Finite Group Databasebeta

← #6795026

Subgroups — #6795026

Order 512.

1914 conjugacy classes of subgroups.

The subgroup lattice has 722 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36176C211yesmaximal, characteristic, complemented
2C2×#2542C211yesmaximal, characteristic, complemented
3#32956C211yesmaximal, characteristic, complemented
4#32446C214yesmaximal, complemented
5#32364C214yesmaximal, complemented
6#29907C212yesmaximal, complemented
7#28541C212yesmaximal, complemented
8#27736C212yesmaximal, complemented
9#27588C212yesmaximal, complemented
10#25811C211yesmaximal, characteristic, complemented
11#25451C212yesmaximal, complemented
12#24676C212yesmaximal, complemented
13#24506C212yesmaximal
14#24298C211yesmaximal, characteristic, complemented
15#22603C211yesmaximal, characteristic, complemented
16#21987C212yesmaximal, complemented
17C4×#447C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C22⋊Q8—21no—
20C22×C22⋊Q8—21no—
21C22×C4×D4C2211yescharacteristic
22C2×#498C2212yescomplemented
23C2×#476C2212yes—
24C2×#474C2212yescomplemented
25C2×#447C2211yescharacteristic
26C2×C4×D8C2212yes—
27C2×#432C2212yes—
28#2857C2211yescharacteristic
29#2854C2212yes—
30#2571C2211yescharacteristic, complemented
31#2563C2211yescharacteristic, complemented
32#2542—21no—
33#2542—21no—
34#2542—22no—
35#2542C2212yes—
36#2542C2212yes—
37#2542C2214yes—
38#2532—22no—
39#2513C2212yescomplemented
40#2508C2212yescomplemented
41#2495—21no—
42#2488—22no—
43#2486—21no—
44C2×#399—22no—
45#2369C2211yescharacteristic, complemented
46#2354C2211yescharacteristic, complemented
47#2352—21no—
48#2346—21no—
49C4×#121C2211yescharacteristic
50C2×#390C2212yes—
51C2×#384C2212yes—
52#2283C2212yes—
53C2×C4×C4⋊C4C2211yescharacteristic
54#2100C2214yescomplemented
55#2094C2214yescomplemented
56#2092C2214yes—
57#2084C2214yescomplemented
58#2073C2214yes—
59#2071C2214yescomplemented
60#2065C2214yes—
61#2058C2214yescomplemented
62#2057C2212yescomplemented
63#2054C2214yes—
64#2023C2212yescomplemented
65#1990C2212yescomplemented
66#1988C2212yescomplemented
67#1972C2214yes—
68#1948C2214yescomplemented
69#1947C2214yescomplemented
70#1939C2212yescomplemented
71#1936C2214yes—
72#1926C2214yes—
73#1925C2214yes—
74#1917C2214yescomplemented
75#1879C2212yes—
76#1874C2212yes—
77#1874C2211yescharacteristic
78#1864C2214yes—
79#1862C2212yes—
80#1853C2212yes—
81#1853C2212yes—
82#1853C2212yes—
83#1852C2212yes—
84#1852C2211yescharacteristic
85#1847C2214yescomplemented
86#1836C2214yescomplemented
87#1817C2212yes—
88#1802C2212yes—
89#1797C2211yescharacteristic
90#1786C2212yes—
91#1786C2211yescharacteristic
92#1770C2212yescomplemented
93C4×#102C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—21no—
96C23×D4C2311yescharacteristic
97C24×C4—21no—
98C22×D8C2312yes—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—44no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2312yes—
105C2×#123—22no—
106C2×C22⋊Q8—21no—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8—24no—
111C2×C22⋊Q8—24no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8—48no—
114C2×C22⋊Q8—48no—
115C2×#121C2311yescharacteristic
116C2×C4×D4—21no—
117C2×C4×D4—21no—
118C2×C4×D4—21no—
119C2×C4×D4—44no—
120C2×C4×D4C2311yescharacteristic
121C2×C4×D4C2312yes—
122C2×C4×D4C2312yes—
123C2×#117—21no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4—22no—
126C22×C4⋊C4—22no—
