Finite Group Databasebeta

← #6433755

Subgroups — #6433755

Order 512.

1945 conjugacy classes of subgroups.

The subgroup lattice has 733 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35308C211yesmaximal, characteristic
2#35279C211yesmaximal, characteristic, complemented
3#30200C212yesmaximal, complemented
4#30199C212yesmaximal, complemented
5#30195C214yesmaximal, complemented
6#30193C214yesmaximal, complemented
7#30188C212yesmaximal, complemented
8#30187C212yesmaximal, complemented
9#23748C212yesmaximal, complemented
10#23629C211yesmaximal, characteristic, complemented
11#23627C211yesmaximal, characteristic, complemented
12#23510C212yesmaximal, complemented
13#23452C211yesmaximal, characteristic, complemented
14C2×#2024C211yesmaximal, characteristic, complemented
15#20976C212yesmaximal
16#20974C212yesmaximal, complemented
17#20336C211yesmaximal, characteristic, complemented
18C22×#131C2212yes—
19C2×#537C2211yescharacteristic
20C2×#536C2211yescharacteristic
21C2×#536C2211yescharacteristic
22C22×C4⋊Q8C2211yescharacteristic
23C2×C4×#41C2211yescharacteristic
24C2×#417C2212yes—
25C2×#419C2212yes—
26C2×#416C2212yes—
27C22×#103C2212yes—
28#2857C2212yes—
29#2857C2212yes—
30#2857C2212yes—
31#2850C2212yes—
32#2628C2214yes—
33#2621C2212yes—
34#2620C2212yes—
35#2568—21no—
36#2567—21no—
37#2563—21no—
38#2542—21no—
39#2487—22no—
40#2429—21no—
41#2429—21no—
42#2427—21no—
43#2427—21no—
44#2425—22no—
45#2425—22no—
46#2409—22no—
47#2317C2212yes—
48#2317C2211yescharacteristic
49#2311C2212yes—
50#2297C2211yescharacteristic
51C2×#391C2211yescharacteristic
52#2285C2211yescharacteristic
53#2101C2214yes—
54#2101C2214yes—
55#2100C2214yes—
56#2099C2214yes—
57#2074C2212yes—
58#2074C2214yes—
59#2073C2212yes—
60#2072C2211yescharacteristic
61#2072C2211yescharacteristic
62#2072C2212yes—
63#2072C2212yes—
64#2071C2211yescharacteristic
65#2071C2211yescharacteristic
66#2043C2214yes—
67#2043C2214yes—
68#2042C2214yes—
69#2042C2214yes—
70#2024C2212yes—
71#2024C2214yes—
72#2024C2212yes—
73#2024C2212yes—
74#2024C2212yes—
75#2023C2212yes—
76#2023C2212yes—
77#1895C2214yes—
78#1894C2214yes—
79#1894C2214yes—
80#1894C2214yes—
81#1893C2214yes—
82#1892C2214yes—
83#1892C2214yes—
84#1892C2214yes—
85#1736C2212yes—
86#1736C2212yes—
87#1735C2212yes—
88C2×#340C2212yes—
89#1729C2212yes—
90#1729C2214yes—
91#1728C2211yescharacteristic
92#1728C2212yes—
93C2×#336C2211yescharacteristic
94C2×#336C2212yes—
95C22×#41C2311yescharacteristic
96C22×#41C2311yescharacteristic
97C23×Q8—21no—
98C23×Q8C2311yescharacteristic
99C2×#131—28no—
100C2×#131—44no—
101C2×#131—44no—
102C22×M16C2312yes—
103C22×M16C2314yes—
104C23×C8C2312yes—
105C23×C8C2314yes—
106#537—28no—
107#536—28no—
108#536—44no—
109C2×C4⋊Q8—21no—
110C2×C4⋊Q8—21no—
111C2×C4⋊Q8—21no—
112C2×C4⋊Q8—44no—
113C2×C4⋊Q8C2312yes—
114C2×C4⋊Q8C2312yes—
115C2×C4⋊Q8C2312yes—
116C2×#124—21no—
117C2×#124C2312yes—
118C2×#123—22no—
119C2×#123—22no—
120C2×C22⋊Q8—21no—
121C2×C22⋊Q8—22no—
122C2×C22⋊Q8—22no—
123C2×C22⋊Q8—21no—
124C2×C22⋊Q8—21no—
125C2×C22⋊Q8C2312yes—
126C2×C22⋊Q8C2312yes—
127C2×C22⋊Q8C2312yes—
128C2×#121—22no—
129C2×#121—21no—
130C2×#121C2312yes—
131C4×#41—44no—
132C2×C4×Q8D411yescharacteristic
133C2×C4×Q8D411yescharacteristic
134C2×C4×D4D411yescharacteristic
135C2×C4×D4D412yes—
136C2×C4×D4D412yes—
137C2×C4×D4D411yescharacteristic
138C2×#117C2312yes—
139C2×#117C2311yescharacteristic
140C2×#117C2311yescharacteristic
