Finite Group Databasebeta

← #6462201

Subgroups — #6462201

Order 512.

2034 conjugacy classes of subgroups.

The subgroup lattice has 773 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36631C211yesmaximal, characteristic, complemented
2C4×#532C211yesmaximal, characteristic, complemented
3#29574C212yesmaximal, complemented
4#29552C212yesmaximal, complemented
5#29513C212yesmaximal, complemented
6#29493C212yesmaximal, complemented
7#29162C212yesmaximal, complemented
8#28578C212yesmaximal, complemented
9#27863C212yesmaximal, complemented
10#25502C212yesmaximal, complemented
11#23839C211yesmaximal, characteristic, complemented
12#23822C211yesmaximal, characteristic, complemented
13#23822C211yesmaximal, characteristic, complemented
14#23802C211yesmaximal, characteristic, complemented
15#21538C214yesmaximal, complemented
16#21122C214yesmaximal, complemented
17#20364C211yesmaximal, characteristic, complemented
18C2×#532C2211yescharacteristic
19C2×#529—21no—
20C2×C4×#41—21no—
21C2×#469C2214yes—
22C2×#442C2214yes—
23C2×C4×M16C2212yes—
24#2861—21no—
25#2860—21no—
26#2860C2212yes—
27#2859C2211yescharacteristic
28#2858C2211yescharacteristic
29#2857C2211yescharacteristic
30#2856C2211yescharacteristic
31#2854—21no—
32#2854C2212yes—
33#2850—21no—
34C4×C4⋊Q8C2211yescharacteristic
35C4×C4⋊Q8C2211yescharacteristic
36C4×C4⋊Q8C2211yescharacteristic
37C4×#127C2211yescharacteristic
38C4×#127C2211yescharacteristic
39C4×#127C2211yescharacteristic
40C4×#124—21no—
41C4×#124—21no—
42C4×#124—21no—
43C4×#121—21no—
44C4×#121—21no—
45C4×#121—21no—
46C4×#121C2212yes—
47C4×#121C2212yes—
48C4×#121C2212yes—
49#2297—21no—
50#2293—21no—
51#2293—21no—
52#2293C2212yes—
53#2293C2212yes—
54#2285C2211yescharacteristic
55C42×D4—21no—
56C42×D4C2212yes—
57C2×C43C2211yescharacteristic
58#2100C2212yes—
59#2100C2212yes—
60#2094C2212yes—
61#2094C2212yes—
62#2084C2212yes—
63#2083C2212yes—
64#2071C2212yes—
65#2071C2212yes—
66#2058C2212yes—
67#2057C2212yes—
68#2041C2212yes—
69#2041C2212yes—
70#2023C2212yes—
71#2023C2212yes—
72#2011C2212yes—
73#2011C2212yes—
74#1970C2214yes—
75#1941C2214yes—
76#1934C2214yes—
77#1934C2214yes—
78#1926C2214yes—
79#1925C2214yes—
80#1920C2214yes—
81#1919C2214yes—
82#1895C2214yes—
83#1895C2214yes—
84#1879C2214yes—
85#1879C2214yes—
86#1864C2214yes—
87#1862C2214yes—
88#1809C2214yes—
89#1776C2214yes—
90#1762C2214yes—
91#1762C2214yes—
92#1745C2211yescharacteristic
93#1745C2211yescharacteristic
94#1745C2212yes—
95#1744C2212yes—
96#1744C2212yes—
97#1728C2212yes—
98#1728C2212yes—
99C22×#41—21no—
100C22×D8C2314yes—
101C22×M16C2314yes—
102#532—28no—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2311yescharacteristic
105C2×#127C2311yescharacteristic
106C2×#127C2311yescharacteristic
107#529—48no—
108C2×#126—22no—
109C2×#125—21no—
110C2×#124—21no—
111C2×#124—21no—
112C2×#124—21no—
113C2×#123—22no—
114C2×C22⋊Q8—21no—
115C2×#121—21no—
116C2×#121—21no—
