Finite Group Databasebeta

← #6848664

Subgroups — #6848664

Order 512.

1651 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36188C211yesmaximal, characteristic, complemented
2#34932C211yesmaximal, characteristic
3#33195C211yesmaximal, characteristic, complemented
4#32203C214yesmaximal, complemented
5#31494C214yesmaximal, complemented
6#29692C212yesmaximal, complemented
7#28717C211yesmaximal, characteristic, complemented
8#28600C212yesmaximal, complemented
9#28284C211yesmaximal, characteristic, complemented
10#27631C212yesmaximal, complemented
11#26443C212yesmaximal, complemented
12#25338C212yesmaximal
13#25275C212yesmaximal, complemented
14#24837C212yesmaximal, complemented
15D4×C4⋊C8C211yesmaximal, characteristic, complemented
16#21887C212yesmaximal, complemented
17C4×#476C211yesmaximal, characteristic, complemented
18C2×#556—21no—
19C2×#529C2212yes—
20C2×#487C2212yes—
21C2×#477C2212yes—
22C2×#476C2211yescharacteristic
23C2×#474C2212yes—
24C2×C8×D4C2212yes—
25C22×C4⋊C8C2212yes—
26#2868C2211yescharacteristic
27#2857C2212yes—
28#2568—21no—
29#2553—22no—
30#2552C2211yescharacteristic
31#2547—21no—
32#2547C2212yes—
33#2547C2211yescharacteristic
34#2542C2211yescharacteristic
35#2537C2211yescharacteristic
36#2499—22no—
37#2496—21no—
38#2496—22no—
39#2496C2214yes—
40#2489C2212yes—
41#2427C2212yes—
42#2422—21no—
43#2395—22no—
44#2365—21no—
45#2365C2212yes—
46C4×C4⋊Q8C2211yescharacteristic
47C4×C4⋊Q8C2211yescharacteristic
48C4×C22⋊Q8C2212yes—
49C4×#121—21no—
50#2285—21no—
51#2285C2212yes—
52C42×D4C2211yescharacteristic
53C42×D4C2211yescharacteristic
54#2087C2214yes—
55#2075C2214yes—
56#2072C2214yes—
57#2068C2214yes—
58#2068C2214yes—
59#2059C2214yes—
60#2042C2214yes—
61#2035C2214yes—
62#2035C2214yes—
63#2033C2214yes—
64#1990C2212yes—
65#1989C2214yes—
66#1988C2212yes—
67#1987C2211yescharacteristic
68#1983C2214yes—
69#1981C2212yes—
70#1948C2211yescharacteristic
71#1947C2212yes—
72#1947C2212yes—
73#1947C2212yes—
74#1939C2212yes—
75#1928C2214yes—
76#1920C2212yes—
77#1919C2214yes—
78#1893C2214yes—
79#1892C2214yes—
80#1892C2214yes—
81#1864C2212yes—
82#1850C2214yes—
83#1842C2211yescharacteristic
84#1808C2214yes—
85#1803C2212yes—
86#1803C2212yes—
87#1803C2212yes—
88#1802C2212yes—
89#1802C2211yescharacteristic
90#1802C2212yes—
91#1797C2212yes—
92C4×#106C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#102C2212yes—
95C23×C8C2314yes—
96#556—48no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2312yes—
102#529—28no—
103#529—44no—
104C2×#126—24no—
105C2×#126—22no—
106C2×#125C2312yes—
107C2×#124—22no—
108C2×#124C2312yes—
109C2×#123—22no—
110C2×C22⋊Q8—22no—
111C2×C22⋊Q8C2312yes—
112C2×C22⋊Q8C2312yes—
113C2×#121—21no—
114C2×#121C2312yes—
115C2×C4×Q8—22no—
116C2×C4×Q8—21no—
117C2×C4×Q8C2312yes—
118C2×C4×D4—22no—
119C2×C4×D4D411yescharacteristic
120C2×C4×D4D411yescharacteristic
121C2×C4×D4C2311yescharacteristic
122C2×C4×D4C2311yescharacteristic
123C2×C4×D4C2311yescharacteristic
124C2×C4×D4C2311yescharacteristic
125C2×#117—22no—
126C2×#117—22no—
127C22×C42—21no—
128C22×C42C2312yes—
129#487—44no—
130#477—44no—
131#476—24no—
132#474—44no—
133C8×D4—44no—
134C2×#107C2314yes—
135C2×#107C2314yes—
136C2×#106C2312yes—
137C2×#106C2312yes—
138C2×C4⋊C8—44no—
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2311yescharacteristic
141C2×C4⋊C8C2311yescharacteristic
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#102C2312yes—
150C2×#102C2314yes—
151C2×#102C2312yes—
152C2×C22⋊C8C2314yes—
153C2×C4×C8C2312yes—
154#400—24no—
155#400—22no—
156#399—22no—
157#398—22no—
158#397—22no—
159#397—24no—
160#397—24no—
161#397—22no—
162#397—22no—
163#394—22no—
164#394—22no—
165#391—22no—
166#391—22no—
167#391—21no—
168#391—22no—
