Finite Group Databasebeta

← #6593617

Subgroups — #6593617

Order 512.

1839 conjugacy classes of subgroups.

The subgroup lattice has 705 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36188C211yesmaximal, characteristic, complemented
2C4×#547C211yesmaximal, characteristic
3#33195C211yesmaximal, characteristic, complemented
4#32958C212yesmaximal, complemented
5#28717C211yesmaximal, characteristic, complemented
6#28284C211yesmaximal, characteristic, complemented
7#27630C212yesmaximal, complemented
8#27531C214yesmaximal, complemented
9#27043C214yesmaximal, complemented
10#26301C212yesmaximal, complemented
11#25882C212yesmaximal
12#24702C212yesmaximal, complemented
13#23214C212yesmaximal, complemented
14#22962C212yesmaximal, complemented
15#22795C211yesmaximal, characteristic, complemented
16C4×#476C211yesmaximal, characteristic, complemented
17#20737C212yesmaximal, complemented
18C2×#556—21no—
19C2×#547C2211yescharacteristic
20C2×C4×#41C2212yes—
21C2×#487C2212yes—
22C2×#477C2212yes—
23C2×#476C2211yescharacteristic
24C2×#474C2212yes—
25C2×#431C2212yes—
26#2868C2211yescharacteristic
27#2857C2212yes—
28#2568—21no—
29#2547—21no—
30#2547C2212yes—
31#2499—22no—
32#2496—22no—
33#2496C2214yes—
34#2365—21no—
35#2365C2212yes—
36#2365C2211yescharacteristic
37#2362C2212yes—
38#2361—21no—
39#2359—22no—
40#2354C2211yescharacteristic
41D4×C4⋊C4C2211yescharacteristic
42#2344—21no—
43#2339C2212yes—
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2211yescharacteristic
46C4×C22⋊Q8C2212yes—
47C4×#123—22no—
48C4×#121—21no—
49#2285C2212yes—
50C42×D4C2211yescharacteristic
51C42×D4C2211yescharacteristic
52C2×C4×C4⋊C4—21no—
53#2077C2214yes—
54#2076C2214yes—
55#2072C2214yes—
56#2068C2214yes—
57#2036C2214yes—
58#2036C2214yes—
59#1990C2212yes—
60#1988C2212yes—
61#1987C2211yescharacteristic
62#1981C2212yes—
63#1954C2212yes—
64#1948C2211yescharacteristic
65#1947C2212yes—
66#1947C2212yes—
67#1947C2212yes—
68#1942C2214yes—
69#1933C2214yes—
70#1927C2214yes—
71#1923C2214yes—
72#1920C2212yes—
73#1918C2214yes—
74#1865C2214yes—
75#1864C2212yes—
76#1862C2214yes—
77#1854C2214yes—
78#1842C2211yescharacteristic
79#1825C2214yes—
80#1803C2212yes—
81#1803C2212yes—
82#1803C2212yes—
83#1802C2211yescharacteristic
84#1802C2212yes—
85#1802C2212yes—
86#1797C2212yes—
87#1788C2214yes—
88#1788C2214yes—
89C4×#106C2212yes—
90C8×C4⋊C4C2212yes—
91C4×C4⋊C8C2211yescharacteristic
92C4×#103C2214yes—
93C4×#102C2214yes—
94C4×#102C2212yes—
95C22×#41C2312yes—
96#556—48no—
97#547—24no—
98#547—24no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×#126—24no—
105C2×#124—22no—
106C2×#123—22no—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8C2312yes—
109C2×#121—21no—
110C4×#41—28no—
111C4×#41—44no—
112C2×C4×Q8—22no—
113C2×C4×Q8—21no—
114C2×C4×Q8C2312yes—
115C2×C4×Q8C2312yes—
116C2×C4×D4—22no—
117C2×C4×D4—22no—
118C2×C4×D4D411yescharacteristic
119C2×C4×D4D411yescharacteristic
120C2×C4×D4C2311yescharacteristic
121C2×C4×D4C2312yes—
122C2×C4×D4C2311yescharacteristic
123C2×C4×D4C2311yescharacteristic
124C2×C4×D4C2311yescharacteristic
125C2×#117—22no—
126C2×#117—22no—
127C2×#117C2312yes—
128C22×C4⋊C4—21no—
129C22×C4⋊C4—21no—
130C22×C42—22no—
131C22×C42—21no—
132C22×C42C2312yes—
133#487—28no—
134#477—44no—
135#476—24no—
136#474—28no—
137#431—44no—
138C2×#107C2314yes—
139C2×#107C2314yes—
140C2×#106C2312yes—
141C2×#106C2312yes—
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2311yescharacteristic
144C2×C4⋊C8C2311yescharacteristic
145C2×C4⋊C8C2312yes—
146C2×C4⋊C8C2312yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#102C2314yes—
152C2×#102C2312yes—
153C2×#102C2312yes—
154C2×C22⋊C8C2314yes—
155C2×C22⋊C8C2314yes—
156C2×C4×C8C2312yes—
157C2×C4×C8C2314yes—
158#400—24no—
159#397—24no—
160#397—24no—
161#391—22no—
162#391—22no—
163#391—21no—
164#391—22no—
165#391C2312yes—
166#391C2312yes—
167#390—21no—
168#389—21no—
169#389—21no—
170#389—21no—
171#389—21no—
172#389—22no—
173#389C2312yes—
174#389C2314yes—