117C2×#121—21no—
118C2×#121C2312yes—
119C2×#121C2312yes—
120C4×#41—48no—
121C2×C4×Q8—21no—
122C2×C4×Q8—21no—
123C2×C4×Q8—21no—
124C2×C4×D4—22no—
125C2×C4×D4—22no—
126C2×C4×D4—22no—
127C2×C4×D4—21no—
128C2×C4×D4—21no—
129C2×C4×D4—21no—
130C2×C4×D4—21no—
131C2×C4×D4—21no—
132C2×C4×D4—21no—
133C2×C4×D4—21no—
134C2×C4×D4C2312yes—
135C2×C4×D4C2312yes—
136C2×C4×D4C2312yes—
137C2×C4×D4C2312yes—
138C2×#117—21no—
139C2×#117—22no—
140C2×#117—22no—
141C22×C42—22no—
142C22×C42D411yescharacteristic
143C22×C42D411yescharacteristic
144C22×C42D411yescharacteristic
145C22×C42D411yescharacteristic
146C22×C42C2311yescharacteristic
147C22×C42C2311yescharacteristic
148C22×C42C2311yescharacteristic
149#469—216no—
150#442—216no—
151C2×#107C2314yes—
152C2×#106C2314yes—
153C2×#106C2314yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×#102C2314yes—
157C2×#102C2314yes—
158C2×#102C2314yes—
159C2×#102C2314yes—
160C4×M16—28no—
161C2×#97C2312yes—
162C2×#97C2312yes—
163C2×C4×C8C2314yes—
164#391—21no—
165#391—21no—
166#391—21no—
167#391—21no—
168#390—21no—
169#390—21no—
170#390—21no—
171#390—21no—
172#390—21no—
173#390—21no—
174#390—21no—
175#390—21no—
176#390—21no—
177#390—21no—
178#390C2312yes—
179#390C2312yes—
180#390C2312yes—
181#390C2312yes—
182#390C2312yes—
183#390C2312yes—
184#389—21no—
185#389—21no—
186#389—21no—
187#389—21no—
188#389—21no—
189#389—21no—
190#389C2312yes—
191#389C2312yes—
192#389C2312yes—
193#389C2312yes—
194#389C2312yes—
195#389C2312yes—
196#388—22no—
197#388—22no—
198#388—22no—
199#388—22no—
200#388—22no—
201#388—22no—
202#388—22no—
203#388—22no—
204#388—22no—
205#388—22no—
206#388—22no—
207#388—22no—
208#386—22no—
209#386—22no—
210#386—22no—
211#386—22no—
212#384C2311yescharacteristic
213#384C2311yescharacteristic
214#384C2311yescharacteristic
215#384C2311yescharacteristic
216#382—21no—
217#382—21no—
218#380—21no—
219#380—21no—
220#380C2312yes—
221#380C2312yes—
222C4×C4⋊C4—21no—
223C4×C4⋊C4—21no—
224C4×C4⋊C4—21no—
225C4×C4⋊C4—21no—
226C4×C4⋊C4C2312yes—
227C4×C4⋊C4C2312yes—
228C4×C4⋊C4C2312yes—
229C4×C4⋊C4C2312yes—
230C4×C22⋊C4—22no—
231C4×C22⋊C4—22no—
232C4×C22⋊C4—22no—
233C4×C22⋊C4—22no—
234C4×C22⋊C4—22no—
235C4×C22⋊C4—22no—
236C4×C22⋊C4—22no—
237C4×C22⋊C4—22no—
238C43D411yescharacteristic
239C43D411yescharacteristic
240C43D411yescharacteristic
241C43D411yescharacteristic
242C43C2311yescharacteristic
243C43C2311yescharacteristic
244C43C2311yescharacteristic
245C43C2311yescharacteristic
246#340C2314yes—
247#340C2314yes—
248#340C2314yes—
249#340C2314yes—
250#340C2314yes—
251#340C2314yes—
252#336C2312yes—
253#336C2312yes—
254#336C2312yes—
255#336C2312yes—
256#336C2314yes—
257#336C2314yes—
258C2×#41—28no—
259C2×#41—44no—
260C2×#41—44no—
261C22×Q8—21no—
262C22×Q8—21no—
263C22×D4—22no—
264C22×D4—22no—
265C22×D4—21no—
266C22×D4—21no—
267C22×D4—21no—
268C22×D4—21no—
269C22×D4C2412yes—
270C22×D4C2412yes—
271C23×C4—22no—
272C23×C4—22no—