141C2×#117C2312yes—
142C22×C4⋊C4—21no—
143C22×C4⋊C4C2312yes—
144C22×C4⋊C4C2311yescharacteristic
145C22×C42C2311yescharacteristic
146C22×C42C2312yes—
147#419—44no—
148#417—44no—
149#416—44no—
150C2×#103—44no—
151C2×#103C2312yes—
152C2×#103C2312yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#102C2312yes—
159C2×#102C2312yes—
160C2×#102C2314yes—
161#400—24no—
162#399—24no—
163#398—22no—
164#398—22no—
165#397—22no—
166#397—22no—
167#396—22no—
168#396—22no—
169#395—24no—
170#394—22no—
171#394—22no—
172#393—24no—
173#391—22no—
174#391—22no—
175#391—22no—
176#391—21no—
177#391—21no—
178#391C2312yes—
179#391C2311yescharacteristic
180#391C2312yes—
181#391C2311yescharacteristic
182#390—22no—
183#390C2311yescharacteristic
184#390C2311yescharacteristic
185#389—21no—
186#389—21no—
187#389—22no—
188#389—22no—
189#389—22no—
190#389—22no—
191#389C2314yes—
192#389C2314yes—
193#388—22no—
194#388—22no—
195#386—24no—
196#386—22no—
197#386—22no—
198#386—21no—
199#386—21no—
200#385—24no—
201#385—22no—
202#384—22no—
203#384C2312yes—
204#384C2311yescharacteristic
205#384C2311yescharacteristic
206#382—22no—
207#382C2312yes—
208#382C2312yes—
209#380—22no—
210#380—22no—
211#380—22no—
212#380—22no—
213#380—21no—
214#380—21no—
215#380C2314yes—
216#380C2314yes—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—21no—
219C4×C22⋊C4—21no—
220C4×C22⋊C4—21no—
221C2×#95—21no—
222C2×#95—21no—
223#340C2312yes—
224#340C2312yes—
225#340C2312yes—
226#340C2312yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#336C2311yescharacteristic
234#336C2311yescharacteristic
235#336C2312yes—
236#336C2312yes—
237#336C2312yes—
238#336C2314yes—
239#336C2314yes—
240C2×#41—24no—
241C2×#41—22no—
242C2×#41—22no—
243C2×#41—24no—
244C2×#41—24no—
245C2×#41—22no—
246C2×#41—22no—
247C2×#41#4118yes—
248C22×Q8—21no—
249C22×Q8—22no—
250C22×Q8—22no—
251C22×Q8—24no—
252C22×Q8—24no—
253C22×Q8—22no—
254C22×Q8—22no—
255C22×Q8—44no—
256C22×Q8—82no—
257C22×Q8—82no—
258C22×Q8C2×D411yescharacteristic
259C22×Q8C2412yes—
260C22×Q8C2411yescharacteristic
261C22×D4—21no—
262C22×D4—22no—
263C22×D4C2×D412yes—
264C22×D4C2×D411yescharacteristic
265C22×D4C2411yescharacteristic
266C23×C4—21no—
267C23×C4—22no—
268C23×C4—21no—
269C23×C4—21no—
270C23×C4C2×D411yescharacteristic
271C23×C4C2×D411yescharacteristic
272C23×C4C2×D412yes—
273C23×C4C2×D412yes—
274C23×C4C2412yes—
275C23×C4C2411yescharacteristic
276#131—432no—
277C2×M16—28no—
278C2×M16—24no—
279C2×M16—24no—
280C2×M16—24no—
281C2×M16—44no—
282C2×M16—44no—
283C22×C8—28no—
284C22×C8—24no—
285C22×C8—24no—
286C22×C8—24no—
287C22×C8—44no—
288C22×C8—44no—
289C22×C8C2412yes—
290C22×C8C2412yes—
291C22×C8C2414yes—
292C4⋊Q8—44no—
293C4⋊Q8—44no—
294C4⋊Q8—44no—
295C4⋊Q8—44no—
296C4⋊Q8—44no—
297C4⋊Q8—44no—
298#124—44no—
299#124—44no—
300#123—48no—
301#123—48no—
302C22⋊Q8—28no—
303C22⋊Q8—28no—
304C22⋊Q8—28no—
305C22⋊Q8—44no—
306C22⋊Q8—44no—
307C22⋊Q8—44no—
308C22⋊Q8—84no—
309C22⋊Q8—84no—
310#121—28no—
311#121—44no—
312#121—48no—
313C4×Q8—42no—
314C4×Q8—42no—
315C4×D4—42no—
316C4×D4—42no—
317C4×D4—44no—
318C4×D4—44no—
319#117—24no—
320#117—24no—
321#117—24no—
322#117—24no—
323#117—44no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—24no—
341C2×C4⋊C4—24no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—44no—
345C2×C4⋊C4C2×D411yescharacteristic
346C2×C4⋊C4C2×D411yescharacteristic
347C2×C4⋊C4C2×D412yes—
348C2×C4⋊C4C2×D412yes—