169#391C2312yes—
170#391C2312yes—
171#390—21no—
172#389—21no—
173#389—21no—
174#389—21no—
175#389—21no—
176#389—22no—
177#389C2312yes—
178#389C2314yes—
179#389C2312yes—
180#388—22no—
181#387—22no—
182#387—24no—
183#387—24no—
184#386—22no—
185#385—22no—
186#384—21no—
187#384—21no—
188#384—21no—
189#384—21no—
190#384C2311yescharacteristic
191#384C2311yescharacteristic
192#384C2312yes—
193#384C2312yes—
194#382—22no—
195#382—22no—
196#380—22no—
197#380—22no—
198#380—21no—
199#380—21no—
200#380—21no—
201#380—22no—
202#380C2312yes—
203#380C2314yes—
204#380C2314yes—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2314yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C22⋊C4—22no—
219C4×C22⋊C4—21no—
220C4×C22⋊C4—21no—
221C4×C22⋊C4—21no—
222C4×C22⋊C4D412yes—
223C4×C22⋊C4D412yes—
224C4×C22⋊C4C2312yes—
225C43C2311yescharacteristic
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#340C2312yes—
233#340C2312yes—
234#340C2312yes—
235#336C2314yes—
236#336C2312yes—
237C22×Q8—21no—
238C22×Q8—22no—
239C22×Q8C2412yes—
240C22×D4—22no—
241C22×D4C2×D411yescharacteristic
242C22×D4C2411yescharacteristic
243C23×C4—21no—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4C2×D412yes—
247C23×C4C2×D412yes—
248C23×C4C2412yes—
249C22×C8—44no—
250C22×C8—44no—
251C22×C8—44no—
252C22×C8C2412yes—
253C22×C8C2414yes—
254C22×C8C2412yes—
255C4⋊Q8—22no—
256C4⋊Q8—24no—
257C4⋊Q8—22no—
258C4⋊Q8—22no—
259C4⋊Q8—24no—
260C4⋊Q8—22no—
261C4⋊Q8—44no—
262C4⋊Q8—44no—
263#126—48no—
264#126—416no—
265#125—24no—
266#125—42no—
267#124—24no—
268#124—42no—
269#124—48no—
270#123—48no—
271C22⋊Q8—24no—
272C22⋊Q8—24no—
273C22⋊Q8—42no—
274C22⋊Q8—42no—
275C22⋊Q8—48no—
276#121—24no—
277#121—42no—
278#121—44no—
279C4×Q8—24no—
280C4×Q8—24no—
281C4×Q8—44no—
282C4×Q8—48no—
283C4×D4—24no—
284C4×D4—22no—
285C4×D4—24no—
286C4×D4—22no—
287C4×D4—22no—
288C4×D4—22no—
289C4×D4—22no—
290C4×D4—22no—
291C4×D4—41no—
292C4×D4—48no—
293C4×D4#4114yes—
294#117—48no—
295#117—48no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2414yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2411yescharacteristic
326C2×C22⋊C4—21no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4C2×D412yes—
338C2×C22⋊C4C2×D412yes—
339C2×C22⋊C4C2×D412yes—
340C2×C42—24no—
341C2×C42—24no—
342C2×C42—22no—
343C2×C42—24no—
344C2×C42—22no—
345C2×C42—21no—
346C2×C42—21no—
347C2×C42—21no—
348C2×C42—44no—
349C2×C42C2×D412yes—
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D412yes—
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42C2411yescharacteristic
356C2×C42C2412yes—
357C2×C42C2411yescharacteristic
358C2×C42C2411yescharacteristic
359#107—44no—
360#107—44no—
361#106—24no—
362#106—24no—
363C4⋊C8—22no—
364C4⋊C8—22no—
365C4⋊C8—24no—
366C4⋊C8—24no—
367C4⋊C8—44no—
368C4⋊C8—42no—
369#103—44no—
370#103—44no—
371#103—44no—
372#103—44no—
373#102—24no—
374#102—42no—
375#102—44no—
376C22⋊C8—44no—
377C4×C8—24no—
378#95—21no—
379#95—21no—
380#95—21no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—24no—
387#95—22no—
388#95—22no—
389#95—24no—
390#95—22no—
391#95—24no—
392#95—24no—
393#95C2×D412yes—
394#95C2×D412yes—
395#95C2×D412yes—
396C24—21no—
397C24C22×D412yes—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—42no—
405C2×Q8—42no—
406C2×Q8—44no—
407C2×Q8—42no—
408C2×Q8—44no—
409C2×Q8—44no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—44no—
418C2×D4—44no—
419C2×D4—82no—
420C2×D4AGL(1,ℤ/8)14yes—
421C2×D4C2×#4112yes—
422C22×C4—24no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—81no—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D412yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C2511yescharacteristic
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—24no—