273C23×C4—22no—
274C23×C4—22no—
275C23×C4C2×D411yescharacteristic
276C23×C4C2×D411yescharacteristic
277C23×C4C2×D411yescharacteristic
278C23×C4C2×D411yescharacteristic
279C23×C4C2×D411yescharacteristic
280C23×C4C2×D411yescharacteristic
281C23×C4C2411yescharacteristic
282C2×D8—28no—
283C2×D8—28no—
284C2×D8—44no—
285C2×M16—28no—
286C2×M16—28no—
287C2×M16—44no—
288C22×C8C2414yes—
289C22×C8C2414yes—
290C4⋊Q8—24no—
291C4⋊Q8—42no—
292#127—24no—
293#127—42no—
294#126—48no—
295#125—44no—
296#124—28no—
297#124—44no—
298#124—44no—
299#123—48no—
300C22⋊Q8—44no—
301#121—28no—
302#121—24no—
303#121—28no—
304#121—44no—
305#121—44no—
306#121#4118yes—
307C4×Q8—44no—
308C4×Q8—44no—
309C4×Q8—44no—
310C4×D4—28no—
311C4×D4—24no—
312C4×D4—24no—
313C4×D4—28no—
314C4×D4—28no—
315C4×D4—44no—
316C4×D4—44no—
317C4×D4—44no—
318C4×D4—44no—
319C4×D4—44no—
320C4×D4—44no—
321C4×D4—48no—
322C4×D4—48no—
323C4×D4—48no—
324C4×D4—44no—
325#117—48no—
326#117—44no—
327#117—48no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4C2412yes—
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2412yes—
348C2×C4⋊C4C2412yes—
349C2×C4⋊C4C2412yes—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C42—24no—
371C2×C42—24no—
372C2×C42—24no—
373C2×C42—24no—
374C2×C42—22no—
375C2×C42—22no—
376C2×C42—22no—
377C2×C42—22no—
378C2×C42—42no—
379C2×C42—42no—
380C2×C42—42no—
381C2×C42—48no—
382C2×C42C2×D411yescharacteristic
383C2×C42C2×D411yescharacteristic
384C2×C42C2×D411yescharacteristic
385C2×C42C2×D411yescharacteristic
386C2×C42C2×D411yescharacteristic
387C2×C42C2×D411yescharacteristic
388C2×C42C2×D411yescharacteristic
389C2×C42C2×D411yescharacteristic
390C2×C42C2×D411yescharacteristic
391C2×C42C2×D411yescharacteristic
392C2×C42C2×D411yescharacteristic
393C2×C42C2×D411yescharacteristic
394C2×C42C2×D411yescharacteristic
395C2×C42C2×D411yescharacteristic
396C2×C42C2×D411yescharacteristic
397C2×C42C2×D411yescharacteristic
398C2×C42C2×D411yescharacteristic
399C2×C42C2×D411yescharacteristic
400C2×C42C2×D411yescharacteristic
401C2×C42C2×D411yescharacteristic
402C2×C42C2×D411yescharacteristic
403C2×C42C2×D411yescharacteristic
404C2×C42C2411yescharacteristic
405C2×C42C2411yescharacteristic
406C2×C42C2411yescharacteristic
407C2×C42C2411yescharacteristic
408C2×C42C2411yescharacteristic
409C2×C42C2411yescharacteristic
410#107—44no—
411#106—28no—
412#106—28no—
413#102—28no—
414#102—28no—
415#102—28no—
416#102—28no—
417#102—44no—
418#102—44no—
419#97—24no—
420#97#4118yescomplemented
421C4×C8—28no—
422#95—22no—
423#95—22no—
424#95—22no—
425#95—22no—
426#95—22no—
427#95—22no—
428#95—22no—
429#95—22no—
430#95—22no—
431#95—22no—
432#95—22no—
433#95—22no—
434C24—22no—
435C24—22no—
436C24C22×D411yescharacteristic
437#41—432no—
438C2×Q8—24no—
439C2×Q8—42no—
440C2×Q8—42no—
441C2×Q8—42no—
442C2×Q8—42no—
443C2×Q8—42no—
444C2×D4—24no—
445C2×D4—24no—
446C2×D4—28no—
447C2×D4—24no—
448C2×D4—22no—
449C2×D4—24no—