127C22×C4⋊C4D411yescharacteristic
128C22×C4⋊C4D411yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2312yes—
131C22×C22⋊C4—21no—
132C22×C22⋊C4—22no—
133C22×C22⋊C4—22no—
134C22×C42C2311yescharacteristic
135C22×C42C2312yes—
136#498—44no—
137#476—44no—
138#474—44no—
139#447—24no—
140C4×D8—44no—
141#432—28no—
142C2×#107C2314yes—
143C2×#107C2312yes—
144C2×#106C2314yes—
145C2×#106C2314yes—
146C2×#106C2314yes—
147C2×C4⋊C8C2312yes—
148C2×C4⋊C8C2314yes—
149C2×#102C2314yes—
150C2×#102C2312yes—
151C2×#102C2312yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×#102C2312yes—
155C2×#102C2312yes—
156C2×C22⋊C8C2312yes—
157C2×C22⋊C8C2311yescharacteristic
158C2×C22⋊C8C2311yescharacteristic
159C2×#97C2314yes—
160C2×C4×C8C2312yes—
161#400—22no—
162#400—22no—
163#399—24no—
164#399—24no—
165#399—24no—
166#399—24no—
167#398—22no—
168#397—22no—
169#397—22no—
170#393—22no—
171#391—21no—
172#391—22no—
173#390—22no—
174#390—22no—
175#390C2312yes—
176#390C2312yes—
177#390C2314yes—
178#390C2314yes—
179#390C2311yescharacteristic
180#389—21no—
181#389—21no—
182#389—21no—
183#389C2311yescharacteristic
184#389C2312yes—
185#389C2312yes—
186#389C2312yes—
187#388—22no—
188#388—22no—
189#388—22no—
190#388—21no—
191#388—22no—
192#388—22no—
193#387—22no—
194#386—22no—
195#386—21no—
196#384—22no—
197#384—22no—
198#384C2311yescharacteristic
199#384C2314yes—
200#384C2314yes—
201#383—22no—
202#382—21no—
203#382—22no—
204#382C2312yes—
205#382C2312yes—
206#381—22no—
207#380—21no—
208#380—21no—
209#380—21no—
210#380—21no—
211#380C2312yes—
212#380C2311yescharacteristic
213C4×C4⋊C4—21no—
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4D411yescharacteristic
216C4×C4⋊C4D412yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2312yes—
219C4×C4⋊C4C2312yes—
220C4×C4⋊C4C2311yescharacteristic
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2312yes—
223C4×C22⋊C4—22no—
224C4×C22⋊C4—21no—
225#376—22no—
226C2×#95D412yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#340C2312yes—
233#340C2312yes—
234#336C2312yes—
235#336C2312yes—
236#336C2314yes—
237#336C2314yes—
238C25—21no—
239C22×Q8—22no—
240C22×Q8—22no—
241C22×Q8—21no—
242C22×Q8—44no—
243C22×Q8—44no—
244C22×Q8—44no—
245C22×D4—24no—
246C22×D4—21no—
247C22×D4—21no—
248C22×D4—22no—
249C22×D4—22no—
250C22×D4—42no—
251C22×D4C2411yescharacteristic
252C22×D4C2412yes—
253C23×C4—22no—
254C23×C4—21no—
255C23×C4—21no—
256C23×C4—22no—
257C23×C4—22no—
258C23×C4—44no—
259C23×C4—44no—
260C23×C4—44no—
261C23×C4C2×D412yes—
262C23×C4C2×D412yes—
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D411yescharacteristic
265C23×C4C2411yescharacteristic
266C2×D8—24no—
267C2×D8—24no—
268C2×D8—42no—
269C22×C8C2412yes—
270C22×C8C2414yes—
271C22×C8C2412yes—
272C4⋊Q8—24no—
273C4⋊Q8—24no—
274C4⋊Q8—24no—
275C4⋊Q8—44no—
276C4⋊Q8—44no—
277C4⋊Q8—44no—
278#123—84no—
279C22⋊Q8—416no—
280C22⋊Q8—48no—
281C22⋊Q8—48no—
282C22⋊Q8—48no—
283C22⋊Q8—416no—
284C22⋊Q8—48no—
285C22⋊Q8—82no—
286#121—22no—
287#121—22no—
288C4×D4—24no—
289C4×D4—24no—
290C4×D4—24no—
291C4×D4—42no—
292C4×D4—42no—
293C4×D4—44no—
294C4×D4—44no—
295C4×D4—44no—
296#117—44no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—42no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4—48no—
317C2×C4⋊C4—48no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×D412yes—
323C2×C4⋊C4C2×D412yes—
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2414yes—