175#389C2312yes—
176#388—22no—
177#388—22no—
178#388—22no—
179#387—22no—
180#387—22no—
181#387—22no—
182#387—24no—
183#387—24no—
184#386—22no—
185#386—22no—
186#385—22no—
187#385—22no—
188#384—21no—
189#384—21no—
190#384C2311yescharacteristic
191#384C2311yescharacteristic
192#384C2312yes—
193#384C2312yes—
194#382—22no—
195#381—22no—
196#380—22no—
197#380—22no—
198#380—21no—
199#380C2312yes—
200#380C2314yes—
201#380C2314yes—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2314yes—
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—
225C2×#95—22no—
226C43C2311yescharacteristic
227#340C2314yes—
228#340C2312yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#340C2312yes—
234#340C2312yes—
235#340C2312yes—
236#336C2314yes—
237#336C2312yes—
238C2×#41—24no—
239C2×#41—24no—
240C2×#41—24no—
241C2×#41—24no—
242C2×#41—42no—
243C2×#41—42no—
244C22×Q8—21no—
245C22×Q8—22no—
246C22×Q8C2412yes—
247C22×D4—22no—
248C22×D4C2×D411yescharacteristic
249C22×D4C2411yescharacteristic
250C23×C4—21no—
251C23×C4—22no—
252C23×C4—21no—
253C23×C4—22no—
254C23×C4—21no—
255C23×C4C2×D412yes—
256C23×C4C2×D412yes—
257C23×C4C2412yes—
258C22×C8C2412yes—
259C22×C8C2412yes—
260C22×C8C2414yes—
261C4⋊Q8—22no—
262C4⋊Q8—24no—
263C4⋊Q8—22no—
264C4⋊Q8—24no—
265C4⋊Q8—22no—
266C4⋊Q8—44no—
267C4⋊Q8—44no—
268C4⋊Q8#4114yes—
269#126—416no—
270#124—48no—
271#123—216no—
272C22⋊Q8—24no—
273C22⋊Q8—24no—
274C22⋊Q8—48no—
275#121—28no—
276C4×Q8—24no—
277C4×Q8—24no—
278C4×Q8—24no—
279C4×Q8—42no—
280C4×Q8—44no—
281C4×Q8—48no—
282C4×D4—24no—
283C4×D4—24no—
284C4×D4—22no—
285C4×D4—24no—
286C4×D4—22no—
287C4×D4—22no—
288C4×D4—22no—
289C4×D4—22no—
290C4×D4—41no—
291C4×D4—42no—
292C4×D4—48no—
293C4×D4—48no—
294C4×D4#4114yes—
295C4×D4#4114yes—
296#117—24no—
297#117—48no—
298#117—42no—
299#117—48no—
300C2×C4⋊C4—28no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
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—24no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2414yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2411yescharacteristic
330C2×C22⋊C4—22no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4C2×D412yes—
340C2×C22⋊C4C2×D412yes—
341C2×C22⋊C4C2×D412yes—
342C2×C42—24no—
343C2×C42—24no—
344C2×C42—22no—
345C2×C42—24no—
346C2×C42—22no—
347C2×C42—22no—
348C2×C42—22no—
349C2×C42—21no—
350C2×C42—21no—
351C2×C42—21no—
352C2×C42—48no—
353C2×C42—44no—
354C2×C42C2×D412yes—
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D412yes—
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D411yescharacteristic
360C2×C42C2412yes—
361C2×C42C2411yescharacteristic
362C2×C42C2411yescharacteristic
363C2×C42C2411yescharacteristic
364#107—28no—
365#107—44no—
366#106—24no—
367#106—42no—
368C4⋊C8—24no—
369C4⋊C8—22no—
370C4⋊C8—22no—
371C4⋊C8—24no—
372C4⋊C8—42no—
373#103—28no—
374#103—28no—
375#103—44no—
376#103—44no—
377#102—28no—
378#102—24no—
379#102—24no—
380C22⋊C8—44no—
381C22⋊C8—44no—
382C4×C8—24no—
383C4×C8—44no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—21no—
388#95—21no—
389#95—21no—
390#95—22no—
391#95—22no—
392#95—24no—
393#95—22no—
394#95—22no—
395#95—24no—
396#95—22no—
397#95—24no—
398#95—24no—
399#95C2×D412yes—
400#95C2×D412yes—
401#95C2×D412yes—
402C24—21no—
403C24C22×D412yes—
404#41—48no—
405#41—48no—
406#41—48no—
407#41—48no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—42no—
414C2×Q8—42no—
415C2×Q8—44no—
416C2×Q8—42no—
417C2×Q8—44no—
418C2×Q8—44no—
419C2×Q8C2×#4114yes—
420C2×D4—28no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4—44no—
428C2×D4—44no—
429C2×D4C2×#4112yes—
430C2×D4C2×#4112yes—
431C2×D42−1+412yes—
432C22×C4—28no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—42no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—44no—
465C22×C4—81no—
466C22×C4—82no—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D412yes—
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C2511yescharacteristic
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—24no—
