Finite Group Databasebeta

← #6852325

Subgroups — #6852325

Order 512.

1835 conjugacy classes of subgroups.

The subgroup lattice has 724 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36561C211yesmaximal, characteristic, complemented
2C2×#2563C211yesmaximal, characteristic, complemented
3#32953C211yesmaximal, characteristic, complemented
4#32371C214yesmaximal, complemented
5#31547C214yesmaximal, complemented
6#29952C212yesmaximal, complemented
7#28541C212yesmaximal, complemented
8#27956C212yesmaximal, complemented
9#27821C212yesmaximal, complemented
10#27684C212yesmaximal, complemented
11#25811C211yesmaximal, characteristic, complemented
12#25452C212yesmaximal, complemented
13#24812C212yesmaximal, complemented
14#24299C211yesmaximal, characteristic, complemented
15#22629C211yesmaximal, characteristic, complemented
16#22046C212yesmaximal
17C4×#447C211yesmaximal, characteristic
18C2×#554—21no—
19C22×C4⋊Q8C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C2×#498C2212yescomplemented
22C2×#476C2212yescomplemented
23C2×#474C2212yes—
24C2×#447C2211yescharacteristic
25C2×C4×D8C2212yes—
26#2857C2212yes—
27#2857C2211yescharacteristic
28#2564—22no—
29#2563—21no—
30#2563—21no—
31#2563—22no—
32#2563C2214yes—
33#2563C2212yes—
34#2563C2212yes—
35#2556—22no—
36#2551C2211yescharacteristic, complemented
37#2542C2211yescharacteristic, complemented
38#2506C2212yescomplemented
39#2492—21no—
40#2452C2212yescomplemented
41#2417—21no—
42C2×#393—22no—
43#2365C2211yescharacteristic, complemented
44#2360—21no—
45D4×C4⋊C4C2211yescharacteristic, complemented
46C4×#121C2212yes—
47C4×#121C2211yescharacteristic
48C2×#390C2212yes—
49C2×#382C2212yes—
50C2×#380—21no—
51C2×#380—21no—
52C2×C4×C4⋊C4C2211yescharacteristic
53#2101C2214yescomplemented
54#2094C2214yes—
55#2092C2214yescomplemented
56#2085C2214yescomplemented
57#2085C2214yescomplemented
58#2065C2214yes—
59#2065C2214yes—
60#2058C2214yescomplemented
61#2058C2212yescomplemented
62#2023C2214yes—
63#2023C2212yescomplemented
64#1993C2214yescomplemented
65#1990C2212yescomplemented
66#1987C2212yescomplemented
67#1972C2214yes—
68#1949C2214yescomplemented
69#1939C2212yescomplemented
70#1936C2214yes—
71#1926C2214yes—
72#1917C2214yescomplemented
73#1879C2214yes—
74#1879C2212yes—
75#1874C2212yes—
76#1874C2211yescharacteristic
77#1862C2214yes—
78#1862C2212yes—
79#1853C2212yes—
80#1853C2212yes—
81#1853C2212yes—
82#1852C2212yes—
83#1852C2211yescharacteristic
84#1847C2214yescomplemented
85#1804C2214yescomplemented
86#1802C2212yes—
87#1797C2212yes—
88#1797C2211yescharacteristic
89#1786C2211yescharacteristic
90#1786C2212yes—
91#1770C2212yescomplemented
92C4×#102C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—21no—
96C23×D4C2311yescharacteristic
97C24×C4—21no—
98C22×D8C2312yes—
99#554—48no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—44no—
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2312yes—
108C2×#125—22no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8—21no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8—21no—
114C2×#121C2311yescharacteristic
115C2×C4×Q8—21no—
116C2×C4×D4—22no—
117C2×C4×D4—21no—
118C2×C4×D4—21no—
119C2×C4×D4—21no—
120C2×C4×D4—44no—
121C2×C4×D4C2312yes—
122C2×C4×D4C2311yescharacteristic
123C2×C4×D4C2312yes—
