
109C2×C4⋊Q8—21no—
110C2×C4⋊Q8—21no—
111C2×C4⋊Q8—21no—
112C2×C4⋊Q8—48no—
113C2×C4⋊Q8D411yescharacteristic
114C2×C4⋊Q8C2312yes—
115C2×C4⋊Q8C2312yes—
116C2×C4⋊Q8C2314yes—
117C2×C22⋊Q8—22no—
118C2×C22⋊Q8—22no—
119C2×C22⋊Q8—24no—
120C2×C4×Q8C2312yes—
121C2×C4×D4C2311yescharacteristic
122C2×C4×D4C2311yescharacteristic
123C2×C4×D4C2314yes—
124C2×C4×D4C2311yescharacteristic
125C2×C4×D4C2311yescharacteristic
126C2×C4×D4C2311yescharacteristic
127C2×C4×D4C2311yescharacteristic
128C2×#117—22no—
129C22×C4⋊C4—22no—
130C22×C4⋊C4—22no—
131C22×C42—21no—
132C22×C42—21no—
133#476—28no—
134#474—44no—
135#446—44no—
136C4×D8—24no—
137C4×D8—24no—
138C4×D8Q818yescomplemented
139C2×#107C2311yescharacteristic
140C2×#107C2311yescharacteristic
141C2×#107C2314yes—
142C2×#107C2311yescharacteristic
143C2×#106C2314yes—
144C2×C4⋊C8C2312yes—
145C2×C4⋊C8C2312yes—
146C2×C4⋊C8C2312yes—
147C2×C4⋊C8C2314yes—
148C2×#102C2314yes—
149C2×#102C2311yescharacteristic
150C2×#102C2311yescharacteristic
151C2×#102C2311yescharacteristic
152C2×#102C2311yescharacteristic
153C2×#102C2314yes—
154C2×#102C2311yescharacteristic
155C2×#102C2311yescharacteristic
156C2×#97C2314yes—
157C2×C4×C8C2311yescharacteristic
158C2×C4×C8C2311yescharacteristic
159C2×C4×C8C2311yescharacteristic
160C2×C4×C8C2311yescharacteristic
161C2×C4×C8C2311yescharacteristic
162C2×C4×C8C2311yescharacteristic
163#400—24no—
164#398—24no—
165#396—24no—
166#395—24no—
167#395—24no—
168#393—24no—
169#391—22no—
170#391—22no—
171#391—24no—
172#391D412yes—
173#390C2314yes—
174#389—22no—
175#389—22no—
176#389—22no—
177#389—22no—
178#389D412yes—
179#389D412yes—
180#389C2314yes—
181#389C2314yes—
182#389C2314yes—
183#388—24no—
184#387—24no—
185#385—24no—
186#385—24no—
187#384—21no—
188#384—21no—
189#384—21no—
190#384—21no—
191#384D411yescharacteristic
192#384C2312yes—
193#384C2312yes—
194#384C2312yes—
195#384C2312yes—
196#382—22no—
197#382C2314yes—
198#380—22no—
199#380—22no—
200#380—22no—
201#380—22no—
202#380—22no—
203#380—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4C2311yescharacteristic
207C4×C4⋊C4C2314yes—
208C4×C4⋊C4C2311yescharacteristic
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—24no—
240C22×Q8—22no—
241C22×Q8—48no—
242C22×Q8—48no—
243C22×Q8—84no—
244C22×Q8C2×D412yes—
245C22×D4C2411yescharacteristic
246C22×D4C2411yescharacteristic
247C23×C4—21no—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4—21no—
251C23×C4—21no—
252C23×C4—21no—
253C23×C4—24no—
254C2×D8—22no—
255C2×D8—22no—
256C2×D8C2×Q814yes—
257C22×C8C2411yescharacteristic
258C22×C8C2414yes—
259C22×C8C2411yescharacteristic
260C22×C8C2411yescharacteristic
261C22×C8C2411yescharacteristic
262C4⋊Q8—28no—
263C4⋊Q8—24no—
264C4⋊Q8—24no—
265C4⋊Q8—24no—
266C4⋊Q8—24no—
267C4⋊Q8—24no—
268C4⋊Q8—44no—
269C4⋊Q8—48no—
270C4⋊Q8—48no—
271C4⋊Q8—48no—
272C4⋊Q8—44no—
273C4⋊Q8—82no—
274C4⋊Q8—82no—
275C4⋊Q8—82no—
276C22⋊Q8—416no—
277C22⋊Q8—48no—
278C22⋊Q8—84no—
279C4×Q8—24no—
280C4×Q8—24no—
281C4×D4—28no—
282C4×D4—22no—