450C2×D4—24no—
451C2×D4—24no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—42no—
455C2×D4—42no—
456C2×D4—42no—
457C2×D4—42no—
458C2×D4—42no—
459C2×D4—42no—
460C2×D4—44no—
461C2×D4—44no—
462C2×D4—44no—
463C2×D4—44no—
464C2×D4—44no—
465C2×D4—42no—
466C2×D4—42no—
467C2×D4—42no—
468C2×D4—42no—
469C2×D4C2×#4114yes—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—28no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—41no—
492C22×C4—41no—
493C22×C4—41no—
494C22×C4—41no—
495C22×C4—41no—
496C22×C4—41no—
497C22×C4—44no—
498C22×C4—44no—
499C22×C4—44no—
500C22×C4—44no—
501C22×C4—44no—
502C22×C4—44no—
503C22×C4—44no—
504C22×C4—44no—
505C22×C4—44no—
506C22×C4—44no—
507C22×C4—44no—
508C22×C4C22×D411yescharacteristic
509C22×C4C22×D411yescharacteristic
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D411yescharacteristic
512C22×C4C22×D411yescharacteristic
513C22×C4C22×D411yescharacteristic
514C22×C4C22×D411yescharacteristic
515C22×C4C22×D411yescharacteristic
516C22×C4C22×D411yescharacteristic
517C22×C4C22×D411yescharacteristic
518C22×C4C22×D411yescharacteristic
519C22×C4C22×D411yescharacteristic
520C22×C4C22×D411yescharacteristic
521C22×C4C2×#4114yes—
522C22×C4C2511yescharacteristic
523D8—416no—
524M16—416no—
525C2×C8—24no—
526C2×C8—24no—
527C2×C8—24no—
528C2×C8—24no—
529C2×C8—24no—
530C2×C8C2×#4118yes—
531C4⋊C4—24no—
532C4⋊C4—22no—
533C4⋊C4—24no—
534C4⋊C4—24no—
535C4⋊C4—24no—
536C4⋊C4—22no—
537C4⋊C4—24no—
538C4⋊C4—22no—
539C4⋊C4—42no—
540C4⋊C4—42no—
541C4⋊C4—44no—
542C4⋊C4—42no—
543C4⋊C4—42no—
544C4⋊C4—42no—
545C4⋊C4—42no—
546C4⋊C4—42no—
547C4⋊C4—44no—
548C4⋊C4—44no—
549C4⋊C4—44no—
550C4⋊C4—42no—
551C4⋊C4—44no—
552C4⋊C4—44no—
553C4⋊C4—42no—
554C4⋊C4C2×#4114yes—
555C22⋊C4—28no—
556C22⋊C4—28no—
557C22⋊C4—44no—
558C22⋊C4—44no—
559C22⋊C4—44no—
560C22⋊C4—44no—
561C22⋊C4—44no—
562C22⋊C4—44no—
563C22⋊C4—44no—
564C22⋊C4—44no—
565C22⋊C4—44no—
566C22⋊C4—44no—
567C22⋊C4—44no—
568C22⋊C4—44no—
569C22⋊C4—44no—
570C22⋊C4—44no—
571C22⋊C4—44no—
572C22⋊C4—44no—
573C22⋊C4—44no—
574C22⋊C4—44no—
575C42—22no—
576C42—22no—
577C42—22no—
578C42—22no—
579C42—22no—
580C42—22no—
581C42—22no—
582C42—22no—
583C42—22no—
584C42—22no—
585C42—22no—
586C42—22no—
587C42—22no—
588C42—22no—
589C42—22no—
590C42—22no—
591C42—22no—
592C42—22no—
593C42—22no—
594C42—41no—
595C42—41no—
596C42—41no—
597C42—41no—
598C42—41no—
599C42—41no—
600C42—44no—
601C42—44no—
602C42—44no—
603C42—44no—
604C42AGL(1,ℤ/8)14yes—
605C42AGL(1,ℤ/8)14yes—
606C42C2×#4114yes—
607C23—21no—
608C23—21no—
609C23—21no—
610C23—21no—
611C23—21no—
612C23—21no—
613C23—24no—
614C23—42no—
615C23—42no—
616C23—42no—
617C23—42no—
618C23—42no—
619C23—42no—
620C23—42no—
621C23—42no—
622C23—42no—
623C23—42no—
624C23—42no—
625C23—42no—
626C23—42no—
627C23#52512yes—
628C23C23×D411yescharacteristic
629Q8—48no—
630Q8—84no—
631D4—28no—
632D4—48no—
633D4—48no—
634D4—416no—
635D4—44no—
636D4—44no—
637D4—44no—