124C2×C4×D4C2312yes—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—21no—
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#447Q818yescomplemented
140C4×D8—44no—
141C2×#107C2312yes—
142C2×#107C2314yes—
143C2×#106C2314yes—
144C2×#106C2314yes—
145C2×#106C2314yes—
146C2×C4⋊C8C2312yes—
147C2×C4⋊C8C2314yes—
148C2×#102C2314yes—
149C2×#102C2312yes—
150C2×#102C2314yes—
151C2×#102C2312yes—
152C2×#102C2312yes—
153C2×#102C2314yes—
154C2×#102C2312yes—
155C2×C22⋊C8C2312yes—
156C2×C22⋊C8C2311yescharacteristic
157C2×C22⋊C8C2311yescharacteristic
158C2×#97C2314yes—
159C2×C4×C8C2312yes—
160#400—22no—
161#399—22no—
162#398—22no—
163#396—22no—
164#395—22no—
165#393—24no—
166#393—24no—
167#393—24no—
168#393—24no—
169#393—22no—
170#391—21no—
171#391—21no—
172#391—22no—
173#390—22no—
174#390—22no—
175#390C2314yes—
176#390C2314yes—
177#390C2312yes—
178#390C2311yescharacteristic
179#389—21no—
180#389—22no—
181#389—21no—
182#389C2312yes—
183#389C2311yescharacteristic
184#389C2312yes—
185#389C2312yes—
186#389C2312yes—
187#388—22no—
188#388—21no—
189#387—22no—
190#387—22no—
191#386—21no—
192#386—22no—
193#385—22no—
194#385—22no—
195#384C2311yescharacteristic
196#384C2312yes—
197#382—22no—
198#382—22no—
199#382C2314yes—
200#382C2314yes—
201#380—21no—
202#380—22no—
203#380—22no—
204#380—24no—
205#380—22no—
206#380—22no—
207#380—22no—
208#380—24no—
209#380—21no—
210#380C2312yes—
211#380C2311yescharacteristic
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4D412yes—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2312yes—
219C4×C4⋊C4C2311yescharacteristic
220C4×C22⋊C4—21no—
221C4×C22⋊C4—22no—
222C2×#95—22no—
223C2×#95—22no—
224C2×#95D412yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#336C2312yes—
233#336C2314yes—
234#336C2312yes—
235#336C2314yes—
236C25—21no—
237C22×Q8—21no—
238C22×Q8—22no—
239C22×Q8—22no—
240C22×Q8—22no—
241C22×Q8—44no—
242C22×Q8—82no—
243C22×Q8—82no—
244C22×D4—21no—
245C22×D4—21no—
246C22×D4—22no—
247C22×D4—42no—
248C22×D4—42no—
249C22×D4C2×Q814yes—
250C22×D4C2411yescharacteristic
251C22×D4C2412yes—
252C23×C4—22no—
253C23×C4—21no—
254C23×C4—21no—
255C23×C4—22no—
256C23×C4—22no—
257C23×C4—22no—
258C23×C4—44no—
259C23×C4—44no—
260C23×C4—44no—
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D412yes—
264C23×C4C2×D412yes—
265C23×C4C2411yescharacteristic
266C2×D8—42no—
267C2×D8—42no—
268C2×D8C2×Q818yescomplemented
269C22×C8C2412yes—
270C22×C8C2414yes—
271C22×C8C2412yes—
272C4⋊Q8—24no—
273C4⋊Q8—24no—
274C4⋊Q8—24no—
275C4⋊Q8—44no—
276C4⋊Q8—44no—
277C4⋊Q8—44no—
278C4⋊Q8—44no—
279C4⋊Q8—44no—
280#125—48no—
281C22⋊Q8—44no—
282C22⋊Q8—44no—
283C22⋊Q8—48no—
284C22⋊Q8—48no—
285C22⋊Q8—44no—
286#121—22no—
287#121C2×Q814yes—
288C4×Q8—44no—
289C4×D4—24no—
290C4×D4—24no—
291C4×D4—24no—
292C4×D4—42no—
293C4×D4—42no—
294C4×D4—44no—
295C4×D4—44no—
296C4×D4—44no—
297C4×D4—48no—
298C4×D4—44no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—42no—
316C2×C4⋊C4—42no—
317C2×C4⋊C4—44no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2×D412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2414yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2411yescharacteristic
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2411yescharacteristic
333C2×C22⋊C4—21no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—24no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—44no—