283C4×D4—22no—
284C4×D4—22no—
285C4×D4—22no—
286C4×D4—22no—
287C4×D4—22no—
288C4×D4—22no—
289C4×D4—22no—
290C4×D4—44no—
291C4×D4C2×Q814yes—
292C4×D4C2×Q814yescomplemented
293C4×D4#4114yes—
294C4×D4#4114yes—
295#117—84no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—48no—
311C2×C4⋊C4—48no—
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2×D412yes—
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2×D412yes—
316C2×C4⋊C4C2×D414yes—
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2414yes—
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2414yes—
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2411yescharacteristic
326C2×C22⋊C4—21no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C42—21no—
336C2×C42—21no—
337C2×C42—21no—
338C2×C42—21no—
339C2×C42—21no—
340C2×C42—21no—
341C2×C42—24no—
342C2×C42—44no—
343C2×C42—44no—
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2414yes—
349C2×C42C2411yescharacteristic
350C2×C42C2411yescharacteristic
351C2×C42C2411yescharacteristic
352#107—22no—
353#107—22no—
354#107—44no—
355#107C2×Q814yescomplemented
356#106—28no—
357C4⋊C8—24no—
358C4⋊C8—24no—
359C4⋊C8—24no—
360C4⋊C8—44no—
361#102—22no—
362#102—22no—
363#102—28no—
364#102—22no—
365#102—22no—
366#102—44no—
367#102C2×Q814yes—
368#102C2×Q814yescomplemented
369#97—44no—
370C4×C8—22no—
371C4×C8—22no—
372C4×C8—22no—
373C4×C8—22no—
374C4×C8—22no—
375C4×C8C2×Q814yescomplemented
376#95—21no—
377#95—21no—
378#95—21no—
379#95—21no—
380#95—21no—
381#95—21no—
382#95—24no—
383#95—24no—
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—48no—
396C2×Q8—48no—
397C2×Q8—44no—
398C2×Q8—44no—
399C2×Q8—44no—
400C2×Q8—44no—
401C2×Q8—84no—
402C2×Q8—82no—
403C2×Q8—82no—
404C2×Q8—832no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4C22×Q812yes—
414C2×D4C22×Q812yes—
415C2×D4C2×#4112yes—
416C2×D4C2×#4112yes—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—48no—
430C22×C4—48no—
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—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—84no—
446C22×C4C22×D414yes—
447C22×C4C22×D414yes—
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C2511yescharacteristic
455D8—28no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—24no—
463C2×C8—21no—
464C2×C8—24no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—24no—
469C2×C8C22×Q812yes—
470C2×C8C22×Q812yes—
471C4⋊C4—24no—
472C4⋊C4—28no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—24no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—24no—
489C4⋊C4—48no—
490C4⋊C4—48no—
491C4⋊C4—48no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—84no—
499C4⋊C4—82no—
500C4⋊C4—82no—
501C4⋊C4—82no—
502C4⋊C4—82no—
503C4⋊C4C22×Q812yes—
504C4⋊C4C22×Q812yes—
505C4⋊C4C2×#4112yes—
506C4⋊C4C2×#4112yes—
507C22⋊C4—24no—
508C22⋊C4—24no—
509C22⋊C4—24no—
510C22⋊C4—24no—
511C22⋊C4—48no—
512C22⋊C4—42no—
513C22⋊C4—42no—
514C22⋊C4—84no—
515C22⋊C4—84no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—24no—
520C42—24no—