349C2×C4⋊C4C2412yes—
350C2×C4⋊C4C2414yes—
351C2×C4⋊C4C2411yescharacteristic
352C2×C4⋊C4C2411yescharacteristic
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2412yes—
355C2×C22⋊C4—21no—
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—21no—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—24no—
366C2×C22⋊C4—24no—
367C2×C22⋊C4—21no—
368C2×C22⋊C4—21no—
369C2×C22⋊C4C2×D412yes—
370C2×C22⋊C4C2×D412yes—
371C2×C22⋊C4C2×D412yes—
372C2×C42—24no—
373C2×C42—21no—
374C2×C42—22no—
375C2×C42—44no—
376C2×C42C2×D411yescharacteristic
377C2×C42C2×D411yescharacteristic
378C2×C42C2×D412yes—
379C2×C42C2×D412yes—
380C2×C42C2412yes—
381C2×C42C2411yescharacteristic
382C2×C42C2411yescharacteristic
383#103—44no—
384#103—44no—
385#103—42no—
386#103—42no—
387#103—44no—
388#103—44no—
389#103—44no—
390#102—42no—
391#102—42no—
392#102—44no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—21no—
398#95—21no—
399#95—21no—
400#95—21no—
401#95—22no—
402#95—22no—
403#95—24no—
404#95—24no—
405#95—24no—
406#95—24no—
407#95—22no—
408#95—22no—
409C24—21no—
410C24C22×D412yes—
411C24C22×D411yescharacteristic
412#41—216no—
413#41—216no—
414#41—48no—
415#41—48no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—24no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—21no—
422C2×Q8—24no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—21no—
426C2×Q8—44no—
427C2×Q8—44no—
428C2×Q8—44no—
429C2×Q8—42no—
430C2×Q8—42no—
431C2×Q8—42no—
432C2×Q8—44no—
433C2×Q8—48no—
434C2×Q8—48no—
435C2×Q8—82no—
436C2×Q8—82no—
437C2×Q8—816no—
438C2×Q8#12414yes—
439C2×D4—24no—
440C2×D4—24no—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—21no—
445C2×D4—22no—
446C2×D4—21no—
447C2×D4—42no—
448C2×D4—42no—
449C2×D4—42no—
450C2×D4—44no—
451C2×D4—82no—
452C2×D4—82no—
453C2×D4#12418yes—
454C2×D4#12414yes—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—21no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—21no—
470C22×C4—24no—
471C22×C4—21no—
472C22×C4—41no—
473C22×C4—41no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—44no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—82no—
487C22×C4—82no—
488C22×C4—81no—
489C22×C4—81no—
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D412yes—
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D412yes—
494C22×C4C22×D412yes—
495C22×C4C22×D411yescharacteristic
496C22×C4C22×D411yescharacteristic
497C22×C4C22×D411yescharacteristic
498C22×C4C2×#4118yes—
499C22×C4C2×#4114yes—
500C22×C4H(5,𝔽2)14yes—
501C22×C4H(5,𝔽2)12yes—
502C22×C42−1+412yes—
503C22×C4C2511yescharacteristic
504M16—416no—
505M16—48no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—24no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—24no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—24no—
515C2×C8—416no—
516C2×C8—48no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—24no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—24no—
525C4⋊C4—24no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—44no—
532C4⋊C4—44no—
533C4⋊C4—44no—
534C4⋊C4—44no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—44no—
538C4⋊C4—44no—
539C4⋊C4—44no—
540C4⋊C4—44no—
541C4⋊C4—42no—
542C4⋊C4—42no—
543C4⋊C4—82no—
544C4⋊C4—82no—
545C4⋊C4—84no—
546C4⋊C4—84no—
547C22⋊C4—24no—
548C22⋊C4—42no—
549C22⋊C4—42no—
550C22⋊C4—44no—
551C22⋊C4—44no—
552C22⋊C4—42no—
553C22⋊C4—44no—
554C22⋊C4—44no—
555C22⋊C4—44no—
556C22⋊C4—44no—
557C22⋊C4—44no—