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2412yes—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—44no—
349C2×C22⋊C4—48no—
350C2×C22⋊C4—48no—
351C2×C42—24no—
352C2×C42—24no—
353C2×C42—24no—
354C2×C42—22no—
355C2×C42—21no—
356C2×C42—22no—
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D412yes—
360C2×C42C2×D412yes—
361C2×C42C2411yescharacteristic
362C2×C42C2412yes—
363C2×C42C2412yes—
364C2×C42C2411yescharacteristic
365#107—42no—
366#107—44no—
367#106—44no—
368#106—44no—
369#106—44no—
370C4⋊C8—28no—
371C4⋊C8—24no—
372#102—28no—
373#102—24no—
374#102—24no—
375#102—42no—
376#102—42no—
377#102—44no—
378#102—44no—
379C22⋊C8—24no—
380C22⋊C8—22no—
381C22⋊C8#4114yes—
382#97—28no—
383C4×C8—24no—
384#95—21no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—21no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—21no—
395#95—22no—
396#95—22no—
397#95C2×D412yes—
398#95C2×D412yes—
399#95C2×D412yes—
400#95C2×D412yes—
401C24—24no—
402C24—22no—
403C24—21no—
404C24—42no—
405C24—42no—
406C24—42no—
407C24—42no—
408C24—42no—
409C24—42no—
410C24C22×D411yescharacteristic
411C2×Q8—44no—
412C2×Q8—44no—
413C2×Q8—44no—
414C2×Q8—42no—
415C2×Q8—44no—
416C2×Q8—44no—
417C2×Q8—44no—
418C2×Q8—81no—
419C2×Q8—81no—
420C2×Q8—816no—
421C2×D4—24no—
422C2×D4—22no—
423C2×D4—21no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—21no—
430C2×D4—22no—
431C2×D4—48no—
432C2×D4—48no—
433C2×D4—42no—
434C2×D4—42no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4—42no—
438C2×D42−1+414yes—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—42no—
458C22×C4—41no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—42no—
462C22×C4—44no—
463C22×C4—42no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—42no—
467C22×C4—44no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—816no—
475C22×C4AGL(1,ℤ/8)14yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D412yes—
484C22×C4C22×D411yescharacteristic
485C22×C4C2×#4112yes—
486C22×C42−1+412yes—
487C22×C4C2511yescharacteristic
488D8—48no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—24no—
492C2×C8—22no—
493C2×C8—24no—
494C2×C8—22no—
495C2×C8—24no—
496C2×C8—22no—
497C2×C8C2×#4114yes—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—21no—
504C4⋊C4—24no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—24no—
508C4⋊C4—22no—
509C4⋊C4—24no—
510C4⋊C4—22no—
511C4⋊C4—24no—
512C4⋊C4—21no—
513C4⋊C4—42no—
514C4⋊C4—48no—
515C4⋊C4—48no—
516C4⋊C4—48no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—42no—
521C4⋊C4—48no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—81no—
525C4⋊C4—82no—
526C4⋊C4—82no—
527C4⋊C4—82no—
528C4⋊C4AGL(1,ℤ/8)14yes—
529C22⋊C4—44no—
530C22⋊C4—48no—
531C22⋊C4—48no—
532C22⋊C4—48no—
533C22⋊C4—42no—
534C22⋊C4—44no—
535C22⋊C4—44no—
536C22⋊C4—42no—
537C22⋊C4—42no—
538C22⋊C4—48no—
539C22⋊C4—44no—
540C22⋊C4—42no—
541C22⋊C4—81no—
542C22⋊C4—82no—
543C22⋊C4—82no—
544C22⋊C4—82no—
545C42—22no—
546C42—22no—
547C42—24no—
548C42—24no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—44no—
555C42—42no—
556C42—82no—
557C42H(5,𝔽2)14yes—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—24no—
563C23—48no—
564C23—48no—
565C23—48no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23—84no—
580C23—84no—
581C23—84no—
582C23—84no—
583C23#54712yes—
584C23C2×AGL(1,ℤ/8)12yes—
585C23#57712yes—
586C23C23×D411yescharacteristic