465C2×C8—24no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—24no—
470C2×C8—88no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—24no—
479C4⋊C4—22no—
480C4⋊C4—21no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—21no—
484C4⋊C4—24no—
485C4⋊C4—22no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—42no—
491C4⋊C4—44no—
492C4⋊C4—48no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—44no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—44no—
499C4⋊C4—42no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—81no—
505C4⋊C4—82no—
506C4⋊C4C2×#4112yes—
507C4⋊C42−1+412yes—
508C22⋊C4—24no—
509C22⋊C4—24no—
510C22⋊C4—24no—
511C22⋊C4—24no—
512C22⋊C4—42no—
513C22⋊C4—44no—
514C22⋊C4—44no—
515C22⋊C4—44no—
516C22⋊C4—44no—
517C22⋊C4—44no—
518C22⋊C4—48no—
519C22⋊C4—44no—
520C22⋊C4—48no—
521C22⋊C4—42no—
522C42—24no—
523C42—24no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42—22no—
528C42—21no—
529C42—21no—
530C42—41no—
531C42—48no—
532C42—44no—
533C42—42no—
534C42—42no—
535C42—42no—
536C42—44no—
537C42#13514yes—
538C42C2×#4112yes—
539C42H(5,𝔽2)12yes—
540C42H(5,𝔽2)12yes—
541C422−1+412yes—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23C2×AGL(1,ℤ/8)14yes—
556C23C23×D411yescharacteristic
557Q8—42no—
558Q8—42no—
559Q8—42no—
560Q8—42no—
561Q8—84no—
562Q8—88no—
563D4—28no—
564D4—22no—
565D4—22no—
566D4—41no—
567D4—41no—
568D4—88no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—48no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—48no—
599C2×C4—48no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—84no—
619C2×C4—84no—
620C2×C4#53712yes—
621C2×C4#53712yes—
622C2×C4#54612yes—
623C2×C4#54612yes—
624C2×C4C2×#13512yes—
625C2×C4C2×#13512yes—
626C2×C4C2×#13512yes—
627C2×C4C2×AGL(1,ℤ/8)12yes—
628C2×C4#57812yes—
629C2×C4C22×#4111yescharacteristic
630C2×C4C2×H(5,𝔽2)11yescharacteristic
631C2×C4C2×H(5,𝔽2)11yescharacteristic
632C2×C4C2×2−1+411yescharacteristic
633C2×C4C2×2−1+411yescharacteristic
634C2×C4#58511yescharacteristic
635C8—42no—
636C8—44no—
637C8—42no—
638C22—24no—
639C22—24no—
640C22—24no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22—42no—
646C22—42no—
647C22—42no—
648C22—81no—
649C22—81no—
650C22—81no—
651C22—81no—
652C22C2×#53711yescharacteristic
653C22C2×#54611yescharacteristic
654C22#348911yescharacteristic
655C22C22×#13511yescharacteristic
656C22C22×AGL(1,ℤ/8)11yescharacteristic
657C22C2×#57811yescharacteristic
658C22#358611yescharacteristic
659C4—22no—
660C4—22no—
661C4—22no—
662C4—42no—
663C4—42no—
664C4—41no—
665C4—42no—
666C4—42no—
667C4—42no—
668C4—82no—
669C4—82no—
670C4—81no—
671C4—84no—
672C4—82no—
673C4—82no—
674C4#324714yes—
675C4#330214yes—
676C4#350711yescharacteristic
677C4#351111yescharacteristic
678C4#354611yescharacteristic
679C4#355011yescharacteristic
680C2—44no—
681C2—82no—
682C2#3946811yescharacteristic
683C2C2×#324711yescharacteristic
684C2#6092511yescharacteristic
685C2C2×#330211yescharacteristic
686C2#6116111yescharacteristic
687C2#6209711yescharacteristic
688C2#6213211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2537C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36188C2semidirect
3. #33195C2semidirect
4. #32203C2semidirect
5. #31494C2semidirect
6. #29692C2semidirect
7. #28717C2semidirect
8. #28600C2semidirect
9. #28284C2semidirect
10. #27631C2semidirect
11. #26443C2semidirect
13. #25275C2semidirect
14. #24837C2semidirect
15. D4×C4⋊C8C2semidirect
16. #21887C2semidirect
17. C4×#476C2semidirect

Maximal subgroups

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

subgroupindex
#361882
#349322
#331952
#322032
#314942
#296922
#287172
#286002
#282842
#276312
#264432
#253382
#252752
#248372
D4×C4⋊C82
#218872
C4×#4762