349C2×C22⋊C4—84no—
350C2×C22⋊C4—84no—
351C2×C42—24no—
352C2×C42—22no—
353C2×C42—21no—
354C2×C42—44no—
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D412yes—
358C2×C42C2×D412yes—
359C2×C42C2411yescharacteristic
360C2×C42C2412yes—
361C2×C42C2412yes—
362C2×C42C2411yescharacteristic
363#107—42no—
364#107—44no—
365#106—44no—
366#106—44no—
367#106—44no—
368C4⋊C8—24no—
369C4⋊C8—44no—
370#102—44no—
371#102—42no—
372#102—42no—
373#102—42no—
374#102—44no—
375#102—44no—
376#102C2×Q818yescomplemented
377C22⋊C8—24no—
378C22⋊C8—22no—
379C22⋊C8C2×Q814yes—
380#97—44no—
381C4×C8—24no—
382#95—24no—
383#95—24no—
384#95—24no—
385#95—24no—
386#95—22no—
387#95—21no—
388#95—21no—
389#95—22no—
390#95—21no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95C2×D412yes—
396#95C2×D412yes—
397#95C2×D412yes—
398#95C2×D412yes—
399C24—24no—
400C24—21no—
401C24—22no—
402C24—42no—
403C24—42no—
404C24—42no—
405C24—42no—
406C24—42no—
407C24—42no—
408C24C22×D411yescharacteristic
409C2×Q8—44no—
410C2×Q8—44no—
411C2×Q8—44no—
412C2×Q8—42no—
413C2×Q8—42no—
414C2×Q8—42no—
415C2×Q8—82no—
416C2×Q8—82no—
417C2×Q8—82no—
418C2×Q8—82no—
419C2×Q8—82no—
420C2×Q8—82no—
421C2×Q8—816no—
422C2×D4—24no—
423C2×D4—24no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—21no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—48no—
433C2×D4—42no—
434C2×D4—42no—
435C2×D4—48no—
436C2×D4—42no—
437C2×D4—42no—
438C2×D4C22×Q814yes—
439C2×D4C22×Q812yes—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—24no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—41no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—42no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—42no—
475C22×C4—44no—
476C22×C4—82no—
477C22×C4—82no—
478C22×C4—816no—
479C22×C4AGL(1,ℤ/8)14yes—
480C22×C4C22×D412yes—
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×Q812yes—
490C22×C42−1+412yes—
491C22×C4C2511yescharacteristic
492D8—48no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—24no—
497C2×C8—22no—
498C2×C8—24no—
499C2×C8—24no—
500C2×C8—22no—
501C2×C8C22×Q814yes—
502C4⋊C4—22no—
503C4⋊C4—24no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—21no—
508C4⋊C4—22no—
509C4⋊C4—24no—
510C4⋊C4—24no—
511C4⋊C4—22no—
512C4⋊C4—24no—
513C4⋊C4—22no—
514C4⋊C4—24no—
515C4⋊C4—22no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—44no—
530C4⋊C4—81no—
531C4⋊C4—82no—
532C4⋊C4C22×Q812yes—
533C22⋊C4—24no—
534C22⋊C4—44no—
535C22⋊C4—44no—
536C22⋊C4—44no—
537C22⋊C4—44no—
538C22⋊C4—42no—
539C22⋊C4—44no—
540C22⋊C4—42no—
541C22⋊C4—42no—
542C22⋊C4—81no—
543C22⋊C4—84no—
544C22⋊C4—84no—
545C22⋊C4—84no—
546C22⋊C4—84no—
547C22⋊C4—82no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—24no—
552C42—24no—
553C42—42no—
554C42—42no—
555C42—44no—
556C42—42no—
557C42H(5,𝔽2)14yes—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—22no—
563C23—24no—
564C23—48no—
565C23—48no—
566C23—42no—
567C23—41no—
568C23—48no—
569C23—42no—
570C23—41no—
571C23—41no—
572C23—42no—
573C23—41no—
574C23—42no—
575C23—41no—
576C23—42no—
577C23—42no—
578C23—41no—
579C23—84no—
580C23—84no—
581C23—84no—
582C23—84no—
583C23D4×Q812yes—
584C23C2×AGL(1,ℤ/8)12yes—
585C23#57712yes—
586C23C23×D411yescharacteristic
587Q8—88no—
588Q8—88no—
589Q8—84no—
590Q8—164no—