521C42—42no—
522C42—42no—
523C42—42no—
524C42—42no—
525C42—42no—
526C42—42no—
527C42—84no—
528C42AGL(1,ℤ/8)14yes—
529C42C22×Q812yes—
530C42C2×#4112yes—
531C42C2×#4112yes—
532C42H(5,𝔽2)12yes—
533C422−1+412yes—
534C422−1+412yes—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23C23×D411yescharacteristic
550Q8—48no—
551Q8—816no—
552Q8—88no—
553Q8—164no—
554D4—22no—
555D4—22no—
556D4—22no—
557D4—22no—
558D4—22no—
559D4—41no—
560D4—41no—
561D4#53314yes—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—48no—
587C2×C4—48no—
588C2×C4—48no—
589C2×C4—48no—
590C2×C4—44no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—44no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—41no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—41no—
602C2×C4—816no—
603C2×C4—84no—
604C2×C4—84no—
605C2×C4#54612yes—
606C2×C4#54612yes—
607C2×C4#54812yes—
608C2×C4#54812yes—
609C2×C4D4×Q812yes—
610C2×C4D4×Q812yes—
611C2×C4C2×AGL(1,ℤ/8)12yes—
612C2×C4C2×AGL(1,ℤ/8)12yes—
613C2×C4C2×AGL(1,ℤ/8)12yes—
614C2×C4#57612yes—
615C2×C4#57712yes—
616C2×C4#57812yes—
617C2×C4C23×Q811yescharacteristic
618C2×C4C22×#4111yescharacteristic
619C2×C4C22×#4111yescharacteristic
620C2×C4C2×H(5,𝔽2)11yescharacteristic
621C2×C4C2×2−1+411yescharacteristic
622C2×C4C2×2−1+411yescharacteristic
623C8—22no—
624C8—22no—
625C8—41no—
626C8—41no—
627C8—44no—
628C22—24no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22—42no—
638C22—81no—
639C22—81no—
640C22—81no—
641C22—81no—
642C22C2×#54611yescharacteristic
643C22C2×#54811yescharacteristic
644C22C2×D4×Q811yescharacteristic
645C22C22×AGL(1,ℤ/8)11yescharacteristic
646C22C2×#57611yescharacteristic
647C22C2×#57711yescharacteristic
648C22C2×#57811yescharacteristic
649C4—22no—
650C4—22no—
651C4—22no—
652C4—44no—
653C4—42no—
654C4—42no—
655C4—42no—
656C4—44no—
657C4—42no—
658C4—81no—
659C4—84no—
660C4—81no—
661C4—84no—
662C4#332514yes—
663C4#336714yes—
664C4#338414yes—
665C4C2×#53311yescharacteristic
666C4#350311yescharacteristic
667C4#350911yescharacteristic
668C4#351111yescharacteristic
669C2—44no—
670C2—82no—
671C2#3612911yescharacteristic
672C2C2×#332511yescharacteristic
673C2#6121711yescharacteristic
674C2C2×#336711yescharacteristic
675C2#6132911yescharacteristic
676C2C2×#338411yescharacteristic
677C2#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. #36194C2semidirect
2. D4×C4⋊Q8C2semidirect
3. #33289C2semidirect
4. #33197C2semidirect
5. #33195C2semidirect
6. #28797C2semidirect
7. #27821C2semidirect
8. #26446C2semidirect
9. #26283C2semidirect
10. #26111C2semidirect
11. #26024C2semidirect
12. #25478C2semidirect
15. #22969C2semidirect
17. #22523C2semidirect
18. #21712C2semidirect
138. C4×D8Q8semidirect
292. C4×D4C2×Q8semidirect
355. #107C2×Q8semidirect
368. #102C2×Q8semidirect
375. C4×C8C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#361942
D4×C4⋊Q82
#332892
#331972
#331952
#287972
#278212
#264462
#262832
#261112
#260242
#254782
#248122
#243262
#229692
#227582
#225232
#217122
C42×D82