587Q8—88no—
588Q8—88no—
589Q8—162no—
590D4—24no—
591D4—24no—
592D4—48no—
593D4—42no—
594D4—42no—
595D4—42no—
596D4—42no—
597D4—84no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—22no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—28no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—22no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—21no—
621C2×C4—48no—
622C2×C4—44no—
623C2×C4—48no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—41no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—42no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—42no—
636C2×C4—41no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—41no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—41no—
645C2×C4—88no—
646C2×C4—84no—
647C2×C4—88no—
648C2×C4—84no—
649C2×C4—88no—
650C2×C4#53512yes—
651C2×C4#53512yes—
652C2×C4#54712yes—
653C2×C4C2×AGL(1,ℤ/8)14yes—
654C2×C4C2×AGL(1,ℤ/8)12yes—
655C2×C4C2×AGL(1,ℤ/8)12yes—
656C2×C4C2×AGL(1,ℤ/8)12yes—
657C2×C4C22×#4111yescharacteristic
658C2×C4C2×H(5,𝔽2)12yes—
659C2×C4C2×2−1+412yes—
660C2×C4C2×2−1+411yescharacteristic
661C8—24no—
662C8—44no—
663C8—42no—
664C22—22no—
665C22—22no—
666C22—22no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—44no—
671C22—44no—
672C22—44no—
673C22—816no—
674C22—82no—
675C22—82no—
676C22—82no—
677C22—82no—
678C22—81no—
679C22—81no—
680C22—81no—
681C22—81no—
682C22—81no—
683C22—81no—
684C22—81no—
685C22#335414yes—
686C22C2×#53511yescharacteristic
687C22C2×#54711yescharacteristic
688C22#348611yescharacteristic
689C22C22×AGL(1,ℤ/8)11yescharacteristic
690C22C22×AGL(1,ℤ/8)11yescharacteristic
691C22C2×#57711yescharacteristic
692C22#358611yescharacteristic
693C4—22no—
694C4—24no—
695C4—22no—
696C4—22no—
697C4—42no—
698C4—42no—
699C4—42no—
700C4—41no—
701C4—81no—
702C4—81no—
703C4—82no—
704C4—84no—
705C4—81no—
706C4—82no—
707C4—82no—
708C4#321714yes—
709C4#352012yes—
710C4#353812yes—
711C2—24no—
712C2—84no—
713C2—161no—
714C2#3891711yescharacteristic
715C2C2×#321711yescharacteristic
716C2#6081311yescharacteristic
717C2C2×#335411yescharacteristic
718C2#6125211yescharacteristic
719C2#6182711yescharacteristic
720C2#6184311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#447C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36176C2semidirect
2. C2×#2542C2semidirect
3. #32956C2semidirect
4. #32446C2semidirect
5. #32364C2semidirect
6. #29907C2semidirect
7. #28541C2semidirect
8. #27736C2semidirect
9. #27588C2semidirect
10. #25811C2semidirect
11. #25451C2semidirect
12. #24676C2semidirect
14. #24298C2semidirect
15. #22603C2semidirect
16. #21987C2semidirect
22. C2×#498C22semidirect
24. C2×#474C22semidirect
30. #2571C22semidirect
31. #2563C22semidirect
39. #2513C22semidirect
40. #2508C22semidirect
45. #2369C22semidirect
46. #2354C22semidirect
54. #2100C22semidirect
55. #2094C22semidirect
57. #2084C22semidirect
59. #2071C22semidirect
61. #2058C22semidirect
62. #2057C22semidirect
64. #2023C22semidirect
65. #1990C22semidirect
66. #1988C22semidirect
68. #1948C22semidirect
69. #1947C22semidirect
70. #1939C22semidirect
74. #1917C22semidirect
85. #1847C22semidirect
86. #1836C22semidirect
92. #1770C22semidirect

Maximal subgroups

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

subgroupindex
#361762
C2×#25422
#329562
#324462
#323642
#299072
#285412
#277362
#275882
#258112
#254512
#246762
#245062
#242982
#226032
#219872
C4×#4472