638D4—84no—
639D4—84no—
640D4—88no—
641C2×C4—28no—
642C2×C4—21no—
643C2×C4—21no—
644C2×C4—21no—
645C2×C4—21no—
646C2×C4—21no—
647C2×C4—21no—
648C2×C4—21no—
649C2×C4—21no—
650C2×C4—21no—
651C2×C4—21no—
652C2×C4—21no—
653C2×C4—21no—
654C2×C4—21no—
655C2×C4—21no—
656C2×C4—21no—
657C2×C4—21no—
658C2×C4—21no—
659C2×C4—21no—
660C2×C4—21no—
661C2×C4—21no—
662C2×C4—21no—
663C2×C4—21no—
664C2×C4—28no—
665C2×C4—21no—
666C2×C4—24no—
667C2×C4—21no—
668C2×C4—21no—
669C2×C4—21no—
670C2×C4—21no—
671C2×C4—21no—
672C2×C4—44no—
673C2×C4—44no—
674C2×C4—44no—
675C2×C4—44no—
676C2×C4—44no—
677C2×C4—416no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—42no—
684C2×C4—42no—
685C2×C4—42no—
686C2×C4—42no—
687C2×C4—42no—
688C2×C4—42no—
689C2×C4—42no—
690C2×C4—42no—
691C2×C4—42no—
692C2×C4—42no—
693C2×C4—42no—
694C2×C4—42no—
695C2×C4—88no—
696C2×C4—88no—
697C2×C4—88no—
698C2×C4#52512yes—
699C2×C4#53212yes—
700C2×C4#53212yes—
701C2×C4C2×AGL(1,ℤ/8)12yes—
702C2×C4C2×AGL(1,ℤ/8)12yes—
703C2×C4C2×AGL(1,ℤ/8)12yes—
704C2×C4C2×AGL(1,ℤ/8)12yes—
705C2×C4C2×AGL(1,ℤ/8)12yes—
706C2×C4C2×AGL(1,ℤ/8)12yes—
707C2×C4#57512yes—
708C2×C4#57512yes—
709C2×C4C22×#4112yes—
710C2×C4C22×#4112yes—
711C2×C4#58512yes—
712C8—28no—
713C8—44no—
714C22—22no—
715C22—22no—
716C22—22no—
717C22—41no—
718C22—41no—
719C22—41no—
720C22—41no—
721C22—44no—
722C22—44no—
723C22—44no—
724C22—82no—
725C22—82no—
726C22—82no—
727C22—82no—
728C22—82no—
729C22—82no—
730C22—82no—
731C22—82no—
732C22—82no—
733C22—82no—
734C22—82no—
735C22C2×#52511yescharacteristic
736C22C2×#53211yescharacteristic
737C22#344911yescharacteristic
738C22C22×AGL(1,ℤ/8)11yescharacteristic
739C22C22×AGL(1,ℤ/8)11yescharacteristic
740C22C2×#57511yescharacteristic
741C22C2×#57511yescharacteristic
742C4—22no—
743C4—22no—
744C4—22no—
745C4—22no—
746C4—22no—
747C4—22no—
748C4—22no—
749C4—22no—
750C4—22no—
751C4—22no—
752C4—22no—
753C4—41no—
754C4—44no—
755C4—41no—
756C4—82no—
757C4—82no—
758C4—82no—
759C4—82no—
760C4—82no—
761C4#347514yes—
762C2—42no—
763C2—84no—
764C2—162no—
765C2#3487611yescharacteristic
766C2#6055811yescharacteristic
767C2#6056111yescharacteristic
768C2#6070011yescharacteristic
769C2#6070011yescharacteristic
770C2#6096811yescharacteristic
771C2#6097011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36631C2semidirect
2. C4×#532C2semidirect
3. #29574C2semidirect
4. #29552C2semidirect
5. #29513C2semidirect
6. #29493C2semidirect
7. #29162C2semidirect
8. #28578C2semidirect
9. #27863C2semidirect
10. #25502C2semidirect
11. #23839C2semidirect
12. #23822C2semidirect
13. #23822C2semidirect
14. #23802C2semidirect
15. #21538C2semidirect
16. #21122C2semidirect
17. #20364C2semidirect
420. #97#41semidirect

Maximal subgroups

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

subgroupindex
#366312
C4×#5322
#295742
#295522
#295132
#294932
#291622
#285782
#278632
#255022
#238392
#238222
#238022
#215382
#211222
#203642