481C2×C8—22no—
482C2×C8—24no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—24no—
486C4⋊C4—28no—
487C4⋊C4—22no—
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—22no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—22no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—21no—
501C4⋊C4—22no—
502C4⋊C4—21no—
503C4⋊C4—44no—
504C4⋊C4—42no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—42no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—42no—
515C4⋊C4—48no—
516C4⋊C4—42no—
517C4⋊C4—81no—
518C4⋊C4AGL(1,ℤ/8)14yes—
519C4⋊C4C2×#4112yes—
520C4⋊C4C2×#4112yes—
521C22⋊C4—28no—
522C22⋊C4—24no—
523C22⋊C4—24no—
524C22⋊C4—24no—
525C22⋊C4—24no—
526C22⋊C4—24no—
527C22⋊C4—44no—
528C22⋊C4—44no—
529C22⋊C4—42no—
530C22⋊C4—48no—
531C22⋊C4—48no—
532C22⋊C4—82no—
533C42—24no—
534C42—24no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—21no—
540C42—21no—
541C42—41no—
542C42—44no—
543C42—44no—
544C42—44no—
545C42—42no—
546C42—42no—
547C42—42no—
548C42—48no—
549C42—44no—
550C42#13514yes—
551C42C2×#4112yes—
552C42C2×#4112yes—
553C42C2×#4112yes—
554C42H(5,𝔽2)12yes—
555C23—22no—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—22no—
562C23—22no—
563C23—22no—
564C23—22no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23C23×D411yescharacteristic
570Q8—24no—
571Q8—24no—
572Q8—42no—
573Q8—42no—
574Q8—84no—
575Q8—88no—
576D4—22no—
577D4—22no—
578D4—22no—
579D4—416no—
580D4—44no—
581D4#55614yescomplemented
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—22no—
602C2×C4—48no—
603C2×C4—416no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—48no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—48no—
616C2×C4—48no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—41no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—84no—
633C2×C4—88no—
634C2×C4#52512yes—
635C2×C4#52512yes—
636C2×C4#54612yes—
637C2×C4#54612yes—
638C2×C4C2×#13512yes—
639C2×C4C2×#13512yes—
640C2×C4C2×#13512yes—
641C2×C4C2×AGL(1,ℤ/8)14yes—
642C2×C4C2×AGL(1,ℤ/8)12yes—
643C2×C4#57812yes—
644C2×C4C22×#4111yescharacteristic
645C2×C4C22×#4111yescharacteristic
646C2×C4C22×#4111yescharacteristic
647C2×C4C2×H(5,𝔽2)11yescharacteristic
648C2×C4C2×2−1+411yescharacteristic
649C2×C4#58511yescharacteristic
650C8—42no—
651C8—44no—
652C8—42no—
653C22—24no—
654C22—42no—
655C22—42no—
656C22—42no—
657C22—42no—
658C22—42no—
659C22—42no—
660C22—42no—
661C22—42no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22—42no—
666C22—42no—
667C22C2×#52511yescharacteristic
668C22C2×#54611yescharacteristic
669C22#348911yescharacteristic
670C22C22×#13511yescharacteristic
671C22C22×AGL(1,ℤ/8)11yescharacteristic
672C22C2×#57811yescharacteristic
673C22#358611yescharacteristic
674C4—22no—
675C4—22no—
676C4—22no—
677C4—24no—
678C4—42no—
679C4—44no—
680C4—42no—
681C4—42no—
682C4—42no—
683C4—41no—
684C4—84no—
685C4—81no—
686C4—82no—
687C4—82no—
688C4—82no—
689C4#310614yes—
690C4#330214yes—
691C4C2×#55611yescharacteristic
692C4#347511yescharacteristic
693C4#350611yescharacteristic
694C4#352011yescharacteristic
695C2—44no—
696C2—82no—
697C2#3616111yescharacteristic
698C2C2×#310611yescharacteristic
699C2#6059411yescharacteristic
700C2C2×#330211yescharacteristic
701C2#6116111yescharacteristic
702C2#6209711yescharacteristic
703C2#6214811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#547C22
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. #32958C2semidirect
5. #28717C2semidirect
6. #28284C2semidirect
7. #27630C2semidirect
8. #27531C2semidirect
9. #27043C2semidirect
10. #26301C2semidirect
12. #24702C2semidirect
13. #23214C2semidirect
14. #22962C2semidirect
15. #22795C2semidirect
16. C4×#476C2semidirect
17. #20737C2semidirect
581. D4#556semidirect

Maximal subgroups

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

subgroupindex
#361882
C4×#5472
#331952
#329582
#287172
#282842
#276302
#275312
#270432
#263012
#258822
#247022
#232142
#229622
#227952
C4×#4762
#207372