591D4—24no—
592D4—24no—
593D4—24no—
594D4—48no—
595D4—48no—
596D4—42no—
597D4—42no—
598D4—42no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—28no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—22no—
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—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—21no—
624C2×C4—44no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—44no—
628C2×C4—41no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—41no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—41no—
636C2×C4—42no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—41no—
643C2×C4—42no—
644C2×C4—44no—
645C2×C4—88no—
646C2×C4—84no—
647C2×C4—84no—
648C2×C4—84no—
649C2×C4—88no—
650C2×C4—84no—
651C2×C4—88no—
652C2×C4—164no—
653C2×C4#53512yes—
654C2×C4#53512yes—
655C2×C4D4×Q812yes—
656C2×C4C2×AGL(1,ℤ/8)12yes—
657C2×C4C2×AGL(1,ℤ/8)12yes—
658C2×C4#57512yes—
659C2×C4C23×Q811yescharacteristic
660C2×C4C2×H(5,𝔽2)12yes—
661C2×C4C2×2−1+411yescharacteristic
662C2×C4#58512yes—
663C8—24no—
664C8—42no—
665C8—44no—
666C22—22no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—22no—
671C22—22no—
672C22—44no—
673C22—42no—
674C22—42no—
675C22—44no—
676C22—42no—
677C22—44no—
678C22—816no—
679C22—81no—
680C22—82no—
681C22—81no—
682C22—82no—
683C22—82no—
684C22—81no—
685C22—81no—
686C22—82no—
687C22#338414yes—
688C22C2×#53511yescharacteristic
689C22C2×D4×Q811yescharacteristic
690C22#348511yescharacteristic
691C22C22×AGL(1,ℤ/8)11yescharacteristic
692C22C2×#57511yescharacteristic
693C22C2×#57711yescharacteristic
694C22#358511yescharacteristic
695C4—22no—
696C4—22no—
697C4—22no—
698C4—42no—
699C4—42no—
700C4—42no—
701C4—42no—
702C4—41no—
703C4—42no—
704C4—82no—
705C4—81no—
706C4—81no—
707C4—82no—
708C4—82no—
709C4—84no—
710C4#321314yes—
711C4#350112yes—
712C4#354112yes—
713C2—24no—
714C2—82no—
715C2—84no—
716C2#3959411yescharacteristic
717C2C2×#321311yescharacteristic
718C2#6081211yescharacteristic
719C2C2×#338411yescharacteristic
720C2#6136711yescharacteristic
721C2#6172011yescharacteristic
722C2#6173111yescharacteristic
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. #36561C2semidirect
2. C2×#2563C2semidirect
3. #32953C2semidirect
4. #32371C2semidirect
5. #31547C2semidirect
6. #29952C2semidirect
7. #28541C2semidirect
8. #27956C2semidirect
9. #27821C2semidirect
10. #27684C2semidirect
11. #25811C2semidirect
12. #25452C2semidirect
13. #24812C2semidirect
14. #24299C2semidirect
15. #22629C2semidirect
21. C2×#498C22semidirect
22. C2×#476C22semidirect
36. #2551C22semidirect
37. #2542C22semidirect
38. #2506C22semidirect
40. #2452C22semidirect
43. #2365C22semidirect
45. D4×C4⋊C4C22semidirect
53. #2101C22semidirect
55. #2092C22semidirect
56. #2085C22semidirect
57. #2085C22semidirect
60. #2058C22semidirect
61. #2058C22semidirect
63. #2023C22semidirect
64. #1993C22semidirect
65. #1990C22semidirect
66. #1987C22semidirect
68. #1949C22semidirect
69. #1939C22semidirect
72. #1917C22semidirect
84. #1847C22semidirect
85. #1804C22semidirect
91. #1770C22semidirect
139. #447Q8semidirect
268. C2×D8C2×Q8semidirect
376. #102C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#365612
C2×#25632
#329532
#323712
#315472
#299522
#285412
#279562
#278212
#276842
#258112
#254522
#248122
#242992
#226292
#220462
C4×#4472