558C22⋊C4—42no—
559C22⋊C4—42no—
560C22⋊C4—81no—
561C22⋊C4—81no—
562C22⋊C4—84no—
563C22⋊C4—84no—
564C42—22no—
565C42—22no—
566C42—22no—
567C42—22no—
568C42—24no—
569C42—42no—
570C42—42no—
571C42—42no—
572C42—44no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—22no—
578C23—22no—
579C23—22no—
580C23—22no—
581C23—22no—
582C23—22no—
583C23—41no—
584C23—41no—
585C23—41no—
586C23—41no—
587C23—41no—
588C23—41no—
589C23—41no—
590C23C2×#12414yes—
591C23C2×#12412yes—
592C23#53412yes—
593C23#53712yes—
594C23C23×D411yescharacteristic
595Q8—28no—
596Q8—44no—
597Q8—42no—
598Q8—44no—
599Q8—42no—
600Q8—88no—
601Q8—88no—
602Q8—84no—
603D4—216no—
604D4—28no—
605D4—42no—
606D4—42no—
607D4—84no—
608D4—164no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—28no—
616C2×C4—28no—
617C2×C4—21no—
618C2×C4—22no—
619C2×C4—21no—
620C2×C4—22no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—22no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—41no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—41no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—42no—
641C2×C4—41no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—44no—
645C2×C4—41no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—41no—
649C2×C4—42no—
650C2×C4—41no—
651C2×C4—42no—
652C2×C4—44no—
653C2×C4—41no—
654C2×C4—42no—
655C2×C4—84no—
656C2×C4—84no—
657C2×C4—88no—
658C2×C4—84no—
659C2×C4—84no—
660C2×C4—162no—
661C2×C4C2×#12414yes—
662C2×C4C2×#12412yes—
663C2×C4#53412yes—
664C2×C4#53712yes—
665C2×C4#53818yes—
666C2×C4#53918yes—
667C2×C4#57712yes—
668C2×C4#57812yes—
669C2×C4C22×#4112yes—
670C2×C4C2×H(5,𝔽2)11yescharacteristic
671C2×C4C2×H(5,𝔽2)12yes—
672C2×C4C2×2−1+411yescharacteristic
673C8—42no—
674C8—42no—
675C8—44no—
676C22—22no—
677C22—22no—
678C22—22no—
679C22—22no—
680C22—22no—
681C22—22no—
682C22—24no—
683C22—24no—
684C22—24no—
685C22—42no—
686C22—42no—
687C22—42no—
688C22—42no—
689C22—81no—
690C22—81no—
691C22—81no—
692C22—81no—
693C22—81no—
694C22—81no—
695C22—81no—
696C22#263914yes—
697C22C22×#12411yescharacteristic
698C22C2×#53411yescharacteristic
699C22C2×#53711yescharacteristic
700C22C2×#57711yescharacteristic
701C22C2×#57811yescharacteristic
702C22#358511yescharacteristic
703C22#358611yescharacteristic
704C4—24no—
705C4—22no—
706C4—22no—
707C4—41no—
708C4—42no—
709C4—41no—
710C4—42no—
711C4—82no—
712C4—81no—
713C4—82no—
714C4—82no—
715C4—84no—
716C4—81no—
717C4—81no—
718C4—82no—
719C4#263914yes—
720C4C2×#53812yes—
721C4C2×#53912yes—
722C2—24no—
723C2—44no—
724C2—161no—
725C2C2×#263911yescharacteristic
726C2#6062311yescharacteristic
727C2#6062511yescharacteristic
728C2#6075711yescharacteristic
729C2#6075911yescharacteristic
730C2#6093011yescharacteristic
731C2#6093111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#537C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #35279C2semidirect
3. #30200C2semidirect
4. #30199C2semidirect
5. #30195C2semidirect
6. #30193C2semidirect
7. #30188C2semidirect
8. #30187C2semidirect
9. #23748C2semidirect
10. #23629C2semidirect
11. #23627C2semidirect
12. #23510C2semidirect
13. #23452C2semidirect
14. C2×#2024C2semidirect
16. #20974C2semidirect
17. #20336C2semidirect

Maximal subgroups

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

subgroupindex
#353082
#352792
#302002
#301992
#301952
#301932
#301882
#301872
#237482
#236292
#236272
#235102
#234522
C2×#20242
#209762
#209742
#203362